Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters

to the editor should be sent to the

editor-in-chief, Chandler Davis.

Can Two Periodic Functions with

Incommensurable Periods Have a

Periodic Sum?

This note adds to the arsenal of coun­terexamples in elementary analysis. It is well known and easily proved that the sum of two periodic functions on the real line is periodic if the periods of the two functions are commensurable, i.e., their quotient is a rational number. It seems reasonable to assume that the sum of two periodic functions with in­commensurable periods must be aperi­odic, as has been done in at least one textbook on differential equations. 1 But this assumption is incorrect.

My examples are everywhere dis­continuous. This suggests the question whether less pathological examples exist. I will give a partial answer to this question at the end.

To be precise, saying thatj: IR � IR is a periodic function of period U means that f(t + U) = .f(t) for all t E ( -x,oo) , and that U is the smallest pos­itive value for which this is true. Note that by this definition a constant func­tion is not periodic, nor is the charac­teristic function of the rationals.

THEOREM 1. Let a > 0 and {3 > 0 be in­commensurable real numbers. Then there exist functions JIIR � IR and g IR � IR with periods a and {3, respec­tively, such that h = f + g is periodic.

Comment. Since the set {qta + Q2f31{qt,Q2} C IQ} is countable, there ex­ist uncountably many real numbers y such that the set { a,{3, y} is linearly in­dependent over IQ. We will need the ex­istence of such a y below. For a simple example, take a = 1, {3 = V2 and y =

Vs. For a class of examples, let a =

VJ;;, {3 = VJ;;., y = �. for distinct primes Pi· Or, let o E IR be transcen­dental, then set a = o, {3 = 82, y = o3. (I thank Basil Gordon for this example and for suggesting improvements to the text.)

Proof Choose y such that a,{3, yare lin­early independent over IQ. Then, if l, m, and n are integers for which la + mf3 + ny = 0, we must have l = m =

n = 0. On this foundation I construct!, g, and h.

Define G = G( a,{3, y) = { lt,m,n = la + m{3 + nyl {l,m,nj C Z}, and let!, g, and h be functions that vanish on the com­plement of G, i.e., for all t ft. G, but are otherwise defined as follows:

.f(t) = mf3 + ny ) g(t) = la- ny t = lt,m,n E G (1.1) h(t) = la + mf3

The number y was chosen to ensure that a non-zero value ofj( t) uniquely de­termines the values of m and n. Similarly, a non-zero value of g(t) or h(t) uniquely determines the values of l and n, or l and m, respectively. Thus, for a fixed pair of integers m and n, not both of which are zero, j(t) = mf3 + ny only at the points t = la + m{3 + ny, for arbitrary l E 1:'. Since j(t) = j(t + a) = 0 for all t ft. G, this shows that .f has period a. Similar remarks apply to g and h. In summary:

1. the period off is a, 2. the period of g is {3, 3. the period of h is y.

But h = .f + g. The theorem is proved.

D

Extending the class of examples The class of triplets f, g, and h con­structed in the proof above can be en­larged. Let <PIN� IQ0 be a bijection, where IQ0 is any infinite subset of the rationals. Now modify the definitions given above in the following way:

f(t) = cfJ(m)f3 + cfJ(n)y g(t) = cfJ(l)a- cfJ(n)y h(t) = cfJ(l)a + cfJ(m)f3

t = lt,m,n E G (1.2)

By reasoning analogous to that in the proof of Theorem 1, a particular non-zero value of .f recurs with period a, with similar remarks holding for g

---- �------�----------· -------------

1Borrelli, Robert L and Courtney S. Coleman, Differential Equations: A Modeling Perspective, 1 st ed., New

York: John Wiley & Sons, Inc., 1 998, Prob. 1 "Periodic Function Facts," 189. This error does not appear in the

second edition�

4 THE MATHEMATICAL INTELLIGENCER © 2005 Springer SC1ence+Bus1ness Media, Inc.

and h. Thus again the periods off, g, and h are a, {3, and y, respectively. By confining lDo to a bounded interval, we see thatf, g, and h themselves can be bounded.

A question remains. Is it possible to have counterexamples in which one of the functions is continuous at a point or on a larger set? (I thank a referee for the question and for finding an error in a previous version of Theorem 2.) Here is a partial answer.

THEOREM 2. Let H(t) = F(t) + G(t), where F and G are bounded and peri­odic with incommensurable periods U and V, andF is continuous everywhere. Then H is not periodic.

Proof I show that the assumption that His periodic, say of period W > 0, leads to a contradiction. Note that W must be incommensurable with both U and V.

The function F attains a maximum F*, and G and H have finite least up­per bounds, say G* and H*, respec­tively. I will show that H* = F* + G*. Because W must be incommensurable with U, the latter equality places high

demands on F, sufficient to prove that F must be constant.

Given s > 0, choose x and y such that F(x) = F* and G(y) > G* - s. Since F is uniformly continuous, there is a 15 > 0 such that IF(t) - F(t ' } < s

for It - t' l < 15. I need two facts concerning a pair

of incommensurable numbers (, TJ E lhbo: (1) 7L( mod TJ is dense in the in­terval [0, TJL and hence (2) for any s1 >

0, there exist positive integers m and n such that

jx + m( - y - nTJI < s1. (1.3)

These facts follow from the well­known fact that, for irrational ( > 0, the set 7L( mod 1 is dense in [0,1].

According to inequality (1.3), we may choose m and n so that

lx + mU - y- nvj < 15. (1.4)

By the choice of 15, IF(x + mU)­F(y + n V)l < s, and therefore

H*::::: H(y + nV) = F(y + nV) + G(y + nV) > F* + G* - 2s. (1.5)

As clearly H* :s: F* + G*, it follows that H* = F* + G*.

We will exploit the fact that at val­ues of the argument for which H is near

its maximal value, so must F and G be

near their maximal values. As evident in (1.5), such a value of the argument is afforded by z = y + n V. Thus, be­cause H is presumed to have period W, and referring again to (1.5),

H(z + jW) = F(z + jW) + G(z + jW) = H(z) > F* + G* - 2s (Vj E 7L).

It follows that

F(z + jW) >F* + G* - G(z + jW)- 2s > F* - 2s (Vj E 7L). (1.6)

Finally, the period ofF is U, U and W are incommensurable, and so the set ltJ = (z + jW) modU IJ E 7L} is dense in [O,U]. And, by periodicity of F, in­equality (1.6) implies that F(t) > F* -2s for all t in the dense set l tJl· This is possible only if F = F*. This contradiction proves that H cannot be periodic. D

Michael R. Raugh

Department of Mathematics

Harvey Mudd College

Claremont, CA 9 17 1 1 -0788

USA

e-mail: raugh@hmc.edu

The Pythagorean Theorem

Extended-and Deflated

In my paper "N-Dimensional Variations on Themes of Pythagoras, Euclid, and Archimedes" (Mathematical Intelli­gencer 26 (2004), no. 3, 43-53), I pro­posed a generalisation of the usual Pythagorean theorem in the form THEOREM OF PYTHAGORAS ND. The square of the (N-1)-dimensional vol­ume of the hypoteneusal face of an N-dimensional orthosimplex is equal to the sum of the squares of the vol­umes of its N orthogonal faces.

This was a rediscovery; I mentioned I had been anticipated by H. S. M. Cox­eter & P. S. Donchian, Math. Gazette 19 (1935), 206.

And by many others! Rajendra Bhatia traced out an In­

dian path-which is, after all, satisfy-

ing, given that ancient Indian mathe­

maticians seem to have known what we call the Pythagorean Theorem well before the Greeks. K. R. Parthasarathy published a proof based on volume in­

tegrals calculated by using Gauss's for­mula for the volume of convex poly­topes: "An n-dimensional Pythagoras Theorem," Math. Scientist 3 (1978), 137-140. This impelled S. Ramanan to give a simpler (unpublished) proof us­ing antisymmetric tensors. After an­other elaborate proof was indepen­dently published by S. Y. Lin and Y. F. Lin in Lin. Multilin. Algebra 26 (1990), 9-13, R. Bhatia sent a letter giving Ra­manan's proof (Lin. Multilin. Algebra 30 (1991), 155), and included it as prob­lem 1.6.6 in his book Matrix Analysis.

More recently, French colleagues also stumbled on the results: J.-P. Quadrat, J. B. Lasserre, and J.-B. Hiriart­Urruty, "Pythagoras' Theorem for Ar­eas," American Mathematical Monthly 108 (2001), 549-551. They pointed out a French connection, at least for the 3-dimensional case, which has been known for quite some time (though its analogy with the standard Pythagorean Theorem was apparently not stressed). The result was very likely known to R. Descartes himself, according to P. Costabel (see his edition of Descartes's Exercices pour la Geometrie des Solides (De Solidorum Elementis), Presses Universitaires de France, Paris, 1987). In any case, the (3-dimen­sional) theorem is found in J.-P. Gua de Malves's memoirs of 17831, and L. N. M. Camot stated the result ( re­ferring to it as already known) in his Geometrie de Position, Crapelet, Paris, 1803. It also found its way into text­books, such as P. Nillus, Ler;ons de cal­cui vectoriel (t. I), Eyrolles, Paris, 1931.

Now the publication of the paper by J.-P. Quadrat et al. brought new refer­ences. The Editor, B. P. Palka, quotes but two comments (see "Editors' End­notes" in the Monthly 109 (2002), 313-314). G. De Marco, from Padova, mentions an equivalent result involving N-dimensional parallelotopes, to be found in F. R. Gantmacher, Theone des Matrices (t. I), Dunod, Paris, 1966.

---------·----·----- -----------------------------1The abbot Gua de Malves is a most interesting character. A typical polymath of the Enlightenment, he was in fact the first editor of the Encyc/opedie, before handing

over the task to Diderot and D'Aiembert.

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005 5

J. Munkres recalls that a more general

result is given in his book Analysis of Manifolds, Westview Press, 1991, pp. 184-187:

THEOREM. Let u be a k-simplex in W'. Then the square of the area of u equals the sum of the squares of the areas of the k-simplices obtained by projecting u orthogonally to the various coordi­nate k-planes of W'.

An elementary proof of a similar result for parallelotopes was published by G. J. Porter in the Monthly 103 (1996), 252-256.

There have been many other publi­cations of the Theorem. A very cursory Google search led me to a note by Eric W. Weisstein on MathWorld [http:// math world. wolfram.comldeGuasTheo­rem.html]; the 3-dimensional case, re­ferred to as "de Gua's Theorem," is said there to be a special case of a general theorem presented by Tinseau to the Paris Academy in 1774 (slightly before de Gua's own publication), quoted in the textbooks by W. F. Osgood and

W. C. Graustein, Solid Analytic Geom­etry, Macmillan, New York, 1930, Th. 2, p. 517, and N. Altshiller-Court, Modern Pure Solid Geometry, Chelsea, New York, 1979, pp. 92 and 300. As for the general case, I found a reference to a pa­per by R. F. Talbot, "Generalizations of Pythagoras Theorem in n Dimensions," Math. Scientist 12 (1987), 117-121, probably following Parthasarathy's 1978 publication in the same journal. A charming sequel is the recent (2002) posting by Willie W. Wong, a Princeton University student, of his proof of "A generalized N-dimensional Pythagorean Theorem" on his site [sep.princeton. edu/papers/gp.pdf].

It may still be that Coxeter and Donchian have the first occurrence in print of the result for N > 3. We may well ponder the significance of the re­curring rediscovery of this result-and of its remaining so little known; its aes­thetic and didactic merits certainly earn it a high place in textbooks or in the oral tradition. The least we can say is that our recording and referencing system clearly shows here its lacunae.

The irony of the situation is that this discussion amounts to much ado about little. Indeed, as pointed out by

6 THE MATHEMATICAL INTELLIGENCER

J. Munkres in the aforementioned book, the theorem holds not only for simplices

and parallelotopes, but (surprisingly at first) also for arbitrary sets lying in a k­

plane of Rn (k < n )! This generalisation is all the more interesting in that it only takes a meaning for higher dimension­alities than the k = 1, n = 2 case of the standard Pythagorean Theorem. How­ever, far from being a deep theorem, it is almost trivial, at least in the case k = n - 1 considered up to now. LetS be an arbitrary set contained in an ( n - 1 )­plane P of Rn, and call its volume A. Let

vp be the unit vector orthogonal to P. Consider now the n projections Si ( i =

1,2, . . . n) of S onto the (n - I)-dimen­sional subspaces orthogonal to the unit vectors vi (i = 1,2, . . . n) of an orthog­onal basis of Rn. Their respective vol­umes Ai are obtained by projection and are given by Ai = (vp. vi)A. Since, by the usual n-dimensional Pythagorean Theo­rem (or the so-called cosine law), one has I11Cvi, Vp)l2 = llvPII2 = 1, we obtain immediately the result announced, to

wit I1 A� = A2• Hardly more than a Lemma!

The crux of the matter is that going

from a !-dimensional segment to a k­dimensional simplex is not the relevant generalisation here. In the present con­text, a !-dimensional segment should be considered as an arbitrary con­nected !-dimensional set.

Here, as so often, a result proved in special cases through rather sophisti­cated means finds an elementary proof showing its intrinsic nature once it is formulated in more general terms. This anticlimax only deepens the question of why the result has not been better understood by its many rediscover­ers-including the present one.

It is a pleasure to thank R. Bhatia, J. Holbrook, and J.-B. Hiriart-Urruty for a first introduction to the literature I had overlooked.

Jean-Marc Levy-Leblond

Physique Theorique

Universite de Nice Sophie-Antipolis

Pare Valrose

061 08 Nice Cedex

France

e-mail: jmll@unice.fr

HOLY GRAIL OF MATHEMATICS FOUND FERMAT'S PROOF TO HIS "LAST THEOREM" (A Restoration]

After some 370 years a 17th-Century proof to the greatest enigma in mathematics is presented as the restoration of Fermat's letter to a dear friend divulging the origin and rationale of both the mathematical AND geometrical proofs as examples of his descent infinite/indefinite discussed in his note on the impossibility of the area of a rectangular triangle being an integer (newly translated) and his August 1659letter to Carcavi (the only translation).

Traces the proof from Euclid and Pythagoras. A MUST FOR EVERY MATHEMATICIAN vii+ 22 pp. +illustrations $12.00 + $2.50 S&H + NJ 6% tax (U.S. $'s only) Institutional checks or money orders only Akerue Publications LLC • PO Box 954 7 Elizabeth, NJ 07202

c.J.ii.U.J.M

Knowledge and Community in Mathematics Jonathan Borwein and

Terry Stanway

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in­chief, Chandler Davis.

Mathematical Knowledge-As We

Knew It

Each society has its regime of truth, its "general politics" of truth: that is, the types of discourse which it accepts and makes function as true; the mech­anisms and instances which enable one to distinguish true and false state­ments, the means by which each is sanctioned; the techniques and proce­dures accorded value in the acquisi­tion of truth; the status of those who are charged with saying what counts as truth.1 (Michel Foucault)

Henri Lebesgue once remarked that "a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy." He went on to add that this was "an opinion, moreover, which has been expressed by many philosophers."2 The idea that mathe­maticians can do mathematics without a precise philosophical understanding of what they are doing is, by observa­tion, mercifully true. However, while a neglect of philosophical issues does not impede mathematical discussion, dis­cussion about mathematics quickly be­comes embroiled in philosophy, and perforce encompasses the question of the nature of mathematical knowledge. Within this discussion, some attention has been paid to the resonance between

the failure of twentieth-century efforts to enunciate a comprehensive, absolute foundation for mathematics and the postmodern deconstruction of meaning and its corresponding banishment of en­compassing philosophical perspectives from the centre fixe.

Of note in this commentary is the contribution of Vladimir Tasic. In his book, Mathematics and the Roots of Postmodern Thought, he comments on the broad range of ideas about the

interrelationship between language, meaning, and society that are com­monly considered to fall under the um­brella of postmodernism. Stating that "attempts to make sense of this elu­sive concept threaten to outnumber at­tempts to square the circle," he focuses his attention on two relatively well­developed aspects of postmodern the­ory: "poststructuralism" and "decon­struction. "3 He argues that the develop­ment of these theories, in the works of Derrida and others, resonates with the debates surrounding foundation­ism which preoccupied the philosophy of mathematics in the early stages of the last century and may even have been partly informed by those debates. Our present purpose is not to revisit the connections between the foundationist debates and the advent of postmodern thought, but rather to describe and dis­cuss some of the ways in which episte­mological relativism and other post­modern perspectives are manifest in the changing ways in which mathematicians do mathematics and express mathemat­ical knowledge. The analysis is not in­tended to be a lament; but it does con­tain an element of warning. It is central to our purpose that the erosion of uni­versally fixed perspectives of acceptable practice in both mathematical activity and its publication be acknowledged as presenting significant challenges to the mathematical community.

Absolutism and Typographic

Mathematics

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations. 4 (G. H. Hardy)

1Michel Foucault, "Truth and Power," Power/Knowledge: Selected Interviews and Other Writings 1972-1977, edited by Colin Gordon.

2Freeman Dyson, "Mathematics 1n the Physical Sciences," Scientific American 21 1 , no. 9 (1964):130.

3VIadimir Tasic.;, Mathematics and the Roots of Postmodern Thought (Oxford: Oxford University Press, 2001 ), 5.

© 2005 Springer Science+ Business Med1a, Inc., Volume 27, Number 2, 2005 7

We follow the example of Paul Ernest and others and cast under the banner of absolutism descriptions of math­ematical knowledge that exclude any element of uncer­tainty or subjectivity. 5 The quote from Hardy is frequently cited as capturing the essence of Mathematical Platonism, a philosophical perspective that accepts any reasonable methodology and places a minimum amount of responsi­bility on the shoulders of the mathematician. An undigested Platonism is commonly viewed to be the default perspec­tive of the research mathematician, and, in locating math­ematical reality outside human thought, ultimately holds the mathematician responsible only for discovery, obser­vations, and explanations, not creations.

Absolutism also encompasses the logico-formalist schools as well as intuitionism and constructivism-in short, any perspective which strictly defines what constitutes mathe­matical knowledge or how mathematical knowledge is cre­ated or uncovered. Few would oppose the assertion that an absolutist perspective, predominately in the de facto Pla­tonist sense, has been the dominant epistemology amongst working mathematicians since antiquity. Perhaps not as ev­ident are the strong connections between epistemological perspective, community structure, and the technologies which support both mathematical activity and mathemati­cal discourse. The media culture of typographic mathe­matics is defined by centres of publication and a system of community elites which determines what, and by extension wlw, is published. The abiding ethic calls upon mathemati­cians to respect academic credentialism and the systems of publication which further refine community hierarchies. Community protocols exalt the published, peer-reviewed ar­ticle as the highest form of mathematical discourse.

The centralized nature of publication and distribution both sustains and is sustained by the community's hierar­chies of knowledge management. Publishing houses, the peer review process, editorial boards, and the subscription­based distribution system require a measure of central con­trol. The centralized protocols of typographic discourse resonate strongly with absolutist notions of mathematical knowledge. The emphasis on an encompassing mathemat­ical truth supports and is supported by a hierarchical com­munity structure possessed of well-defined methods of knowledge validation and publication. These norms sup­port a system of community elites to which ascension is granted through a successful history with community pub­lication media, most importantly the refereed article.

The interrelationships between community practice, structure, and epistemology are deep-rooted. Rigid episte­mologies require centralized protocols of knowledge vali­dation, and these protocols are only sustainable in media environments which embrace centralized modes of publi­cation and distribution. As an aside, we emphasize that this is not meant as an indictment of publishers as bestowers of possibly unmerited authority-though the present dis-

junct between digitally "published" eprints which are read and typographically published reprints which are cited is quite striking. Rather, it is a description of a time-honoured and robust definition of merit in a typographical publish­ing environment. In the latter part of the twentieth century, a critique of absolutist notions of mathematical knowledge emerged in the form of the experimental mathematics methodology and the social constructivist perspective.

In the next section, we consider how evolving notions of mathematical knowledge and new media are combining to change not only the way mathematicians do and publish math­ematics, but also the nature of the mathematical community.

Towards Mathematical Fallibilism

This new approach to mathematics-the utilization of ad­vanced computing technology in mathematical research­is often called experimental mathematics. The computer provides the mathematician with a laboratory in which he or she can perform experiments: analyzing examples, testing out new ideas, or searching for patterns. 6 (David Bailey and Jonathan Borwein)

The experimental methodology embraces digital com­putation as a means of discovery and verification. De­scribed in detail in two recently published volumes, Math­ematics by Experiment: Plausible Reasoning in the 21st Century and Experimentation in Mathematics: Compu­tational Paths to Discovery, the methodology as outlined by the authors Uoined by Roland Girgensohn in the later work) accepts, as part of the experimental process, stan­dards of certainty in mathematical knowledge which are more akin to the empirical sciences than they are to math­ematics. As an experimental tool, the computer can pro­vide strong, but typically not conclusive, evidence regard­ing the validity of an assertion. While with appropriate validity checking, confidence levels can in many cases be made arbitrarily high, it is notable that the concept of a

"confidence level" has traditionally been a property of sta­tistically oriented fields. It is important to note that the au­thors are not calling for a new standard of certainty in mathematical knowledge but rather the appropriate use of a methodology which may produce, as a product of its methods, definably uncertain transitional knowledge.

What the authors do advocate is closer attention to and acceptance of degrees of certainty in mathematical knowl­edge. This recommendation is made on the basis of argued assertions such as:

1. Almost certain mathematical knowledge is valid if treated appropriately;

2. In some cases "almost certain" is as good as it gets; 3. In some cases an almost certain computationally derived

assertion is at least as strong as a complex formal as­sertion.

4G. H. Hardy, A Mathematician's Apology (London: Cambridge University Press, 1 967), 21.

5Paul Ernest, Social Constructivism As a Philosophy of Mathematk:;s (Albany: State University of New York Press, 1998), 13.

6J. M. Borwein and D. H. Bailey, Mathematics by Expenment: Plausible Reason1ng 1n the 21st Century, A. K. Peters Ltd, 2003. ISBN: 1-56881 -21 1 -6, 2-3.

8 THE MATHEMATICAL INTELLIGENCER

The first assertion is addressed by the methodology it­self, and in Mathematics by Experiment, the authors dis­cuss in detail and by way of example the appropriate treat­ment of "almost certain" knowledge. The second assertion is a recognition of the limitations imposed by Gi:idel's In­completeness Theorem, not to mention human frailty. The third is more challenging, for it addresses the idea that cer­tainty is in part a function of the community's knowledge validation protocols. By way of example, the authors write,

. . . perhaps only 200 people alive can, given enough time, digest all of Andrew Wiles' extraordinarily sophisticated proof of Fermat's Last Theorem. If there is even a one per­cent chance that each has overlooked the same subtle er­ror (and they may be psychologically predisposed so to do, given the numerous earlier results that Wiles' result relies on), then we must conclude that computational re­sults are in many cases actually more secure than the proof of Fermat's Last Theorem. 7

Three mathematical examples

Our first and pithiest example answers a question set by Donald Knuth,8 who asked for a closed form evaluation of the expression below.

Example 1 : Evaluate

00 { kk 1 } I --,-::k -, ;c;-;-2

= -o.o840695o872765599646t . . .

k�l k. (::" " v �7Tk

It is currently easy to compute 20 or 200 digits of this sum. Using the "smart lookup" facility in the Inverse Sym­bolic Calculator9 rapidly returns

0.0840695087276559964 = � + � . We thus have a prediction which Maple 9.5 on a laptop con­firms to 100 places in under 6 seconds and to 500 in 40 sec­onds. Arguably we are done. 0

The second example originates with a multiple integral which arises in Gaussian and spherical models of ferromag­netism and in the theory of random walks. This leads to an impressive closed form evaluation due to G. N. Watson:

Example 2:

� ITT ITT ITT 1 %= ��� -TT -TT -TT 3 - cos(x) - cos(y) - cos(z)

= cV3 - 1 r2 ( _!__ ) r2 ( .!.!. )

96 24 24 "

The most self-contained derivation of this very subtle Green's function result is recent and is due to Joyce and

7Borwein and Bailey, p. 10.

Zucker. 1° Computational confirmation to very high preci­sion is, however, easy.

Further experimental analysis involved writing w3 as a product of only r-values. This form of the answer is then susceptible to integer relation techniques. To high preci­sion, an Integer Relation algorithm returns:

0= -1.* log[w3] + -1.* log[gamma[l/24]] + 4. *log[gamma[3/24ll

+ -8. *log[gamma[5/24ll + l.* log[gamma[7/24]] + l4.*log[gamma[9/24]]

+ -6.*log[gamma[ll/24ll + -9.*log[gamma[l3/24]] + 18. *log[gamma[ 15 I 24]]

+ -2.*log[gamma[l7/24]]-7.*log[gamma[l9/24]] Proving this discovery is achieved by comparing the out­

come with Watson's result and establishing the implicit[­representation of c\13- 1)2/96.

Similar searches suggest there is no similar four-dimen­sional closed form for W4. Fortunately, a one-variable inte­gral representation is at hand in W4 := fo exp( -4t)Ig(t)dt, where I0 is the Bessel integral of the first kind. The high cost of four-dimensional numeric integration is thus avoided. A numerical search for identities then involves the careful computation of exp( -t) I0(t), using

t2n exp( -t) Io(t) = exp( -t) I

n�o 22n(n!)2

for t up to roughly 1.2 · d, where d is the number of signif­icant digits needed, and

( ) ( ) 1 � II��1 (2k- 1)2

exp -t Io t = -- L V2m n�o (8t)nn!

for larger t, where the limit N of the second summation is chosen to be the first index n such that the summand is less than w-d. (This is an asymptotic expansion, so taking more terms than N may increase, not decrease the error.)

Bailey and Borwein found that W4 is not expressible as a product of powers of f(k/120) (for 0 < k < 120) with co­efficients of less than 12 digits. This result does not, of course, rule out the possibility of a larger relation, but it does cast experimental doubt that such a relation exists­more than enough to stop one from looking. 0

The third example emphasizes the growing role of visual discovery.

Example 3: Recent continued fraction work by Borwein and Crandall illustrates the methodology's embracing of computer-aided visualization as a means of discovery. They

8Posed as MAA Problem 1 0832, November 2002. Solution details are given on pages 1 5-1 7 of Borwein, Bailey, and Girgensohn.

9At www.cecm.sfu.ca/projects/ISC/ISCmain.html 10See pages 1 1 7-1 21 of J. M. Borwein, D. H. Bailey, and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A.K. Peters Ltd, 2003.

ISBN: 1 -56881 - 1 36·5.

© 2005 Springer Science+ Business Mecia, Inc., Volume 27, Number 2, 2005 9

...

.01

Fig. 1 . The starting point depends on the choice of unit vectors, a

and b.

investigated the dynamical system defined by: to : = t1 := 1 and

tn � � tn�l + Wn�l (1- �) tn�2,

where wn = a2,b2 are distinct unit vectors, for n even, odd, respectively-that occur in the original continued fraction. Treated as a black box, all that can be verified numerically is that tn � 0 slowly. Pictorially one learns more, as illus­trated by Figure 1.

Figure 2 illustrates the fine structure that appears when the system is scaled by Vn and odd and even iterates are coloured distinctly.

With a lot of work, everything in these pictures is now explained. Indeed from these four cases one is compelled to conjecture that the attractor is finite of cardinality N ex­actly when the input, a or b, is an Nth root of unity; other­wise it is a circle. Which conjecture one then repeatedly may test. D

The idea that what is accepted as mathematical knowl­edge is, to some degree, dependent upon a community's methods of knowledge acceptance is an idea that is cen­tral to the social constructivist school of mathematical phi­losophy.

The social constructivist thesis is that mathematics is a social construction, a cultural product, fallible like any other branch of knowledge. 11 (Paul Ernest)

Associated most notably with the writing of Paul Ernest, an English mathematician and Professor in the Philosophy of Mathematics Education, social constructivism seeks to define mathematical knowledge and epistemology through the social structure and interactions of the mathematical community and society as a whole. In Social Construc­tivism As a Philosophy of Mathematics, Ernest carefully traces the intellectual pedigree for his thesis, a pedigree that encompasses the writings of Wittgenstein, Lakatos, Davis, and Hersh among others. 12

11 Ernest, p. 39ff.

12Ernest, p. 39ff. 13Quoted from The Influence of Darwin on Philosophy, 1 9 1 0.

10 THE MATHEMATICAL INTELLIGENCER

For our purpose, it is useful to note that the philosoph­ical aspects of the experimental methodology combined with the social constructivist perspective provide a prag­matic alternative to Platonism-an alternative which fur­thermore avoids the Platonist pitfalls. The apparent para­dox in suggesting that the dominant community view of mathematics-Platonism-is at odds with a social con­structivist accounting is at least partially countered by the observation that we and our critics have inhabited quite distinct communities. The impact of one on the other was well described by Dewey a century ago:

Old ideas give way slowly; for they are more than ab­stract logical forms and categories. They are habits, pre­dispositions, deeply engrained attitudes of aversion and preference . . .. Old questions are solved by disappearing, evaporating, while new questions corresponding to the changed attitude of endeavor and preference take their place. Doubtless the greatest dissolvent in contemporary thought of old questions, the greatest precipitant of new methods, new intentions, new problems, is the one effected by the scientific revolution that found its climax in the "Origin of Species. "13 (John Dewey)

New mathematics, new media, and

new community protocols

With a proclivity towards centralized modes of knowledge validation, absolutist epistemologies are supported by well­defined community structures and publication protocols. In contrast, both the experimental methodology and social constructivist perspective resonate with a more fluid com­munity structure in which communities, along with their implicit and explicit hierarchies, form and dissolve in re­sponse to the establishment of common purposes. The ex­perimental methodology, with its embracing of computa­tional methods, de-emphasizes individual accomplishment by encouraging collaboration not only between mathe­maticians but between mathematicians and researchers from various branches of computer science.

Conceiving of mathematical knowledge as a function of the social structure and interactions of mathematical com­munities, the social constructivist perspective is inherently accepting of a realignment of community authority away from easily identified elites and in the direction of those who can most effectively harness the potential for collab­oration and publication afforded by new media. The ca­pacity for mass publication no longer resides exclusively in the hands of publishing houses; any workstation equipped with a LATEX compiler and the appropriate in­terpreters is all that is needed. The changes that are oc­curring in the ways we do mathematics, the ways we publish mathematical research, and the nature of the math­ematical community leave little opportunity for resistance

Fig. 2. The attractors for various ial = lbl = 1.

or nostalgia. From a purely pragmatic perspective, the com­munity has little choice but to accept a broader definition of valid mathematical knowledge and valid mathematical publication. In fact, in the transition between publishing protocols based upon mechanical typesetting to protocols supported by digital media, we are already witnessing the beginnings of a realignment of elites and hierarchies and a corresponding re-evaluation of the mathematical skill-set. Before considering more carefully the changes that are oc­curring in mathematics, we turn our attention to some per­haps immutable aspects of mathematical knowledge.

Some Societal Aspects of Mathematical

Knowledge

The question of the ultimate foundations and the ultimate meaning of mathematics remains open: we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalisa­tion.14 (Hermann Weyl)

Membership in a community implies mutual identification with other members which is manifest in an assumption of some level of shared language, knowledge, attitudes, and practices. Deeply woven into the sensibilities of mathemati­cal research communities, and to varying degrees the sensi­bilities of society as a whole, are some assumptions about the

role of mathematical knowledge in a society and what con­stitutes essential mathematical knowledge. These assump­tions are part of the mythology of mathematical communities and the larger society, and it is reasonable to assume that they will not be readily surrendered in the face of evolving ideas about the epistemology of mathematics or changes in the methods of practicing and publishing mathematics.

Mathematics as fundamental knowledge

Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.15 (Paul Dirac)

In the epistemological universe, mathematics is con­ceived as a large mass about which orbit many other bod­ies of knowledge and whose gravity exerts influence throughout. The medieval recognition of the centrality of mathematics was reflected in the quadrivium, which as­cribed to the sciences of number-arithmetic, geometry, astronomy, and music-four out of the seven designated liberal arts. Today, mathematics is viewed by many as an impenetrable, but essential, subject that is at the founda­tion of much of the knowledge that informs our under­standing of the scientific universe and human affairs. We are somehow reassured by the idea of a Federal Reserve Chairman who purportedly solves differential equations in his spare time.

The high value that society places on an understanding of basic mathematics is reflected in UNESCO's specifica­tion of numeracy, along with literacy and essential life skills, as a fundamental educational objective. This place of privilege bestows upon the mathematical research com­munity some unique responsibilities. Among them, the ar­ticulation of mathematical ideas to research, business, and public policy communities whose prime objective is not the furthering of mathematical knowledge. As well, as con­cerns are raised in many jurisdictions about poor perfor­mance in mathematics at the grade-school level, research communities are asked to participate in the general dis­cussion about mathematical education.

The mathematical canon

I will be glad if I have succeeded in impressing the idea that it is not only pleasant to read at times the works of the old mathematical authors, but this may occasionally be of use for the actual advancement of science. 16 (Con­stantin Caratheodory)

The mathematical community is the custodian of an ex­tensive collection of core knowledge to a larger degree than any other basic discipline with the arguable exception of the combined fields of rhetoric and literature. Preserved largely by the high degree of harmonization of grade-school and undergraduate university curricula, this mathematical canon is at once a touchstone of shared experience of com-

14Cited in: Obituary: David Hilbert 1862-1943, RSBIOS, 4, 1 944, pp. 547-553.

15Dirac writing in the preface to The Principles of Quantum Mechanics (Oxford, 1 930).

16Speaking to an MAA meeting in 1 936.

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005 1 1

munity members and an imposing barrier to anyone who

might seek to participate in the discourse of the commu­

nity without having some understanding of the various re­

lationships between the topics of core knowledge. While

the exact definition of the canon is far from precise, to vary­

ing degrees of mastery it certainly includes Euclidean

geometry, differential equations, elementary algebra, num­

ber theory, combinatorics, and probability. It is worth not­

ing parenthetically that while mathematical notation can

act as a barrier to mathematical discourse, its universality

helps promote the universality of the canon.

At the level of individual works and specific problems,

mathematicians display a high degree of respect for histor­

ical antecedent. Mathematics has advanced largely through

the careful aggregation of a mathematical literature whose

reliability has been established by a slow but thorough

process of formal and informal scrutiny. Unlike the other

sciences, mathematical works and problems need not be re­

cent to be pertinent. Tom Hales's recent computer-assisted

solution of Kepler's problem makes this point and many oth­

ers. Kepler's col\iecture-that the densest way to stack

spheres is in a pyramid-is perhaps the oldest problem in

discrete geometry. It is also the most interesting recent ex­

ample of computer-assisted proof. The publication of

Hales's result in the Annals of Mathematics, with an "only

ggo,.b checked" disclaimer, has triggered varied reactions. 17

The mathematical aesthetic

The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. 18 (G. H. Hardy)

Another distinguishing preoccupation of the mathemat­

ical community is the notion of a mathematical aesthetic.

It is commonly held that good mathematics reflects this

aesthetic and that a developed sense of the mathematical

aesthetic is an attribute of a good mathematician. The fol­

lowing exemplifies the "infinity in the palm of your hand" encapsulation of complexity which is one aspect of the aes­

thetic sense in mathematics.

1 1 1 11 1 1 + 22 + 33 + 44 + .

. . = 0 :ff dx

Discovered in 1697 by Johannes Bernoulli, this formula has

been dubbed the Sophomore's Dream in recognition of the

surprising similarities it reveals between a series and its in­

tegral equivalent. Its proof is not too simple and not too

hard, and the formula offers the mix of surprise and sim­

plicity that seems central to the mathematical aesthetic. By

contrast several of the recent very long proofs are neither

simple nor beautiful.

To see a World in a Grain of Sand; and a Heaven in a Wild Flower; Hold Infinity in the palm of your hand; And Eternity in an hour. (William Blake)

Freedom and Discipline

In this section, we make some observations about the ten­

sion between conformity and diversity which is present in

the protocols of both typographically and digitally oriented

communities.

The only avenue towards wisdom is by freedom in the presence of knowledge. But the only avenue towards knowledge is by discipline in the acquirement of ordered fact.19 (Alfred North Whitehead)

Included in the introduction to his essay The Rhythmic Claims of Freedom and Discipline, Whitehead's comments

about the importance of the give and take between free­

dom and discipline in education can be extended to more

general domains. In the discourse of mathematical re­

search, tendencies towards freedom and discipline, decen­

tralization and centralization, the organic and the ordered,

coexist in both typographic and digital environments. While

it may be true that typographic norms are characterized by

centralized nodes of publication and authority and the com­

munity order that they impose, an examination of the math­

ematical landscape in the mid-twentieth century reveals

strong tendencies towards decentralization occurring in­

dependently of the influence of digital media. Mutually re­

inforcing trends, including an increase in the number of

PhD's, an increase in the number of journals and published

articles, and the application of advanced mathematical

methods to fields outside the domain of the traditional

mathematical sciences combined to challenge the tendency

to maintain centralized community structures. The result

was, and continues to be, a replication of a centralized com­

munity structure in increasingly specialized domains of in­

terest. In mathematics more than in any other field of re­

search, the knowledge explosion has led to increased

specialization, with new fields giving birth to new journals

and the organizational structures which support them.

While the structures and protocols which describe the

digital mathematical community are still taking shape, it

would be inaccurate to suggest that the tendency of digital

media to promote freedom and decentralized norms of

knowledge-sharing is unmatched by tendencies to impose

control and order. If the natively centralized norms of

typographic mathematics manifest decentralization as

knowledge fragmentation, we are presently observing ten­

dencies emerging from digital mathematics communities to

find order and control in the knowledge atomization that

results from the codification of mathematical knowledge at

the level of micro-ontologies. The World Wide Web Con-

17See "In Math, Computers Don't Lie. Or Do They?", The New York Times, April 6, 2004.

18G. H. Hardy, A Mathematician's Apology (London: Cambridge University Press, 1 967), 21. 19Aifred North Whitehead, The Aims of Education (New York: The Free Press, 1957), 30.

12 THE MATHEMATICAL INTELLIGENCER

sortium (W3C) MathML initiative and the European

Union's OpenMath project are complementary efforts to construct a comprehensive, fine-grained codification of mathematical knowledge that binds semantics to notation and the context in which the notation is used. 20 The tongue­in-cheek indictment of typographic subject specialization as producing experts who learn more and more about less and less until achieving complete knowledge of nothing-at­all becomes, under the digital norms, the increasingly de­tailed description of increasingly restricted concepts until one arrives at a complete description of nothing-at-all. On­tologies become micro-ontologies and risk becoming "non­tologies. " If typographic modes of knowledge validation and publication are collapsing under the weight of subject spe­cialization, the digital ideal of a comprehensive meta-math­ematical descriptive and semantic framework which em­braces all mathematics may also prove to be overreaching.

Some Implications

Communication of mathematical research and scholar­ship is undergoing profound change as new technology creates new ways to disseminate and access the litera­ture. More than technology is changing, however, the cul­ture and practices of those who create, disseminate, and archive the mathematical literature are changing as well. For the sake of present and future mathematicians, we should shape those changes to make them suit the needs of the discipline. 21 (International Math Union Committee on Electronic Information and Communication)

. . . to suggest that the normal processes of scholarship work well on the whole and in the long run is in no way contradictory to the view that the processes of selection and sifting which are essential to the scholarly process are filled with error and sometimes prejudice. 22 (Kenneth Arrow)

Our present idea of a mathematical research community is built on the foundation of the protocols and hierarchies which define the practices of typographic mathematics. At this point, how the combined effects of digital media will affect the nature of the community remains an open ques­tion; however, some trends are emerging:

1. Changing modes of collaboration: With the facilita­tion of collaboration afforded by digital networks, indi­vidual authorship is increasingly ceding place to joint authorship. It is possible that forms of community au-

thorship, such as are common in the Open Source pro­gramming community, may find a place in mathemati­cal research. Michael Kohlhase and Romeo Anghelache have proposed a version-based content management system for mathematical communities which would per­mit multiple users to make joint contributions to a com­mon research effort. 23 The system facilitates collabora­tion by attaching version control to electronic document management. Such systems, should they be adopted, challenge not only the notion of authorship but also the idea of what constitutes a valid form of publication.

2. The ascendancy of gray literature: Under typo­graphic norms, mathematical research has traditionally been conducted with reference to j ournals and through informal consultation with colleagues. Digital media, with its non-discriminating capacity for facilitating in­stantaneous p ublication, has placed a wide range of sources at the disposal of the research mathematician. Ranging from Computer Algebra System routines to Home Pages and conference programmes, these sources all provide information that may support math­ematical research. In particular, it is possible that a published paper may not be the most appropriate form of publication to emerge from a multi-user content management such as proposed by Kohlhase and Anghelache. It may be that the contributors deem it more appropriate to let the result of their efforts stand with its organic development exposed through a his­tory of its versions.

3. Changing modes of knowledge authentication: The refereeing process, already under overload-induced stress, depends upon a highly controlled publication process. In the distributed p ublication environment af­forded by digital media, new methods of knowledge au­thentication will necessarily emerge. By necessity, the idea of authentication based on the ethics of referees will be replaced by authentication based on various types of valuation p arameters. Services that track ci­tations are currently being used for this purpose by the Web document servers CiteSeer and citebase, among others. 24 Certainly the ability to compute informedly with formulae in a preprint can dramatically reduce the reader's or referee's concern about whether the result is reliable. More than we typically admit or teach our students, mathematicians work without proof if they feel secure in the correctness of their thought processes.

4. Shifts in epistemology: The increasing acceptance of the experimental methodology and social constructivist

2°For background on these projects, see: www.w3.org/Math/ and www.openmath.org, respectively. 21The IMU's Committee on Electronic Information and Communication (CEJC) reports to the IMU on matters concerning the digital publication of mathematics. See www.ceic.math.ca/Publications/Recommendations/3_best_practices.shtml

22E. Roy Weintraub and Ted Gayer, "Equilibrium Proofmaking," Journal of the History of Economic Thought, 23 (Dec. 2001), 421--442. This provides a remarkably de­tailed analysis of the genesis and publication of the Arrow-Debreu theorem. 23Michael Kohlhase and Romeo Anghelache, "Towards Collaborative Content Management and Version Control for Structured Mathematical Knowledge," Lecture Notes

in Computer Science no. 2594: Mathematical Knowledge Management: Proceedings of The Second International Conference, Andrea Asperti, Bruno Buchberger, and James C. Davenport editors, (Berlin: Springer-Verlag, 2003) 45.

24citeseer.ist.psu.edu and citebase.eprints.org, respectively.

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005 13

· �

', ' ,, ' :

., .·

, ) ,I Fig. 3. What you draw is what you see. Roots of polynomials with coefficients 1 or -1 up to degree 18. The coloration is determined by a

normalized sensitivity of the coefficients of the polynomials to slight variations around the values of the zeros, with red indicating low sen­

sitivity and violet indicating high sensitivity. The bands visible in the last picture are unexplained, but believed to be real-not an artifact.

perspective is leading to a broader definition of valid knowledge and valid forms of knowledge representa­tion. The rapidly expanding capacity of computers to fa­cilitate visualization and perform symbolic computa­tions is placing increased emphasis on visual arguments and interactive interfaces, thereby making practicable the call by Philip Davis and others a quarter-century ago to admit visual proofs more fully into our canon.

The price of metaphor is eternal vigilance (Arturo Rosenblueth & Norbert Wiener)

For example, experimentation with various ways of rep­resenting stability of computation led to the four images in Figure 3. They rely on perturbing some quantity and recomputing the image, then coloring to reflect the change. Some features are ubiquitous while some, like the bands, only show up in certain settings. Nonethe­less, they are thought not to be an artifact of roundoff or other error but to be a real yet unexplained phe­nomenon.

5. Re-evaluation of valued skills and knowledge: Com­plementing a reassessment of assumptions about math­ematical knowledge, there will be a corresponding reassessment of core mathematical knowledge and methods. Mathematical creativity may evolve to depend less upon the type of virtuosity which characterized twentieth-century mathematicians and more upon an ability to use a variety of approaches and draw together and synthesize materials from a range of sources. This is as much a transfer of attitudes as a transfer of skill sets; the experimental method presupposes an experi­mental mind-set.

6. Increased community dynamism: Relative to com­puter- and network-mediated research, the static social entities which intermesh with the typographic research environment extend the timeline for research and pub­lication and support stability in inter-personal relation-

25Folkmar Bornemann, Dirk Laurie, Stan Wagon, Jbrg Waldvogel, SIAM 2004. 26See Borwein and Bailey, Chapter 3.

ships. Collaborations, when they arise, are often career­long, if not life-long, in their duration. The highly pro­ductive friendship between G. H. Hardy and J. E. Little­wood provides a perhaps extreme example. While long-term collaborations are not excluded, the form of collaboration supported by digital media tends to admit a much more fluid community dynamic. Collaborations and coalitions will form as needed and dissolve just as quickly. The four authors of The SIAM 100-digit Chal­lenge: A Study In High-accuracy Numerical Comput­ing25 never met while solving Nick Trefethen's 2002 ten challenge problems which form the basis for their lovely book.

At the extreme end of the scale, distributed com­puting can facilitate virtually anonymous collaboration. In 2000, Colin Percival used the Bailey-Borwein-Plouffe algorithm and connected 1, 734 machines from 56 coun­tries to determine the quadrillionth bits of 7T. Accessing an equivalent of more than 250 cpu years, this calcula­tion (along with Toy Story Two and other recent movies) ranks as one of the largest computations ever. The computation was based on the computer-discov­ered identity

which allows binary digits to be computed indepen­dently.26

A Temporary Epilogue

The plural of "anecdote" is not "evidence." 27 (Alan L. Lesh­ner)

These trends are presently combining to shape a new community ethic. Under the dictates of typographic norms, ethical behaviour in mathematical research involves ad­hering to well-established protocols of research and publi-

27The publisher of Science speaking at the Canadian Federal Science and Technology Forum, Oct 2, 2002.

14 THE MATHEMATICAL INTELLIGENCER

cation. While the balance of personal freedom against com­munity order which defines the ethic of digitally oriented mathematical research communities may never be as firm or as enforceable by community protocols, some principles are emerging. The CEIC's statement of best current prac­tices for mathematicians provides a snapshot of the de­veloping consensus on this question. Stating that "those who write, disseminate, and store mathematical litera­ture should act in ways that serve the interests of math­ematics, first and foremost, " the recommendations advo­cate that mathematicians take full advantage of digital media by publishing structured documents which are ap­propriately linked and marked-up with meta-data.28 Re­searchers are also advised to maintain personal homepages with links to their articles and to submit their work to pre­print and archive servers.

Acknowledging the complexity of the issue, the final CEIC recommendation concerns the question of copyright: it makes no attempt to recommend a set course of action, but rather simply advises mathematicians to be aware of copyright law and custom and consider carefully the op­tions. Extending back to Britain's first copyright law, The Statute of Anne, enacted in 1710, the idea of copyright is historically bound to typographic publication and the pro­tocols of typographic society. Digital copyright law is an emerging field; it is presently unclear how copyright, and the economic models of knowledge distribution that de­pend upon it, will adapt to the emerging digital publishing environment. The relatively liberal epistemology offered by the experimental method and the social constructivist per-spective and the potential for distributed research and publication afforded by digital media will reshape the pro­tocols and hierarchies of mathematical research commu­nities. Along with long-held beliefs about what constitutes mathematical knowledge and how it is validated and pub­lished, at stake are our personal assumptions about the nature of mathematical communities and mathematical knowledge. 29

While the norms of typographic mathematics are not without faults and weaknesses, we are familiar with them to the point that they instill in us a form of faith; a faith

that if we play along, on balance we will be granted fair ac­cess to opportunity. As the centralized protocols of typo­graphic mathematics give way to the weakly defined pro­tocols of digital mathematics, it may seem that we are ceding a system that provided a way to agree upon math­ematical truth for an environment undermined by rela­tivism that will mix verifiably true statements with state­ments that guarantee only the probability of truth and an environment which furthermore is bereft of reliable sys­tems for assessing the validity of publications. The simul-

28CEIC Recommendations. See: http://www.ceic.math.ca

taneous weakening of community authority structures as typographic elites are rendered increasingly irrelevant by digital publishing protocols may make it seem as though the social imperatives that bind the mathematical commu­nity have been weakened. Any sense of loss is the mathe­matician's version of postmodern malaise; we hope and predict that, as the community incorporates these changes, the malaise will be short-lived. That incorporation is tak­ing place, there can be no doubt. In higher education, we now assume that our students can access and share infor­mation via the Web, and we require that they learn how to use reliably vast mathematical software packages whose internal algorithms are not necessarily accessible to them even in principle.

One reason that, in the mathematical case, the "unbear­able lightness" may prove to be bearable after all is that while fundamental assumptions about mathematical knowledge may be reinterpreted, they will survive. In par­ticular, the idea of mathematical knowledge as being cen­tral to the advancement of science and human affairs, the idea of a mathematical canon and its components, and the idea of a mathematical aesthetic will each find expression in the context of the emerging epistemology and protocols of research and publication. In closing, we note that to the extent that there may be an opportunity to shape the epis­temology, protocols, and fundamental assumptions that guide the mathematical research communities of the future, that opportunity is most effectively seized upon during these initial stages of digital mathematical research and publishing.

Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation de­pends on particular philosophical background assump­tions. This hidden influence is an acute embarrassment to many researchers, and it is therefore not often ac­knowledged. Such fundamental notions as reality, space, time, and causality-notions found at the core of the sci­entific enterprise-all rely on particular metaphysical as­sumptions about the world. 30 (Christof Koch)

The assumptions that we have sought to address in this article are those that define how mathematical reality is in­vestigated, created, and shared by mathematicians work­ing within the social context of the mathematical commu­nity and its many sub-communities. We have maintained that those assumptions are strongly guided by technology and epistemology, and furthermore that technological and epistemological change are revealing the assumptions to be more fragile than, until recently, we might have reasonably assumed.

29As one of our referees has noted, "The law is clearly 25 years behind info· technology. " He continues, "What is at stake here is not only intellectual property but the

whole system of priorities, fees, royalties, accolades, recognition of accomplishments, jobs."

301n "Thinking About the Conscious Mind," a review of John R. Searle's Mind. A Brief Introduction, Oxford University Press, 2004.

© 2005 Springer Science+ Bus,ness Media, Inc., Volume 27, Number 2, 2005 15

Jonal

DellOUSie o 8 canada ResearCh Chaw Wl Collaboral Technolo· �n Computer Science

'RRAY STANWAY Cen re lor

Teny SlallWay Is 8 nama 1CS teacher, rades 8 o 12, lrod IS assocaa ed h Simon Fraser's Cen re Ol Experimen and Con· s ructiV8 lhemaiiCS. In spare tme he enjoys hoc:key. and cycling

��scient if ic WorkPlace· Mathematical Word Processing • U 1i Typesetting • Computer Algebra

Version 5 Sharing Your Work Is Easier

16 THE MATHEMATICAL INTELLIGENCER

c.m;.u.u;

Evolution and Design Inside and Outside Mathematics Eric Grunwald

Let me start with three dichotomies.

Although the first one (at least) is very old, I will refer to them for conve­nience by the names of twentieth­century mathematicians who have dis­cussed them.

Hardy's Dichotomy

. . . there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is 'mental' and that in some sense we construct it ourselves, others that it is outside and independent of us [ 1}.

Gowers's Dichotomy

Loosely speaking . . . the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and under­standing theories [2}.

Atiyah's Dichotomy

Geometry and algebra are the two for-mal pillars of mathematics . . . . Geom-etry is, of course, about space . . . . Al-gebra, on the other hand . . . is concerned essentially with time. Whatever kind of algebra you are do­ing, a sequence of operations is per­formed one after the other and 'one af­ter the other' means you have got to have time. In a static universe you cannot imagine algebra, but geometry is essentiaUy static [3}.

My questions in this paper are these: are the three dichotomies related, and are they special cases of a wider di­chotomy that operates outside as well as inside mathematics? My answer to both questions is yes.

The Three Dichotomies

Notice the following about these di­chotomies:

1. Cowers's and Atiyah's dichotomies describe extreme points on a spec­

trum. These extreme points can probably not be attained, but like "left-wing" and "right-wing," over­simplifications of highly complex

political views, they are a useful lan­

guage for describing real-life atti­tudes and positions.

Both Gowers and Atiyah recog­nize this explicitly. Gowers points out that most mathematicians would say that there is truth in both points of view. The distinction is, as Gowers says, between the priori­ties of mathematicians. Some math­ematicians prefer to develop a gen­eral understanding of mathematics and mathematical theories; others, to solve specific problems; but it's hard to imagine an intelligent math­ematician thinking that theories are bunk or that problems don't matter.

The extreme points of Atiyah's dichotomy also tend not to exist in isolation. One of the great glories of mathematics is the interconnected­ness of algebra and geometry, to the extent that it's sometimes hard to tell whether a piece of mathematics is actually geometry or algebra. Nevertheless, some mathematicians certainly prefer geometry and think geometrically, whereas others are happier with algebra.

It does seem possible, on the other hand, for an intelligent person to take an extreme position on Hardy's dichotomy. Indeed, Hardy himself goes on to say, " . . . I will state my own position dogmatically. I believe that mathematical reality lies outside us, that our fimction is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply the notes of our observations." For an al­ternative view, that mathematics is a social and cultural creation, see [ 4].

2. The dichotomies operate on different levels. Hardy's dichotomy is at the top level. It is concerned with mathemat­ics as a whole and the question, what is mathematics? Gowers's dichotomy acts at the level of mathematicians and their preferences and motiva­tions. And Atiyah's dichotomy deals with the structure of mathematics. When we do mathematics we move

© 2005 Spnnger Sc1ence+ Business Media, Inc., Volume 27, Number 2, 2005 1 7

around between these levels. Mathe­maticians may only rarely think ex­plicitly about the Hardy question, but at a deep level their attitude to it must affect everything they do. Atiyah, for example, likes to "move around in the mathematical waters" [5]: but in or­der to do that, he must think of the waters as being there for him to move around in. When mathematicians do mathematics their choices about what to do and the ways they go about doing it are shaped by their po­sition on Gowers's spectrum, and when we analyse the mathematics they produce we are deep in Atiyah's dichotomy.

3. The three dichotomists are not neu­tral dichotomy-observers. They make their own positions pretty clear. Hardy is dogmatic. Atiyah speaks from the heart:

. . . algebra is to the geometer what you might call the 'Faustian Offer'. As you know, Faust in Goethe's story was offered whatever he wanted . . . by the devil in return for selling his soul. Algebra is the offer made by the devil to the math­ematician. The devil says: 'I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine' . . . . the danger to our soul is there, be­cause when you pass over into al­gebraic calculation, essentially you stop thinking; you stop thinking geometrically, you stop thinking about the meaning {3}.

And Gowers calls his paper The Two Cultures of Mathematics, and makes a plea on behalf of the prob­lem-solvers:

. . . the subjects that appeal to the­ory-builders are, at the moment, much more fashionable than the ones that appeal to problem­solvers. Moreover, mathematicians in the theory-building areas often regard what they are doing as the central core (Atiyah uses this ex­act phrase) of mathematics, with subjects such as combinatorics thought of as peripheral and not

18 THE MATHEMATICAL INTELLIGENCER

particularly relevant to the main

aims of mathematics {2}.

Evolution and Design

For thousands of years, the stories through which we made sense of the world were founded on the idea of a solid design. Heroes did great deeds; vil­lains tried to stop them. The gods, or the one God, duly rewarded or punished them, and the world stayed much the same. Right was right, wrong was wrong, nothing fundamentally changed, and we all knew where we stood.

Zeus spoke, and nodded with his dark­ish brows, and immortal locks feU for­wardfrom the lord's deathless head, and he made great Olympus tremble { 6}.

When Zeus nodded you were well ad­vised to jump to it. As late as the sev­enteenth century, Newton seemed to have shown that the world operated me­chanically, like a clock. There may have been superficial variations as planets moved and apples fell, but the funda­mental design was unchanging, perma­nent. Newton himself was never a fully committed Newtonian, by the way. Like some other great scientists, he felt un­comfortable with the consequences of his own discoveries. He wanted to look through nature to see God. He believed that God, like Zeus, acted in the world and retained a perpetual involvement with and control over His creation [ 19].

Newton's God didn't merely set up the machine and press the "start" button: He was always fiddling with the world and adjusting it in ways that mortals can't predict. Newton's world was de­signed, redesigned, and re-redesigned.

Then, in the eighteenth century, Adam Smith introduced the idea of the invisible hand. According to Smith, people's desire for self-betterment, guided by their reason, together with the forces of competition that ensure ever greater efficiency, ensure the well­being of all:

It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard to their own interest [7}.

The story Adam Smith was telling, unlike Homer's, had no Zeus making us

tremble as he placed us all in his death­less design. There was no geometry, fixed in space, in Smith's story. Rather, it was about rules of procedure. In or­der to get rich you need to maximise your sales and minimise your costs. The butcher needs to cut as much meat as he can of a type that people are prepared to pay a high price for, using the most efficient equipment he can afford, and to beat down as much as possible the price of the carcasses he buys. In this way, the farmers, the slaughterers, and the dis­tributors will have to become more effi­cient, and the butcher's customers will enjoy good nourishing food, and, so for­tified, go out and enrich themselves and the rest of us. And so on. There is no need at all for Zeus to tell us what to do-all he needs to do is to set up the rules of competition, make sure we all want to better ourselves, and let rip. We are guided, not by Zeus, but by the in­visible hand detected by Adam Smith. And this hand will make sure that the world won't stay the same: it will get richer. Smith's story is based on the al­gebra of Atiyah's dichotomy: a process that moves over time. In this algebraic story there is no end-point, no objective, only rules of procedure.

In the nineteenth century, Charles Darwin did the same trick for biology. His story told how life on earth, with its astonishing variety and beauty, was the result of random variations to­gether with a rule of procedure called natural selection. This ensures that or­ganisms that manage to survive and re­produce will produce more organisms much like themselves, while the ones that can't feed themselves or find mates will simply die out. Despite ap­pearances, we are, according to Dar­win, not the result of a design at all, but the product of a battle for survival and resources, rather like Adam Smith's competitive game involving bakers, farmers, and bread-eaters. We are not, in Darwin's story, going anywhere, let alone trying to achieve perfection in or­der to deflect the nods from a god's darkish brows. We are merely trying to compete: to be more attractive than the next man or woman, and to survive, if necessary at the expense of others.

And in the twentieth century, Karl Popper told an evolutionary story about science itself. According to his

ideas, scientists do not use a technique

called "induction" to analyse data and

approach ever closer to the true design

of nature. Rather, they create theories

about the world that compete with

each other according to certain crite­

ria, the winner emerging to face chal­

lenges from new rivals.

The evolutionary stories, like alge­

bra, tell of processes through time, op­

erating according to rules of proce­

dure. The design stories are based on

the assumption of a timeless geomet­

ric structure, often a goal, an ideal of

perfection towards which we are en­

joined to strive.

Just as algebra is a more recent dis­

covery (to use the language of Hardy's

dogma) or invention than geometry, so

the algebraic, evolutionary stories,

such as Adam Smith's invisible hand

and Charles Darwin's natural selection,

are much more recent than the ancient

design stories. Betty S. Flowers [8] tells

the story of how American society has

developed under the influence of a se­

quence of myths (where the word

"myth" is not intended to imply either

truth or falsity), from the hero myths

of Greek warriors and gun-slinging

white-hatted cowboys through the "re­

ligious myth," the "enlightenment

myth" or "democratic myth," to what

she believes to be the currently pre­

dominant "economic myth." In the lan­

guage of Atiyah's dichotomy, the old

hero myths are geometric in nature,

based on a fixed design of right and

wrong, good and evil. The religious

myth is still geometric. The enlighten­

ment/democratic myth also involves an

element of geometry through the idea

of a search for pre-Popper scientific

"truth" and the "best" solutions, al­

though it also contains a considerable

element of algebra: democracy is a

process of votes, elections, changes of

policy and rulers in the light of experi­

ence, more elections and so on, with

no fixed geometric end in view. The

economic myth, which is so influential

nowadays, is pure algebra, pure evolu­

tionary process. There is no end-point:

the imperatives are to get more effi­

cient, get bigger, get richer!

The evolutionary stories have a pe­

culiar power to discomfort us. Many of

us are still trying to come to terms with

their real meaning. We feel uncomfort-

able with stories that don't give us the

comfort of clear moral rules, elegant de­

signs, and God-given systems of rewards

and punishments. The initial objections

to Darwin's evolutionary theory were

not principally against the notion that

species vary or even that we are de­

scended from apes, ideas that had been

current in the scientific community for

some time. Darwin's predecessors had

assumed that evolution was a goal-di­

rected process, that each stage of evo­

lutionary development was a more per­

fect realisation of a plan that had always

been in existence: a plan, perhaps, pres­

ent in the mind of God. Darwin disposed

of this comforting idea [9] . It was the

lack of a fixed goal that shocked Dar­

win's contemporaries, and continues to

shock today. The fundamental objection

is that Darwin moved us from geometry

to algebra.

Protesters who disrupt interna­

tional meetings to agitate against

something called "globalisation" are

also shocked by a shift from geometry

to algebra. They are protesting against

the economic myth. Statesmen and ex­

ecutives of multi-national companies

may explain and illustrate the benefits

of joining the global economy, but they

find their motives questioned. From a

geometric point of view, it is necessary

to describe one's overall design, which

soon acquires the status of a moral

code. To a geometer, the algebraist's

simple answer "The process works!"

cuts no ice.

Language The linguistic scientist David Crystal has

written a book called The Stories of English [ 10]. The plural in the title de­

liberately suggests his viewpoint: that

there is no such thing as "standard" Eng­

lish, spoken in England, with somewhat

defective versions in the USA, India,

Nigeria, and so on. Rather, English is an

evolutionary system with different vari­

ants spoken in different places and at

different times. The rules of procedure

that underlie this evolution are not

clearly understood: as yet, we haven't

found the linguistic equivalent of Smith's

invisible hand or Darwin's natural se­

lection. We don't know why, for exam­

ple, the second person plural has be­

come youse in Liverpool, ye and yiz in

parts of Ireland and Scotland, and y'aU

in the southern states of the USA, where

it can be spelled you aU, you-aU, ya 'U, yawl, or yo-aU [ 10, p.449]. Nevertheless,

in the linguistic equivalent of Hardy's di­

chotomy, Crystal stands firmly on the

side of evolution. There is no fixed de­

sign for the English language, he says, to

which we should all aspire. The lan­

guage is constantly changing through

complex social processes of which we

have only a limited understanding, and

it has no end to aim at.

Crystal tells how English has gone

through phases when it has split into

different species, and other periods

when dialects have been brought

closer together, sometimes prescrip­

tively. In mediaeval times, for example,

regional variation was rife in England,

but the introduction of printing, to­

gether with the establishment of writ­

ten laws, made standardisation impor­

tant: for the rule of law to work, it was

essential that people could understand

what was written and interpret it as far

as possible in exactly the same way.

The most highly prescriptive period

reached its zenith in the eighteenth and

nineteenth centuries, when English

dictionaries were first published, and

people could write things like

harsh as the sentence may seem, those at a considerable distance from the capital, do not only mispronounce many words taken separately, but they scarcely pronounce, with purity, a single word, syllable, or letter [ 1 1 j.

According to Crystal, English is un­

dergoing a period of rapid evolution,

with many new species developing

round the world out of British and Amer­

ican English, the first two to emerge.

The proponents of design in English

tend to conflate their own strict views

with a moral code. The following letter

to a newspaper, with its morally loaded

words, is a typical example:

. . . Oh, and the phrase "dumbing down. " A wholesale and quite horrible corruption of the verb, filtered in from the US, I accept, but no more forgiv­able for that. Dumb is a perfectly ho­nourable word, meaning an inability to speak, and its modern usage is very lazy, unacceptable and a sign of ver­bal and literate degeneracy [ 12].

© 2005 Spnnger Sc1ence+Business Med1a, Inc., Volume 27. Number 2, 2005 1 9

The novelist Kingsley Amis has

identified the two extreme points on

Gowers's spectrum as they apply to the

use of the English language. He calls

them berks and wankers ( 13] . Berks

don't care about the language's design,

they are content to let it evolve hap­

hazardly. They are "careless, coarse,

crass, gross. . . . They speak in a slip­

shod way." Left to berks, a language

would "die of impurity. " They are ex­

treme problem-solvers in Gowers's

language, using whatever expressions

come to mind: they certainly don't care

about grammar. Wankers, on the other

hand, are obsessed with the mainte­

nance of what they mistakenly believe

to be timeless rules of English. They

are "prissy, fussy, priggish, prim. . . .

They speak in an over-precise way . . . . "

They write letters to newspapers, such

as the one above, complaining about

new usages. Left to wankers, a lan­

guage would "die of purity." Wankers

are on the theory-building extreme of

Gowers's spectrum: mathematical

wankers (if they existed) would take

so much care of the structural unity of

mathematics that it would become too

inward-looking to have any relevance

outside itself.

What about Atiyah's dichotomy? Can

we fmd linguistic equivalents of fixed

geometric space and an algebraic

process over time? There is some evi­

dence that our brains do operate in

time-like and space-like ways when we

use regular and irregular verbs. (Regu­

lar verbs in English are those, such as

"continue" or "risk," that add -d or -ed

to form the past tense and past partici­

ple, as opposed to irregular verbs such

as go-went-gone and ring-rang-rung.)

The experimental psychologist

Steven Pinker reckons that the brain

produces regular and irregular verbs in

two quite different ways [ 14]. Accord­

ing to him, regular verbs are produced

through a series of rules, which the

brain knows. One such rule is: "add

-ed to make the past tense and the past

participle." There's no need to look up

the past forms in your memory: merely

follow the rules.

Irregular verbs, on the other hand,

are retrieved from memory. But, ac­

cording to Pinker, this memory isn't a

mere string of words, it includes a net­

work of connections between them.

20 THE MATHEMATICAL INTELLIGENCER

That's why irregular verbs show so

many patterns: blow-blew, grow-grew,

throw-threw, and so on. If irregular

verbs were stored as an unconnected

list, we would expect them to be quite

random. It also explains why many

verbs have become irregular. Ring-rang

used to be ring-ringed, but it became at­

tracted to ing-ang-ung by analogy with

words like "sing." Quit-quit became pop­

ular only in the nineteenth century: Jane

Austen used "quitted." Light-lit, creep­

crept, kneel-knelt, dive-dove, catch­

caught are other examples. And sneak­

snuck is in the process of becoming

standard in America. We can also see

this process at work in experiments with

meaningless verbs. For instance, 800/o of

people suggest that the past tense of

"spling'' is "splang" or "splung."

I like to think of the irregular verb

forms as crystals. These crystals have

formed, somehow or other, out of the

surrounding flow of general rules.

Once a crystal has become established,

it can grow by attracting other similar

forms. Bear-bore attracted wear-wore,

for example. A crystal will form more

easily, and an irregular verb enter the

mental dictionary more easily, if its

pattern is repeated more frequently.

That's why the irregular verbs tend to

be frequently used: of the most com­

monly used verbs in English, the first

regular ones are "use" (fourteenth

place) and "seem" (sixteenth place)

[ 15]. Most of the crystals that spring

into existence by the sporadic use of a

new irregular form of a verb will die

away. In order to grow and survive,

these crystals need a lot of use.

The crystals are examples of design. Of course the crystals originally sprang

up as the result of some sort of evolu­

tionary process, but once they have been

created, their designs are so powerful

that they become fixed and draw other

words towards them, making them more

like themselves. Like design stories,

such as religions or Zeus with his im­

mortal locks, these designs are hard to

resist. People who say that the past

tense of "spling" is "splung" are partici­

pants in an old and gripping design story.

So our brains use two ways of work­

ing-at least, as far as producing past

tenses of verbs is concerned. Either we

take an evolutionary path, consult our

rule book, follow its instructions, and

add -ed or -d to the end of the verb; or

we look up our special dictionary of de­

signs, with its network of connections,

for the irregular verbs. These two modes

of operation can both be highly creative.

As we have seen, the networked dictio­

nary creates new irregular verbs all the

time. And, although the word "rule" may

sound dull and constricting, the general

rule for past tenses can, and often does,

lead to the creation of completely new

expressions. Pinker has noted "he out­

Clintoned Clinton," and made-up chil­

dren's words such as "spidered" and

"lightninged." Living in Malaysia, I was

delighted when people "onned" and

"offed" the lights.

There seems, therefore, to be a "di­

chotomorphism" between mathemat­

ics and language under which the six

dichotomists are mapped onto each

other (Fig. 1).

Discovery and Synthesis

Leaders in business organisations can,

and do, use two kinds of approach. They

can set up a series of rules to determine,

for example, the processes by which a

business should run. They might specify

the circumstances under which parts of

the business should be divested or, al­

ternatively, given extra resources. These

rules might be in terms of return on cap­

ital, or profit, for example. And then they

can simply allow the different parts of

the business to have their heads but sub­

ject them to the rigour of the rules. This

is an evolutionary approach: there is no

objective, merely a process. The overall

business is broken into separate parts,

each of which is free to go its own way

within the rules. This approach is simi­

lar to the way we make regular verbs:

we break the word up into its con­

stituent parts, and apply a series of rules

to each part.

An extreme kind of holding com­

pany would use an evolutionary ap­

proach, an algebraic process though

time. The Chief Executive Officer

(CEO) would, in principle, say to each

of the business managers something

like, "Make a business plan that brings

in a return on capital of at least 12%.

Then go away and implement the plan.

Come and see me again next year with

your 12% return and with the next plan.

If during the course of the year you can

see that you won't make your 12%, let

Level Mathematics Language

Whole Hardy B Crystal

People Gowers B Arnis

Structure Atiyah B Pinker

Fig. 1. The "dichotomorphism" between mathematics and language.

when great chunks of the British econ­omy were moved from the geometric central control of government owner­ship and made to ply their trades ac­cording to the algebraic rules of the mar­ket, with no destination in view. It was a time when government gradually stopped trying to pick the economic sec­tors in which Britain should be a big player-steel or car manufacture, for ex­ample-and started trying instead to set a framework, a process within which the players would determine their own fu­ture. The Department of Trade and In­dustry was transformed (to a large ex­tent) from the British economy's owner into its regulator. It used to do geome­try: to try, at least, to determine the con­figuration of Britain's industrial econ­omy. But now it does algebra: it sets the rules and allows the economy to develop over time. In Amis's linguistic terms, one might say that where wankers once tried unsuccessfully to shape the British economy, now berks don't care what it looks like as long as it works.

me know. But, unless there's a very good reason for it, watch out!"

Such a CEO would have little idea of the overall shape of the business in three or four years' time. He or she might not even care: as long as we're making a good return, why worry about what we're selling? This is a highly Dar­winian, evolutionary approach. Nature doesn't care where each species is go­ing. And, like the rule of natural selec­tion and the rules which shape regular verbs, the rules of the market bring us extraordinary new species.

On the other hand, business leaders can use a design approach. They can view the overall business as a whole and impose a structure on it, fixed in space like a geometrical object. This kind of approach leads to concepts like the "core business." CEOs say, in effect, "This is what we do, trying out other lines of business diverts too much man­agement effort, we should stick to what we do best." The parts of such a busi­ness come together. It is reminiscent of the way our brains deal with irregular verbs. First, you spot a crystalline pat­tern: widget factories in West Europe are good business. Then you focus on that pattern: our core business is widget manufacture in West Europe. Then you grow the crystal slowly: we will invest a little in splodget factories in West Eu­rope and, maybe, widget factories in East Europe.

The two approaches--evolution and design-are quite different. But, just as the brain needs both a set of rules for regular verbs and a dictionary with con­nections for irregular verbs, corpora­tions need to use both evolutionary and design approaches in different parts of

the business, and at different times. The skill is to know when to break up and al­low to evolve and when to bring together into a known crystalline pattern which will grow in a relatively controlled way. Many companies go through phases of evolution, of breaking apart existing structures, when they give individual business managers freedom to diversify into new, apparently profitable, busi­nesses. They generally do this in good times, when the risk of a few loss-mak­ing operations can be borne in view of the possibility of some bonanzas. And they also go through phases of design, of bringing together. In harder times, when companies decide they can no longer af­ford to wait for experimental businesses to become profitable, businesses are sold off or shut down, and the company retreats to its "core business."

Gary Hamel, well-known in the world of business as a strategy expert, has de­scribed strategy as a process of discov­enJ and synthesis [16] . Discovery is the algebraic process of moving through time, of following clear rules of engage­ment that allow businesses freedom to evolve and determine, on the basis of their performance, whether they are to be shut down or expanded. Synthesis is the geometric redesign in space, the bringing together of current knowledge and insights into what appears to be the best business configuration at a partic­ular point in time.

Right and Left

Another complex system, the British economy, made a significant move from design towards evolution in the 1980s, under Margaret Thatcher. This was the beginning of the era of privatisation,

Just as Atiyah sees algebra as the devil's Faustian offer, many people felt that some sort of Faustian bargain was being struck in 1980s Britain: that the country was giving something up, per­haps social cohesion or even decency and kindness, by its pursuit of wealth in the turbulent marketplace. It felt at the time like a turning point, and with the benefit of hindsight we can see that indeed it was. Britain's long relative economic decline stopped. In 2002, for the first time, the UK rate of unem­ployment was the lowest of all the G 7 countries. And for the first time since

the 1970s, the UK economy overtook that of France. This may explain why Britain has, ever since, voted for more or less Thatcherite governments led by Margaret Thatcher herself, John Major, and Tony Blair. If the British people have sold their soul, at least the devil has kept his or her side of the bargain.

It's worth pointing out that, although the British economy moved towards an evolutionary story, through privatisa­tions and increased labour-market flexi­bility, in another respect the country be­came more centralised, more geometric. By 2002, government in Britain raised only 4% of taxes through local govern­ment, compared with 1ZO/o in the US and 100/o in France. Mrs Thatcher didn't trust

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc., Volume 27, Number 2, 2005 21

local councils to push the evolutionary

revolution through-probably rightly.

Neither did her successors [ 1 7].

Vince Cable, a Liberal Democrat

Member of Parliament in the UK, has

written about the distinction between

left and right in politics, how it has

blurred and lost meaning and signifi­

cance in recent years [ 18]. Following the

end of the Cold War, the issues that ex­

cite and divide people don't fall within

the framework of left-wing and right­

wing. Such questions as regional inte­

gration and loss of sovereignty (in the

EU, for example), minority rights, and

immigration are not left-right matters,

and the existing political parties, having

grown up in the old left-right era, are of­

ten thoroughly divided when it comes to

these issues. Many of the "new fissures"

in the world, as Cable terms them, are

along questions of identity. What nation

do I feel I belong to? In what minority

group, linguistic or racial or sexual or

professional, do I feel most at home?

We can now think of the old left-right

distinction as a question of evolution

versus design. The Left had a solid, so­

cialist design in mind. They knew what

they wanted the world to look like, and

were less clear on how to get there. The

Right, on the other hand, wanted a cer­

tain process through time, a world op­

erating largely through the evolutionary

processes of free markets, but they were

unclear about where the world should

be heading. The Left did geometry. The Right did algebra. One might say that

while the Left sometimes thought the

ends justified the means, the Right be­

lieves the means justify the ends. By the

end of the twentieth century, in the post­

Cold War world, this conflict essentially

ended in victory for algebra. (A big sur­

prise for many people who, like me, en­

tered university in 1968, the Year of Rev­

olutions.)

Following the victory of evolution,

people are yearning for the comfort of a

solid design. An evolutionary world is a

cold, soulless place. The unease people

feel with evolutionary theories like Dar­

win's natural selection, and the repulsion

with which many view the evolution of

free markets under Smith's invisible

hand, leads them to search for a new de­

sign, an identity through nationhood or

language or colour or sexual orientation,

in an algebraic, ever-changing, evolu-

22 THE MATHEMATICAL INTELLIGENCER

tionary world. The politics of evolution

versus design is being replaced by the

search for design in an evolutionary

world. And we can see in the aftermath

of the twentieth century's great design

theories-communism, socialism, and

fascism-the beginnings of some design

theories for the twenty-first century: re­

ligious fundamentalism and a strong

form of environmentalism known to its

detractors as "eco-fascism."

Back to Mathematics

The dichotomy of evolution and design

provides different points of view on a

number of complex systems including

mathematics, science, language, busi­

ness organisations, economics, and

politics. The design point of view, like

geometry, centres on an image fixed in

space, often one towards which we

strive. The evolutionary point of view,

on the other hand, has no end-point,

and deals with a process through time.

In the case of mathematics, the two

points of view are reflected in the three

dichotomies with which this paper was

introduced: they describe the distinc­

tion between evolution and design as

it impinges upon the overall philoso­

phy of mathematics, the motives and

interests of mathematicians, and the

structure of mathematics itself.

The evolutionary point of view is

more recent than the design idea. Al­

gebra was invented/discovered long af­

ter geometry, and the apparently sim­ple evolutionary ideas of Smith and

Darwin emerged thousands of years

later than the religious and mythical

theories and stories. Science managed

things an order of magnitude more

quickly: the gap from Bacon to Popper

was merely hundreds of years.

Evolutionary stories have the power

to discomfort us. Darwin's ideas are

still the subject of heated debate, and

"globalisation" invokes fierce protests.

Design theories often become associ­

ated with moral codes, so that Dar­

win's theory is seen as "anti-religious,"

capitalism can be thought to be im­

moral, algebra can be said to be a Faus­

tian offer, and Gowers writes a paper

imploring theory-builders not to look

down on problem-solvers.

Geometry/design, on the other hand,

while generally more comforting than

algebra/evolution, can, if we're not

careful, degenerate into stagnation. In­

deed, geometry is sometimes used as a

metaphor in this very sense:

The impression is of two men at the height of their abilities but exhausted and immobilised by the fixed geome­try of their power [21].

Large, overly bureaucratic compa­

nies that concentrate on synthesis to the

exclusion of discovery become resis­

tant to change, and are eventually

picked off by nimbler competitors.

Crystal [10] observed that the vast ma­

jority of comments he received from lis­

teners to his radio series about English

were complaints about linguistic evolu­

tion: nobody seemed to be worried that

the language might be too static.

People move between the evolution­

ary and design points of view, just as

mathematicians switch between algebra

and geometry: we all use both regular

and irregular verbs and have moments of

linguistic experimentation and pangs of

irritation when other people push the

boundaries of language. A mathemati­

cian need not be on the same side of each

of the three dichotomies. Complex busi­

ness organisations can operate in evolu­

tionary, discovery mode in one part of

the business while retreating to the core

business, in synthesis mode, in another.

The British economy moved towards

evolution when state industries were

privatised while simultaneously moving

towards design when power moved

from local to central government. We

shouldn't think of evolution and design

as mutually exclusive modes of opera­

tion, rather as two distinct points of view

that are often relevant in understanding

and describing what's going on.

In some fields, the theory of biolog­

ical species for example, the two

points of view do not have equal sta­

tus: as evidence is accumulated one

point of view becomes preferable to

the other. But in many cases, such as

mathematics, the existence of two

points of view enriches our under­

standing of the whole.

Atiyah's dichotomy describes the

most exquisite conjunction between

evolution and design that has ever been

achieved in any complex system: the

interplay between algebra and geome­

try. Although geometry existed for a

long time before the evolutionary in­

terloper appeared, and although there

are no doubt many local skirmishes, peace has essentially broken out be­tween algebra and geometry to the benefit of everyone: many of the liveli­est parts of mathematics have names like algebraic topology, differential geometry, and analytic number theory.

Gower is talking about evolution and design as they affect the politics of math­ematics. Who sneers at whom? Who gets the plum jobs? Who gets the money? The situation he discusses seems eerily fa­miliar to observers of complex business organisations. The people running the core businesses are trying to perfect a design. One senior executive even told me once that he thought of himself as the steward of a stately home: his role was to hand his successor the business in inunaculate condition. The evolution­ists in big companies are the people dri­ving the diversifications, people in an en­ergy company trying to develop a timber business, for example, or those in a med­ical insurance company developing a care homes business. Like Gowers's problem-solvers, they don't want to per­fect the current design, they want to strike out in new directions, not caring much about the connections with the present design: they merely want to solve the problem of making more money for the business. To adapt Amis's words, left to the stately home stewards, the business would die of purity; left to the diversifiers it would die of impurity.

The power in large companies usu­ally lies with the big battalions of the core business, because, inevitably, that's where most of the money and the people are. Most of the new ventures fail, so the safest thing to do when times are hard is to abandon them al­together. (I might add that it's gener­ally easier to have an intelligent con­versation about these matters in a business context than in mathematics, because in business there are more or less agreed financial criteria against which to judge proposals.) I find it hard to see how the situation could be much different. The evolutionists, mutating on the fringes, are in an uncomfortable and dangerous region. Most of them will probably fail. The designists in the core business of mathematics may make life difficult for them, and per-

haps one should view this as part of the competitive, evolutionary environment

that ensures that the unsuccessful mu­tations die quickly.

In [22], Atiyah recognises the need for a balance between ensuring the continuity and unity of mathematics (i.e., taking care of its design) and al­lowing the evolutionary possibility of exciting new discoveries that might at first appear to be disjoint from the mathematical core. In practice, when, for example, specific funding decisions have to be made that might affect this balance, the outcome may well depend on whether the decision-makers are natural evolutionists or designists.

On the two sides of Hardy's di­chotomy are people who believe in a timeless design of mathematics that it is our duty to discover, and those who think of mathematics as a human creation evolving through complex processes over time. Of the three di­chotomies, this is the least capable of discussion on the basis of evidence or facts. It is truly a matter of point of view: what inspires you to do mathe­matics? To what extent is it to under­stand and explain the design of the world outside us, including mathemat­ics itself; and to what extent is it to cre­ate something beautiful and remark­able that evolves and grows over time?

Hardy's dichotomy generates a good deal of heat because, when people ar­gue about it, they are arguing not about mathematics or even prestige, power, or money; they are arguing about them­selves. The issues are their own deeply held beliefs. The stakes are really much higher than in debates on the other two levels. Lose the argument and what's left of you? Thus Hardy, sensibly enough, is merely dogmatic, while Mar­tin Gardner (referring to the mathe­matics of elementary particles) ad­monishes, "To imagine that these awesomely complicated and beautiful patterns are not 'out there,' indepen­dent of you and me, but somehow cob­bled by our minds in the way we write poetry and compose music, is surely the ultimate in hubris'' [20]. Some be­lievers in the design theory of mathe­matics may want to batter the evolu­tionists into submission. Judging from the infiltration of algebra into geome­try's domain, and the successes of the

evolutionists in economics, politics, bi­ology, epistemology, and linguistics,

perhaps they have a point.

REFERENCES

[ 1 ] G. H. Hardy, A Mathematician's Apology,

Cambridge University Press, 1 969, 1 23.

[2] W T. Gowers, The Two Cultures of Math­

ematics, Mathematics: frontiers and per­

spectives, 65-78, Amer. Math. Soc. Provi­

dence, R . I . , 2000.

[3] M. F. Atiyah, Mathematics in the Twentieth

Century, Am. Math. Monthly (200 1 ) , 1 08 ,

no. 7 , 654-666.

[4] R. Hersh, What is Mathematics, Really?,

Jonathan Cape, 1 997.

[5] M . F. Atiyah, An interview with Michael

Atiyah , Math. l ntell igencer 6 (1 984), no.

9-1 9.

[6] Homer, The Iliad.

[7] A. Smith, Wealth of Nations.

[8] B. S. Flowers, The Economic Myth, Center

for International Business Education and

Research, University of Texas at Austin .

[9] T. Kuhn, The Structure of Scientific Revo­

lutions, 1 962, 17 1 .

[1 0] D. Crystal, The Stories of English, Allen

Lane, 2004.

[1 1 ] J. Walker, Pronouncing Dictionary of Eng­

lish, 1 774, quoted in [1 0] , 408.

[1 2] D. Taylor, The Times, letter, 21 Septem­

ber 2004, 32.

[1 3] K. Amis , The King's English, Harper

Collins, 1 997.

[1 4] S. Pinker, Words and Rules, Weidenfeld & Nicholson, 1 999.

[1 5] N. Francis and H . Kucera, Frequency

Analysis of English Usage; Lexicon and

Grammar, 1 984.

[ 1 6] G. Hamel, Masterclass at CBI Conference,

November 1 999.

[1 7] The Economist, 4 May 2002, 29.

[1 8] V. Cable, The World's New Fissures: Iden­

tities in Crisis , Demos, 1 994.

[1 9] B. J . T. Dobbs, The Janus Faces of Ge­

nius, Cambridge University Press, 1 99 1 .

[20] M. Gardner, Math. lntelligencer 23 (2001 ) ,

no. 1 , 7 .

[2 1 ] S. Jenkins, The Times, 2 9 September

2004.

[22] M. Raussen and C. Skau, Interview with

Michael Atiyah and Isadore Singer, EMS

September 2004.

Perhelion Ltd.

1 87 Sheen Lane

London SW1 4 8LE

UK

e-mail: ericGrunwald@compuserve.com

© 2005 Springer Sc1ence+ Business Media. Inc., Volume 27, Number 2, 2005 23

Mathematic a l ly Bent

The proof is in the pudding.

Opening a copy of The Mathematical

Intelligencer you may ask yourself uneasily, "lt?tat is this anyway-a mathematical journal, or what?" Or you may ask, "lt?tere am I?" Or even "lt?to am I?" This sense of disorienta­tion is at its most acute when you open to Colin Adams's column.

Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

Column editor's address: Colin Adams, Department of Mathematics, Bronfman

Science Center, Williams College,

Wil liamstown, MA 01 267 USA

e-mail: Colin .C.Adams@williams.edu

Col i n Adams, Editor

The Mathematical Ethicist Colin Adams

Dear Dr. Brad,

I have gotten in the habit of throwing a lavish banquet for my students the evening before I hand out the student course evaluations each semester. My question is whether or not it is appro­priate to have the students fill out the forms at the banquet. There is a break in the festivities after the dinner but preceding the floor show, which could serve for this purpose.

Best,

Waldo Wendt University of Westport

Dear Waldo

I am assuming that the course evalua­tions play a substantial role in the tenure process at your institution, and further, that you are junior faculty. It is a rare senior faculty member who throws a banquet with floor show for his or her students.

Given these assumptions, it be­hooves you to behave in a manner that cannot possibly be interpreted to sug­gest, in the slightest way, that you are attempting to influence the outcome of the student course surveys. Even the hint of such impropriety could be­smirch your career permanently. In other words, do not give the surveys out between dinner and the floor show. Wait until the next day. Besides, if they did fill them out right after the banquet, the students would not yet have expe-

24 THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media, Inc.

rienced the floor show. So it could not influence their survey responses, a complete waste of what must be a ma­jor component of your budget. Com­mon sense, fella?

-Dr. Brad

()()()

Dear Dr. Brad,

I was recently invited to give a talk at the prestigious Oberwolfach Tricen­tennial Conference on Number Theory. This was quite an honor for me, par­ticularly since I am not a number the­orist. However, when it was time for me to give my talk, the conference or­ganizer introduced me by saying, "And here is the man who proved Fermat's Last Theorem, Andrew Wiles." Unfor­tunately, I am not Andrew Wiles, and it was at this point that I realized that a mistake had been made. It was never their intent to invite me at all. I looked out at the sea of expectant faces, and did the only thing I could do. I pre­tended to be Andrew Wiles for the next hour, receiving a standing ovation at the end. Did I do the right thing?

Andre Wilson Prinsetown University

Dear Andre,

Under the circumstances, you did the right thing. You certainly wouldn't want to disappoint the audience, many of whom had come a long way to hear Dr. Wiles. Luckily, mathematics is eso­teric enough that you can make terms up on the fly, and the audience mem­bers will be too embarrassed to ac­knowledge that they have no idea what the heck you are talking about. So they applaud at the end, even though they haven't understood. Would you con­sider giving a talk at my institution? We can't afford Wiles.

-Dr. Brad

()()()

Dear Dr. Brad,

A paper of mine was recently published in the Journal of Algebra. It was a translation of a paper by an obscure Bulgarian author that appeared in 1972

in Bioavtomatika, the Journal of the Bulgarian Academy of Science. How­ever, I did make a few changes in the spacing and in the numbering of the theorems. Unfortunately, the Bulgarian author discovered my translation (turns out he speaks English, who knew?) and he seems to be upset. Should I have ref­erenced his paper in the bibliography?

Hortense Galbloddy College of St. Geronimo

Dear Hortense,

Do ideas transcend the language in which they are stated? And is it the ideas with which we credit a creator? These are questions that I ask myself some­times, when I am in the shower, and I don't feel like going to work After sev­eral hours of consideration, I step out of the shower, a complete prune, and think about what to have for lunch. But enough about me and my day.

The short answer is yes, ideas do transcend the language in which they are created. Hence, if the ideas in your paper are exactly the same as the Bul­garian's ideas, you must credit him with their discovery. But what is an idea? How does one decide if two ideas are identical? These are questions I reserve for my time in the tub. My con­clusion is that there is no idea yardstick which can be utilized to determine the size of an idea, and to compare it with the size of other ideas. So who is to say if your translation actually captures the same ideas that the Bulgarian was at­tempting to express? How can you know what was in his head? You can't. So sleep easy. You need not feel guilty for neglecting to include his paper in your references.

-Dr. Brad

P. S. Of course, the Ethics Committee at your institution may see it differ­ently. You might want to send some ex-

pensive presents to your new Bulgar­ian friend.

()()() Dear Dr. Brad,

At a recent conference, I saw a very nice talk on laminated deck transfor­mations. Afterward, I suggested to the speaker that he might want to extend his results to poly laminated deck trans­formations. I was chagrined, six weeks later, when the editor of a prestigious journal asked me to referee a paper by this same speaker, in which he ex­plained laminated deck transforma­tions and the extension to polylami­nated deck transformations, with no mention of me whatsoever. I see that I have three alternatives.

1. I could contact the author and let him know I am the referee, making it clear there is no way I will rec­ommend the paper for publication without my own name on it as co­author.

2. I could submit my referee's report, recommend that the editor reject the paper, and in the meantime, write up my own version and sub­mit it elsewhere.

3. I could send out a blanket e-mail to everyone in the field explaining how this cretin tried to steal my idea.

But whatever happens, I want to make sure that my actions are completely ethical and above reproach. I look for­ward to hearing from you as soon as possible.

With great respect,

Dr. Donald Dumpstead Ullalah U.

Dear Don,

The question of what constitutes a suf­ficient contribution to a paper to jus­tify inclusion as a co-author is one of the most difficult and slippery in all of mathematics. It is a question that oc­cupies my thought processes when I brush my teeth every morning. I reserve that time to consider it. Can a single lemma be enough? Brush, brush. One

theorem, two theorems? Brush. And what about a corollary? Spit. Some­times I find myself lost in thought, froth dribbling from my open mouth, the sound of banging on the bathroom door from desperate members of my family echoing in the background.

But in your case, the time it takes to

floss should suffice. It boils down to a single word. Polylaminated.

In fact, your contribution wasn't even the whole word. It was actually just the prefix. Does a four-letter pre­fix justify inclusion as a co-author in a paper. It turns out that there is a prece­dent. In its landmark ruling of 1967,

the Ethics Committee of the Canadian Mathematical Society determined that a prefix of three or fewer letters does not suffice to presume co-authorship. Hence the prefixes sub-, dis-, ir-, bi-, in-, co-, and non- do not cut the mustard.

However, in an intricate argument I will not attempt to recreate here, they determined that a prefix of four or more letters, as long as at least one letter is from the last ten letters of the alphabet, does suffice. Hence, any of para-, trans-, null-, pseudo-, semi-, ortho-, or quasi­will do nicely. But endo- doesn't quite make the cut. Of course, if the prefix contains a Greek letter, such as a-, there is no lower limit on the number of let­ters necessary to warrant co-authorship. For God's sake, it's a Greek letter.

In your case, poly- does the trick. This means that any of the three alter­natives you outlined above would be fully justified, and you can rest assured, if nowhere else, you will find support on the Ethics Committee of the Cana­dian Math Society.

-Dr. Brad

()()() This concludes another column. But re­member, when you find yourself tan­gled in the morass of mathematical morality, you are only an e-mail away. I hope you don't have to write often. But a letter once in a while wouldn't hurt.

Conscientiously yours, Dr. Brad Dearborn, Ph.D.

© 2005 Springer Science+Business Media, Inc , Volume 27, Number 2 , 2005 25

GOVE EFFINGER, KENNETH HICKS, AND GARY L. MULLEN

I ntegers and Po ynom ia s · Com pari ng the Close Cous i ns Z and Fq [x]

umber theory is an enigmatic discipline; its fundamental simplicity is

tempered by a very rich complexity. Few areas of human inquiry give rise

to questions which are so easy to ask but so difficult to answer. The sim-

plicity is obvious in the straightforward, even intuitive, definition of the

positive integers. The complexity arises when, among other

things, one attempts to isolate the fundamental building

blocks (the prime numbers) and their distribution. How­

ever, the integers are not the only ring with this enigmatic

property. In this paper, we contrast and compare the ring

of integers and the ring of polynomials in a single variable

over a finite field. We explore a number of questions which

have analogous versions in both settings. While these ques­

tions are quite easy to state, they are at the same time not

easily answered in either setting.

Why Fq[x] Is Special

As is standard, we denote the integers as Z and the posi­

tive integers as z+. Viewed algebraically, Z is a commuta­

tive ring, and, in fact, Z is a very nice ring in the sense that

it is a unique factorization domain, which means that any

integer can be written uniquely (up to the ordering of the

factors) as the product of a unit and powers of prime num­

bers (the units, i.e., elements possessing multiplicative in­

verses in the ring, of Z are simply 1 and - 1 ). This unique

factorization in Z is precisely the content of the Funda­

mental Theorem of Arithmetic. The basic idea in any unique factorization domain is that there are certain (non­

unit) elements, called irreducible elements or, in the case

26 THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media, Inc

of Z, primes, which form the "multiplicative building

blocks" for the ring. Notice, by the way, that z+ is exactly

those elements of Z which have 1 as the unit in their mul­

tiplicative representation.

Because they do form the foundation for the multi­

plicative structure of Z, prime numbers play a central role

in many of the questions which arise in classical number

theory. For example, we know the role of primes in multi­

plication, but what about in addition? Is it true, for exam­

ple, that there are enough primes so that every even posi­

tive integer can be written as sum of two of them? This is

precisely the famous Goldbach Conjecture, first formulated

in 1 7 42. It is an easy question to ask, but it remains un­

solved to this day. For another example, two primes are

called twins if they are two apart (e.g., 3 and 5, 5 and 7, 1 1

and 13, etc). Easy question: Are there infinitely many such

pairs? The answer to this "Twin Primes Question" is also

unknown.

Generalizing now, we note that while Z is one example

of a unique factorization domain, we have many other ex­

amples which may be less fundamental than Z but nonethe­

less are interesting and important in their own right. So let

us investigate one class of such examples: polynomials in

one variable x with coefficients in a field K. We denote this

set by K{x] and observe that it is in fact, like Z, a unique factorization domain (see, for example, [14] , pages 289 and 3 19). What are the units of this ring? Well, they are pre­cisely the (non-zero) elements of the field, i.e., the non-zero constant polynomials. This tells us then that the correct analogue to z+ in this domain is precisely the set of monic polynomials, i.e., polynomials whose leading coefficient is 1. Now, we see that the analogues to prime numbers in Z are simply monic irreducible polynomials in K{x] .

All right, can we now do "number theory" in this new domain? For example, what would be a reasonable ana­logue to the Goldbach Conjecture in this setting? Maybe one should ask, "Can every even monic polynomial be writ­ten as a sum of two monic irreducible polynomials?" Though the meaning of this question is not completely clear (for example, what is an "even" polynomial?), it does seem to make some sense. If, however, we try the Twin Primes Conjecture in this setting, it's not even clear what "twin irre­ducible polynomials" might be. Hence we see that questions which are asked in classical number theory about the inte­gers may or may not even make sense in our new setting.

If we want to raise or, even better, answer questions like the two just presented about polynomials, we need to know something about the irreducible ones. The basic idea would be: If irreducibles are more dense among the polynomials than primes are among the integers, then the answers should be yes; if irreducibles are less dense than primes, the answers may be no; and if the densities are similar, then the questions may be comparably hard to answer. But the nature and density of irreducible polynomials depends completely on the coefficient field K, and so we need to consider some different, common fields of coefficients. Those which come to mind immediately are the complex numbers C, the real numbers R, and the rational numbers Q. Let's think a little about irreducibles in the domains of polynomials over these three fields.

The Fundamental Theorem of Algebra quickly settles the issue of irreducible polynomials in C[x] . Since every poly­nomial of degree n over C has n roots in C, we see that every polynomial over C factors completely into linear polynomials, that is, the only irreducibles over C are the linear polynomials. Turning now to the reals R, we note that since complex conjugation (i.e., the function which takes a complex number a + bi to a - bi) is a field homo­morphism, its natural extension to C[x] will take a poly­nomialf(x) in R[x] to itself. But now factoringj(x) over C and applying conjugation, we see that any non-real roots f(x) might possess must occur in conjugate pairs. If such a pair is a + bi and a - bi, then (x - (a + bi))(x - (a ­bi)) = x2 - 2ax + a2 + b2 is irreducible over R. We see then that over R,j(x) factors into a product of linear and/or quadratic polynomials; that is, all irreducibles over R are either linear or quadratic.

Thus we see that, in some sense, irreducibles in C[x] and R[x] are relatively scarce. In particular, it is obvious that no analogue to the Goldbach Conjecture could possi­bly be true in these domains. More generally, the obviously wide gap between the multiplicative structures of Z on the

one hand and C[x] and R[x] on the other make any com­parable "number theory" of those domains unlikely.

The situation with Q[x] is not so clear. There exist few criteria for determining the irreducibility of polynomials over Q, the Eisenstein Criterion being one of those few. It states that ifj(x) has integer coefficients and if there ex­ists a prime number p which does not divide the leading coefficient of f(x), which does divide all the other coeffi­cients, but whose square does not divide the constant co­efficient, then}t:x) is irreducible. For example, x" + p is ir­reducible for all exponents k and any prime p. In particular then, the Eisenstein Criterion shows us that, unlike the cases of C[x] and R[x] , there are in Q[x] irreducibles of ar­bitrarily high degree. But are they more or less dense in the set of all polynomials in Q[x] than the primes in Z? This is difficult to answer precisely, but a strong hint is the fol­lowing result of D. R. Hayes [20] , whose proof is quite short and uses primarily the Eisenstein Criterion:

Theorem 1 (A Goldbach Theorem for Q[x]) Every poly­nomial over Q can be written as a sum of two irreducible polynomials.

The ease with which this result comes forth, in contrast to the great difficulty of the classical Goldbach Conjecture, certainly leads us to suspect that irreducibles are quite dense in Q[x] and that any "number theory" of Q[x] would (as in the cases of C[x] and R[x] but, in a sense, for the op­posite reason) be quite different from that of Z.

Though perhaps not as "famous" as the fields C, R, and Q, another important collection of fields is the finite fields. If q is any power of a prime number p, then there exists a (unique up to field isomorphism) finite field, denoted F q, which has q elements. Once again, for any such q, the ring Fq[X] is a unique factorization domain. What about its ir­reducibles? We shall discuss these at some length in this paper, but let us start by saying that although recognizing irreducible polynomials over F q can be tricky, counting them is not hard. Restricting our attention to monic poly­nomials (as previously justified), it is obvious that the num­ber of such of degree r over F q is exactly qT since there are r coefficient slots and q choices for each. Of these qr poly­nomials, how many are irreducible? It turns out not to be difficult to see that there are precisely

Nq(r) = _!_ ) JL(d)qTid r d[';. (1 . 1)

monic irreducibles of degree r over F q, where f.L is the so­called Mobius function, whose values are 0, - 1 and + 1 . See, for example, Theorem 3.25 of [23] . The order (i.e., largest term) of the sum is qr, and hence we see that the order of the number of monic irreducibles of degree r over Fq is simply qr/r. In other words, about one out of every r monic polynomials of degree r over F q is irreducible.

How does this density compare with the density of primes in z+? According to the famous Prime Number The­orem of classical number theory, the number of primes less than a positive integer n is of order nllog(n), where log in-

© 2005 Springer Sc1ence+Business Media, Inc . Volume 27, Number 2, 2005 27

dicates the natural logarithmic function. In fact, a more ac­curate estimate for the number of primes below a real num­ber y is the so-called logarithmic integral

. (Y dt Ll(y) = Jz log(t) ' (1.2)

which makes it clear that "near" an integer n, the density of primes is approximately 1/log(n). For an excellent dis­cussion of these ideas, see Section I of Chapter 4 of [29].

Now, back in F q [x] , we would like to have a way to mea­sure the "size" of a monic polynomial of degree r. A nat­ural such measure would be the following:

Definition 2 The absolute value of a polynomial A of de­gree r in F q[x], denoted !AI, is qr.

Note that whereas only one positive integer n has (in the standard sense) absolute value n, numerous (in fact qry monic polynomials of degree r over F q have (in this new sense) absolute value qr. Using this notion and observing that r = logq(q1, we see the following fascinating and im­portant fact:

In both Z and F q[X], the density of irreducible elements "near" an element is approximately 1 over the log of the absolute value of that element. In the case of Z the log is natural; in the case of Fq[x] the log is base q.

This close connection between these two quite different looking unique factorization domains Z and F q[X] means that their number theories may very well be quite similar. In the remainder of this paper we shall investigate these similarities, together with the inevitable differences which arise. Can we state and prove a precise "Goldbach Con­jecture for F q[x]"? What about a precise "Twin Irreducibles Conjecture for Fq[x]"? In what follows we consider these and the analogues of various other questions, answered and unanswered, from classical number theory.

Factorization

Both Z and Fq[X] being unique factorization domains, every element in each can be factored in a unique way, apart from order of the factors, into "prime" elements. In the ring Z this factorization is into a product of prime numbers while in F q[X] it is into a product of irreducible polynomials. Given any unique factorization domain, some basic ques­tions arise quite naturally: How can we determine if a given element is irreducible, and are there efficient algo­rithms to make this determination? More generally, we can ask the harder questions: Given an element of our do­main, how can we determine its unique factors, and are there efficient algorithms to make this determination? In this section we address these questions briefly in the cases Z and Fq[x], saying more about the polynomial case since it is less well known.

A partial answer to all of our questions in the case of F q[X] is as follows: There are at least two efficient linear algebraic methods which can be used to factor polynomi­als over F q (and hence, of course, determine irreducibility),

28 THE MATHEMATICAL INTELLIGENCER

at least in the case when the field is "small," i.e., when the cardinality q of the field is smaller than the degree r of the polynomial. We deal here primarily with polynomial fac­torization over small fields since in most applications the field size is small (e.g., in algebraic coding theory for the error-free transmission of information and in cryptography for the secure transmission of information, q is usually 2). The first of these two methods is known as Berlekamp 's algorithm, and is described in detail in Section 4.1 of [23] . The second method, developed by Niederreiter in [24] , uses non-linear ordinary differential equations in characteristic p. In both cases, in order to factor a polynomial A(x) of de­gree r, one sets up an r X r matrix MA over Fq. One then uses linear algebra to reduce the matrix MA - I, where I is the identity matrix. It turns out that the rank of MA - I is r - k, where k is the number of irreducible polynomials in the factorization of A(x). Note then that the polynomial A(x) is irreducible if and only if the rank of the matrix MA - I is r - 1. Moreover, with further work we can ob­tain the actual irreducible factors using the vectors in a ba­sis of the nullspace of MA - I. Again, these algorithms are efficient; i.e., their running times are bounded by a poly­nomial function of r.

To contrast now with the case of Z, we remark also that the Berlekamp and Niederreiter algorithms are not only ef­ficient but are also deterministic (as opposed to proba­bilistic) in the sense that they are guaranteed to give a def­inite and correct output upon any appropriate input. Thus, as far as Fq[x] with small q goes, things are as good as they can be in terms of irreducibility testing and factorization. This however, is not the case for integers. First, for pri­mality testing, up until very recently the existence of a comparable algorithm for integers had eluded mathe­maticians, though there are various deterministic but not efficient algorithms as well as efficient but probabilistic al­gorithms for proving primality (see for example [1]). How­ever, a deterministic polynomial time algorithm for prov­ing primality was announced in August 2002 and will be published in [2] , providing a significant break-through in the area of computational number theory. Nonetheless, it would appear that irreducibility testing in F q[x] is a somewhat easier problem than primality testing in Z. The phenomenon of problems seeming to be a bit easier in the polynomial setting than in the integer setting is a theme we shall see repeatedly in this paper.

Turning to factorization, the problem for Z is a notori­ously difficult one; there is no known efficient algorithm to factor integers. This difficulty lies at the heart of modem cryptography, for current cryptographic systems such as RSA depend on the fact that multiplication is easy but fac­torization is hard. The fastest methods known are based upon number field sieves or on elliptic curves; see [7], [22] , and [33] , for example. If one wishes to speculate about the use of quantum computers (not a current reality), then Shor has provided a polynomial time algorithm in his 1994 pa­per [32] . However, there are not even any efficient proba­bilistic algorithms for integer factoring as yet! On the poly-

nomial side, we have seen that for small q, all is well, but if we remove restrictions on q, then there are again no known efficient deterministic polynomial factorization al­gorithms, even using the generalized Riemann Hypothesis (see for example [ 15]). However, there are efficient prob­abilistic polynomial time algorithms for polynomial factor­ing, some with near quadratic running times (see [ 16]). In other words, factorization appears to be, in general, a hard problem in both Z and Fq[x], but, again, probably "less hard" for polynomials than for integers, and definitely not hard if q is small, i.e., less than r.

The Distribution of Primes and Irreducible&

When studying the distribution of primes in Z or of irre­ducible polynomials in Fq[x], one can be concerned with "large-scale" or "small-scale" distribution. As noted in the opening section, the large-scale distribution is essentially understood in both domains. The Prime Number Theorem tells us that the number of primes below an integer n is asymptotically

n 1T(n) � -- ,

log n

and the number of monic irreducible polynomials at or be­low a given degree r is asymptotically

Nq(r) � qr!r.

We observed there the remarkable similarity of these esti­mates. We should also emphasize, in support of one of our themes, that whereas the proof of the Prime Number The­orem is extremely deep and complex, the estimate for Nq(r) is quite easy to establish using basic facts from the field theory.

The small-scale distribution of primes and irreducibles seems to be a much more difficult problem in both domains. In this and the next section we shall look at two problems which illustrate this difficulty: the classical and polynomial versions of the Twin Primes Conjecture and the Goldbach Conjecture. In each of these cases, it is the small-scale dis­tribution which is at issue. Here the matter of the Riemann Hypothesis, which ties the small-scale distribution of primes to the zeros of the Riemann Zeta-function, and the analogue of that hypothesis in the polynomial setting must come into play.

So let us consider now twin primes and irreducibles. The Twin Prime Conjecture (see, e.g., [ 19]) states that there are an infinite number of consecutive primes with a difference of two, that is, Pm+ 1 - Pm == 2 for infinitely many m. More specifically, it is conjectured that the number 1r2( n) of such twin prime pairs less than n is asymptotically

(3.3)

The product is known as the "twin prime constant," de­noted C2, whose value is about 0.66016. In other words then, the density of twin primes near an integer n is ap­proximately 1 .32/(log n)2. The Twin Primes Conjecture has

never been proved, but there is excellent numerical evi­dence in support of its truth (see, e.g., [25]).

It is thought-provoking to note that if the probability of finding a prime near n is of order 1/log(n), then the prob­ability of finding a twin prime pair near n should be of or­der 1/(log n)2• A wonderful discussion of heuristic reason­ing in the theory of numbers and the apparent connection (perhaps midleading) between these probabilities is given by P6lya [28] .

It may seem remarkable that while the reciprocal sum of the primes diverges, the reciprocal sum of the twin primes converges, whether or not there are an infinite num­ber of twin primes. This was proved by Viggo Brun in 1919 [4]; the sum, known as Brun's constant and denoted Bz has value approximately 1.90216 and has been tabulated to con­siderable precision [25] . This result expresses the scarcity of twin primes compared with the primes, much like the divergence of the harmonic series as compared to the con­vergence of the reciprocal sum of squares.

We tum now to an analogous question in Fq[x]. Can one define "twin" irreducible polynomials, and if so, does the dis­tribution of twin irreducibles mimic the distribution of twin primes? The answers to these questions are "yes" and have been presented in some detail in our recent paper [ 13]. Again, we will consider only monic polynomials, for these are the correct polynomial analogues to positive integers. In anal­ogy to the integer case, we define two irreducible polyno­mials to be "twins" if they differ by as little as possible.

Definition 3 Two polynomials P1 and Pz, both of degree r over F q, are said to be twin irreducible polynomials, or simply twin irreducibles, provided that IPz - P11 == 4 if q = 2 or IPz - P11 = 1 otherwise.

In fact, it is easy to show [ 13] that for any fixed degree r, there are infinitely many twin irreducibles as the field or­der q goes to infinity. However, the true polynomial ana­logue of the Twin Primes Conjecture is when the order q is fixed and the degree r goes to infinity. Using a line of reasoning similar to that used by Hardy and Wright [ 19], it is possible to derive an analogue of the Twin Primes Con­jecture for irreducible polynomials. From [ 13] , this con­jecture is: ( q - 1 ) qr ( 1 ) Nz,q(r) � 0 -

2- r2 I) 1

- ciPI - 1)2 '

where Nz,q(r) is the number of twin irreducible pairs of de­gree r over F q[x], and the product extends over all irre­ducibles P provided q > 2 but does not include linear irre­ducibles if q = 2. Also, o = 4 if q = 2 and o = 1 otherwise. The infinite product on the right converges to a number which depends on q. The reader should note the intriguing similarity between N2,q(r) and 1r2(n) (Equation 3.3), giving another nice example of the closeness of Fq[x] and Z. Though well supported by numerical evidence (see [ 13] for the polynomial case), both of these results remain conjec­tures for now.

© 2005 Springer Science+ Bus1ness Med1a, Inc., Volume 27, Number 2, 2005 29

We remark that it is necessary to separate out the case for the field Fz because only for this field is the smallest difference between irreducibles not a simple constant term of the polynomial. For example, when q = 2 and r = 3, the irreducibles :r3 + x + 1 and :r3 + x2 + 1 differ by x2 + x, the smallest difference between irreducibles for all r > 2. In contrast, for q = 3 and r = 3, :r3 + 2x + 1 and :r3 + 2x + 2 are twins differing by the constant 1 .

Finally, for any given q, one can estimate an appropri­ate analogue B2,q of Brun's constant. For example, in [ 13] it is shown that B2,2 = 1.0591 . . . . Thus it is true that whereas both the sum over reciprocal primes and the sum over reciprocal absolute values of irreducibles over Fq[x] (which the reader should check is essentially the harmonic series) diverge, both the sum over reciprocal twin primes and the sum over reciprocal absolute values of twin irre­ducibles converge. This is another indication of the close connection between the distribution of primes in Z and the distribution of irreducibles in Fq[X]. One wonders if the Twin Irreducibles Conjecture may be easier to settle than the notorious unsolved Twin Primes Conjecture.

The Goldbach Conjecture and the

Riemann Hypothesis

In 17 42 the German mathematician Christian Goldbach conjectured in a letter to Leonard Euler that every positive integer (greater than 5) is a sum of three prime numbers. Euler observed in reply that this is equivalent to every even positive integer (above 2) being a sum of two primes (the reader should check this equivalence), and it is this latter formulation which is now commonly referred to as the Goldbach Conjecture. There are many excellent discus­sions of the complicated mathematical history of this con­jecture (see, for example, [29]); we will focus here on a spe­cific strand of that history which is then tied to the corresponding problem for polynomials.

In 1912, 170 years after Goldbach's letter, the distin­guished German number theorist Edmund Landau declared in a lecture in London that the Goldbach problem was "beim gegenwtirtigen Stande der Wissenschaft unan­greijbar"-"in the current state of knowledge intractable." Two young British mathematicians, G. H. Hardy and J. E. Littlewood, took up the challenge and, over the next 15 years, produced a seminal series of papers entitled "Some Problems of 'Partitio Numerorem.' " In the third of these, subtitled "On the Expression of a Number as a Sum of Primes," they attacked the Goldbach problem with a brand new method and, though still failing to solve it, were able to " . . . show, however, that the problem is not 'unan­

greijbar', and bring it into contact with the recognized methods of the Analytic Theory of Numbers" ([18], page 2). This method, known today as the "Circle Method" or (ap­propriately) the "Hardy-Littlewood Method" (first used by Hardy and Ramanujan, see page 698 of [31]), uses complex analysis (more specifically, complex line integrals around the unit circle in the complex plane) to obtain asymptotic formulas for the number of representations of integers in various forms, including as sums of primes. It turns out, as

30 THE MATHEMATICAL INTELLIGENCER

we shall see, that the method can be fruitfully applied to other domains by using appropriate analogues to the com­plex plane, its compact unit circle, and line integrals.

Saving some details for a bit later, here is what Hardy and Littlewood discovered. Let us denote by Mk(n) the number of representations of a positive integer n as a sum of k odd primes. Assuming what they called "Hypothesis R," a generalization of the Riemann Hypothesis to Dirich­let L-functions, they obtain an asymptotic formula for Mk(n), meaning a formula which contains a main term, which measures Mk( n) with greater and greater relative ac­curacy as n � oo, and an error term, whose magnitude is given using "Big-0" notation. That formula can be summa­rized as follows:

nk- 1 M (n) C (n) + O(nk/2 + nk- 1 -<\

, k = k (log n)k ;

where Ck(n) is a positive bounded function whose exact value depends on k and on n's prime factorization, where n and k have the same parity (otherwise Mk( n) is evidently 0), and where 0 < E :=::; 1/4.

The key to success for any asymptotic analysis is that the main term must grow at a rate strictly greater than that of the error term as the key parameter (in this case n) goes to infinity. First let's set k = 2, that is, let's consider the case with which the Goldbach Conjecture deals. Here, since 2/2 = 1 > 1 - E = 2 - 1 - E, we see that the formula says that

n M2(n) = C2(n)

(log n)Z + O(n),

from which, unfortunately, we learn nothing, for the growth rates of the main and error terms are the same. Hence Hardy and Littlewood's analysis, even assuming the un­proven Hypothesis R, fails to solve the Goldbach Conjec­ture. However, if k 2: 3, then

k > 2 + 2E, 2k - 2 - 2E > k k - 1 - E > k/2,

and so we obtain, for k 2: 3,

nk-1 Mk(n) = Ck(n)

(log n)k + O(nk-1 -E).

Now the asymptotics have "kicked in," and we can use Hardy and Littlewood's analysis to prove the existence of representations of positive integers as the sum of k primes provided k 2: 3, and provided, of course, that Hypothesis R is true.

The case k = 3 is of particular interest. We shall call this the Odd Goldbach or 3-Primes Conjecture: Every odd num­ber greater than 5 is a sum of three primes. Let us write down here exactly what Hardy and Littlewood proved for this important case:

Theorem 4 (Asymptotic 3-Primes) If Hypothesis R is true, then every sufficiently large odd number can be repre­sented as a sum of three odd primes, and the number of such representations is given asymptotically by:

M3(n) � (lo;2

n)3 Jl ( 1 + (p � 1)3 )

· !] (1 - p2 - �p + 3 ) where p runs over prime numbers as specified.

The latter two products here are the explicit form of C3( n ). The first product does not depend on n and has value about 1. 15. The second does depend on n's prime factor­ization and has a value at or just below .67 if 3 1 n, and just below 1 otherwise, provided n is odd. Notice also that if n is even, this factor is 0 and so M3(n) = 0, as it must be.

One can use a simple heuristic to see in general that the basic order of the main term for Mk(n) should be nk- 1/(log n)k. Here is the argument for the case k = 3. To write an odd number n as a sum of three odd primes, by the Prime Number Theorem we have about nllog n choices for primes below n in each of the 3 slots. Each of these n3/(log n)3 combinations adds up to an odd number below 3n, so if these sums are uniformly distributed, about 1 out of every 3n/2, or 2n2/3(log n )3 of them, will add up to n itself. The 2/3 part of this obviously needs closer analysis, but the n2/(log n)3 part is now clear.

We now tum to the connection between Hardy and Lit­tlewood's analysis and the Riemann Hypothesis, which is the source of the E in the error terms above. If x is a "nu­merical character" mapping Z to C (see, for example, [3], page 418), then the Dirichlet L-function associated with x, denoted L(s,x), is defined, for a complex number s sat­isfying ffi(s) > 1, by

L(s,x) = I x(�).

n � l n

Like their close relative the Riemann Zeta-function, the L­functions can be extended analytically to the whole com­plex plane with a unique pole at (1,0) and with "non-triv­ial" zeroes in the strip 0 ::S ffi(s) ::S 1. The Generalized Riemann Hypothesis (GRH) is that these functions all have the property that all their non-trivial zeros z actually sat­isfy ffi(z) = 112. Hardy and Littlewood's "Hypothesis R" (or a Weak Generalized Riemann Hypothesis) is that there ex­ists a ®, 1/2 ::S ® < 3/4, such that every such zero z satis­fies ffi(z) ::S ®. Finally, E above is simply 3/4 - ®.

The connection between the locations of the zeros of the Dirichlet L-functions and the numbers Mk( n) is far from transparent and can only be hinted at here. The interested reader can profitably consult the original work [ 18] or sec­ondary sources such as [34]. We say here only that in the course of the difficult but beautiful analysis done by Hardy and Littlewood, the L-functions and in particular their log­arithmic derivatives L'(s,x)IL(s,x) arise in a natural way. The locations of the zeros of L(s,x) bear directly on the es­timates being made, and what is of crucial importance is that the real parts of those zeros stay away from 1, in fact stay strictly to the left of 3/4. Unfortunately, this was un­proved in 1922, and it remains unproved today.

Before moving to the analogous problem for polynomi-

als, we very briefly summarize the current state of knowl­edge about the Goldbach Conjecture. In 1937 Vinogradov, using a modification of the Hardy-Littlewood method to­gether with numerous ingenious estimates of trigonomet­ric sums, proved without hypothesis that every sufficiently large odd number is a sum of three primes [35]. His analy­sis has been refined to the point that "sufficiently large" to­day means greater than about 1043000 [6], which is of course still far beyond the possibility of checking the cases below it using computation. In 1997 it was shown that if the full GRH holds, then every odd number above 5 is a sum of three primes [8]; that is, assuming the GRH allows us to eliminate the "sufficiently large" part of the Odd Goldbach Conjecture-this will be significant when we discuss the polynomial analogue below. Finally, concerning the origi­nal Goldbach Conjecture, the best current result, obtained in 1973 by Chen [5] using sieve methods, is that every suf­ficiently large even number is a sum of a prime and an "al­most-prime" (a number which is either prime or the prod­uct of two primes).

Let us tum now to the polynomial case. If we hope to write a monic polynomial of degree r as a sum of two or three irreducible monic polynomials, then, in general, one of the irreducibles must also be of degree r and the other( s) must be of lesser degree. It is not clear that the concepts of "even" and "odd" make any sense in the polynomial con­text, but, surprisingly, they not only make sense but also come into play, much as in the integer case. We make the following definition:

Definition 5 A polynomial A over Fa is called even if it is divisible by an irreducible whose absolute value is 2.

Otherwise A is called odd.

It is evident from the definition of absolute value that even polynomials exist only over the field F2 and are pre­cisely those polynomials which are divisible by x or x + 1, the two even irreducibles. Hence a polynomial over F2 is even if it lacks a constant term or if it has an even number of non-zero terms (check). Unlike Z, it is not true, for ex­ample, that "even plus odd is odd." However like Z, the reader should check that no even polynomial can be the sum of three odd irreducibles, and so oddness is a neces­sary condition for our desired "3-irreducibles representa­tion." In a similar vein, polynomials which are too "small," e.g., linear ones, won't have such a representation, and in fact it turns out that if q is even, then A = x2 + a (a E F q) also won't have such a representation (check), being also a little too "small." But those tum out to be the only ex­ceptions:

Theorem 6 (A Complete "3-Irreducibles" Theorem) Every odd monic polynomial of degree r 2: 2 over every finite field F q (except for the case of x2 + a when q is even) is (t sum of three monic irreducible polynomials.

The proof of this result, which is somewhat lengthy, is contained mostly in [ 12] and completed in [9], [ 10], and [ 1 1 ] .

© 2005 Springer Science+Bus1ness Media, Inc., Volume 27, Number 2 , 2005 31

Actually, the first asymptotic result in this direction was David Hayes's 1966 paper [21], but a drawback there was that it dealt with representations of A (of degree r) of the form aP1 + f3P2 + yP3, where P1, P2, and P3 are all of de­gree r and a + f3 + y = 1. Nonetheless, it pointed the way to the analysis done in [12] , which we now discuss briefly.

To apply the Hardy-Littlewood method in the polyno­mial setting, one first needs an appropriate analogue to the unit circle in C on which to carry out the analysis. In [12], that analogue is the compact adele class group Aklk, where k is Fq(x), the field of rational functions over the finite field F q, and Ak is the adele ring. This latter object Ak is a re­stricted direct product of the completions of k at all its places, including the infinite place. The reader should con­sult Chapter 4 of [12] for the details. Now the complex path integral around the unit circle used by Hardy and Little­wood is replaced by the Haar integral on A�, and the analysis can go forward in a similar fashion. The classical Dirichlet £-functions come over into this setting in a way which is natural but somewhat complicated (see Chapter 5 of [ 12]). But now a major advantage occurs because of the pioneering work of Andre Weil, who in 1948 proved an analogue of the Generalized Riemann Hypothesis in this function field setting [36]. We write now the implication of his remarkable result for our analysis:

Theorem 7 (GRH for Function Fields) Given suitable re­strictions on the character x, the function field £-func­tion L(s, x) is a complex polynomial Px in q-s, and when factored into

each n satisfies

lril = q112• Though the details are many, this result allows the analy­

sis to proceed very smoothly and eventually yields not only the asymptotic result below but also very sharp numerical estimates for the error terms (see Chapters 6 and 7 and the Appendix of [ 12]). These estimates then lead to the "com­plete" Theorem 6. The asymptotic result is

Theorem 8 (Asymptotic "3-Irreducibles") If either the field order q or the degree r is sufficiently large, then every odd polynomial A of degree r over F q is a sum of three odd ir­reducible polynomials, and the number M3(A) of such rep­resentations is given asymptoticaUy by

q� ( 1 ( 1 ) M3CA) - � J}2 1 + c�l - 1)3 TI 1- �12 - 3�1 + 3 ,

where P runs over irreducible polynomials over F q as specified.

Note once again that if A were an even polynomial, then the second product, and hence M3(A), would be 0, as ex­pected.

32 THE MATHEMATICAL INTELLIGENCER

We hope the reader, in comparing this result to Hardy and Littlewood's Theorem 4, cannot help but be impressed with their remarkable similarities. Z and Fq[X] are indeed close cousins!

Some Concluding Thoughts on Z and Fq[x] Despite the many similarities between Z and Fq[x], there are differences ( 1) in structure and (2) in the depth of analysis needed to uncover that structure. Two obvious examples of the former are that Z has characteristic 0 while F q[x] has characteristic p (where q = pk for some k), and that whereas each positive integer has its unique ab­solute value, qr monic polynomials over F q have absolute value qr. Two examples of the latter which we have seen are the Prime Number Theorem (very deep) versus for­mula (1.1) for the number Nq(r) of monic irreducibles in Fq[x] of degree r (not deep), and the Riemann Hypothesis (unsolved) versus Weil's Riemann Hypothesis for function fields (deep but solved). We present now another exam­ple where things are a little bit easier to understand in Fq[x] than in Z.

For d 2: 1, let P(x) = PaXd + · · · + Po E Z[x] be a poly­nomial of degree d with Pa = 1. If Q = Z[x]I(P(x)) denotes the quotient or factor ring of Z[x] modulo the ideal gener­ated by the polynomial P(x), then each a E Q has a repre­sentation of the form a = ao + a1x + · · · + aa- 1xd- l with ai E Z, 0 :::; i :::; d - 1. The pair (P(x), M) with M = (0, 1, . . . , !Pol - 1}, is called a canonical number system if each

a E Q admits a unique representation in the form a = a0 + a1x + · · · + ahxh for some h 2: 0 with each ai E M, 0 :::; i :::; h and ah =t- 0 for h =F 0.

The problem of characterizing canonical number sys­tems over Z is a very difficult one. As an indication of this difficulty, for quadratic polynomials over Z the problem has been solved, but only partial results are known for cubic polynomials; see [30]. On the other hand, the following re­sult from [30] provides a complete characterization of anal­ogous digit system polynomials in the setting of F q[x].

For the finite field analogue, let x, y be transcendental

over Fq and let P(x,y) = bnyn + bn- lyn- l + · · · + bo E Fq[x,y] with bi E Fq[x], bn =F 0, deg bn = 0, and deg bo > 0. Let N = (P E Fq[X] : deg p < deg b0} and let R denote the quotient ring Fq[x,y]I(P(x,y)). Then each r E R can be rep­resented as r = r0 + r1y + · · · + rn- l Yn- l with ri E Fq[x]. Further, we say that r E R has a finite y-adic representa­tion if r admits a representation of the form r = r0 + r1y + · · · + rhyh for some h 2: 0 with ri E N for 0 :::; i :::; h and rh =F 0 for h =F 0. Finally the pair (P(x,y), N) is called a digit system in R if each r E R has a unique finite y-adic repre­sentation. A complete characterization of finite field digit systems is given as follows in [30]: A polynomial P(x,y) as above is a digit system polynomial if and only if

n n:tax{deg bil < deg bo.

t � l

Despite examples like this one where the gap between Z and F q[x] seems somewhat wide, we hope that by this time the reader is struck more by the similarities between

these two domains than by their differences. Though we have gone into some detail on certain points, we have not attempted to be comprehensive in our comparisons, and so the reader can certainly pursue further comparisons on his or her own-in fact such studies might make excellent un­

dergraduate research topics. For example, what about a "Waring Problem" for polynomials-can a given monic polynomial over F q be written as a sum of a fixed number of k-th powers of polynomials? What about a polynomial analogue to Dirichlet's Theorem on primes in arithmetic progression? And so on. Here we have simply tried to shed a little light on the wonderful and mysterious relationship between these two domains.

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© 2005 Springer SC1ence+Bus1ness Media, Inc. , Volume 27, Number 2, 2005 33

A U T H O R S

QOVE EFFINGER

USA

34 THE MATHEMATICAL INTELLIGENCER

KENNETH HICK

Depart ol rOilOf'llV OM lkWersi1y

, OH 45701 USA

IIXEIS out spare t

l)iafiO �rod doing com·

Parallel Histories

M?•ffiJ•i§::Gihfii@i§#fii.i,i§lid M ichael Kleber and Ravi Vaki l , Editors

Cartographiana Michael Kleber

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on.

Contributions are most welcome.

Please send all submissions to the

Mathematical Entertainments Editor,

Ravi Vakil, Stanford University,

Department of Mathematics, Bldg. 380,

Stanford, CA 94305-2 1 25, USA

e-mail: vakil@math.stanford .edu

Maps maps maps, we love maps.

Herein a collection of carto­

graphical miscellania, each of which

appeals to us mathematically for its

own reason.

Not Looking Over a

Four-Leaf Clover

Item one is courtesy of the "Knotted

Objects" section of University of

Toronto professor Dror Bar-Natan's

Image Gallery [ 1 ] .

American drivers can generally tum

right easily: at a standard intersection

of two-way roads, a right tum does not

require crossing any lanes of traffic.

Turning left is more difficult; the

"cloverleaf' highway interchange, for

example, makes up for this by replac­

ing 90° left turns with 270° right turns.

Dror points out that "in England, on

the other hand, it is easier to make left

turns than it is to make right turns. The

lovely interchange of I-95 and I-695,

northeast of Baltimore (Fig. 1), com­

bines the advantages of the two sys­

tems-it has an outer layer of Ameri­

can (right) turns, followed by a

braiding of the lanes, followed by an in­

ner layer of English (left) turns!"

Tom Hicks, the Director of Traffic

and Safety for the Maryland State High­

way Administration, says this delight­

ful interchange probably would not be

built today, primarily because it takes

up too much space. The reliance on the

unusual left exit ramp is also a bit of a

strike against it, according to the Hu­

man Factors people (and couldn't we

do with a few more of those in mathe­

matics?).

But the interchange does work well.

Geometrically, Hicks tells me that the

curves of the roadway and the ramps

are gentle enough that cars have to

slow down only slightly from full high­

way speeds. It's also a win topologi­

cally. In a cloverleaf, two lanes of traf­

fic must cross one another, as cars

coming off one ramp need to switch

places with ones entering another

ramp. This interchange avoids that

problem: drivers entering from any di­

rection can tum either left or right

without crossing any other lanes of

traffic. (Unlike the cloverleaf, though,

one cannot use multiple exit ramps to

make a free U-tum.)

But I like it purely aesthetically. So

does Dror, who writes, "Notice that the

lanes of I-95 and of 1-695 are braided in

a non-trivial way, proving that the de­

signer of the interchange cared about el­

egance." We're not the only ones: this

same interchange put in an appearance

as the "Michael G. Koerner Highway

Feature of the Week" in January 1999

[4], and from Koerner I also learned that

there is at least one more instance of

this design, where 1-2011-59 crosses 1-65

just west of Birmingham, Alabama [5].

(In Birmingham, though, the lanes of the

individual highways are unbraided.)

Hicks tells me that you can drive

through the Baltimore interchange

without ever noticing there's some­

thing unusual going on: to the driver it

just looks like a left exit and a few ex­

tra bridges, and only the sharp-eyed

will notice that the first bridge's cross

traffic is going the wrong way. Hooray

for maps.

Cartograms

Item two is courtesy of the 2004 U.S.

election.

In the past several years, the media

have taken to referring to "blue states"

and "red states," those which respec­

tively vote Democrat or Republican;

televised election-night returns fea­

tured maps which converged (albeit

slowly!) to one like Figure 2a. Robert

Vanderbei, a professor of Operations

Research and Financial Engineering at

Princeton, has championed in re­

sponse a Purple America map, linearly

interpolating the colors on a county-by­

county level to show that we're not so

divided after all. (His map differs from

the one reproduced in Figure 2b in that

this one is higher contrast: a county

here is colored pure red or blue if its

vote split is 700h-300,6 or more.)

© 2005 Springer Science+ Business Media. Inc. , Volume 27, Number 2, 2005 35

36 THE MATHEMATICAL INTELLIGENCER

Fig. 1 . Cartographic and photographic views of an inter­

change northeast of Baltimore.

Fig. 2a. The standard red/blue map of the election.

Fig. 2b. Purple America, with county colors interpolated.

© 2005 Springer Science+ Business Media, lnc., Volume 27, Number 2, 2005 37

j t

Fig. 3. Name this city. (For a hint, see "Where in the World," below.)

This coloration is more informative than the winner-take-all version, but it

still looks quite unbalanced, for an elec­

tion where only a few percentage points

separated the popular vote for the two

candidates. The problem, of course, is

that the map is an equal-area projection,

while we would need one with counties

sized in proportion to population.

Such a map is called a cartogram. Given a map and a population density

function on it, a cartogram is a trans­

formation of the map whose Jacobian

at each point is proportional to the den­

sity function-so after transformation,

the population is uniformly distributed.

Of course, this is not enough to deter­

mine the map, and the art of car­

togramy to date has been in trying to

develop a way to find such a transfor­

mation which also looks good.

Figure 2c is the work of Michael

Gastner, Cosma Shalizi, and Mark

Newman, of the Center for the Study

of Complex Systems and Department

of Physics at the University of Michi­

gan. The technique is a new algorithm,

by Gastner and Newman [3], which al­

lows the population density to equili­

brate by flowing according to basic

linear diffusion, with the map bound­

aries being dragged along for the ride.

The physics does a remarkably good

job; I think their map is beautiful.

Their Web page [2] contains more

pictures.

Dror (see item one) tells me he has

heard, from both Bill Thurston and

Yael Karshon, that "given a smooth density function on S2 there is a

canonical (up to rotations) diffeomor­

phism of 82. that takes it to the uniform

area density." So there is, for example,

a canonical way to smoothly redraw

the globe with each country's area

proportional to its population-well,

once you figure out how to populate

the oceans. I wonder how it compares

to Gastner and Newman's diffusion

technique.

Where in the World?

And finally courtesy of University of

Toronto professor Balint Virag: what

city is shown in the map in Figure 3?

The answer appears below, slightly ob­

scured.

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005 39

Credits, thanks, answer

Thanks to Greg Slater of the Maryland

State Highway Administration for the

aerial photograph in Figure lb and the

data behind Figure la, and to Lisa

Sweeney, Head of GIS Services at

MIT, for help in handling it. See the

Web sites listed in the references for

more interchange maps and aerial

photos.

The maps in Figure 2 are by Michael

Gastner, Cosma Shalizi, and Mark

Newman, and are reproduced here

with their permission. Moreover, the

work is licensed under a Creative Com­

mons License, and the images, includ­

ing the ones which appear here, may

t \ I t I I I ">

Calculus: The Elements MICHAEL COMENETZ

537 pp $46 softcover (981 -02-4904-7) $82 hardcover (981 -02-4903-9)

Both editions have sewn bindings

40 THE MATHEMATICAL INTELLIGENCER

be freely distributed and used to make

derivative works for any purpose, as

long as the original authors are given

proper credit and any redistribution

passes on these terms.

The city pictured in Figure 3 is of

Lpmohdnrth ( eoyj drbrm ntofurd), if

you type the city's name and notable

mathematical attribute with your fin­

gers shifted one key to the right. The

map is copyright Ardis Media Group,

and used with permission.

REFERENCES

[1 ] Dror Bar-Natan. "Dror Bar-Natan's Image

Gallery." http://www.math.toronto.edu/�

drorbn/Gallery/

[2] Michael Gastner, Cosma Shalizi, and Mark

Newman, "Maps and cartograms of the

2004 US presidential election results."

http://www-personal . umich .edu/� mejn/

election/

[3] Michael T. Gastner and M. E. J. Newman,

Diffusion-based method for producing den­

sity equalizing maps, Proc. Nat/. Acad. Sci.

USA 1 01 (2004), 7499-7504.

[4] Michael Koerner, "The Michael G. Koerner

Highway Feature of the Week, 9 January

1 999." http://www.gribblenation.corn/hfotw/

exit_ 40.html

[5] Michael Koerner, "The Michael G. Koerner

Highway Feature of the Week, 21 Novem­

ber 1 998." http://www.gribblenation.com/

hfotw/exit_33.html

A CALCULUS BOOK WORTH READING • Clear narrative style • Thorough explanations and accurate proofs • Physical interpretations and applications

"Unlike any other calculus book I have seen . . . Meticulously written for the intel ligent person who wants to understand the subject. . . Not only more intuitive in its approach

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beginner . . . A course based on it would in my opinion definitely have a much greater chance of producing students who understand the structure, uses, and arguments of calculus, than is usually the case . . . Many recent and popular works on the topic will appear intellectually sterile after exposure to this one." -Roy Smith, Professor of

Mathematics, University of Georgia (complete review at publisher's website)

"One has the feeli ng that it is a work by a mathematician still in close touch with physics . . . The author succeeds well in giving an excel lent intuitive introduction while

ultimately maintaining a healthy respect for rigor." -Zentralblatt MATH (online)

A selection of the Scientific American Book Club

World Scientific Publishing Company http://www. worldscientific.com 1 -800-227-7562

Leray in Edelbach Anna Maria Sigmund, Peter Michor,

and Karl Sigmund

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium

e-mail: dirk.huylebrouck@ping.be

This is a most unlikely place for the

mathematical tourist to visit In fact,

it is off-limits for tourists of any kind.

Photographing, filming, even drawing,

is prohibited by law, as signposts tell

you sternly, and trespassers will be

punished. If they survive at all, that is.

Indeed, the signposts also warn you of

LEBENSGEFAHR, meaning mortal

danger. You are in a military zone, and

had better watch out. Don't step on any

mines, and avoid getting shot, says an

urgent inner voice.

But this is ridiculous. We are in Aus­

tria, after all, with almost sixty years of

peace and prosperity behind us. No­

body wants any trouble. Let's not get

caught, that's all.

Welcome to Edelbach, or what is

left of it. The place is not easily found

on a map: it ceased to exist many years

ago, during the darkest days of Aus­

trian history. Nobody lives here any

longer. The main road between Vienna

and Prague is a couple of miles to the

north, but it can be neither seen nor

heard. An eerie silence hangs over the

place. All that remains of the former

village are a few stone-heaps between

thickets of fir trees, and a small, aban­

doned graveyard. To the north of it, a

modem fence surrounds a vast ammu­

nition depot. It is very well guarded,

and you can be sure, by now, that binoculars are fixed on you.

This place was once a camp for pris­

oners of war, mostly French officers.

An "Offizierslager"-or Oflag for short:

the bureaucrats of the Third Reich

were fond of abbreviations. Oflag

XVIIA was the birthplace of a substan­

tial part of algebraic topology. Spectral

sequences and the theory of sheaves

were fathered here by an artillery lieu­

tenant named Jean Leray, during an in­

ternment lasting from July 1940 to May

1945 ([Sch 1990] [Eke 1999] [Gaz 2000]).

In the annals of science one finds

several examples of first-rate mathe­

matical research conducted by prison­

ers of war. The Austrian Eduard Helly,

for instance, wrote a seminal paper

on functional analysis in the Siberian

camp of Nikolsk-Ussurisk, during World

War I; and a century before, the Napo­

leonic officer Jean-Victor Poncelet de­

veloped projective geometry while in

Russian captivity for five years. This

may sound as if the monastic reclusion

and monotonic regularity of confmed

life provided ideal conditions for con­

centrating the mind. And indeed, An­

dre Weil wrote that "nothing is more

favourable than prison for the abstract

sciences" [Weil 1991 ] . He wrote this

while he was in prison, and managed,

during his months of captivity, to find

some of his major theorems. But he

had a prison cell for himself, could re­

ceive visits from his family, and knew

assuredly, to use his words, "captivity

from its most benign side only." The

physical and psychic deprivations of

years in a POW camp, with its over­

crowding, sickness, hunger, and biting

cold, on top of the boredom and un­

certainty, were something else: in

these conditions, intense intellectual

pursuit must have been a desperate

means for keeping hold of sanity.

The prisoners of Edelbach founded

a "University in Captivity." Of the 5,000

inmates of the camp, of which a few

hundred were Polish and the rest

French, almost 500 got degrees, and

their diplomas were all officially con­firmed in France after the war. The fact that Jean Leray had been the director,

or recteur, of this impromptu univer­

sity must have helped with the French

authorities. His academic credentials

were impressive: he had received his

doctorate at the elite Ecole Normale Superieure in Paris, and had been pro­

fessor at the Universite de Nancy before

being drafted into the war. His joint

work with the Polish mathematician

Juliusz Schauder Oater a victim of the

Holocaust) developed a topological in­

variant to prove the existence of solu­

tions of partial-differential equations.

This earned him in 1940 the Grand Prix in mathematics from the Acadbnie des Sciences de Paris.

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005 41

M I LITX R I S C H E S S PE R R G E B I ET 1

L e b e n s g e f a h r !

Betreten und B efahren,

Fotografieren, Fi lmen u nd Zeichnen

gesetzlich verboten und strafbar!

Fig. 1. Tourists are not exactly welcome in Edelbach nowadays, but what can you expect from an

ammunition depot?

But Leray was not the only distin­

guished scientist in the Oflag. There

was the embryologist Etienne Wolff, by

all testimonies a driving force behind

the university, but obliged, for racial

reasons, to keep discreetly in the back­

ground. Etienne Wolff later became

professor at the College de France, and member of the Academie des Sciences de Paris as well as of the Academie Franr;aise. Another luminary was

Fran<;ois Ellenberger, a future presi­

dent of the Societe Geologique de France. The geologists at Oflag XVII

had to content themselves with the

stones they could find in the prison

yard. Their laboratory was an old

kitchen which they could use for a few

hours daily.

Eventually, friends and relatives

from France were permitted to send

books. Over the years, Leray received

a small library from his former teacher

Henri Villat [Sch 1990], [Ell 1948].

From eight in the morning to eight

in the evening, Barrack 19 housed lec­

tures on law and biology, on psychol­

ogy and Arab language, on music and

moral theology, on horse-raising (by a

42 THE MATHEMATICAL INTELLIGENCER

Polish fellow-officer, bien sur!), on

public finances, and on astronomy. The

course on probability was given by

Lieutenant Jean Ville, who had pub­

lished, just before the war, an inge­

nious elementary proof of von Neu­

mann's minimax theorem [Poll 1989] .

Recteur Leray lectured mostly on cal­culus and topology. He had succeeded

in hiding from the Germans the fact that

he was a leading expert in fluid dynam­

ics and mechanics (a mecanicien, as

he liked to say). He turned, instead, to

algebraic topology, a field which he

deemed unlikely to spawn war-like ap­

plications. This led, first, to some notes

in the Comptes Rendus de l'Academie des Sciences de Paris, and eventually to

a three-part work "Algebraic topology

taught in captivity," which was submit­

ted in 1944 to the Journal des Mathe­matiques Pures et Appliquees, through

the good offices of Heinz Hopf from

neutral Switzerland, who endorsed it

enthusiastically. It was published, after

Leray's release, in 1945 [CRAS 1942]

[JMPA 1945].

The university's curriculum shows

that on Sunday nights, the prisoners

could listen to a lecture giving "practi­

cal advice for constructing an inex­

pensive house," before having to return

to their cheerless cold quarters. The

barracks consisted of two rooms hous­

ing 100 inmates each, one small kitchen,

and one toilet with eight wash-basins.

There was a special building for the showers: each officer could use it

twice a month. Half of one barrack was

used as a chapel. More than seventy of

the prisoners were priests, and each

could say mass daily if he wished. The

captives founded a first-rate choir and

a theatre group, and soon set up their

own sports stadium, named stade Pe­tain. The prisoners even managed to

produce, behind the back of their

guards, a documentary film of about

thirty minutes' length, entitled Sous le Manteau ("Beneath the Cloak," be­

cause the camera had always to remain

hidden). Three versions of it have

survived to this day ([Poll 1989] [Kus

2003]).

As in many other POW camps, the

captives printed their own newspaper,

a weekly called Le Canard . . . en KG. KG is W ehrmacht shortspeak meaning

Kriegsgefangener, or prisoner of war,

and the French would pronounce it as

Le canard encage (The caged duck), a

pun referring to the celebrated Le Ca­nard Enchaine (The duck in chains),

which was, and still is, a hugely popu­

lar satirical journal in France. The pris­

oners' version was not permitted to comment on politics, satirically or oth­

erwise: it was filled with harmless cari­

catures, theatre bills, sports news, cross­

word puzzles, and announcements of

special lectures. Nothing about the war,

or about the conflicts dividing the

French community into what, with

hindsight, was simply the issue of col­laboration vs. resistance, but seemed

much more confusing at the time. The

Vichy regime tried to foster a network

of "hommes de confiance," but an un­

derground resistance group, who called

themselves the mafia, eventually be­

came the dominating force in the camp.

For many of the prisoners, the dilemma

was whether to become a civilian

worker in Germany, with a freedom . . .

of sorts, or to stick it out behind the

barbed wire, in the hope that the legal

status of a captive officer would pro­

tect them from the worst. For Leray,

who in 1933 had witnessed in Berlin

the accession of Hitler to power, col­

laboration was never an issue.

When Leray later spoke about Edel­

bach, he located it "near Austerlitz, in

Austria" [Sch 1990]. Actually, Austerlitz

is across the border, in Czechia, and

not really nearby (some 83 kilometres

away). Edelbach is closer to Vienna than

to Austerlitz, but for the defeated French

officers, the thought of being near the

site of the great Napoleonic victory-"a

portee de canon d'Austerlitz," as some

liked to say-must have been a solace.

At first, they all had hoped to be back

in France by the end of 1940. The war

seemed over. When this proved an il­

lusion, many fell prey to depression

and to homesickness. Leray and his

academic colleagues used to meet

every evening in the highest, southern­

most comer of the camp, and watch,

weather permitting, the sunset over "Ia

petite France."

Needless to say, the French did not

merely bemoan their fate. Some tried

to change it. The prison guards became

experts at discovering tunnel en­

trances beneath the barracks. They

were so good at it that they overlooked

a tunnel entrance which was out in the

open, right under their noses. It was

through this 90-meters-long tunnel that

on the nights of September 17 and 18,

1943, no fewer than 132 prisoners de­

camped. It was the greatest escape

from a POW camp in World War II, and

its story is almost unknown [Kus 2004).

The prisoners had established an

open-air theatre, called Theatre de la Verdure. They were allowed to deco­

rate it with twigs and greenery, hiding

it partially from the guard towers. Be­

cause delegates of the International

Red Cross had found that the camp

lacked protection against Allied air

raids, the POWs were told to dig a few

trenches, and were even provided with

shovels and wheel-barrows. Under a

plank bridging one of the dug-outs, they

started burrowing in earnest. The tun­

nel grew quickly, by almost a metre per

day, although water kept flooding in.

After some time, ventilation became a

problem: through a hose made from tin

cans, fresh air had to be pumped into

the gallery, which was less than two

feet wide and three feet high. In paral­

lel, a tailor shop produced civilian

clothes, and the printing press pre­

pared maps and forged documents.

Canned food was hoarded in hidden

depots.

The first group left on a Saturday

night. Their escape went unnoticed

during Sunday, because some of the

guards were on holiday. The second

group left on the following night. Most

of the runaways hoped to pass for

Fig. 2. Lieutenant Jean Leray, POW, became the rector of the "University in Captivity." The picture

on the right shows him with his Edelbach colleagues. Some would later join him at the Sorbonne or

the College de France [Gaz 2000].

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2. 2005 43

Fig. 3. The curriculum of the Universite en Captivite [Poll 1989]. As Leray later said, "students had

no other distraction than their studies. They had little to eat, and little to keep warm; but they were

courageous." [Sch 1990]

French civilians, of whom there were many working in Germany at that time. The first escapees were arrested and returned to the camp by the police even before the break-out was discov­ered by the military guards. EventuaJly, only two fugitives managed to reach France.

Soon after, a panel of agitated Ger­man officers, including severaJ gener­aJs, visited the Oflag, where they were filmed surreptitiously by the French prisoners. The commission decided to play down the escape-it did not show the Wehnnacht in a favourable light. The prisoners were sternly told that they should not try it again. Handbills were distributed warning that "break­ing out is no longer a sport" and that death zones were waiting for the run­aways. Half a year later, 76 British fly­ers escaped from Oflag III Luft in Sagan. This time, the Wehrmacht could no longer keep it a secret from Hitler and Rimmler. Only three of the fugi­tives reached England; 50 were shot.

During the five years that Leray spent in Oflag XVII, battles raged from one end of Europe to the other, at no

44 THE MATHEMATICAL INTELLIGENCER

time touching Edelbach. Nevertheless, the booming of great guns and the an­gry buzz of Stukas could be heard at aJl times by the inmates of the camp. In­deed, Oflag XVII was located within an evacuated zone, strictly off-limits for civilians, the Truppenilbungsplatz

"Breaking out is

no longer a sport. "

Dollersheim. This was the largest mil­itary training ground in centraJ Europe, twenty kilometres in diameter, larger than the dukedom of Liechtenstein. A few months after the Anschluss, Hitler's annexation of Austria in 1938, the German army had taken over the ground. Forty-five villages with more than seven thousand inhabitants were hastily evacuated, and huge mechanised forces rattled across the fields, taking little notice of the fact that the harvest was not yet in. The Wehrmacht had to live up to its new and as yet untested doctrine of the Blitzkrieg. The barracks which Leray and his fellow-prisoners

were soon to use were erected origi­nally to house the first German soldiers claiming the exercise grounds. Very soon, the Truppenilbungsplatz proved an ideaJ stepping-stone for the armies which were assembling to invade and dismember nearby Czechoslovakia, in spring 1939, and for preparing the as­sault on Poland during the following summer months [Poll 1989].

The fact that both the father and the mother of Adolf Hitler had been born in the region, which was so suddenly and ruthlessly evacuated, gave rise to spec­ulations. One of the closest as­sociates of the FUhrer, Hans Frank, would later write, in his death cell in Nuremberg, that Hitler intended there­by to erase all traces of his origins [Frank 1953]. He reported that these traces could reveaJ a dark secret, the shame and scandaJ of the Third Reich: namely, that Hitler had a Jewish grand­father. This rumour, which had been widespread in Nazi Germany and still finds adherents today, has been de­bunked by scores of historians since. Hitler's father had been born out of wedlock, as Alois Schicklgruber, and

was later to change his name, but the

FUhrer was far too powerful to have felt

threatened by slurs concerning his an­

cestry. In fact, when the villages around

Dollersheim were evacuated, all church

archives were properly stored. They are

preserved to this day. For years, the

Wehnnacht had been looking for a king­

sized training ground to accommodate

its frantic growth, and to manoeuvre

with its new weapons, whose range

would not fit into existing exercise ar­

eas. The W aldviertel (or woods district),

with its poor soil and its sparse, lowly

population was perfectly suited: a hilly

plateau, some 600 metres above sea

level, with long, bitterly cold winters,

and no reputation for hospitality.

It is clear that Hitler had no emo­

tional ties to the Waldviertel. The prop­

aganda from the Goebbels ministry had

hailed it as the Ahnengau, the cradle of

the ancestors, and the humble dwellers

of the tiny hamlet of Grosspoppen, led

by their inn-keeper, had conferred hon­

orary citizenship on Hitler in 1932,

when he was a rising young politician

and demagogue. In return, they first got

scolded by the authorities of Lower

Austria (who pointed out that the ac­

tion was legally void because Hitler

was no longer an Austrian citizen),

then frowned upon by the Viennese

regime, which was engaged in a hope­

less struggle against illegal Nazis, and

fmally, right after the revels of the an­

nexation, expelled from their land

without further ado. No account was

taken of the fact that 220 out of the 220

"Today only

the church of

Dol lersheim ' "

surv1ves . . .

citizens of Grosspoppen had voted for

the Anschluss. In fact their hamlet,

which obstructed a planned artillery

range, was the first to become men­schenrein (the callous Nazi expression

for "evacuated") and be knocked down.

A fortunate few were compensated

with hastily built ersatz farms, not too

far away. Others were given provi­

sional quarters and the promise of a

settlement after the war. In 1942, all

evacuees were offered a special re­

duction on a richly produced coffee­

table book, Die alte Heimat, complete

with pictures of their empty villages, and

Hitler's family tree as a keepsake [Heim

1942). In the ensuing years, Nazi au-

thorities had other things on their minds.

Eventually, the district of Lower Austria

was occupied by the Red Army, which

could find good use for the vast training

opportunities filled with bunkers and

artillery ranges. By 1955, the Allied oc­

cupation troops left Austria, but the

evacuated region was not returned to

its former dwellers. They had been

scattered all over the district and were

far too weak to succeed in their de­

mands for a return. The small new Aus­

trian army managed to keep the over­

sized training grounds for itself. Those

abandoned houses which were still

standing, after the years of Nazi and

Soviet occupation, including Edelbach,

were now flattened in a remarkably

short time. The Austrian army had in­

herited an amazing amount of ammuni­

tion, and made a point of spending it lav­

ishly by shelling the empty settlements.

Today only the church of Dollersheim

survives: its spire serves as a conve­

nient mark for ranging artillery sights.

But during Leray's years of intern­

ment he was daily faced with the

vacant houses of a seemingly intact,

menschenfrei Edelbach behind the

barbed-wire fence. The chimneys did

not smoke and the doors never opened.

The window-panes had been replaced

by planks. A poem on the front page of

Fig. 4. Notes from captivity. KG Jean Leray reports, in this Comptes Rendus note from 1942, that in

his present condition, he is unable to guarantee the originality of his results [Gaz 2000].

© 2005 Springer Sci811Ce+ Business Media, Inc., Volume 27, Number 2, 2005 45

Fig. 5. No open university, but a closed universe of 440 x 530 meters. The camp, and campus, of

Oflag XVII housed some 5,000 prisoners. Today, the barracks are gone: in their place one finds con­

crete, earth-covered ammunition dumps. The village of Edelbach is a rubble of stones covered by a

dense forest.

the Canard en KG, with the title "Le viUage ignore," describes the mute

bell-tower of the deserted hamlet, and

the silence broken only by the wind [Poll 1989]. And while the Nazi picture

book acknowledges that when Edel­

bach had to be cleared out, some left

it with a bleeding heart, the captive

French poet imagines how his heart,

far from bleeding, "jumps with joy on

the day, known only to destiny," when

he is released and the forsaken village

vanishes behind the firs.

The day known only to destiny was April 17, 1945. The camp had to be

evacuated because the Red Army was

perilously close. The Wehrmacht was

by now out of gas and lorries. The Blitzkrieg days were over. The prison­

ers had to march, carrying their be-

longings on their backs. Some of the

guards used bicycles, and their officers

sat on underfed horses. The trek aimed

for Linz, some 128 kilometers away to the west. The group covered, on aver­

age, less than ten kilometres a day, and

dwindled rapidly in size. The marching

column was long, the forest dense. Un­

derfed Fran<;ois Ellenberger schlepped a rucksack half his own weight: he had

Fig. 6. A room with a view. The barracks were originally built for the Wehrmacht soldiers claiming

the grounds. Fences and watchtowers were added later [Poll 1989].

46 THE MATHEMATICAL INTELUGENCER

Fig. 7. Underground film. These clandestine stills show the escape tunnel, also known as metro pour

Ia liberte. The clandestine movie Sous le manteau, shot in real time, is a French alternative to Holly­

wood's The Great Escape [Corr 1954].

insisted on taking along his volumi­

nous mineralogical notes, a hand-made

telescope, and his rock samples, some

of which had come from the tunnel. He

still found the strength to sketch the

lines of the hills in his notebook, and

the interiors of rural chapels. The pris­

oners had to look after their own food;

some managed to get it from old wives

and barefoot children, in exchange for

soap, which they had produced in their

camp. By May 10, the column had been

reduced by half. This was the day the

Wehrmacht surrendered.

After his liberation, Jean Leray be­

came professor, first at the University

of Paris (which had appointed him in

1942), and then, in 1947, at the presti­

gious College de France. In 1953 he was

elected to the Academie des Sciences de Paris (which had made him a cor­

responding member in 1944). He was

showered with prizes: among them, the

prix Ormoy in 1950, the Feltrinelli prize

in 1971, the Lomonosov gold medal in

1988 Gointly with Sobolev), and in

1979, the Wolf prize, jointly with Andre

W eil (who, incidentally, had also been

I I

a candidate for that same chair at the

College de France). In an obituary writ­

ten for Nature, Ivar Ekeland called

Leray "the first modem analyst," and

compared him with Weil, "the first

modem algebraist" [Eke 1999].

The parallels, which also were

stressed by Jean-Michel Kantor [Gaz

2000], are indeed intriguing: the two

men share their year of birth, 1906, and

their year of death, 1998. They both

were among the very select few to at­

tend the Ecole Normale Superieure, and both did some of their best work

Fig. 8. Cold feet and frosty advice. Unaware of being filmed, a Wehrmacht delegation decided to

keep the news of the escape under wraps. But posters warned the French that henceforth, s'evader

n'est plus un sport.

© 2005 Spnnger Science+Bus1ness Media, Inc., Volume 27, Number 2, 2005 47

Fig. 9. The church of Edelbach, in an already deserted village. The poem "Le village ignore" laments

that in the humble church, no bell ever rings. In 1957 the church was flattened by Austrian artillery.

in prison. But the differences are even more striking. Weil followed his dharma (that is to say, he was a con­scientious objector) and therefore took

hair-raising risks to avoid waging war against Hitler. Leray served as a patri­otic officer and remained stolidly at his post to the end, during the swift Ger-

man assault and throughout the pro­tracted years of confinement. Whereas W eil studied abstract algebraic struc­tures and shunned anything even re-

Fig. 10. Forty years after. This stone commemorates a visit in 1985 by some former inmates of the

Oflag. The French prisoners had their own graveyard in Edelbach, complete with funeral statue.

48 THE MATHEMA1lCAL INTELLIGENCER

motely smacking of applications or

physical intuition, Leray was deeply

steeped in physics and geometry. This

makes all the more remarkable the fact

that he switched to algebraic topology

in the prison camp, and laid the basis

for what soon became a main item on

Bourbaki's menu, although he had left

the Bourbaki group in 1935.

Changing direction seems to have

posed no problem for Leray. "The es­

sential characteristic of my publica­

tions is their diversity, " he later said,

simply. "It was my interest in mechan­

ics that obliged me to give new devel­

opments to mathematical analysis and

algebraic topology" [Sch 1990). Indeed,

Leray had been interested in topology

even before the war, but as a tool

rather than as an end in itself. The ho­

motopy invariant now known as the

Leray-Schauder degree was created in

order to prove the existence of solu­

tions to non-linear partial-differential

equations. Such equations, particularly

those which stemmed from mathemat­

ical physics, were at the centre of

Leray's work. In 1936, he published a

truly pioneering paper investigating

the existence, uniqueness, and smooth­

ness of solutions of the initial-value

problem for the three-dimensional

Navier-Stokes equations for incom­

pressible fluids. He showed, in partic­

ular, that non-stationary solutions for

smooth initial data remain smooth for

a finite time only; beyond this, they

may only be continued in a weak sense

(giving rise to what are called weak so­

lutions nowadays). Leray called such solutions turbulent, thereby suggest­

ing that the onset of turbulence is

caused by the breakdown of smooth­

ness. He certainly had good reasons

not to wish the Germans to learn of his

work It is interesting to speculate

what he would have done if he had

been given an opportunity to do scien­

tific work for the Allies.

As it was, he "turned his minor into

his major interest" and started working

on algebraic topology as an end in it­

self-Weil-style, as it were. He worked

in great, but not total scientific isola­

tion, avoiding contacts with German

mathematicians. Apart from some

reprints provided by Heinz Hopf, from

neutral Switzerland, Leray was cut off

from ongoing research, in particular

from contemporary, related work by

Eilenberg and Steenrod, and had to

start from scratch.

As Armand Borel later wrote,

Leray's original concepts, based on a

language of his own making, have been

strongly modified or have not survived

[BHL 2000). Leray's aim was to create

something similar to differential forms,

keeping their multiplicative algebraic

structure, but in a purely topological

framework His cohomology was simi­

lar to that created by Cech, and his re­

sults did not, as Borel wrote, "seem to

go drastically beyond those of main­

stream algebraic topology." But the in­

tention behind them was different:

Leray aimed at studying, not only the

topology of a space, but the topology

of a representation, i.e., topological in­

variants for continuous maps. He took

as starting point his notes on a course

by Elie Cartan on differential forms,

published in 1935 [Cart 1935]. He

aimed to understand cohomology

(which he persistently called homol­

ogy) in a way similar to the de Rham

cohomology, with its multiplicative

structure.

From his work with Schauder on

fixed-point theorems, he was used to

the relative viewpoint. He considered

mappings between two spaces as

the basic object. This was a lasting

achievement. The Leray-Serre spectral

sequence of a filtration is still in gen­

eral use today. Grothendieck would

also stress the importance of the rela­

tive point of view in algebraic topology [Jack 2004].

Soon after his release, Leray found a

way to define cohomology with respect

to sheaves, and introduced the spectral

sequence of a continuous map, which

relates the cohomology of the domain to that of the range and of the fibre. His

original ideas, intended to be as general

as possible, were still not general

enough, however, for three young

Frenchmen named Henri Cartan, Jean­

Louis Koszul, and Jean-Pierre Serre.

They extended his concepts to obtain

spectacular applications to analytic

spaces and algebraic geometry. In the

late forties, the development became al­

most breathless [Gaz 2000). The two

Fields medallists of 1954, Serre and Ko-

daira, both based their work on Leray's

sheaves and spectral sequences.

In the hands of Cartan and Oka,

sheaves became an essential tool for

the theory of several complex vari­

ables. Weil used sheaf cohomology and

spectral sequences on real manifolds

to give a lucid proof of de Rham's the­

orem, generalising the Mayer-Vietoris

sequence from an open cover of two

sets to one of infinitely many sets.

Godement wrote the definitive treat­

ment of sheaves and their cohomol­

ogy for algebraic topology. Serre and

Grothendieck adapted the notion of

sheaves for algebraic geometry. Even

the (still unfinished) theory of motives

concerns a category of sheaves. The

central problem, on which Voevodsky

made some recent inroads, is to find

enough injective resolutions for coho­

mology to work. With the papers of Ko­

daira and Spencer, and the Habilita­

tionsschrift of Hirzebruch [Hirz 1956],

sheaf cohomology crossed the French

borders. Sato used complex analytic

sheaf cohomology to define hyperfunc­

tions as generalised boundary values of

holomorphic functions, and investigated

microlocal analysis on the cotangent

bundle. Sato's microfunctions are more

powerful than Hormander's wave-front

sets, which in turn were inspired by

Maslov. Later, Leray would devote a

whole book to the role of Planck's con­

stant in mathematics, again in an at­

tempt to understand Maslov [Ler 1981) .

Leray's concept of spectral se­

quences appeared first as a compli­

cated set of relations among various

cohomologies of double complexes. They allowed Leray to compute the co­

homology of compact Lie groups and

flag manifolds. Serre used spectral se­

quences, already in their modern form,

to determine the dimensions in which

the higher homotopy groups of the

n-sphere are not finite, namely n and

2n-1. Massey made spectral sequences

more easily accessible via the notion of

exact couples.

Leray himself, after 1950, returned to

partial-differential equations. He stud­

ied the Cauchy problem, its connection

with multidimensional complex analy­

sis, residue theory on complex mani­

folds, and integral representations. Al­gebraic topology became a tool again

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005 49

A U T H O II S

for Jean Leray. The interlude which

had begun in the POW camp of Edel­

bach, as a kind of camouflage, was

over. But generations of pure mathe­

maticians would exploit the ideas

which had germinated in Oflag XVIIA.

Acknowledgments

We thank Hofrat Dr. Andreas Kusternig for a wealth of information

on Oflag XVIIA. Jean-Michel Kantor,

Reinhard Siegmund-Schultze, and Han­

nelore Brandt provided considerable

help in preparing this article.

REFERENCES

(Eke 1 999] Ekeland, I . : Jean Leray (1 906-

1 998), Nature 397, 482.

[Gaz 2000] Gazette des mathematicians , Sup­

plement au no. 84, 1 -88 is entirely dedicated

to Jean Leray, and includes articles by J. M .

Kantor, Y . Choquet-Bruhat, J . Y. Chemin, H .

50 THE MATHEMATICAL INTELUGENCER

Miller, J . Serrin, R. Siegmund-Schultze, A Yger, C. Houze!, and P. Malliavin .

[Sch 1 990] Schmidt, M . : Hommes de science,

Hermann, Paris.

(Wei! 1 99 1 ] Wei!, A: Souvenirs d'apprentis­

sage, Birkhauser, Basel.

[Ell 1 948] Ellenberger, F . : La geologie a I 'Oflag

XVIIA, Annates Scientifiques de Franche­

Comte 3, 2 1 -24.

(Carr 1 954] Carre, M . : Defense de photogra­

phier, reportage photographique clandestin

sur Ia vie d'un camp de prisonniers francais,

Oflag XVII A(Autriche)

[Poll 1 989] Palleross, F. : 1938 Davor-Oanach:

Beitrage zur Zeitgeschichte des Waldviertels,

Horn-Krems.

[CRAS 1 942] Leray, J . : Comptes rendus de

I'Academie des Sciences de Paris 214,

781 -783, 839-841 , 897-899, 938-940.

[JMPA 1 945] Leray, J . : Cours d'algebre

topologique enseigne en captivite, J. Math.

Pures Appl. 24, 95-1 67, 1 69-1 99, 201 -248.

[Kus 2003] Kusternig, A: Grosse Flucht aus

dem Oflag XVIIA, Nieder6sterreich Perspek­

tiven 3, 22-25.

[Fr 1 953] Frank, H . : lm Angesicht des Galgens,

Munchen/Grafelfing.

[Heim 1 942] Die a/te Heimat: Sudetendeutsche

Verlagsdruckerei, Berl in.

[BHL 2000] Borel, A , Henkin, G.M. , and Lax, P D . : Jean Leray (1 906-1 998), Notices AMS 47, 350-359.

[Cart 1 935] Carlan, E . : La methode des reperes

mobiles, Ia theorie des groupes continus et

/es espaces generalises, Notes written by J .

Leray, Hermann, Paris.

[Jack 2004] Jackson, A: As if summoned from

the void, the life of Alexandre Grothendieck, No­

tices of the AMS 51 , 1 038-1 056, 1 1 96-1 2 1 2.

[Hirz 1 956] Hirzebruch, F. : Neue topologische

Methoden in der algebraischen Geometrie,

Ergebnisse 9, Springer-Verlag, Berlin.

[Ler 1 98 1 ] Leray, J . : Langrangian analysis and

quantum mechanics, MIT Press, Mass.

CHRISTIAN BOYER

Some Notes on the Mag ic Squares of Sq uares Prob em

Permettez-moi, Monsieur, que je vous parle encore d'un probleme qui me parait fort curieux et digne de toute attention.

-Leonhard Euler, 1 770, sending his 4 X 4 magic square of squares to Joseph Lagrange.

Can a 3 X 3 magic square be constructed with nine distinct square numbers? This short question asked by Martin LaBar [38] in 1984 became famous when Martin Gardner repub­lished it in 1996 [25] [26] and offered $100 to the first per­son to construct such a square. Two years later, Gardner wrote [28]:

So far no one has come forward with a "square of squares" -but no one has proved its impossibility either. If it exists, its numbers would be huge, perhaps beyond the reach of today's fastest computers.

Today, this problem is not yet solved. Several other arti­cles in various magazines have been published [ 10] [ 1 1 ] [ 12] [27] [29] [30] [49] [ 5 1 ] [52]. John P. Robertson [51] showed that the problem is equivalent to other mathe­matical problems on arithmetic progressions, on Pythagorean right triangles, and on congruent numbers and elliptic curves y2 = x3 - n2x. Lee Sallows [52] dis­cussed the subject in The Mathematical InteUigencer, pre­senting the nice (LSJ) square, a near-solution with only one bad sum.

52 THE MATHEMATICAL INTELLIGENCER © 2005 Springer Scierce+Business Media, Inc.

1272 � 582

� 1� 94

74 822 g72

L81. Three rows, three columns, and one diagonal have the same

magic sum 82 = 21609. But unfortunately the other diagonal has a

different sum 82 = 38307.

In the present article I add both old forgotten European works of the XVIIIth and XIXth centuries that I am proud to revive (and to numerically complete for the first time [9]) after years of oblivion, and very recent developments of the very last months on the problem-and more gener­ally on multimagic squares, cubes, and hypercubes. And I have highlighted 10 open subjects. An open invitation to number-lovers!

The magic square of squares problem is an important part of unsolved problem D 15 of Richard K. Guy's Unsolved Problems in Number Theory [30], third edition, 2004, sum­marizing the main published articles on this subject since

1984. I have organized my exposition around nine quota­

tions from Guy's text.

1 . "Martin LaBar asked for a proof or disproof that

a 3 X 3 magic square can be constructed from

nine distinct integer squares."

Martin LaBar [38] is represented as the original prepounder

of the problem in 1984: by Andrew Bremner [ 10], Martin

Gardner [25], Richard Guy [29] [30], Landon Rabem [49],

John Robertson [51] . But I have discovered that this prob­

lem was posed, more than one century before LaBar, by

the French mathematician Edouard Lucas: in 1876, in the

magazine Nouvelle Correspondance Mathematique [40],

edited by the Belgian mathematician Eugene Catalan.

Edouard Lucas (Amiens 1842-Paris 1891) stated the 3 X 3 magic

square of squares problem in 1876.

In his article, Lucas particularly studies this parametric

family (ELl) of semi-magic squares, "semi-magic" meaning

that all the row sums and column sums are the same, but

not the two diagonals:

EL 1 . Lucas's 3 X 3 semi-magic square of squares family. The 3 rows

and 3 columns have the same magic sum S2 = (p2 + q2 + r2 + s2)2.

e&2 "

7&2 1&2 232

2SZ .. 17

EL2. The example of a 3 X 3 semi-magic square of squares by E. Lu­

cas in 1876. Generated with (p, q, r, s) = (6, 5, 4, 2) and moving

rows and columns. Or generated, for example, with (2, 6, 5, -4) and

just inverting columns 2 and 3. S2 = 812 = 38• This square has other

characteristics: 1 + 16 + 64 = 34, with 1 + 68 + 44 + 76 + 16 + 23 +

28 +41 + 64 = 192, and 12 · 682 · 442 + 762 · 162 · 232 + 282 · 412 · 542 =

12 . 7S2 . 282 + 682 . 162 . 412 + 442 . 232 . 642.

He presents a numerical example (EL2), using (p, q, r, s) = (6, 5, 4, 2), with other interesting characteristics:

• The three rows and the three columns have the same

magic sum, which is an eighth power of an integer.

• The main diagonal, when its numbers are not squared, has

a sum which is the fourth power of the same integer.

• The nine numbers, when they are not squared, have a

sum which is a squared integer.

• The sum of the products of the rows is equal to the sum

of the products of the columns.

Lucas does not remark that this last characteristic is also

true for all 3 X 3 magic and semi-magic squares. He also

does not remark that, with a good choice of (p, q, r, s), his

family allows seven of the eight sums to agree, as I will

show in part 4. But he proves that his family cannot pro­

vide a solution with the eight magic sums, adding that "cela ne prouve pas, il est vrai, que le probleme soit insoluble": "it's not a proof that the problem has no solution."

See also below, part 4, part 5, and the supplement [9]

for a complete numerical study of Lucas's family.

Citing Legendre [39], Lucas mentions the 4 X 4 magic

square of squares problem proposed by Leonhard Euler

(see part 5), but seems to be unaware-like Legendre-that

Euler had published in 1770 [ 18] this very similar family of

3 X 3 matrices, but with non-integer entries:

A = rp2 + q2 - r2 - s'2)!u, B = [2(qr + ps)]lu, C = [2(qs - pr)]/u,

0 = [2(qr - ps)]lu, E = (p2 - q2 + r2 - s2)1u, F = [2(rs + pq)]lu,

G = [2(qs + pr)]lu, H = [2(rs - pq)]lu, I = (p2 - q2 - r2 + s'2)1u,

where u = p2 + q2 + r2 + s2• Euler used directly his works

on mechanics about the rotation of a solid body around a

fixed point [ 19] . Joseph Lagrange, then later Arthur Cayley,

worked on the same subject of physics, and used similar

matrices.

Euler announces that his solution has the 12 following

characteristics:

A2 + B2 + C2 = 1

02 + E2 + F2 = 1

G2 + H2 + J2 = 1

A2 + 02 + G2 = 1

82 + E2 + H2 = 1

C2 + F2 + J 2 = 1

AB + DE + GH = O

AC + OF + Gl = 0

BC + EF + HI = 0

AD + BE + CF = 0

AG + BH + Cl = 0

DG + EH + FI = 0

Euler gives four examples, his smallest being (LEl):

47157 28157 181&7

4157 23157 521157

32157 -44157 1 7157 G

LE1 . The smallest example published by Euler in 1770, generated by

(p, q, r, s) = (6, 4, 2, 1), solving 6 identities like A2 + B2 + C2 = 1 , and

6 identities like AB + DE + GH = 0.

We can very easily rewrite his result, removing signs and

denominators and squaring the integers, and say that Euler

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005 53

has found the first 3 X 3 semi-magic square of squares (LElcb):

472 2fl 11'

4 '¥Jl 52'

32' 44 72

LE1 cb. The smallest semi-magic square of squares, directly coming

from (LE1). Magic sum S2 = 3249 = 572•

And even better: Euler has produced the smallest pos­sible semi-magic example. There is no smaller example, even outside of this family. But because Euler never men­tioned the problem to get (in his notation) the two further properties

A2 + E2 + J2 = 1 C2 + E2 + G2 = 1 ,

we can say that the earliest known author to propose the problem of a 3 X 3 magic square of squares is Lucas in [41].

Several years later, Lucas presents (ELl) again, but very briefly, without any comments [42] [43] [44].

2. "Duncan Buell [1 2] searched for a 'magic hour­

glass'

a - b

a + c

a + b + c

a

a - b - c

a - c

a + b

with all seven entries squares, but found none

with . < 25 . 1 024.

Bremner [1 0] [29] found a magic square with seven

square entries: 3732 2892 380721 4252 2052 5272

5652 (AB1)

222 1 2 1 " It seems that the (ABl) square was found independently by Lee Sallows [29].

As well analyzed by Bremner [ 1 1 ] , it is easy to get a lot of examples with six square entries. But it is strange to re­mark that, excluding rotations, symmetries, and multiples of the (ABl) square, no other example has been found with seven square entries, after long efforts of computing by dif­ferent people . . . including me [8] . . . .

And no example is known with eight square entries.

54 THE MATHEMAnCAL INnELLIGENCER

3. "Michael Schweitzer showed that any such square

must have entries with at least 9 decimal digits"

Michael Schweitzer showed in 1996 [26) that if a 3 X 3

magic square of squares exists, the centre entry is bigger than 109, not all the entries. Because such a magic square of squares needs to be at least a "magic hour-glass," the re­sult of Duncan Buell done in 1998 [ 12] is far better: it im­plies that if a 3 X 3 magic square of squares exists, then its centre cell is > 25

·

1024.

Martin Gardner was right in saying [28]: "If it exists, its numbers would be huge."

4. "He [Michael Schweitzer] gives the following

specimens in which only one diagonal fails.

1 272 462 582 22 1 1 32 942 742 822 972

1 882 1 942 1 1 62 42 1 482 2542 2262 1 642 922

2822 291 2 1 742 62 2222 381 2 3392 2462 1 382

(MS 1)

(MS2)

(MS3)

The magic totals are squares in each case: 1 472, 2942, 441 2• Does this have to happen?"

Lee Sallows also found in 1996 [26] the first specimen (MSJ): it is the (LSi) already presented in the introduction. It is the smallest possible example in which only one diagonal fails. But I remark that it is simply a Lucas magic square! Apply (ELl) with (p, q, r, s) = (1, 3, 4, 1 1), and you will get exactly the same square. And it explains why the magic total is a square:

(p2 + q2 + r2 + s2)2 = ( 12 + 32 + 42 + 1 12)2 = 1472.

The second specimen (MS2) also belongs to Lucas's fam­ily! With (p, q, r, s) = (2, 7, 4, 15), columns 2 and 3 being inverted. It explains also why the magic total is a square:

(pz + q2 + rz + s2)2 = (22 + 7z + 42 + 152)2 = 2942.

The third specimen (MS3) does not belong to Lucas's family. The same magic sum of 4412 is, however, possible, for example with (1 , 2, 6, 20), (1 , 4, 10, 18), (1 , 10, 12, 14),

(2, 4, 14, 15), . . . but giving squares in which the two diag­onals fail, instead of only one with this third Schweitzer ex­ample. The solution (3, 12, 12, 12) gives a square in which only one diagonal fails, but because q = r = s, the integers are of course not all distinct.

Bremner's first specimen (AB2), found before the above squares and published by Guy and Nowakowski in 1995 [29],

in which the two diagonals fail, belongs to Lucas's family.

AB2. This semi-magic square of squares by Andrew Bremner in 1995,

is a 3 X 3 Lucas square with (p, q, r, s) = (2, 6, 4, 3) and inverting

columns 2 and 3. S2 = 4225 = 652•

Michael Schweitzer [26] was the first to fmd an example (MS4) in which again one diagonal fails but having a non­square magic sum:

3'9 34951 29582

36422 21251 1785'

2n!¥Z 2()582 3()05?

MS4. Example with a non-square magic sum 52 = 20966014, in

which one diagonal fails.

Because its magic sum is not a square, this example is of course not a member of the Lucas family.

5. "He [Andrew Bremner] [1 1 ] also gave the 4 X 4 magic square of squares: 372 232 212 222 1 2 1 82 472 1 72 (AB3, with S2 = 2823)

382 1 1 2 1 32 332 32 432 � 3 1 � As for the 3 X 3 magic square of squares problem in part 1 , there is a forgotten work of an old and prestigious au­thor on this subject. The first 4 X 4 magic square of squares was constructed by Leonhard Euler, in a letter [56] written in French that he sent in 1770 to Joseph Lagrange, not giv­ing any method. This letter with the (LE2) square was known by Legendre [39) in 1830 and by Lucas [40] in 1876,

but not the method.

fliP 2'¥1 41 371

1 7' 31 ..

tiP 'Ill' zt' 81

1 1 .,. at

LE2. The first known magic square of squares, sent in 1770 by Leon­

hard Euler to Joseph Lagrange. The 4 rows, 4 columns, and 2 diag­

onals have the same sum 52 = 8515.

Euler gave his method (LE3) at the St Petersburg Acad­emy also in 1770 [ 18]:

LE3. Euler's 4 X 4 magic square of squares family. Magic sum 52 =

(a2 + b2 + c2 + d2)(p2 + q2 + r2 + s2).

He needs two supplemental conditions in order to get the two diagonals to sum to S2:

• pr + qs = 0,

• a I c = [ -d(pq + rs) - b(ps + qr)] I [b(pq + rs) + d(ps + qr)] .

The work of Euler is linked to the theory of quaternions [2) [ 15] [36] [37], developed later in 1843 by William Hamil­ton. In his (LE3) square, Euler reuses an identity that he found and sent to Christian Goldbach in 1748 [21 ) :

(a2 + bz + c2 + d2)(p2 + q2 + r2 + s2) = (ap + bq + cr + ds)2 + (aq - bp - cs + dr)2

+ (ar + bs - cp - dq)2 + (as - br + cq - dp)Z.

This identity also follows from the fact that the norm of the product of two quaternions is the product of the norms. Euler first used this identity in 1754 [ 17) in a partial proof that every positive integer is the sum of at most four square integers, an old co{\jecture announced by Diophantus, Ba­chet, and Fermat. Using as a basis these partial results of Euler's, Lagrange published in 1770 (55] the first complete proof of this four square theorem, the same year as the let­ter from Euler with the first 4 X 4 magic square of squares.

The magic square received by Lagrange is a member of this family of squares with (a, b, c, d, p, q, r, s) = (5, 5, 9,

0, 6, 4, 2, -3).

In 1942, Gaston Benneton [ 1) [2) published another 4 X 4 square of squares using Euler's method. Its magic sum is 7150, a smaller sum than the Euler example. In 2005, we can now say that the smallest member of Euler's family producing distinct numbers is one not found by Euler, (GEl). And it is a member of a nice sub-family (CB2).

48" zt' at 18'

21 • 33' 322

1 3111 13' 422

222 � 44' 8'

CB1 . The smallest magic square of squares of Euler's family, gen­

erated by (2, 3, 5, 0, 1 , 2, 8, -4), giving 52 = 3230.

CB2. Sub-family of Euler's magic squares of squares, 52 = 85(k2 +

29). k = 0, 1, 2 do not produce distinct numbers. k = 3 produces the

(CB1) square. See [9] for another sub-family producing Euler's (LE2)

and Benneton's squares.

Its magic sum is smaller than in Euler's example (LE2), smaller than Benneton's square, but bigger than the Brem­ner example (AB3), which is not a member of Euler's fam­ily. I can confirm that the solution with the smallest sum

© 2005 Springer Science+ Business Media. Inc., Volume 27, Number 2, 2005 55

Leonhard Euler (Basel 1 107-5t. Petersburg 1783) sent in 1770 a 4 X 4 magic square of squares to Joseph-Louis Lagrange

(Euler's original letter: Bibliotheque de l'lnstitut de France, photo C. Boyer).

56 THE MATHEMATICAL INTELLIGENCER

Euler's method published in Latin at St. Petersburg

(Bibliotheque de I'Ecole Polytechnique, photos C. Boyer).

is the Bremner sample (AB3) (and all its permutations). I

can also add that it is impossible to construct another ex­

ample with a sum smaller than 3230: it means that the sec­

ond smallest solution of this problem is my square (CBI), coming from Euler's family. See the supplement [9] for a

complete numerical study of Euler's family.

Warning: both Lucas's and Euler's families often gen­

erate incorrect squares, because all their numbers are not

always distinct. For example, the square ( CB3) is smaller

than the Bremner example, but it unfortunately contains

the same number twice: a small game, will you quickly lo­

cate it?

CB3. A member of Euler's family, generated by {3, 2, 4 - 1 , 2, 8, -4),

but unfortunately an incorrect magic square!

Note that with a = p, b = q, c = -r and d = -s, Euler's

family of squares becomes (LE3cb ). It is a disguised and

permuted version of Lucas's family (ELl) of 3 X 3 squares:

LE3cb. Transition from Euler's 4 X 4 to Lucas's 3 X 3 magic square

of squares, generated by a = p, b = q, c = -r, d = -s.

And what about 5 X 5 magic squares of squares? They

are also possible. Here are two examples, (CB4) (CB5). They seem to be the first published 5 X 5 magic squares of

squares, and the smallest possible examples.

Some related squares are given below (CB9) and in the

supplement [9].

CB4. The smallest 5 X 5 magic square of squares. 82 = 1375.

CB5. The second smallest 5 X 5 magic square of squares. 82 = 1831.

Bimagic squares

Now, let us switch to bimagic squares: magic squares stay­

ing magic when their entries are squared-or, if you pre­

fer, magic squares of squares which stay magic when their

entries are not squared. The previous examples in this ar­

ticle were not bimagic, because they do not stay magic

when their entries are not squared. The first published

bimagic square was an 8 X 8 square made by G. Pfeffer­

mann in 1890, and published in January 1891 [4] [7] [45]

[47] [62] .

Amazed, Edouard Lucas immediately published in

1891 [42] , just before his accidental death, a compliment

to Pfeffermann for his achievement, together with a proof

that a 3 X 3 magic square using distinct integers cannot

be bimagic. John R. Hendricks published in 1998 [32] a

different, long proof. I propose here a third-and far

easier-proof of the impossibility, proving also the new

result that even 3 X 3 semi-bimagic squares are not pos­

sible.

=S1 =S2 CB6. A putative 3 X 3 semi-bimagic square.

If such a square exists (CB6), then:

And:

Sl = a + b + c = a + d + e e = b + c - d

S2 = a2 + b2 + c2 = a2 + d2 + e2 bz + c2 = d2 + e2 b2 + c2 = d2 + (b + c - d?

0 = 2d2 + 2bc - 2bd - 2cd (d - b)(d - c) = 0.

This implies that d = b, or d = c. But a magic square has

to use distinct numbers: without this requirement we

would have bimagic, trimagic, . . . squares with, for ex­

ample, 1 in each cell! So a 3 X 3 semi-bimagic square can­

not exist, implying of course that a 3 X 3 bimagic square

cannot exist.

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2. 2005 57

Now increment the size, searching for a 4 X 4 bimagic

square. Cauchy worked in 1812 on a related subject. After

partial proofs of Gauss (triangular case in 1796) and La­

grange (square case in 1770, as already mentioned above),

Cauchy was the first to prove completely what Fermat pro­

posed: every positive integer is a sum of at most three tri­

angular numbers, four square numbers, five pentagonal

numbers, and n n-gonal numbers . . . . In his proof [ 14],

Cauchy used the same system of two equations that we

have to use in the search for a bimagic square:

• k = t2 + u2 + v2 + w2

• s = t + u + v + w

It is not too difficult to find samples of semi-bimagic

squares ( 4 bimagic rows, 4 bimagic columns), the smallest

being ( CB7). See also an example of a semi-bimagic square

using only prime numbers in [9].

1 35 46 61 1 2 352 462 61 2 = 1 43 =7063

37 71 13 22 372 712 132 222 = 143 =7063

43 26 67 7 432 262 672 72 = 1 43 ..... =7063

62 1 1 1 7 53 622 1 12 1 72 532 =143 =7063

= 1 43 = 143 = 1 43 = 1 43 = 7063 =7063 =7063 = 7063

CB7. The smallest 4 X 4 semi-bimagic square without magic diago­

nal. 51 = 143, 52 = 7063.

It is also possible to get one magic diagonal. For example

(CBS),

9 55 1 05 36

69 1 00 21 15

28 49 1 9 109

99 1 60 45

CBS. The smallest 4 X 4 semi-bimagic square with one magic diag­

onal. 51 = 205, 52 = 1 5427.

But in a 4 X 4 square having 4 bimagic rows and 4

bimagic columns, it seems very difficult-or impossible­

to get two simply magic diagonals; or one bimagic diago­

nal. And it is proved impossible to get two bimagic diago­

nals: answering puzzle 287, posed by Carlos Rivera and me

in October 2004 [50], Dr. Luke Pebody (Trinity College,

Cambridge, England) and Jean-Claude Rosa (Cluny,

France) proved independently that a 4 X 4 bimagic square

is impossible.

It is unknown if a 5 X 5 magic square using distinct in­

tegers can be bimagic. The following (CB9) magic square

has 5 bimagic rows and 5 bimagic columns. The two diag­

onals are magic but not bimagic: not yet a bimagic square,

but a better result than the (CBS) square.

58 THE MATHEMATICAL INTELLIGENCER

3 37 20 44 1 6

34 35 1 1 2 38

41 8 24 40 7

1 0 36 47 13 14

32 4 28 1 1 45

CB9. The smallest 5 X 5 semi-bimagic square with two magic diag­

onals. 51 = 1 20, 52 = 3970.

The 6 X 6 magic square (GPJ) was published by G.

Pfefferman in 1894 [ 46]. It has 6 bimagic rows and 6 bimagic

columns. The two diagonals are magic, but not bimagic, as

in (CB9). 6 X 6 magic squares with the same characteris­

tics were published by Huber in 1891 [35], Planck in 1931

[48], Planck modified by Lieubray also in 1931 [48], Venkat­

achalam Iyer in 1961 (RVIJ) [61 ] , Collison in 1992, page 147

of [31 ] .

6 42 29 3 40 30

8 44 47 21 20 10

33 31 4 1 37 1 7

19 1 7 1 3 9 43 49

36 2 5 35 34 38

48 14 15 45 1 2 1 6

GP1. A 6 X 6 semi-bimagic square with two magic diagonals, pub­

lished by G. Pfeffermann in 1894. 51 = 1 50, 52 = 5150.

4x+y 5x+6y 2y 6x+2y x+6y 2x+y

5x+2y y 4x+6y 2x+6y 6x+y x+2y

6y 4x+ 2y 5x+y x+y 2x+2y 6x+6y

0 4x+4y 5x+5y x+5y 2x+4y 6x

5x+4y 5y 4x 2x 6x+5y x+4y

4x+5y 5x 4y 6x+4y X 2x+5y

RVI1 . A 6 X 6 semi-bimagic square family with two magic diagonals,

published by R. Venkatachalam lyer in 1961. 51 = 18x + 18y, 52 =

82x2 + 108xy + 82y2•

Open problem 3. What is the smallest bimagic square us­ing distinct integers? Its size is unknown: 5 x 5, 6 x 6,

or 7 x 7? My feeling is that 5 x 5 bimagic squares do not

exist. Bimagic squares of izes 8 X 8 and above are al­

ready known, see part 7.

Open problem 4. Construct a bimagic square using dis­

tinct prime numbers. [9) [50].

6. "It's implicit in the work of Carmichael that

there can be no 3 X 3 magic squares with entries

which are cubes or are fourth powers"

The work of Euler implies already that there can be no 3 X

3 magic square with entries which are cubes. If z3 is the

number in the centre cell, then any line going through the

centre should have x3 + y3 = 2z3. Euler and Legendre [39]

demonstrated that x3 + y3 = kz3 is impossible with dis­

tinct integers, for k = 1, 2, 3, 4, 5. Adrien-Marie Legendre

mistakenly announced that k = 6 is also impossible: Edouard

Lucas published the general solution for k = 6 in the Amer­ican Journal of Mathematics Pure and Applied of J. J.

Sylvester [41 ] , and gave the example 173 + 373 = 6 · 213. The

equation x3 + y3 = 7z3 has been known to be possible since

Fermat, one of his examples being 43 + 53 = 7 · 33.

Legendre showed also that x4 + y4 = 2z2 is impossible

if x i= y. Because z4 = (z2)2, this implies that there can be

no 3 X 3 magic square with entries which are fourth pow­

ers. It's also implicit in the later work of Carmichael [ 13]

that there can be no 3 X 3 magic square with entries which

are cubes, or are fourth powers or 4k-th powers. Noam

Elkies [26] points out that with Andrew Wiles's proof it can

be shown that an + bn = 2cn has no solution for n greater

than 2, and thus that there can be no 3 X 3 magic square

with entries which are powers greater than 2.

And as said in the D2 problem, "The Fermat problem,"

page 219 of Guy's book [30], "It follows from the work of

Ribet via Mazur & Kamienny and Darmon & Merel that the

equation xn + yn = 2zn has no solution for n > 2 apart

from the trivial x = y = z."

So, 3 X 3 magic squares of cubes are impossible. I think

that 4 X 4 are also impossible with distinct positive inte­

gers. The 12 X 12 (WTl) trimagic square of part 7 below,

when its numbers are cubed, is a magic square of cubes.

If we accept negative integers, and using the interesting

but obvious remark that n3 and ( -n3) are not equal (the rule

in a magic square is to use "distinct" integers, and the trick

is that they are distinct!), (CBJO) and (CEl l) are magic

squares of cubes having a nuU magic sum. They seem to be

the first published 4 X 4 and 5 X 5 magic squares of cubes. If you do not like the terminological trick I used, then Open

problem 5 is for you! And the (CB12) square is a first step.

Open problem 5. Construct the smallest possible magic square of cubes: 5a) using integers having different ab­

solute values, 5b) using only positive integers.

Open problem 6. Construct a magic square of cubes of

prime numbers [9].

1g3 (-3)3 (- 1 0)3 (- 1 8)3

(-42)3 213 283 353

423 (-21)3 (-28)3 (-35)3

(- 19)3 33 1 03 183

CB10. A 4 X 4 magic square of cubes. 53 = 0.

1 1 3 (-20)3 123 1 33 143

(- 1 5)3 213 33 (- 1 0)3 (- 1 7)3

(-5)3 (-4)3 03 43 sa --1-

1 73 103 (-3)3 (-21 )3 1 53

(- 1 4)3 (- 1 3)3 (- 12)3 203 (- 1 1)3 CB1 1 . A 5 X 5 magic square of cubes. 53 = 0. A special property:

four consecutive integers are used in the first row.

g3 473 543 643 963

233 973 63 483 (723

1 03 1 43 673 1013 423

1 103 363 213 33 283

403 703 983 1 83 383

CB12. The smallest 5 X 5 semi-magic square of cubes using posi­

tive integers. 53 = 1 408896.

7. "The following example [30] [34] , due to David

Collison, is 'trimagic' in the sense that it is magic

and stays so when you either square or cube the

entries"

Collison's 16 X 16 trimagic square uses distinct but non­consecutive integers. Bimagic and trimagic squares using

consecutive integers, so more restricted than squares us­

ing non-consecutive integers, were published before this

Collison square.

Wanting. In parts 1 to 6, we spoke about magic squares

using distinct integers, but generally non-consecutive in­

tegers. ow, in parts 7 to 9, we speak about magic

squares and cubes using consecutive integers.

The first published n X n bimagic square using consec­

utive integers from 1 to n2 is an 8 X 8 square made by G.

Pfeffermann in 1890, and published in January 1891 [45] .

Various other bimagic squares are currently known with

various sizes n ::=:: 8. Walter Trump and I showed in 2002

that an n X n bimagic square using consecutive integers is

impossible for n < 8.

The first published trimagic square, staying magic when

you either square or cube the entries, is a 128 X 128 magic

square made by Gaston Tarry in 1905 [59]. Later, smaller

trimagic squares were found: 64 X 64 by General E. Caza­

las in 1933 [16] , 32 X 32 by William H. Benson in 1976 [3],

and 12 X 12 (WTl) by Walter Trump in 2002 [60] .

Even using consecutive integers, this trimagic square is

smaller than Collison's square. Walter Trump and I showed in

2002 that an n X n trimagic square is impossible for n < 12.

Nobody will ever construct a trimagic square smaller than

Trump's.

© 2005 Springer Science+Business Media, Inc. , Volume 27, Number 2, 2005 59

1 22 33 41 62 66 79 83 104 1 1 2 123 144

9 1 1 9 45 1 1 5 1 07 93 52 38 30 1 00 26 136 1--

75 141 35 48 57 14 131 88 97 1 1 0 4 70

74 8 1 06 49 1 2 43 102 133 96 39 137 71 1--1- - --

140 101 124 42 60 37 108 85 103 21 44 5

1 22 76 1 42 86 67 126 19 78 59 3 69 23

55 27 95 1 35 130 89 56 1 5 1 0 50 1 1 8 90 - - --

132 1 1 7 68 91 1 1 99 46 134 54 77 28 1 3 i- - -

73 64 2 1 2 1 1 09 32 1 1 3 36 24 1 43 81 72

58 98 84 1 16 138 1 6 129 7 29 61 47 87 -

80 34 105 6 92 1 27 1 8 53 1 39 40 1 1 1 65 -- -

51 63 31 20 25 1 28 1 7 120 125 1 14 82 94

WT1 . The first 12 X 12 trimagic square, constructed by Walter Trump in 2002.

This square using consecutive integers from 1 to 144 is magic, and it is again

magic when its numbers are squared or cubed. 51 = 870, 52 = 83810, and

53 = 9082800. It is impossible to construct a trimagic square of a smaller size

(and using consecutive integers).

In 2001, I published the first known tetramagic and pen­tamagic squares, staying magic when you square, cube, or raise to the fourth or fifth power the entries, in an article of Pour La Science (the French edition of Scientific Amer­ican) [4] after a joint work with Andre Viricel: 512 x 512 tetra and 1024 X 1024 penta. Later, I constructed a smaller tetramagic square (256 X 256), and Li Wen a smaller pen­tamagic square (729 X 729). It is unknown if n X n tetra­magic and pentamagic squares are possible for 12 < n < 256 and 12 < n < 729. The first hexamagic square, magic until the 6th power, was constructed by Pan Fengchu in 2003 ( 4096 X 4096).

For more about all these multimagic squares, see refer­ences [4] [7] [ 16] [47] [57] [62] . Because they are at least bimagic, all the multimagic squares are "magic squares of squares."

8. "Has anyone constructed a 5 X 5 X 5 magic

cube, or proved its impossibility?"

Martin Gardner already asked this question in 1976 [22] [23] and again in 1988 [24].

Yes, a 5 X 5 X 5 magic cube has been constructed, re­cently, in 2003.

What should we mean here by "magic cube"? Richard Guy speaks about n X n X n perfect magic cubes using consecu­tive integers from 1 to n3, and having 3n2 + 6n + 4 magic lines: their n2 rows, n2 columns, n2 pillars, 3 · 2n plane diag­onals, and 4 space diagonals. A standard magic cube ( = non­perfect) does not have its 3 · 2n plane diagonals magic.

A 3 X 3 X 3 magic cube is possible, but a 3 X 3 X 3 per­fect magic cube is impossible. In 1640, Fermat [58] sent to Mersenne a 4 X 4 X 4 nearly perfect magic cube (PFl) with 64 magic rows: Fermat mistakenly announced 72 magic

60 THE MATHEMATICAL INTELLIGENCER

lines in his cube, but 64 remains an excellent result com­pared to the needed 76 theoretical magic rows.

More than three centuries after Fermat's 4 X 4 X 4 cube, Richard Schroeppel showed in 1972 [53] that a 4 X 4 X 4 per­fect magic cube is impossible, and in 1976 [54] that, if a 5 X 5 X 5 perfect magic cube is possible, then its centre cell is 63.

Walter Trump and I constructed in 2003 [6] [7] [63] the first known 5 X 5 X 5 perfect magic cube (WT2CB13): of course, its centre cell is 63.

Bigger magic cubes of various sizes were previously known, the first known perfect magic cube being a 7 X 7 X 7 cube constructed in 1866 by Reverend Andrew H. Frost [20].

The first published bimagic cube is a 25 X 25 X 25 cube created in 2000 by John R. Hendricks [33]. Coincidence? A

bimagic cube of the same size was announced several years before by David Collison Gust before his death, but unfor­tunately never published) to the same John R. Hendricks: news announced by John R. Hendricks himself in 1992 [31] .

In 2003 [5], I succeeded i n constructing various multimagic cubes including the currently smallest known bimagic cube, a 16 X 16 X 16 cube, smaller than Hendricks's. I also con­structed the first known perfect bimagic cubes, and the first known standard and perfect trimagic and tetramagic cubes. My smallest perfect bimagic cube is 32 X 32 X 32.

I showed that n X n X n bimagic and perfect bimagic cubes are impossible for n < 8. It is unknown if n X n X n bimagic cubes are possible for 8 ::::: n < 16, and perfect bimagic for 8 ::::: n < 32.

Open problem 7. Construct the smallest possible magic cube of squares (the 16 x 16 X 16 bimagic cube, when its numbers are squared, is already a magic cube of squares).

/ ...r.J? / .!7...r 7 .-7 ...r<Y7 / -- / .26" / .2.T / .J?.T7 / .24" / .?4" / .S1:9 / .2.!7 / / -- / ...r.:F / ...r- / ..£? / r / 6".:7 / 6" / .T 1/ .:F.T / / ...r.T / -.T / - / ...?,&7 / / -.!7 / ..Z'.9 / ...rB' / 4 IS" / [/ 4" / -/ .:F.9 / .:r 17 I

/ 4I:SI" / ...:z:.r / ...r.t? I/ __. / / .Sf:.? / _,., / .J?-!7 / .2..9 / / .9'6" / ..?.:77 .szf 7 -:5 W7 / ..9 / .:F-5" / � / ..z:.? /

I / - / <!£P / 6U1' v ...r / / -...r / .23 / ....;;;?!? / -7 / .;...r / 4/CY / -- / .::; t.r 7

1/ - / ..2 / ..? / «r 1/ PF1 . Pierre de Fermat (Beaumont de Lomagnes 1601-Castres 1665) sent in 1640 a 4 X 4 X 4 nearly perfect magic cube to Mersenne.

Among its 76 lines, 64 lines have the magic sum 51 = 130, but 12 lines have a different sum.

/ .2.!7 / ...r6" / - 7 ...r-7 ..9.:7 7 / .L.r.:F / .94" / - / ...r / ..9.T / / -<JC2 /�/ � / .2 / .T.!r / / 6"6" 7 .?.::? 7 .2.T /...ra? / - / / 6'".T / ...rB' / ...:z:.r..? / ...rPI6" / .!7 7

/ ..?...r / .T.T / .?-r / / .T.t? / / � / 6"- /..z.r.T7 -17 .r..Y"7 / .?.:7 /'...:z:.rB' / � /..z:.?..Y'7 12.?7 / .26'" / .?.9 / .R.? / - /1 -/ 1/'...:z:.rd" / ...r ..T / ...r- / .?:Y / ..9.!7 I/

/ -..T / 6"...r / � / � liP7 4"6'" 7 /...r.t?.T/ 4/CY / � / ..Y'J v --7 / 4".9 / 6"4" -- / .:?..T / / .?.2 / ..9.? / - / 4CY / ..z 19 / v -- / .!7.:7 / «r / 6:? 7 ..T..9 [7 / .J?...r / � /�/...r. �/ ...r.t? / / ...z:.2 / - / _,_ / B'..T 1/..,r,e?,e?/ /...r�/ ..Y' /...r�7 4" / ..96" / / .L.r.JI'"/ .!7..T / .9 / dP / / 101"7 / ..5"6'" / ...z:.2.:7 / � / 11<9 / ..Y'-!7 v /"..z:.?...r /...r6'4" / ..T / � !? / .!7..9 /

/ .2..9 / .24" 7 -Z.2? /...z:.2� ...:z:.r / / .£r / ...r.:F / ..c:r /�/ --7 / ..TB' / .5"- / .9:9 / .2- / ,6 � / v .?6'" /'...:z:.r.t? / -6'" / ....;;;?!? /...r.a: I/

WT2CB13. The first 5 x 5 x 5 perfect magic cube, constructed in 2003 by Walter Trump and Christian Boyer. Its 109 lines have the

magic sum 51 = 315. The centre cell is the average cell value: 63. A 3 X 3 X 3 or 4 X 4 X 4 perfect magic cube is impossible.

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005 61

Open problem 8. Construct a bimagic cube smaller than 16 X 16 X 16.

Open problem 9. Construct a perlect bimagic cube mailer than 32 X 32 X 32.

My smallest perfect trimagic cube is a 256 X 256 X 256 cube. I thank Yves Gallot, Renaud Lifchitz, Walter Trump, and Eric Weisstein, the author of the excellent Math World ency­clopaedia http://mathworld.woifram.coml, who each inde­pendently checked this cube and confirmed its properties.

My best multimagic cube is a 8192 X 8192 X 8192 per­fect tetramagic cube, staying perfectly magic when you square, cube, or raise to the fourth power the entries.

The main challenge in magic squares or cubes is to have the maximum of characteristics in the minimum of space. That's why most of the open problems of this article are to fmd the smaUest squares or cubes having the prescribed properties. A big object has no interest if it has poor char­acteristics. But we fmd that each time we want to add char­acteristics, i.e., a new multimagic degree, we are "forced" to work with a bigger object. I am unable, at least today, to get the perfect tetramagic characteristics in a smaller cube!

This 8192 X 8192 X 8192 cube is monstrous. All its 201 million possible lines of 8192 numbers are tetra-magic: its 67, 108,864 lines, 67, 108,864 columns, 67, 108,864 pillars, 4 main diagonals, and 49,152 small diagonals. All the sums of these 201 million lines are equal to the same number S1 =

2251799813681 152. When its 550 billion numbers are raised

CB14. The 8192 X 8192 X 8192 perfect tetramagic cube compared

to the cathedral of Notre-Dame de Paris!

51 = 2251799813681 152

52 = 825293359521335050119065600

53 = 340282366919700523424090353056775929856

S4 = 14965776766200389409021627523658015558475388872704{)

62 THE MATHEMATICAL INTELLIGENCER

to the 4th power, all the sums of these 201 millions of lines are equal to the same number:

S4 = 149657767662003894090216275236580155584753888727040.

Similarly, the lines have the same sums, S2 and S3, when their numbers are squared and cubes, respectively. This cube is big:

• So big that, if you built it, with cells of 2 em X 2 em X 2 em on each of which a number of 12 digits maximum is engraved, you would get a cube bigger than Notre­Dame de Paris. (CB14)

• So big that, if you check 1000 numbers per second, you will need more than 17 years to check the whole cube.

It is very difficult to compute and to check such a huge cube. Eric Weisstein was able to check the 256th-order per­fect trimagic cube, but he said that he and Mathematica were not able to check 512th-order or bigger cubes, mainly for memory reasons.

I thank Yves Gallot, the author of the famous Proth pro­gram used worldwide by searchers for big prime numbers, who checked this tetramagic cube and confirmed its prop­erties. He wrote a nice piece of specific assembly and C + +

code to check the cube, different from my code: the com­puting work behind this big cube is comparable to the com­puting work behind a big prime number. One needs vast memory and multiprecision routines.

Again in 2003, I constructed the first known multimagic hypercubes of dimension 4. For example, a 256 X 256 X

256 X 256 trimagic hypercube, also checked by Yves Gallot. About these magic and multimagic cubes and hyper­

cubes, see [5] [6] [7] [47] [62] [63].

9. "Rich Schroeppel notes that the centre cell of a

magic 95 is always the average cell value, and

that a corollary is that there is no magic 96." This sentence needs an explanation: magic 95 and magic 96 mean perfect magic hypercubes of dimension 5 and 6, us­ing consecutive integers.

Schroeppel's demonstration referred to is similar to his demonstration in part 8: the centre cell of what he called a "magic 53" ( = a perfect magic cube 5 X 5 X 5)-if it exists-is the average cell value [54]. And a corollary is that there is no "magic 54" ( = perfect magic hypercube 5 X 5 X

5 X 5). The current list of the conclusions of Richard Schroep­

pel are these:

• centre cell of magic 53 is average value � there is no magic 54.

• centre cell of magic 74 is average value � there is no magic 75.

• centre cell of magic 95 is average value � there is no magic 96.

We can easily fill in the simple case:

• centre cell of magic 32 is average value � there is no magic 33

The general case would apparently be that for all k :2: 1 , • centre cell of magic (2k + 1 )k+ 1 is average value � no

magic (2k + 1)k+Z.

REFERENCES

(1 ] Gaston Benneton, Sur un probleme d 'Euler, Comptes-Rendus

Hebdomadaires des Seances de I'Academie des Sciences

214(1 942), 459-461 .

[2] Gaston Benneton, Arithmetique des Ouaternions, Bulletin de Ia So­

ciete Mathematique de France 71 (1 943), 78-1 1 1 .

[3] William Benson and Oswald Jacoby, New recreations with magic

squares, Dover, New York, 1 976, 84-92

(4] Christian Boyer, Les premiers carres tetra et pentamagiques, Pour

La Science N°286, August 2001 , 98-1 02 .

[5] Christian Boyer, Les cubes magiques, Pour La Science N°31 1 ,

September 2003, 90-95.

(6] Christian Boyer, Le plus petit cube magique parfait, La Recherche,

N°373, March 2004, 48-50.

[7] Christian Boyer, Multimagic squares, cubes and hypercubes web

site, www.multimagie.com/indexengl.htm

[8] Christian Boyer, A search for 3 x 3 magic squares having more

than six square integers among their nine distinct integers, preprint,

September 2004

[9] Christian Boyer, Supplement to the article " Some notes on the

magic squares of squares problem" , downloadable from (7], 2005

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A U T H O R

CHRISTIAN BOYER

53, rue De Mora 95880 Enghien les Bains

France e-mail: cboyer@club-internet.fr

Christian Boyer was born near Bordeaux, and graduated from

two of the "grandes eccles" of engineering . He has worked

for Microsoft France, and more recently co-founded a suc­

cessful start-up in software. {He even offers advice to others

creating new companies.) Beside mathematical research and

popularizing of mathematics, he is a sports-car aficionado . He

is married, with three daughters.

Villars, Paris, 2(1 894), pp. 1 86-1 94 (partial reprint of the letter at

www.multimagie.com/Francais/Fermat.htm)

[59] Gaston Tarry, Le carre trirnagique de 1 28, Compte-Rendu de /'As­

sociation Fran9aise pour /'Avancement des Sciences, 34eme ses­

sion Cherbourg ( 1 905), 34-45

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www. multimagie. com/Eng/ish/Tri 12Story. htm

[61 ] R. Venkatachalam lyer, A six-cell bimagic square, The Mathemat­

ics Student 29(1 961 ), 29-31

[62] Eric Weisstein, Magic figures, MathWorld, http://mathworld.

wolfram. comltopics/MagicFigures. html

[63] Eric Weisstein, Perfect magic cube, MathWorld, http:!/

math world. wolfram. com/PerfectMagicCube.html

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MARTIN J. MOHLENKAMP AND LUCAS MONZON

Trigonometric dent it ies and Sums of Separab e Funct ions

For what value(s) of a, {3, 'Y does the equality

. . sin(y + {3- a) sin(z + 'Y - a) sm(x + y + z) = sm(x) . ( ) . ) sm {3 - a sm( 'Y - a

+ sin(x + a - {3) . ( )

sin(z + 'Y - {3) sm y .

( ) ,

sin(a - {3) sm 'Y - {3

+ sin(x + a - y) sin(y + {3 - y)

sin(z) sin(a - y) sin(f3 - y)

hold for all values of x, y, and z?

Motivation

(1)

Modem computers have made commonplace many calcu­lations that were impossible to imagine a few years ago. Still, when you face a problem with a high physical di­mension, you immediately encounter the Curse of Dimen­sionality [ 1 , p.94]. This curse is that the amount of com­puting power that you need grows exponentially with the dimension. The simplest manifestation appears when you try to represent a function by its sample values on a grid. If a function of one variable requires N samples, then an analogous function of n variables will need a grid of N" samples. Thus, even relatively small problems in high di­mensions are still unreasonably expensive.

A method has been proposed in [2] to address this prob­lem, based on approximating a function by a sum of sepa­rable functions:

r

f(xl, · · . , xn) = L ¢{(xi)¢�(x2) · · · ¢�(xn). (2) j� I

This representation would require only r n N samples, so if the approximation can be made sufficiently accurate while keeping the separation rank r small, we can bypass the curse.

We describe here a particular test of (2), when the "straight­forward" approximation is exact but has very large separation rank. Although it may not be directly useful in applications, the result of this test is surprising, positive, and, we believe, cute. It illustrates a richness of structure that invites future study. Other mechanisms that allow representations of the form (2) with low separation rank are described in [2] .

A Test Function

Our test function is the sine of the sum of n variables, sin(�.f� 1 Xj), which is a wave oriented in the "diagonal" di­rection in n-dimensional space. One could use complex ex­ponentials to express it as the sum of two separable func­tions, ( n ) 1 n . 1 n .

sin Ixj = ---; fl eur:J - ---; fl e-ur:J, j� l 2t j� l 2t j� l

but in our test we allowed only real functions. You can use ordinary trigonometric identities to find

such a representation. When n = 2 we have

sin(x + y) = sin(x) cos(y) + cos(x) sin(y), (3)

which expresses sin(x + y) as a sum of two separable func­tions. When n = 3 we have

sin(x + y + z) = sin(x) cos(y) cos(z) + cos(x) cos(y) sin(z) + cos(x) sin(y) cos(z) - sin(x) sin(y) sin(z), (4)

This material is based in part on work supported by the National Science Foundation under grant DMS-9902365, and by University of Virginia subcontract MDA-972-

00-1-0016.

© 2005 Springer Sc1ence+ Business Media, Inc . . Volume 27, Number 2, 2005 65

Figure 1. left: Graphical separated representation of sin(K + y + z) using the usual trigonometric identity (4). Each of the four rows gives the

factors of a separable function. For example, the first row corresponds to sin(K) cos(y) cos(z). The separable functions from each row are

then added. right: Graphical separated representation of sin(K + y + z) using (1) with a = 0, {3 = Tr/3, and y = 2Tr/3. The amplitude has been

equidistributed. (The original illustration uses three colors, blue, red, and green, for the three curves in each graph.)

which uses four terms. The drawback to this approach is that, for n variables, the number of terms is 2n- l. This exponen­tial growth in the number of terms negates the benefit of us­ing the form (2). Indeed, if this really is the minimal number of terms needed, then the entire approach is doomed.

We then asked what the minimal number of terms is, and our program replied "n" and produced graphs such as that shown in Figure 1 (right). After some investigation, we de­termined the trigonometric identity that our program had un­covered. What is most remarkable is that the program was numerical, not symbolic, and so uncovered a trigonometric identity without even knowing it was doing trigonometry!

Any representation of a function of a sum of n variables will have n - 1 free parameters, because one can include n shifts Xj � Xj + ai and one linear constraint �J= 1 ai = 0

and have �J= l (Xj + aj) = �J=l Xj. The identity that we pre­sent below in Theorem 2 has n - 1 additional independent parameters, which play a structural role in our represen­tation. When n = 3, it provides an answer for our opening teaser. The identity (1) holds for arbitrary a, {3, and y, as long as sin( a - {3) * 0, sin( a - y) * 0, and sin({3 - y) * 0.

Since these three parameters occur only as differences, only two of them are independent. One can introduce two additional parameters as phase shifts to make versions of (1) with different symmetries.

The Identity

Lemma 1 The function s(x) = sin(x) satisfies the equation

s(A + B) = s(A)s(B + {3 - a)

+ s(A + a - {3)s(B)

s({3 - a) s(a - {3) (5)

66 THE MATHEMATICAL INTELLIGENCER

for all values of A, B, a, and {3 such that s( a - {3) * 0 (and s({3 - a) * 0).

Proof With the notation c(x) = cos(x) and y = {3 - a, par­tially expand the right-hand side using the usual trigono­metric identity (3) to obtain

s(A)

s( y) (s(B)c( y) + c(B)s( y))

s(B) + -

(-

) (s(A)c(- y) + c(A)s( - y))

s - y

Multiplying out and using that s(x) is odd and c(x) is even, all terms cancel except for s(A)c(B) + c(A)s(B), which we recognize as s(A + B). D

Theorem 2 Any junction s(x) that satisfies (5) also sat­isfies ( � ) - � ( ) nn s(xk + ak - aj)

s L Xj - L s Xj . j= l j= l k=l,k¥j s(ak - aj) (6)

for aU choices of { llJ} such that s( ak - aj) * 0 for aU j * k.

The proof is by induction and is given in the Appendix. We can generate a more general form by introducing n shifts a/

s Ct Xj +it ai) n n

= I s(Xj + aj) n j=l k= l,k¥j s(xk + ak + ak - aj)

s(ak - aj)

By choosing different ways to satisfy the linear constraint �J= l ai = 0, we can produce a variety of identities similar

to (6) without modifying the parameters a1, which are the structural elements of our representation. Note that in the set { ak - a1}Mj only n - 1 parameters are linearly inde­pendent, say { a1 - ai1J=2·

Other Functions that Satisfy the Same Identity

Because of Lemma 1 and Theorem 2, we know that sin(�J= l x1) is exactly separated with separation rank n. Moreover, this function is peculiar in that sin(·) is the only function used in the separated representation. We now consider the problem of finding other functions s(x) sat­isfying (6). Since the general case (6) is equivalent to the n = 2 case, it is enough to describe all functions that sat­isfy (5).

Lemma 3 The function s(x) = x satisfies the identity (5).

This and the following lemma may be verified directly.

Lemma 4 If s(x) satisfies (5), then so does

a exp(bx)s(cx)

for all complex a =F 0, b, and c =F 0.

Starting with our two basic functions sin(x) and x, we can use Lemma 4 to construct other functions that satisfy (5), and then ask if we have missed any others. We only wish to consider reasonably nice functions. The appropri­ate condition is that s(x) be meromorphic.

Theorem 5 A meromorphic function s(x) satisfies (5) if and only if

s(x) = a exp(bx)x or s(x) = a exp(bx) sin(cx)

for some complex constants a =F 0, b, and c =F 0.

The proof is given in the Appendix.

Extensions and Relationships with Other Identities

If in Theorem 2 we set x1 = a for all j, we obtain the fol­lowing corollary.

Corollary 6 Under the same conditions as in Theorem 2,

s(na) = I rl s(a + ak - a1)

s(a) j= l k= ! ,k #j s(ak - aj) When s(x) = sin(x), this result is presented in [4] . For a proof using Lagrangian interpolation, see [5, page 272]. The approach of [4] and [5], however, does not produce the gen­eral results of Theorems 2 and 5. Conversely, our results can be used to derive only a few of the identities listed in [5, Section 2.4.5.3].

The situation is different if we consider Milne's identity [6, 3]

n n n

1 - I1 Yi = I (1 - yj) I1 j= l j= l k= l,k #j

1 - Yk8k/81 1 - 8kJe1 ·

(7)

We can obtain another proof of this identity by setting s(x) = 1 - exp(x), ai = In 8j, and Xj = In Yi in (6). Theorem 5 applies to this function because 1 - exp(x) = -2i exp(�) sin(�). Conversely, (6) for s(x) = sin(x) can be obtained

by setting y1 = exp( -2ix1) and e1 = exp( - 2ia1) in Milne's identity, and then multiplying by exp(i�J= l x1)!2i.

A "multiplicative" version of the identities that we have discussed can be derived by generalizing this observation. Simply note that the identity

f(C D) = f(C)f(D¢/8) + f(C8/¢)f(D)

f(¢/8) !(8/¢)

is equivalent to (5) with the substitutions C = exp(A), D =

exp(B), 8 = exp(a), ¢ = exp(/3) , and s(x) = f(exp(x)). Sim­ilarly, (6) is equivalent to

f(Il Yi) = If(yj) Il f(Yk8k181), (8)

j= l j= l k=l,k#j fC8k/8j)

In analogy to Lemma 4, from the particular solutions f(x) =

ln(x) and f(x) = 1 - x to (8) we can generate other solu­tions to (8), namely

axb f(XC)

for constants a, b, and c. In this way we obtain a general­ization of Milne's identity.

Remarks and Conclusions

It is easy to extend our results to find similar identities for f(�J= l Xj), where f(x) could be cos(x), cos2(x), or sin2(x), for example.

We also tested the function of six variables sin( u + v + w) sin(x + y + z). Using ( 1) on each factor and then mul­tiplying out yields a representation of the form (2) with 9

terms, but our program found a representation with 8

terms. After considerable effort, we have still not been able to find the formula analogous to (6) for this case.

A survey on the problem of exact separated represen­tations is the book [7] by Rassias and Sim8a. As they pointed out in Problem 4 on page 158, to find a minimal rank rep­resentation for a separated representation is still an open problem. We believe that our Theorems 2 and 5 are an ex­ample of such minimal representations.

Lemma 1 can be proven geometrically, in a way similar to the geometric proof of the usual identity (3). We have not been able to find a geometric interpretation of (6).

Appendix: Proofs

Proof of Theorem 2 The case n = 2 is Lemma 1 with A = x1, B = x2, a = a1, and {3 = a2. The proof will be by induction in n, so we as­sume (6) has been proven for n - 1. We will use (5) to sep­arate out the variable Xn, then cancel like terms and reduce the n case to the n - 1 case.

First, expand the left-hand side of (6) using (5) with A =

r;: f Xj, B = Xn, (X = O'.n- 1 , and {3 = O'.n to obtain

S(Xn + O'.n - O'.n- 1) s( O'.n - O'.n- 1)

+ S (nfl Xj + O'.n- 1 - an) s(Xn) . (9)

J = l s(an- l - an)

© 2005 Springer Sc1ence+Business Media, Inc., Volume 27, Number 2, 2005 67

On the right-hand side of (6), first separate off the j =

n term in the sum. When j i= n, we expand the k = n term

in the product using (5) with A = a, - a1, B = Xn, a = an-1, and {3 = an. Explicitly, the k = n term is

s(Xn + a, - aj) =

1

s(an - aj) s(a, - aj)

X ( S(a,, - aj)S(Xn + Cin - an- 1) + S(Cin- 1 - aj)S(Xn) ) s(an - Cin-1) S(Cin- 1 - a,)

s(xn + a, - Cin- 1) ( s( Cin- 1 - aJ) ) ( s(xn) ) =

s(a, - Cin- 1) + s(an - aj) s(an-1 - an) ·

Note that the first term does not depend onj, and that when

j = n - 1 the second term is absent. Combining these ex­

pansions, we can express the right-hand side of (6) as (I1 s(Xj)

fY s(xk + ak - aJ) ) s(xn + � - Cin- 1)

J=1 k=1 ,k"'J s(ak - aj) s(an Cin- 1)

(�2 )

S(Cin- 1 - aj) nn-1 S(Xk + ak - aj) ) + L s(Xj

j= 1 s(an - aj) k= 1,k#J s(ak - aj)

Now compare our expansions (9) and (10) of the two sides

of (6). Using the induction hypothesis at n - 1, we can see

that the first terms in (9) and (10) are equal, and so cancel.

The remaining terms all have a factor of s(xn) in the numer­

ator and s( a,- 1 - a,) in the denominator, which we can also

cancel. Thus we have reduced the proof to showing that (n-1 ) S � Xj + Cin- 1 - an

J= l

Now make the substitutions Xn-1 = Xn-1 + an- 1 - an and

an - 1 = an and rearrange to obtain (n-2 - ) n-2 s(in- 1 + Un-1 - aj)

s I Xj + Xn- 1 = I s(xj) C _ ) X j=l j=l S an- 1 aJ

We recognize this equation as the n - 1 case of (6), which

is true by the induction hypothesis. D The proof of Theorem 5 depends on two lemmas.

Lemma 7 If a meromorphic function s(x) satisfies (5), then there exists a complex constant b such that exp( - bx)s(x) is an odd function.

Lemma 8 An odd meromorphicfunction s(x) satisfies (5) if and only if

s(x) = ax or s(x) = a sin(cx)

for some complex constants a i= 0 and c i= 0.

68 THE MATHEMATICAL INTELLIGENCER

Proof of Theorem 5, given Lemmas 7 and 8 We have already shown that these functions satisfy (5), so

we need only show there are no more solutions. We now assume s(x) satisfies (5) and will deduce its properties.

Using Lemma 7, we know that h(x) = exp( -bx)s(x) is

an odd function. By Lemma 4, h(x) also satisfies (5). Then,

by Lemma 8, h(x) is either ax or a sin(cx), so s(x) = a exp(bx)x or s(x) = a exp(bx) sin(x), which completes the

proof. D The proofs of Lemmas 7 and 8 use the fact that s(O) =

0. By setting {3 - a = A in (5) and subtracting s(A + B) from both sides we obtain

s(O)s(B) 0 =

s(-A) '

valid for all A such that s(-A) i= 0 and for all B. Choosing

B such that s(B) i= 0 implies that s(O) = 0.

Proof of Lemma 7 We define the auxiliary meromorphic function

s(x) F(x) = - -­

s( -x) ' (11)

which cannot be identically zero, and show that it satisfies

the functional equation

F(x + w) = F(x)F(w). (12)

This functional equation is satisfied only by exponentials,

so we can conclude that F(x) = exp(2bx) for some con­

stant b. Rewriting this condition in terms of s, we have

exp( -bx)s(x) = -exp(bx)s( -x), which is what we are try­

ing to show.

To show (12), we substitute in (11) and manipulate to

form the equivalent equation

0 = s(x)s(w)

+ s( -w)s( -x)

. (13) s(x + w) s(-x - w)

Using (5) with A = x, B = -x, a= -x, and {3 = w, we con­

clude that the right-hand side of (13) is equal to s(O) = 0. D

Proof of Lemma 8 Taking a derivative with respect to A in (5), using the fact

that s is odd, and setting A = - a, B = a and {3 = -a, we obtain

s'(O)s(2a) = 2s(a)s '(a) .

Thus, s' (0) i= 0, and because of the invariance with respect

to multiplication by constants, we can assume s '(O) = 1.

We have the system

{ s'(O) s(2a)

Since s(O) = 0, we know that s is analytic around zero.

We can write s(z) = kk=O akzZk+ 1 and use the previous con­

ditions to obtain a recurrence for the sequence an, { a0 = 1

22n+ 1an = (2n + 2) k�=O an-kak. (14)

A U T H O R S

MARTIN MOHLENKAMP

Department of MathematiCs Ohio University

Athens, OH 45701 USA

e-mail: mjm@math.ohiOu.edu

Martin Mohlenkamp received his Ph.D. in 1 997 from Yale Univer­

sity. He spent a semester at MSRI in Berkeley and several years

at the University of Colorado in Boulder before moving to Ohio Uni­

versity. He works mainly in numerical analysis.

The value of an for n > 1 is uniquely determined by the

value of a1, which is arbitrary. Setting A = 6a1 we claim

An an = (2n + 1)! · (15)

When A = 0 we have s(x) = x and when A =!= 0 we have s(x) =

sin( Ax) and the Lemma follows. We prove the claim by gen­

eralized induction on the variable n. Thus we assume (15) for

0 ::::; n ::::; N - 1, and show it for n = N. Using (14) with n = N, N

22N+laN = (2N + 2) L aN-kak k�O

N-1 AN-k = 2(2N + 2)aoaN + (2N + 2)k� (2(N - k) + 1)! (2k + 1) ! '

and thus

AN 1 N- 1 (2N + 2) aN= (2N + 1)! 22N+ l - 2(2N + 2) k2:.l 2k + 1 '

and the result follows because 2f�o (::�) = 22N+ l.

Acknowledgments

D

Thanks to Gregory Beylkin for leading us to separate sine

in the first place. Thanks to Richard Askey for pointing out

LUCAS MONZ6N

Department of Applied Mathematics

University of Colorado

Boulder, CO 80309-0526 USA

e-mail: monzon@colorado.edu

Lucas Monz6n, following his undergraduate degree from the Uni­

versity of Buenos Aires, obtained his Ph.D. from Yale University.

Besides computational harmonic analysis, he enjoys theater, po­

etry, and the visual arts. His partner is Mariana lurcovich, an in­

ternational consultant in public health.

that the identity (6) is valid for s(x) = x, and so inspiring

Theorem 5.

REFERENCES

[ 1 ] Richard Bellman. Adaptive Control Processes: A Guided Tour.

Princeton University Press, Princeton, New Jersey, 1 961 .

[2] Gregory Beylkin and Martin J. Mohlenkamp. Numerical operator cal­

culus in higher dimensions. Proc. Nat!. Acad. Sci. USA, 99(1 6):

1 0246-1 0251 , August 2002. University of Colorado, APPM preprint

#476, August 2001 ; http://www.pnas.org/cgi/content!abstract/

1 1 2329799v1 .

[3] Gaurav Bhatnagar. A short proof of an identity of Sylvester. Int. J.

Math. Math. Sci. , 22(2) :43 1 -435, 1 999.

[4] F. Calogero. Remarkable matrices and trigonometric identities I I . Commun. Appl. Anal. , 3(2) :267-270, 1 999.

[5] F. Calogero. Classical many-body problems amenable to exact

treatments. Lecture Notes in Physics, monographs m66. Springer­

Verlag, 2001 .

[6] S. C. Milne. A q-analog of the Gauss summation theorem for hy­

pergeometric series in u(n). Adv. in Math. , 72(1 ) :59--1 31 , 1 988.

[7] Themistocles M. Rassias and Jaromir Smsa. Finite sums decom­

positions in mathematical analysis. Pure and Applied Mathematics.

John Wiley & Sons Ltd. , Chichester, 1 995.

© 2005 Springer Science+ Business Media, Inc., Volume 27. Number 2, 2005 69

GERALD L. ALEXANDERSON AND LEONARD F. KLOSINS

Mathemat ic ians and O d Books

hose interested in book collecting, mathematical books in particular, have observed

remarkable phenomenon over the past few years: the prices of rare mathematical m

terials have not been driven downward by the recent recession and the bursting of t.

dot-com bubble. While prices for fine art definitely slumped and prices in other are'

of book collecting have declined or held steady, reflecting

the state of the economy, prices of books in mathematics

and science, particularly at the high end, have risen, even

after extraordinary price increases in the 1980s and 1990s.

In 1982 we wrote a short article on collecting rare math­

ematics books [1 ) . Recently we learned that the article was

scheduled to go into a mathematical anthology and, when

we reread it, we found that it was very out of date, not only the prices but other aspects as well. We decided more could

be said on the subject. Here we'll be talking mainly about

first editions of mathematical classics, unless otherwise

noted.

Probably the most extraordinary price increases in mod­em times appeared in the famous auction of the Haskell F.

Norman collection at Christie's (New York) in 1998. Nor­

man had been a psychiatrist in San Francisco and collected

rare books of high quality in science and medicine. His son,

Jeremy Norman, the prominent San Francisco rare book

dealer, upon his father's death, put the collection up for auc­

tion, and Christie's, aware that the collection would attract

lots of attention, took some of the best pieces in the sale

out on the road to show to collectors in Milan, Paris, Lon­

don, New York, Chicago, San Jose (!), and Tokyo, among

other cities. The sale included many items that had been ac­

quired some years ago and are now seldom seen on the mar­

ket. There were unique items like a first edition of Euler's

VoUstdndige Anleitung zur Algebra (1770) that Euler had

presented to Lagrange, with an inscription-very desirable

to a knowledgeable collector. In the 1980s one could find a

copy of this book for about $1000; this very special copy in

70 THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media. Inc.

the Norman sale went for $19,550. The whole Norman c

lection (632 lots) sold for almost $8,000,000.

After the Norman sale the conventional wisdom arne collectors and dealers seemed to be that the high pri<

Fig. 1 . The cover of the Christie's catalogue for the Norman s;

decorated with the title page of Newton's Principia of 1687.

moUjidn�ige � n l e t t u n g

,. 1 g''t 6 t a t>on

�rn. 2ton�orb �u l er. • ·•

��·�����������·� e5 t. '}>ctu6&t1 r g.

srbnufr �19 Cn' .l'i•9f· 'll:cob. tlf 'll:hff•nfcll• rm mo.

Fig. 2. The cover page of Euler's Algebra, with the inscription to Lagrange.

The writing is almost certainly that of an assistant since, by the time this was

published, Euler was almost totally blind.

could have been a local maximum and resulted from two factors: (1) Norman was a well-known collector, and (2) Christie's created hype surrounding the sale by arrang­ing the world tour of the principal books. Provenance is important. Coming from a famous collection gives a book

cachet, and the best copies, just like great paintings, come with a long list of the names of previous, and preferably prominent, owners.

The surprise is that prices have held at the high level and are actually increasing. If prices were driven up by the

© 2005 Springer Science+ Business Media, Inc., Volume 27. Number 2, 2005 71

Norman sale and by a lot of dot-com millionaires with an

interest in mathematics and science, why have the prices

held after many of those dot-com fortunes vanished? We

don't have an answer.

As we write this, the huge and important Macclesfield

sale is going on at Sotheby's in London. This extensive li­

brary in the history of science was housed in Shirbum Cas­

tle near Oxford, and, according to the catalogue, encapsu­

lated "almost everything published on the astronomical and

mathematical sciences up to about 1750: from Regiomon­

tanus and Peurbach to Copernicus and his contemporaries,

and from there through Tycho Brahe, Giordano Bruno,

Gilbert, Galileo and Hevelius to Newton, Leibnitz, Whiston

and Euler . . . . There are many books famous in the annals

of science, such as the copy of Copernicus's De revolu­tionibus . . . which Owen Gingerich saw in situ and on a

well-chosen day (as he relates in The Book Nobody Read, he went on a Thursday which was, according to Lady Mac­

clesfield, 'the only day we have a cook, and we can invite

you to lunch') . . . . The genesis of this remarkable collec­

tion . . . lies in the biographies and interests of two men,

. . . John Collins, the mathematician [and friend of Newton]

whose collection was acquired by William Jones about

twenty-five years after his death, and Jones himself, who

bequeathed his library to the second Earl [of Macclesfield]."

O B S E R V A T I O D O M I N I P E T R I DE F E R M A T.

C J'/,.,. ••l<m ;, "••s ••h•s , •111 f"•Jr•t•fuJr•u• "' ""'' '!••Jr4111J11Urlfl•l 6 s•••r•lltiT """•• "' ;,,,;,.,. .,,,. , •• J,.,,.,,. p•�<ft•u• •• ""'' •l•/·

J,., .,,.;,;, f•s tjl J;.u,,. t�shu rei tk•••J1r•ll•"'"' •ir•biltlll flu tl.mtl. H411t "'"'"'" ""SIIIIM ,., .. ,.,., 0

Q.Y A: S T l O l X.

Rv • • ' • oport�t quadr3tUm 16 diuidere in duos qu,adruos. Pon3·

rur rurl"us primi latus 1 N. altcnus •cro quorcunquc nulll<rorum cum dcfca&& tot Yoit:atum, quot confbt latus diuideodi. Eflo itaque � N. - + erunrqaudr.ui, hJc ,.aidem • Q,_ ille vero 4 � .. :+ r.s:. - r6 • �m volo vrrumque limul a:quari �� r6. lgitur S q:.:+ J6.- r6 • 1!f!l¥W Yo,icacibus r6. & lie 1 N.'l czit

The first Macclesfield sale of mathematics books took

place June 10, 2004, and covered only the books by authors

with names beginning with A-C. (A second sale in the se­

ries was to take place November 4, 2004.) The first 251 lots

sold in June brought in almost $6.5 million, and that's for

only three letters of the alphabet! Prices in the sale so far

indicate that they are not only holding their own but in­

creasing. The Copernicus of 1543 went for $1.2 million. It was a nice copy, heavily annotated by the astronomer John

Greaves. Still, for most people that's a lot of money to spend

on a book (In 1964 Ernst Weil in London listed the 1543

Copernicus in his catalogue for $2,380. In the Honeyman

sale of 1978, there were two copies-one sold for $35,000,

the other for $19,000.) This book and the great works of

Galileo, the first edition of Newton's Principia of 1687, the

1482 Ratdolt Euclid (the first printed Euclid, produced

roughly 40 years after Gutenberg's invention of moveable

type), and other such rarities now go for very high prices;

few copies remain in private hands.

As with any other items of interest to collectors, prices

fluctuate over time, moving upward with overall inflation,

and both upward and downward depending on shifting

fashion. Fermat's 1670 edition of Bachet's Diophantus (this

edition was supervised by Fermat's son and published

shortly after Fermat's death) contains the printed version

\U' A: S T ! O I X.

E:!T n ,. •>• «, ,; -,.o-.• Mi- .. �:�

. ,. ..... ,..,... (( H.rl'ittr,w �· ·"-�· f.., A w.,.,..., ,; T'l�) .. ,, " :, .n ,....._, � �·

.... .. r.. '"' '"'"'� ·on;�+!<!l'l ,; . lw.4r..c � i ,i ,;. � ,;,.

Fig. 3. The page from the 1670 edition of Sachet's Diophantus, with the famous remark by Fermat on having a miraculous proof but inade­

quate space to write it in the margin (under "Observatio Domini Petri de Fermat").

72 THE MATHEMATICAL INTELLIGENCER

of what Fermat supposedly wrote in the margin about hav­

ing a proof but not having enough room in the margin to

write it down. Not long after Andrew Wiles's 1995 proof of

"Fermat's Last Theorem," a copy of the Diophantus ap­

peared in a London dealer's catalogue for $9,075. Then in

the Norman sale in 1998 it sold for $25,300 (after an esti­

mate of $2,000-$3,000 by the auction house!), and later a

New York dealer had a copy listed at $45,000. One of the

authors of this article bought a copy in San Francisco in

1965 for $484. At a June 2004 auction in Paris a copy sold

for $31 ,020. The increase is far beyond the rate of inflation.

But, of course, not every book enjoys the stimulus of a

widely publicized proof after 350 years, one that made

"Nova" and the front page of the New York Times. Perhaps another book belongs in the category above:

Gauss's Disquisitiones Arithmeticae of 1801, a later book

than the Euclid and the Newton but one of considerable

mathematical interest and rarity. One of us bought a copy

in 1967 from the dean of West Coast book dealers, Warren

Howell in San Francisco, for $1,350. In 2003 a special copy

that had been owned by C. J. Doppler was listed by the

London dealer, Bernard Quaritch, for $60,800, but a plain,

ordinary copy sold at a recent Christie's sale in Paris for

$49,350. It is perhaps not surprising that this, probably the

greatest of Gauss's works, should have this amazing in­

crease in selling price, but a copy of his Theoria motus cor­porum coelestium, a less well-known book, was purchased

from Howell for $190 in 1963 and a London dealer listed a

copy in 2001 for $6,800. So even lesser works by great math­

ematicians have shown a significant increase.

What can explain prices like this? After all, the text of

many of these works is available in modem editions or on

the Internet. The splendid 1847 edition of Oliver Byrne's

Euclid in brilliant colors printed by woodblock and still

available on the market for something like $7,500 (one of

us owns a copy bought in a London bookshop in 1960

for less than $13) can now be seen on the Internet at

http://www.sunsite.ubc.ca/DigitalMathArchive/Euclidlbym

e.html. It's a beautiful book either way, but looking at pages

on the screen is not quite the same experience as holding a copy in your hand and flipping through the pages. Byrne,

incidentally, seems to have had time on his hands; he is

identified on the title page as "Surveyor of Her Majesty's

Settlements in the Falkland Islands."

Here are a few examples of the increases over the years

for some important books in mathematics:

• the editio princeps of the works of Archimedes of 1544:

Honeyman sale of 1978, $1,280; W. P. Watson in London,

$ 1 12,000 in 2002;

• Jacques Bernoulli's Ars Conjectandi: Honeyman sale,

$900 in 1978; Norman sale, $12,650 in 1998; W. P. Wat­

son, $21,000 in 2004;

• Jean Bernoulli's Opera Omnia: Quaritch, $75 in 1962; the

same dealer, $7,200 in 1997;

• Euler's Introductio in Analysin Injinitorum: Zeitlin &

Ver Brugge in Los Angeles, $75 in 1961; Jonathan Hill in

New York, $13,500 in 2000;

• Galileo's Dialogo of 1632: E. Weil, $294 in 1958; Zeitlin &

Ver Brugge, $5,500 in 1978; Norman sale, $27,600 in 1998;

Christie's in Paris, $49,761 in 2004;

• Lagrange's Mechanique Analitique: Joseph Rubinstein in

Berkeley, $250 in 1973; Norman sale, $ 13,800 in 1998;

Watson, $20,000 in 2002;

• Newton's 1687 Principia: E. Weil, $490 in 1948; Zeitlin

& Ver Brugge, $15,000 in 1978; Norman sale, $321,500 in

1998; W. P. Watson, $356,000 in 2004.

Even the third edition of the Newton (1726), of interest

because it was the last edition published in Newton's life­

time, was recently listed at $35,000 by a Los Angeles dealer.

One of us paid $236 for a copy in 1969.

Why does a person want to collect old and rare mathe­

matics books? Why indeed? They are often in Latin, a

language not understood by many these days, and they are

often filled with notation unfamiliar to modem mathemati­

cians. But there is a certain thrill for someone who loves

mathematics to pull a volume off the shelf and see, for ex­

ample, the first appearance in print of Euler's formula for

polyhedra in "Elementa doctrinre solidorum," in the Novi Commentarii Academiae Scientiarum Imperialis Petro­politanae (1758), 109-60, or the first appearance of the dif­

ferential calculus as most of us know it, Leibniz's "Nova

method us pro maximis et minimis," in the Acta Eruditorum of Leipzig for 1684. Sometimes one gets lucky and comes

across something totally unexpected, like a short manuscript

we bought from a New York dealer that was a report signed

by Laplace and Legendre and sent to the Academie des Sci­

ences, Paris, saying in French, but here translated: "We have

examined at the request of the Academy the description of

a vessel capable of traveling under water at a determined

depth for whatever amount of time, and diving to any depth

for a certain amount of time. The means that the author pro­

posed did not seem to us to be practicable or worthy of the

attention of the Academy." So much for the idea of a sub­

marine. On another occasion, a letter turned up, signed by

Gauss. There was nothing so unusual about that except that

it was in English. At first glance that might appear suspi­cious, but perhaps less so when one realizes that many of

Gauss's descendants ended up in the United States, some re­

putedly because they couldn't get along with him.

Potential collectors should be concerned about the pos­

sibility of buying a fake. Obviously this is a problem in the

art world, where skilled and successful forgers often have

fooled even the museum experts. Autographs and manu­

scripts are also susceptible to forgery. Books have been rel­

atively safe, at least until the prices went so high, because

the amount of work involved in typesetting with antique

typefaces and printing on paper of the correct period,

would be prohibitively time-consuming and expensive. Still,

a collector has to be careful. In 1976 one of us bought a

copy of Gauss's doctoral dissertation, published in Helm­

stadt in 1799 and containing an important result, the first

generally accepted proof of the Fundamental Theorem of

Algebra. The book was purchased for $1,000 from a dealer

of impeccable reputation in San Francisco, acquired from

© 2005 Springer Sc1ence+ Business Media, Inc., Volume 27, Number 2, 2005 73

A P X I M H � OY � T 0'1 :l Y P /1. K OV % I or, T tr. �� r. K p I r..QVII(.,a.c,n-,

A R C H i d E D I S S Y R A C V S A N I P H I L 0 0 P H I t-A C G E 0 M E. T R t.AE. E. X· ccUauifiimi Opaa . qua:quidcm canr,omnu,muln iam f«ulisd r,.

dcrata, acq� :\ q�m pauciflimi$ hadmu uifa, nunc� primt'un & rxci& urincm lu­ccm tdila.

(Sorum C:ualogumucrf� �a r<pcdes .

..A eli ell" 1110'/> font E V T O C II ..A C ..A L O NI T<.Alt

l � E. 0 $ D B ,\\ A. 1\ C U I I ! I) I $ . L 1 . brosCommcnt1n":�,ir<m Gr.tcc�uon ',

nunqu:un �rca =fa.

Cum C.rJMaicfl .gmtia (!)' priuilcgto ad 1111nq11cMiu«.

B c.A $ I L l!. c.A1!, 1tM1111ts Hr:rullg/NS cxnulifol/,

An. 1-1 .0 X L 1 J l 1 .

Fig. 4. The title page of the first printed collection of the works of Archimedes, in Greek and Latin (1544).

a collection ostensibly brought by a refugee family from

Europe when fleeing from the Nazis in the 1930s. It was

only some years later that it was discovered that this col­

lection had been stolen, book by book, from the John

Crerar Library in Chicago by a person who was to become

well known as one of the cleverest book thieves of the cen­

tury. So the Gauss dissertation-really quite rare, the pre­

vious sale of a copy having taken place in 1928-was "re­

quested" by the U.S. Attorney in Chicago as evidence in the

trial of the alleged thief. There it stayed for several years,

during which time the remainder of the Crerar Library was

acquired by the University of Chicago. Finally, at the end

of the trial (the suspect was convicted), a letter came from

7 4 THE MATHEMATICAL INTELLIGENCER

the University asking whether they should return the $1,000

or the book The choice was clear: the book There was no

copy of this book in the Honeyman or Norman sales, and

it's unlikely there will be one in the Macclesfield sale, for

the book came along about fifty years late for that collec­

tion. It is certainly not a common book, so it is difficult to

guess what copies would sell for today. But $1,000 did seem

like a bargain. A New York dealer subsequently listed a

copy in his catalogue at $39,500 in 1997.

Occasionally one finds interesting copies of books that

have been owned by other mathematicians. One or the other

of us has acquired over time Cayley's annotated copy of Rie­

mann's Gesammelte Mathematische Werke; Dedekind's own

M.ENSIS OCTOBRIS .A,jM DC LXXXIV. 467 NOVA METHODUS PRO MAXTMIS ET Ml-11imM, ittm¥'" tllngmtifmJ1 gu" n<r fmllu1 nu irflflio•Aiu ¥111111fifPu monJt11r, (J' jingNIAre /" illil tAkllli

gmsu; ptt G, G. L.

Sltaxis AX, & curvz plurts, utVV,WW, Y Y, Z Z, quarum erdJ.TABJUI. natz, ad axtm normales, VX, WX, Y X, ZX, qllZvoctntur r<fpe·

llivt, P, w, y, z ; &ipfia A X abfcilfa ab axt,vocttut x. Tang<ntt5 fint V B, W ('� Y D, Z Eni occurrcntes r<fp<dive in pundis B, C,D, E. Jam r<daaliqua proarbitrio a!fumta vocetur dx, & rttla quot fir 1d dx,ut• ( vrlw,vdy, vd z ) dhd V B (vt) W C, vciYD, vtl Z E ) vo-ccturd • ( vel d w, vrl dpd dz) fiH dilltrtnua ipf.rum P (vel ip&-rum w, aut y, aut z) His pofitiscalruU rrgulzcrunt tales :

Sit a quanutasdata confbru,trit dastquaUso, & d u trit zqu• • dx : fi fit y "''u • ( ICu ordinataquzvilcurYII:Y Y, zqualis cuivis or­dinatz rtfpondcnticurYIO: V V) crit dystqu.d• , Jam .At/Jt�i•6' Sub­,,..n;, ; fi6tz-y>fow+x zqu.P1trit d z - y+ w +x feu d p, zqn d z - d Y+dw >fod x. M•/11!/iwi•,d-;:; zqo. x d •+• d x, ftD polito yzqu.x•, li<td y zqu x d •+•dx. In arbitrso tnimdl vrl formulam, Ut X •• vtl compendia pro •• uttram, Ut y, odlubtre. Notandam & X & d x tod<m modo in hoc ulcolo ua&.ri, ut y & dy, vel ali am tit cram ind<tcrmlnatam cum fua dilf.rtntiali. Notandum ttiam non dari fcmpcr r<grtlfum a dilftrtnriali .£quat ion<, nifi cum quadam cautio-n<, de quo alibi. Porro Di>ifo1 d �vel ( pofitoz zqu.�) d z zqu, ±•dy � y d • y y

y y QJ!_oadSif-.o hoc probt not&ndum, cum in calculo pro littta

fubllituiturfimphcittr <JUI dilftrcntialis, ftrvari quldcm eadem C.gnJ, & pro+zfcribi + dz, pro -z fc:ribi -dz, utcx addition< & fubrra­tuone paulo ante pofito appartt ; fed quando ad extgcfin V>lorum vcnitur, feu cum confidcra.tur ipfius z ctlatio ad z, rune :appartrrj an nlor ipGusd z fit quantitas al!irmativa, an nihilo minor feu ncgativo: quodpollcnuscomlit, tunc t>ngcns Z E ducitur a pvndo Z non vcr­fils A, fed in partes conrr.uias feu infr> X,id tl\ tunc cum ipfzordinm•

N n n J • deere-

Fig. 5. The first page of Leibniz's article of 1684 on differential cal­

culus. Note the addition, subtraction, product, and quotient rules for

differentials in the second paragraph.

copies of the works of Dirichlet; Darboux's copy of Gauss's

Werke (only nine volumes, because the remaining volumes

appeared after Darboux's death, after which he stopped col­

lecting-to borrow a phrase from H. W. Lenstra, Jr.); Hardy's

copy of Waring's Meditationes Algebraicae (the volume with

the first appearance of Wilson's theorem, and of special in­

terest because of the contributions of Hardy and Littlewood

to the solution of Waring's problem); a manuscript by Fatio

de Duillier on alchemy, with "corrections" in longhand by his

close friend Isaac Newton; Charles Babbage's copy (pre­

sented to him by the translator) of the four volumes of

Nathaniel Bowditch's translation into English of Laplace's

Traite de Mecanique Celeste, where, because Laplace's work

was so unclear, Bowditch would give a few lines of Laplace

with the rest of the page devoted to notes in small type to

explain what Laplace had just said; and offprints of George

Green's that he had presented to Jacobi. These last offprints,

bought from the Honeyman collection, were valuable to Mary

Carmell in her biography of Green, in showing that he had

been more in touch with continental mathematicians than

had been previously thought.

These things are out there to be collected. A person just

has to read auction house and dealers' catalogues or Web

sites assiduously. Manuscript materials are, in general,

w....:- .:. J'-i'fi

'!!�·+ p-/...d",dr=r-r ,w . .r:. ]4 � r?w-'t�---,­� ' � . �· I

_f>ir I' ,.,_,...., �.

ai. -r r t:...fu.,. /'� h.L4wl<? ,,.r;rn..t-�I''ZJJ� ...u:. .tr et.. tar�,._ .�- t�rAJ .. � .

¥'"' / ..=:7�

Fig. 6. A report by Laplace and Legendre recommending against fur­

ther investigations by the Academie des Sciences on an idea for a

submarine.

TIIEORE)IATIS 01\1 El\1 FV ' C T I ,' E M A L G E B R I C A M

£l ,\ TJ Q, ' ALF.:i\1 I T E G R A I VNJ\'S \'ARlAOtLIS

lN FACTORES RE;\LES PRI\11 VEL S£CV, 01 GRAD

llESOLVt POSSE

A V CT O R E.

C A R O L O F R I O E R I C O

HCLMSTADII Ar\'D C. Q. t t.&Ct._&f, • J �9

G A V S S.

Fig. 7. Title page of Gauss's doctoral dissertation of 1 799.

© 2005 Springer Science+- Business Media. Inc .. Volume 27, Number 2. 2005 75

MECAN IQUE CELESTE.

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Fig. 8. The title page of Bowditch's translation of Laplace's Mecanique Celeste, with a typical page showing Laplace's text in the first five

lines with Bowditch's explication below.

harder to find than good books. When asked about the pos­

sibility of getting a couple of manuscripts from the Stanitz

collection when that came up for auction at Sotheby's in

1984, the dealer who would be doing the bidding warned that

there was little chance of getting them. "The institutions will

be out in force bidding for those"-one a manuscript of a

paper on elliptic functions by Jacobi, the other a collection

of manuscripts by Cauchy written when he was a student.

We were assured that it would be much easier to get books in the sale. As it turns out, we got no books but got both of

the manuscripts, and at a good price. Not every manuscript

attracts the sort of attention the Archimedes palimpsest did

in 1998 at Christie's. It sold for $2 million. That price was

predictable, but generally at auctions a person just never

knows how things will turn out. It can be exciting.

The discoverer of Vitamin E, Herbert McLean Evans at

the University of California, Berkeley, led the way for sci­

ence collectors in 1934 when he issued a small booklet list­

ing the books that he thought were the "epochal achieve­

ments in the history of science." He built seven great

science collections himself and is generally accepted as the

pioneer in science collecting, which then received another

boost when the Grolier Club (an organization of book col­

lectors in New York) published a handsome volume in 1964

called One Hundred Books Famous in Science, by Harri­

son Horblit, another great science collector. (Even this

modem book, that just lists books, had gone up in price

from $50 to $750 when it appeared in a recent catalogue of

a Los Angeles dealer.)

76 THE MATHEMATICAL INTELLIGENCER

Some who may know rather little about mathematics can

find for themselves a little bit of immortality by assembling

a great collection in the history of mathematics or science.

Who would remember the name Robert Honeyman today

had he not accumulated an enormous book collection? Hon­

eyman was an engineer, a graduate of Lehigh University, to

which he gave his great literature library and his extraordi­

nary Darwin collection in the 1950s. His Western Americana

collection went to the Bancroft Library at the University of

California, Berkeley. For some reason his science collection

went on the auction block at Sotheby's Parke Bernet in Lon­

don in 1978, and, for some of us, it's a good thing it did: we

enjoy seeing on our bookshelves the telltale bright red mo­

rocco slipcases that Honeyman commissioned for his books.

This auction may have stimulated the first major increase in

prices for mathematics and science books, for it put on the

market many books previously not of great interest to a large

group of collectors.

There were some young dealers who were looking for

stock and who moved into the science area: Roger Gaskell,

Jonathan Hill, Michael Phelps, W. P. Watson, and Jeff

Weber. Prior to that, the principal dealers in science were

Bernard Quaritch, Dawson's of Pall Mall, Ernst Well, and R.

D. Gurney in London; Alain Brieux, Paris; Rosenkilde & Bag­

ger, Copenhagen; E. Offenbacher, H. P. Kraus, and Lathrop

C. Harper in New York; Jacob Zeitlin, the Rootenbergs, and

Harry Levinson in Los Angeles; and Warren R. Howell (John

Howell-Books) and Jeremy Norman in San Francisco. Only

five of the firms in this second list survive: Quaritch, Daw-

son's in the form of Pickering & Chatto, Brieux, B. & L.

Rootenberg, and Jeremy Norman. Once when scouring Lon­

don for a rare book in mathematics and having no luck, we

were told that we really should be looking in Los Angeles.

The legendary Jake Zeitlin there was a pioneer in making

scientific books attractive to collectors. He, with E. T. Bell,

advised William Marshall Bullitt, a prominent Louisville, Ken­

tucky, lawyer and U. S. Solicitor General under William

Howard Taft, on building an extraordinary collection of rare

mathematics books now in the library of the University of

Louisville. Bullitt was prompted to build a collection by a

parlor game suggested by his friend, G. H. Hardy. He was a

good friend of mathematics and was instrumental, along with

John von Neumann, in arranging for Charles Loewner's first

appointment in the United States at Louisville. Bullitt may

have heard the answer to our question of why prices of math­

ematics books continue to rise: the most renowned of Amer­

ican book dealers, A. S. W. Rosenbach of Philadelphia, ad­

vised him that "science and mathematics books [are] likely

to prove more valuable as investments than books on

Napoleon or Henry Clay." For more information on this fas­

cinating collector see [2].

Like Warren Howell, Jake Zeitlin was a scholar. A visit

to either of these booksellers was like going to a private

club. It involved a move into their private offices, where

the good things were kept in safes, and one could settle

down for an hour of good conversation about books, han­

dle some important pieces, and often meet science histo­

rians of note. We remember meeting Stillman Drake for the

first time in Zeitlin's office. Haskell Norman writes glow­

ingly in his introduction to the catalogue of his collection

about his Saturday afternoon visits to Warren Howell's of­

fice where he would often find Herbert Evans. The group

of science dealers, experts, and collectors was small in

those days and everyone knew each other.

Someone wanting to see the great books and manu­

scripts in mathematics, but unable to afford them, can view

them at some of the great libraries of the world. For a de­

scription of some of the European libraries with important

mathematical holdings-the British Library, the Vatican Li­brary, the Bodleian at Oxford, and a great California col­

lection, the Huntington near Los Angeles, see [3]. Other im­

portant collections are at the Bundy Library of the Dibner

Institute at MIT, the Institut Mittag-Leffler near Stockholm,

the Linda Hall Library in Kansas City, MO, the Thomas

Fisher Library at the University of Toronto, the Lownes Col­

lection at Brown University, the Bullitt collection men­

tioned above, the DeGolyer Collection at the University of

Oklahoma, and the David Eugene Smith Collection at Co­

lumbia University, among others. These collections were

rarely assembled book by book by the institutions; they

were usually gifts from generous patrons.

How does one get started in this highly entertaining and

rewarding (but often expensive) hobby? It would help to

have been born many years ago and to have started col­

lecting in the 1950s at the latest. But that's hard to arrange.

Given the market these days, most have to give up any as­

pirations to acquire a 1543 Copernicus or a major Newton

or even a major Gauss. It's probably best to pick a subdis­

cipline, perhaps one's own specialty, and look for the best

available in the field that one can afford. Then there are the

books slightly out of the mainstream: books on mathemat­

ics and music (there are affordable works by Descartes,

Euler, d'Alembert), or books on art and perspective, say.

These are really quite cheap, as long as a person is not try­

ing for something like Pacioli's great work with the

Leonardo drawings of the polyhedra. Some things get to

be expensive not for their mathematical content but for

their association with someone well-known, in this case,

Leonardo. Perhaps mathematics and poetry could be pur­

sued. D. E. Smith wrote on this subject. The French poet

Paul Valery was more or less a mathematician. Sylvester

wrote poetry, rather bad poetry, actually. Cajori says in his

A History of Mathematics (Macmillan, 1919) that: "At the

reading, at the Peabody Institute in Baltimore, of his Ros­

alind poem, consisting of about 400 lines all rhyming with

'Rosalind,' he [Sylvester] first read all his explanatory foot­

notes, so as not to interrupt the poem; these took one hour

and a-half. Then he read the poem itself to the remnant

of his audience." Warren Howell in 1977 had a copy of

Sylvester's Spring's Debut, a Town ldylVin Two Centuries of Continuous Verse, printed "for Private Circulation Only"

in 1880. It too has each line ending in the same sound. This

TENT AMEN NOVAE THEORIAE

VS CAE EX

CERT ISSIMIS HAR 0 IAE PRINCIPIIS

DU.VODE EXPOSITAE. .AYCTORJt

LEONHARDO EVLERO:

PE:fROPOU , EX TYI'OGRAPHIA ACADEI'IlAE SCIE.:mAJ.\'&1. obbO: Zl<llllro

Fig. 9. The title page of Euler's treatise on music of 1 739.

© 2005 Springer Science+Business Media, Inc., Volume 27. Number 2. 2005 77

slim volume sold for $80, the price of a good dinner now

and a very good dinner then. That the price was not higher

may be surprising, since it is a presentation copy with an

inscription by Sylvester to "Professor Child" and with hand­

written corrections by Sylvester. But it perhaps shares the

fate of Hamilton's Lectures on Quaternions, very com­

monly inscribed by Hamilton. They didn't sell, so the au­

thor had to give them away-ours was presented to Sir

George Biddell Airy, onetime English astronomer royal.

If a collector's resources are limited (and whose are

not?) another way of building an interesting collection is

to acquire reprints. They're often presented by the authors

with interesting inscriptions. For example, Mark Kac,

knowing of P6lya's work on the problem, "can you hear the

shape of a drum?", sent P6lya a reprint of his own paper

on the subject and inscribed it "From another drummer,

Mark Kac." A person can pick up things like this when col­

leagues retire and vacate their offices. Dealers who carry

such things are Malcolm Kottler at Scientia, Arlington, MA;

Ray Giordano at Antiquarian Scientist, Southampton, MA;

Elgen Books, Rockville Centre, NY; and Jeff Weber, Los

Angeles. Weber's catalogues are filled with interesting and

modestly priced books in mathematics.

How does a person find such things, besides reading cat­

alogues and Web sites? One haunts bookstores, and there

are still lots of wonderful bookstores around. Fewer seem

to be able to afford street-level shops anymore, so one may

have to climb a few stairs. But they're there, and the deal­

ers are usually eager to educate the interested novice. A

Cecil Court bookseller in London once gave very good ad-

S P R I G ' S D E B UT.

A Town Idyll

J. J. Y L V •; T E R, F. R . . , .dodltor •I Uto £4.,. •I Yerk.

O P ! B A ae V & Jt. n�o.

vice that we've both tried to follow: "It's never the things

you buy that you regret, only the things you don't buy!"

A few years back, a beginning collector of science wrote

in the magazine Biblio that he visited the Quaritch shop in

London and asked about rare books in science, adding

rather apologetically that he realized, of course, that he

would never be able to afford a Copernicus or a Galileo Di­alogo. The attendant asked gently whether he would like

to look at their copies. The days of being able to walk into

a shop and browse through a first edition of Copernicus

are probably over. Even a shop like Quaritch, at those

Olympian heights, probably would not keep a copy in stock.

With the recent Owen Gingerich census of copies of Coper­

nicus, we are aware of the location of every known copy

of both the first and second editions.

New York and other major cities like Leipzig, Amster­

dam, Paris, and Milan, have book fairs. California has a

huge annual fair, held in alternate years in San Francisco

and Los Angeles. These are good ways of visiting dealers

from all over the world. The last fair in San Francisco had

almost 250 dealers present. And they often bring their best

things to fairs, particularly if they know there are interested

collectors for specific material in the area.

Auctions are tricky. In person one can get carried away

bidding and end up paying more than is prudent. Absentee

bidders can submit a bid to the auction house directly or

have a dealer bid for them for a small fee. The dealer or

the dealer's agent can check in advance on condition, col­

lation, and other details to fill in gaps in the catalogue de­

scription. At auctions, though, one must always keep in

PoET : Who i he so blithe of 1nien

Airy n�"trCbarlcs Slr<!et """"•'

ight � to u sair 'ecn,"

Light as elves that trip the green Thread deep d II or ).,.fy deoc, Led by Oberon' faerie Queen, .Freob "-'! dew-drenched JellS&min, Or on Pennington'& storied screen� t

Or where Tait paints cattle io •

t Charle� Str t. ti tbe nd Stre-et, the Eternal StrMt or e.lt"g��neo and (,..hion1 of Baltimoro. lo America the moN piC\UrfllqUf!l (Otm Of tJ:prfiNiOD n On thi!l llrHL" (& r.u� wbeN in Engluul w• ahould MY 11 in t.be 1treet. u OR tOn­

veyl the idea or a.n unoneloMd ·� s xr. nar�t P�on\ogton at. tba a,:e of 18 painted on a reeD for .ll!• llar1 Oal'TCtt a .Dorby Day of CupldJ

moon\.011 ou Dragi)Q.ft.iet1 whkb w ubibited u lh• lata art. loan �zblLhion in ll ltlmore wh� h anr.o\ed gtMl and d(.'Mrved admiration.

* �lr. J, R. T•it, at.o of llahimor"", U a pt.inler of et.ltla piifl«jj aod landseapea and bleb. f111lr lO takt rlllik liOmo d•y N the. .AJnel'i�o Tro,.on. I own II man piiiKlt of hll (C•ttle and SuDMt.) wbe.ro the. rocoding tonllill or t'b IUITU� llgbL It!.

5

Fig. 1 0. Sylvester's Spring's Debut, both the title page and a page of verse (corrected by Sylvester).

78 THE MATHEMATICAL INTELLIGENCER

mind that the buyer pays the house an additional fee, the

premium on hammer price. So the outlay can end up con­

siderably higher than the bid. On the other hand one can

get bargains if one is shrewd and informed. And you know

that if you're bidding against a dealer, that person is going

to sell the book for a much higher price in the shop. Auc­

tion houses to check out are Sotheby's in London, New

York, and Paris (sothebys.com); Christie's in those same

cities (christies.com); Swann's in New York (swanngal­

leries.com); Bonham's and Buttertield's in San Francisco

and London (buttertields.com); the Dorotheum in Vienna

(dorotheum.com); Tajan in Paris (tajan.com); Reiss und

Sohn in Konigstein, Germany ( reiss-sohn.de ); Zisska & Kist­

ner in Munich (zisska.de); and Pacific Book Auctions in San

Francisco (pbagalleries.com).

Can one make a lot of money investing in rare books?

Probably not. Like the stock market, prices fluctuate; and

just because mathematics books have been "hot" over the

past few decades, there's no guarantee they'll stay that way.

Over the long term, art has largely kept up with Standard

& Poor's, but art, books, stamps, and other fields of col­

lecting pose special problems. When a person wants to dis­

pense with a collection, it's not so easy. An auction house

will sell the books but will charge the seller (as well as the

buyers) a percentage. And dealers in rare books, as a rule

of thumb, mark books up by 10(}0;\J over what they paid for

them. So don't expect to get more than 50% of the current

retail price. Most dealers do not have the wherewithal to

buy a whole collection, so they will often agree to take

A U T H O R S

GERALD L. ALEXANDERSON

Department of Mathematics & Computer Science Santa Clara University

Santa Clara, CA 95053-0290 USA

e-mail: gaJexand@math.scu.edu

Gerald L. Alexanderson has been collecting rare mathematics

books, Lewis Carroll works, and press books since 1 961 . He was

chair of his department at Santa Clara University for 35 years and

has been a member of the faculty there since 1 958. He has served

as Secretary and President of the Mathematical Association of

America and most recently received the MAA's Haimo Award for

teaching and the Yu-Gin Gung and Dr. Charles Y. Hu Award for

Distinguished Service in 2005. He does not like to travel.

something on consignment and then charge the seller a cer­

tain percentage of the sale price. By contrast, selling stocks

or bonds is relatively easy. But as an expert pointed out re­

cently in the New York Times, when stock prices collapse,

the investor is left with a stack of worthless paper, whereas

the wine collector can at least drink the wine and a book

collector can ef\ioy the books.

So collecting is not a way to get rich. A collector has to

like the books themselves, and the discoveries made in col­

lecting them. Further, if you have a good collection, rare

book librarians at major institutions will be very nice to

you if they find out what you have. If the collection is good

enough, you may get your name attached to a library col­

lection. That's immortality of a sort.

Here's to good hunting!

Acknowledgement

The authors wish to express their gratitude to Ellen Hef­

felfinger, librarian and bibliographer of the American In­

stitute of Mathematics, for her extraordinarily helpful and

perceptive comments on an earlier draft of this article.

REFERENCES

[ 1 ) G. L. Alexanderson and L. F. Klosinski , On the value of mathemat­

ics (books) . Mathematics Magazine 55 (1 982), pp. 98-1 03.

[2) R. M. Davitt, William Marshall Bullitt and his amazing mathematical

collection, Mathematical lntelligencer 1 1 :4 (1 989), pp. 26-33.

[3) S. I. B. Gray, A mathematics treasure in California, Mathematical ln­

telligencer 20:2 (1 998), pp. 4 1-46.

LEONARD F. KLOSINSKI

Department of Mathematics & Computer Science

Santa Clara, University

Santa Clara, CA 95053-0290 USA

e-mail: lklosinski@math.scu.edu

Leonard F. Klosinski collects sculpture, drawings, maps, and paint­

ings, along with books if they're old enough, preferably incunab­

ula. He has taught mathematics and computer science at Santa

Clara University since 1 964 and has directed the William Lowell

Putnam Mathematical Competition since 1 975, longer than any

previous director. In 2001 he won the MAA's Haimo Award for Dis­

tinguished College or University Teaching of Mathematics. He has

traveled to all seven continents.

© 2005 Springer Science+Business Media. Inc . Volume 27. Number 2. 2005 79

i;l§iil§ldJ Osmo Pekonen , Editor I

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

Column Editor: Osmo Pekonen, Agora

Center, University of Jyvaskyla, Jyvaskyla,

40351 Finland

e-mail: pekonen@mit.jyu.fi

Using the Mathematics Literature edited by Kristine K. Fowler

NEW YORK, MARCEL DEKKER, 2004. 389 PP., US $165,

ISBN 0-8247-5035-7

REVIEWED BY J. PARKER LADWIG AND

E. BRUCE WILLIAMS

''This book deals with the basic

tools and skills needed in the

mathematical laboratory." It is written

not only for librarians, but more im­

portantly for undergraduates doing

mathematical research, for graduate

students, and for faculty exploring new

areas.

This is the 66th volume of Dekker's

Books in Library and Information Sci­

ence. It is divided into two parts,

"Tools and Strategies," and "Recom­

mended Reading by Subject," and it has

two indexes: author and subject. The

first place a seasoned mathematician is

likely to browse is Part II, "Recom­

mended Readings by Subject," just to

check the list of resources for his or

her subject. The subjects included are:

History of mathematics

Number theory

Combinatorics

Abstract algebra

Algebraic and differential geometry

Real and complex analysis

Differential equations

Topology

Probability theory and stochastic

processes

Numerical analysis

Mathematical biology

Mathematics education

NOTE: We understand from the editor

that contributors for other subjects

(like mathematical logic) would have

been welcome.

Each subject is given a chapter writ­

ten by a mathematician and/or a math­

ematics librarian. Each entry gives

80 THE MATHEMATICAL INTELLIGENCER © 2005 Springer Sc1ence+ Business Media, Inc.

enough information to locate and even

order the resource, and often a one- or

two-sentence description. The empha­

sis is on books, but key journals and

on-line resources are also indicated.

The chapters generally contain an

introduction, a section on general

sources, and then sections for the ma­

jor subdivisions of a field. Sections of­

ten refer to general texts, and then

further refine the subdivision. For ex­

ample, the chapter on topology by Alan

Hatcher contains a section on intro­

ductory books, then sections on al­

gebraic topology, manifold theory,

low-dimensional topology, history, and

other resources. The section on mani­

fold theory, for example, discusses

differential topology, piecewise-linear

topology, topological manifolds, and

surgery theory.

Here are a few additional sugges­

tions for the chapter on topology:

II. Algebraic Topology

A. Introduction

I. Madsen and J. Tornehave.

From calculus to cohomol­ogy: de Rham cohomology and characteristic classes. Cambridge: Cambridge Uni­

versity Press, 1997. Chern classes are constructed via

curvature forms and via pure

algebraic topology. This is

done after the reader is given

the necessary background in

topology and de Rahm the­

ory.

III. Manifold Theory

A. Differential Topology

V. Guillemin and A. Pollack.

Differential topology. Engle­

wood Cliffs, N.J.: Prentice­

Hall, Inc. , 1974.

This is a very readable book.

The main topics are trans­

versality, intersection the­

ory, and differential forms.

T. Brocker and K. Janich.

Introduction to differential topology. Translated from the

German by C. Thomas and M.

Thomas. Cambridge: Cam­

bridge University Press, 1982

This is an excellent, concise

introduction to basic mater­

ial in algebraic topology.

Dundas, Bjorn, Differential topology, http://www.mi.uib.

no/�dundas/

This is a very reader-friendly

expansion of the Brocker­

Janich book.

D. Surgery theory

W. Luck. A basic introduc­tion to surgery theory, http://

wwwmath. uni-muenster.

de/index.en.html

The main topics are s-cobor­

dism theorem, Whitehead tor­

sion, surgery exact sequence,

the Farrell-Jones cor\iecture,

and computations of K- and

L-groups.

Part I, "Tools and Strategies," con­

tains three chapters. The first is a very

interesting one on the culture of math­

ematics. For undergraduates who are

thinking about advanced study (or for

friends and family who are puzzled

about what a mathematician does), this

is a concise and even elegant overview.

"Tools" continues with chapters

on "Finding Mathematics Information"

and on "Searching the Research Liter­

ature." Both chapters are written by

experienced mathematics librarians

and answer questions asked by those

learning and studying mathematics.

"Finding Mathematics Information"

contains sixteen sections (too many to

enumerate )-two of our favorites are

"Locating Definitions and Basic Expla­

nations" and "Finding or Verifying Quo­

tations and Anecdotes. " As with Part II,

each entry contains complete biblio­

graphic information with a one- or two­

sentence abstract.

"Searching the Research Literature"

contains five sections: introduction,

strategies, finding journal articles us­

ing indexes, finding papers on the Web,

and obtaining the resources found.

This chapter is more of a discussion

than a list of resources, but like the

chapter on "Finding Mathematics In­

formation" would be helpful for some-

one just beginning research in mathe­

matics.

Because this work is primarily

arranged by discipline, it offers a dif­

ferent perspective than Nancy D. An­

derson and Lois M. Pausch, editors, A Guide to Library Service in Mathe­matics (Greenwich, CT: JAI Press, Inc.),

1993. One might also consult the $65

book by Martha Tucker and Nancy An­

derson, Guide to Information Sources in Mathematics and Statistics (West­

port, CT: Libraries Unlimited), 2004.

The book's major drawback is its

price-42¢ per page vs. 19¢ for Tucker

and Anderson. However, it is still an

important addition to your library's

collection, a relevant resource for un­

dergraduate and graduate student ad­

visors, and perhaps a gift for the new

librarian who will be working with

your department.

Mathematics Library

University of Notre Dame

Notre Dame, IN 46556-5641

USA

e-mail: ladwig . 1 @nd.edu

Mathematics Department

University of Notre Dame

Notre Dame, IN 46556-5641

USA

e-mail: williams.4@nd.edu

Mathematics in Nature. Modeling Patterns in the Natura l World by John A. Adam

PRINCETON UNIVERSITY PRESS, 2004. 354 PP.,

ISBN 0-691 - 1 1 429-3, US $39.50

REVIEWED BY THOMAS GARRITY

Once in the early evening in high

school, on the flat desolate plains

of the Texas panhandle, I was on the

back of a horse with a girl that I was

trying to impress. As we rode along,

she pointed out to me the beauty of the

sunset, which was spectacular. It so

happened that earlier that day I had

read about the underlying mathematics

behind sunsets. Of course I had to tell

her my newly acquired knowledge, in

part to impress my riding partner but

also to share my newfound apprecia­

tion and awe of the sunset. Unfortu­

nately, she looked at me not with any

sort of admiration but with contemp­

tuous disdain. I realized that I had ru­

ined any chance of a romantic moment.

Still, even in high school, I could not

fathom how understanding the under­

lying science and mathematics of a

physical phenomenon could at all de­

tract from its immediate aesthetic, if

not sensual, appeal. Surely insight into

causes could only add to the delight of

our experience. While I suspect that

that girl of long ago would still find my

viewpoint a bit ridiculous, I am confi­

dent that John Adam would whole­

heartedly agree with me. In fact, the

point of his Mathematics in Nature: Modeling Patterns in the Natural World is that the underlying math can

add immeasurably to our delight in

merely walking about and looking at

the world around us. In the first chap­

ter, in talking about this very issue,

Adam states, "I have always found, for

example, that my appreciation for a

rainbow is greatly enhanced by my un­

derstanding of the mathematics and

physics that undergird it. . . . "

More generally, Adam states in the

first chapter, "The idea for this book

was driven by a fascination on my part

for the way in which so many of the

beautiful phenomena observable in the

natural realm around us can be de­

scribed in mathematical terms . . . . " He

describes how much he er\ioys observ­

ing the world as he walks to work each

day, and he wants to share this joy.

Thus his goal is to foster in the reader

the skills needed to model the phe­

nomena in nature that we encounter in

our day-to-day lives. He wants to ex­

plain how a few simple principles can

be used to understand many seemingly

disparate physical phenomena.

It is not surprising that Adam's ini­

tial stab at modeling involves Fermi

problems. This is another name for tra­

ditional "back of the envelope" calcu­

lations. An example of a Fermi prob­

lem would be to approximate how

many Snickers bars are eaten each year

in, say, Chicago. The goal is to use a

few pieces of numerical information

© 2005 Springer Sc1ence+ Business Media, Inc., Volume 27, Number 2, 2005 81

(in the above you would need to have

a rough guess as to the population of

Chicago). You win with these problems

if you can get an answer that is accu­

rate up to an order of magnitude. As Adam states, this chapter owes a lot to

Paulos's Innumeracy and Harte's Con­sider a Spherical Cow.

In a similar vein, in the next chapter

he turns to the basics of scaling, or in

other words, dimensional analysis.

Here he discusses how to make quick

estimates of sizes of area, volumes, etc.

For example, this chapter would pro­

vide the reader with the ability to see

that a mouse will shiver in the cold far

more quickly than an elephant. The rea­

soning is as follows. The amount of

heat in an animal should be propor­

tional to its volume. The amount of heat

that an animal loses, on the other hand,

will be proportional to its surface area,

as it is only through our surfaces that

we can lose heat. Crudely, volume goes

up by radius cubed while surface area

goes up by radius squared. The mouse

shivers because its ratio of surface area

to volume is far larger than the corre­

sponding ratio for the pachyderm. (This

also suggests why Harte titled his book

Consider a Spherical Cow.) The rest of the book develops mod­

els for specific physical phenomena,

from the meandering of rivers to the

formation of hurricanes. You will be

able to understand the clouds in the

sky as well as the dunes at the beach.

After spending some time on waves

(both linear and nonlinear), you will be

able to see how a bird flies. Adam

closes with dispersion relations, em­

phasizing the now standard explana­

tion for how a leopard gets its spots.

As can be imagined, there is a lot more

that is discussed. In fact, with an ad­

mitted bit of exaggeration, the author

claims, " . . . if you can see it outside,

and a human didn't make it, it's proba­

bly described here!"

There is the question of who is the

intended audience. For example, sup­

pose I have a neighbor who readily ad­

mits to not liking math but who also

asks me for a book that will show the

importance of math in everyday life. I

would recommend a book like Innu­meracy by Paulos. Mathematics in Nature requires greater mathematical

82 THE MATHEMATICAL INTELLIGENCER

maturity. This is not quite what the au­

thor intends; as he states in the pref­

ace, "Anyone interested in the beauty

of nature, regardless of mathematical

background will also (I trust) ef\ioy

much of this book. Although the math­

ematical level ranges across a broad

swathe, from 'applied arithmetic' to

partial-differential equations, there is a

measure of nonmathematical discus­

sion of the basic science behind the

equations that I hope will also appeal

to many others who might wish to ig­

nore the equations (but not at their

peril). Thus those who have no formal

mathematical background will find

much of value in the descriptive mate­

rial contained here." I am skeptical.

While certainly anyone would profit

from paging briefly through the book,

I have a difficult time imagining any­

one spending serious time reading the

text who was not at the least a fairly

strong undergraduate math or hard sci­

ence major. It would take some effort

for any of us to work out the details be­

hind the equations in the book.

This minor fault is no doubt due to

the author's enthusiasm in trying to

convey to others his awe at the math­

ematics of the world. Few readers of

The Intelligencer will have these prob­

lems. For this audience, the book can

be both dipped into to get a feel of the

underlying math behind nature and

also studied with paper and pen. Both

approaches will be ef\ioyable.

I liked this book a lot.

Department of Mathematics and Statistics

Williams College

Williamstown, MA 01 267

USA

e-mail: tgarrity@williams.edu

Model ing Differential Equations in B iology by Clifford Henry Taubes

UPPER SADDLE RIVER, NEW JERSEY, PRENTICE-HALL,

2000. 479 PAGES, $58.99, ISBN: 0 1 301 73258

REVIEWED BY THOMAS HILLEN

Shall we expose undergraduate stu­

dents to original research articles?

Taubes answers this question with

a boldface YES. In this introductory

textbook for Mathematical Biology, he

presents a large number of original re­

search articles which are interwoven

with the main text. Most of these arti­

cles are from Nature or Science, and

the collection shows an amazing vari­

ety over many important parts of

Biology.

Before I comment more on the pa­

pers in this book, let me talk about the

real contents. The text deals with ordi­

nary and partial differential equations

applied to biological systems. The level

of exposition is very basic, directed

mainly to students in biology. A student

in mathematics will not be satisfied

with the exposition, because many de­

tails are omitted in favor of "rule-of­

thumb" statements. The text introduces

the biology student to modeling with

0 DEs (ordinary differential equations)

and PDEs (partial-differential equa­

tions). Indeed, one strength of the book

is the painless treatment of PDEs. If

needed, a student can continue in the

study of differential equations using a

more mathematical textbook.

I mentioned the research articles of

this textbook already, and you might

get the impression that they are dis­

cussed in detail and that the corre­

sponding models are analyzed with the

methods just introduced. But this is not

the case, which I consider a weak part

of the book. The research articles

merely parallel what has been done in

the corresponding sections. Some of

them are only loosely, or not at all, con­

nected to the contents of the section at

hand. For example, Chapter 2, on "Ex­

ponential Growth," concludes with

four research articles: first an article

on HIV virus load, which goes way over

the head of a beginning student; sec­

ond, the discussion of right-handed and

left-handed snail shells, which is en­

tertaining and also understandable on

the undergraduate level; third, an arti­

cle on gene control of kidney develop­

ment-way too difficult; finally, one

answer to the symmetry problem of the

snail shells.

The set-up of Chapter 2 forms a

good example of the overall text. Very

basic mathematics is followed by

about four research articles of mixed

level. This has the advantage that an in­

structor can choose articles depending

on the level of the students.

Another example is the choice of pa­

pers concerned with reaction-diffusion

equations and pattern formation in

Chapter 18. We all know that the Tur­

ing mechanism creates wonderful pat­

terns, but it is still unclear if this mech­

anism is responsible for animal skin

patterns, for example. Taubes's selec­

tion of papers shows the controversy

quite nicely. An initial publication on

fish-pattern is opposed by a second ar­

ticle, which then is commented on by

the authors of the first article. This

leaves the true impression that this dis­

cussion is still open.

Other topics of the text: ODEs,

phase-plane analysis, linearization,

vector-matrix notation, advection, dif­

fusion, separation of variables, reaction­

diffusion equations, pattern formation,

traveling waves, periodic solutions,

fast and slow dynamics, and chaos.

Although the text is not suitable for

a course in mathematics, the enormous

number of well-chosen references

makes it a useful addition for the shelf

of a generally interested researcher. As Taubes says in his preface, his "goal is

to introduce to future experimental bi­

ologists some potentially useful tools

and modes of thought."

Department of Mathematical and Statistical

Sciences

University of Alberta

Edmonton, Alberta T6G 2G1

Canada

e-mail: thillen@math.ualberta.ca

Probabi l ity Theory : The Logic of Science by E. T. Jaynes

CAMBRIDGE, CAMBRIDGE UNIVERSITY PRESS, 2003.

727 PP., $65.00, HARDBACK ISBN 0-521 -59271-2

The Fundamenta ls of Risk Measurement by Chris Marrison

BOSTON, McGRAW-HILL, 2002. 4 1 5 PP. $44.95

HARDBACK ISBN 0-07-1 38627-0

The Elements of Statistical Learning: Data M ining, Inference and Prediction by Trevor Hastie, Robert Tibshirani,

and Jerome Friedman

NEW YORK, SPRINGER-VERLAG, 200 1 . 533 PP. $82.95

HARDBACK ISBN 0-387-95284-5

REVIEWED BY JAMES FRANKLIN

A standard view of probability and

statistics centres on distributions

and hypothesis testing. To solve a real

problem, say in the spread of disease,

one chooses a "model," a distribution

or process that is believed from tradi­

tion or intuition to be appropriate to

the class of problems in question. One

uses data to estimate the parameters of

the model, and then delivers the re­

sulting exactly specified model to the

customer for use in prediction and

classification. As a gateway to these

mysteries, the combinatorics of dice

and coins are recommended; the ener­

getic youth who invest heavily in the

calculation of relative frequencies will

be inclined to protect their investment

through faith in the frequentist philos­

ophy that probabilities are all really rel­

ative frequencies. Those with a taste

for foundational questions are referred

to measure theory, an excursion from

which few return.

That picture, standardised by Fisher

and Neyman in the 1930s, has proved in many ways remarkably serviceable.

It is especially reasonable where it is

known that the data are generated by

a physical process that conforms to

the model. It is not so useful where the

data is a large and little-understood

mess, as is typical in, for example, in­

surance data being investigated for

fraud. Nor is it suitable where one has

several speculations about possible

models and wishes to compare them,

or where the data is sparse and there

is a need to argue about prior knowl­

edge. It is also weak philosophically,

in failing to explain why information

on relative frequencies should be rele­

vant to belief revision and decision­

making.

Like the Incredible Hulk, statistics

has burst out of its constricting gar­

ments in several directions. In the

foundational direction, Bayesians, es­

pecially those of an objectivist stamp

like E. T. Jaynes, have reconnected sta­

tistics with inference under uncertainty,

or rational degree of belief on non-con­

clusive evidence. In the direction of en­

gagement with the large and messy

data sets thrown up by the computer

revolution, the disciplines of data min­

ing and risk measurement, represented

by the books of Hastie et al. and Mar­

rison, have developed data analysis

and tools well outside the traditional

boundaries.

The essence of Jaynes's position is

that (some) probability is logic, a rela­

tion of partial implication between ev­

idence and conclusion. According to

this point of view, statistical inference

is in the same line of business as "proof

beyond reasonable doubt" in law and

the evaluation of scientific hypotheses

in the light of experimental evidence.

Just as "all ravens are black and this is

a raven" makes it logically certain that

this is black, so "99% of ravens are

black and this is a raven" makes it log­

ically highly probable that this is black

(in the absence of further relevant ev­

idence). That is why the results of drug

trials give rational confidence in the ef­

fects of drugs. Galileo and Kepler used

the language of objective probability

about the way evidence supported

their theories, and in the last hundred

years a number of books have filled out

the theory of logical probability­Keynes's Treatise on Probability (the

great work of his early years, before he

went on to easier pickings in econom­

ics), D. C. Williams's The Ground of In­duction, George P6lya's Mathematics and Plausible Reasoning, and now

E. T. Jaynes's posthumous master­

piece, Probability Theory: The Logic of Science.

Jaynes's school are called "objective

Bayesians" or "maxent Bayesians,"

to distinguish them not only from

frequentists but from "subjective

Bayesians," who think that any degrees

of belief are allowable, provided they

are consistent (that is, obey the axioms

of probability such as that the proba­

bility of a proposition and its negation

© 2005 Springer Science+Bus1ness Media, Inc., Volume 27, Number 2, 2005 83

must sum to one). The objectivists em­

phasise, on the contrary, that one's de­

gree of belief ought to conform to the

degree to which one's evidence does

logically support the conclusion. In

asking why evidential support should

satisfy the axioms of probability the­

ory, objectivists have been much im­

pressed by the proof of R. T. Cox

(American Journal of Physics, 1946)

that any assignment of numbers to the

relation of support between proposi­

tions which satisfies very minimal and

natural logical requirements must obey

the standard axioms of conditional

probability. They have been corre­

spondingly unimpressed by supposed

paradoxes of logical probability that

purport to demonstrate that one can­

not consistently assign initial probabil­

ities. In some of his most entertaining

pages, Jaynes exposes these "para­

doxes" as exercises in pretending not

to know what everyone really does

know. His reliance on symmetry prin­

ciples to assign initial probabilities

shows its worth, however, well beyond

such philosophical polemics. In in­

verse problems like image reconstruc­

tion, where the data grossly under­

determines the answer, it is essential

to assign initial probabilities as non­

dogmatically as possible, in order to

give maximum room for the data to

speak and point towards the truth.

Jaynes's maximum entropy formalism

allows that to be done. In the business world, there is the

same need as in science to learn from

data and make true predictions. But

among other forces driving the expan­

sion of commercial statistics are the

new compliance regimes in banking

and accounting. Following a number

of corporate scandals and unexpected

collapses, the world governing bodies

in banking and accounting have de­

cided on standards that include,

among other things, risk measure­

ment. The Basel II standard in banking

says, in effect, that banks may use any

sophisticated statistical methodology

to measure their overall risk position

(in order to determine the necessary reserves), provided they disclose their

methods to their national banking reg­

ulator (the Federal Reserve in the U.S.,

84 THE MATHEMATICAL INTELLIGENCER

the Bank of England in the U.K.). Mar­

rison's book is an excellently written

introduction to the standard ideas in

the field. It avoids the unnecessary el­

ements in usual statistics courses and

goes immediately to the most applica­

ble concepts. These include the "value­

at-risk" formalism, which measures

the loss such that worse losses occur

exactly 1% (say) of the time, and the

concepts needed for precision in han­

dling rare losses, such as heavy-tailed

distributions and correlations be­

tween the losses of different financial

instruments. It is significant, for ex­

ample, that foreign exchange rate

changes resemble a random walk, but

are heavy-tailed, are heteroskedastic

Mathematicians ,

pure and appl ied ,

th ink there is

someth i ng wei rd ly

d ifferent about

stat istics .

(variable in "volatility," that is, stan­

dard deviation), and have some ten­

dency to revert to the mean. It is per­

haps surprising to learn that credit

ratings are intended to mean absolute probabilities-a AAA rating means

one chance in 10,000 of failure within

a year; naturally it is hard to ground so

small a probability in data, so one pre­

sumes that credit rating agencies will

need to use priors and qualitative evi­

dence (a euphemism for market ru­

mours?) in the style of Jaynes. Marri­

son's insights into how bank risk

teams really work is enlivened by oc­

casional dry humour: in pointing out

that profits from risky trades need to

be discounted, he adds, "Convincing

traders that their bonuses should be

reduced according to Allocated Capi­tal X Hr is left as an exercise for the

reader."

In accounting, the forthcoming IFRS

(International Financial Reporting

Standards) compliance standard will

play a role similar to Basel II in bank­

ing, in enforcing higher standards of

mathematical competence. It will be

necessary to price options reasonably

in the interests of their truthful display

on balance sheets, for example. The

book for accountants corresponding to

Marrison's appears not yet to be writ­

ten, so there may be a gap in the mar­

ket for an ambitious textbook writer

who would like to become very rich

very quickly.

If there is a dispute in statistics as

heated as that between frequentists

and Bayesians, it is that between tra­

ditional statisticians and data miners.

Data mining, with its roots in the neural

networks and decision trees developed

by computer scientists in the 1980s, is

a collection of methods aiming to un­

derstand and make money from the

massive data sets being collected by

supermarket scanners, weather buoys,

intelligence satellites, and so on.

"Drink from the firehose of data," says

the science journalist M. Mitchell Wal­

drop. It is not easy-and especially not

with the model-based methods devel­

oped by twentieth-century statisticians

for small and expensive data sets. With

a large data set, there is a need for very

flexible forms to model the possibly

complicated structure of the data, but

also for appropriate methods of

smoothing so that one does not "over­

fit," that is, learn the idiosyncracies of

the particular data set in a way that will

not generalise to other sets of the same kind. Are specialists in data mining (or

"analytics" as they now often prefer)

pioneers of new and exciting statistical

methodologies, or dangerous cowboys

lacking elementary knowledge of sta­

tistical models? Those who enjoy vig­

orous intellectual debate will want

to read data miner Leo Breiman's pug­

nacious "Statistical modelling: the

two cultures," Statistical Science 16

(2001), 199-219, with a marvellously

supercilious reply on behalf of the tra­

ditionalists by Sir David Cox. As an in­

troduction to the field for practitioners

in the business world, Michael Berry's

Mastering Data Mining (New York,

Wiley, 2000) is often recommended,

but for mathematicians interested in

understanding the field, Hastie et al.'s

Elements of Statistical Learning is the

ideal introduction. Assuming basic sta­

tistical concepts and an ability to read

formulas, it runs through the methods

of supervised learning (that is, gener­

alisation from data) that have come

from many sources: neural networks,

kernel smoothing, smoothed splines,

nearest-neighbour techniques, logistic

regression and newer techniques like

bagging and boosting. The unified

treatment and illustration with well­

chosen (and well-graphed) real data­

sets makes for efficient understanding

of the whole field. It is possible to

appreciate how different methods are

really attempting the same task-for

example, that classification trees de­

veloped by computer scientists to suit

their discrete mindset are really per­

forming non-linear regression. But the

differences between methods are well

laid out too: the table on p. 313 com­

pares the methods with respect to such

crucial qualities as scalability to large

data sets, robustness to outliers, han­

dling of missing values, and inter­

pretability. The less-tamed territory of

unsupervised learning, such as cluster

analysis, is also well covered. One

topic of current interest missing is the

attempt to infer causes from data, but,

as is clear from Richard Neapolitan's

Learning Bayesian Networks (Har­

low, Prentice Hall, 2004), that theory is

still in a primitive state. Spatial statis­

tics and text mining are not covered ei­

ther; they too await readable textbooks

of their own.

Mathematicians, pure and applied,

think there is something weirdly dif­

ferent about statistics. They are right.

It is not part of combinatorics or mea­

sure theory but an alien science with

its own modes of thinking. Inference

is essential to it, so it is, as Jaynes

says, more a form of (non-deductive)

logic. And, unlike mathematics, it

does have a nice line in colourful

polemic.

School of Mathematics

University of New South Wales

Sydney 2052

Australia

e-mail: j .franklin@unsw.edu.au

Mathematics Across Cultures Helaine Selin and Ubiratan

D'Ambrosio, editors

KLUWER ACADEMIC PUBLISHERS, 2000, 479 PAGES

HARDBOUND, ISBN 0·7923·6481 ·3, € 1 95.50

PAPERBACK, ISBN 1 ·4020·0260·2, €63.00

REVIEWED BY HELENE BELLOSTA

This book is meant as a supplement

to the Encyclopaedia of the His­tory of Science, Technology and Med­icine in Non-Western Cultures (Kluwer

Academic Publishers, 1997) and is

aimed at a more scholarly audience;

the aim is to explore the same topics

in greater depth.

The book is divided into two parts:

the authors of the six essays in the first

section try to define the field of ethno­

mathematics and to make a general

study of the connection between math­

ematics and culture as well as the vari­

ability of the concept of rationality,

while the second part is devoted to the

description of fifteen individual cul­

tures and their mathematics in various

"non Euro-American" areas: the Middle

East, America (native cultures), the Pa­

cific and Australia, Africa, and the Far

East.

If the intention behind the book-to

rehabilitate the so-called non-Western

cultures and to denounce the damag­

ing effects of cultural imperialism and

eurocentrism, the consequence of

which is a certain contemptuous dis­

regard for these cultures-is highly

laudable, this enterprise is not entirely

free from danger. The main difficulty is

defining and naming the field of study:

how should we divide sciences into

Western and non-Western, or Euro­

pean and non-European? The criterion

is not geographical but cultural (H.

Selin, p. v), for the studies in this book

deal with mathematics in the Far and

Middle East, as well as mathematics in

Aboriginal, Amerindian, or African so­

cieties. Should we, as some authors do,

speak of "non-modem" or "traditional"

sciences, even though this mixes up

different eras, from the 3rd millennium

Be to today? Should we then group

these sciences together under the

heading "ethno-sciences"? But what

then are the criteria that include sci­

ence in Mesopotamia or ancient Egypt

in ethno-sciences, but exclude Greek

science, although all the authors in

Greco-Hellenistic Antiquity regard sci­

ence in ancient Egypt as the origin of

Greek science? Why should Arab math­

ematics, the heir to Greek mathemat­

ics, whose contribution is essential to

understand the constitution of classical

mathematics in 17th-century Europe, be

included in ethno-mathematics?

This book seems to make a rather

strange division. On one side we have

ethno-sciences, bringing together sci­

ences as different as science in ancient

China and science in present-day Abo­

riginal societies, these being viewed as

sciences of unusual societies, the pe­

culiarities of which, together with their

incommunicability, some papers dili­

gently stress; and on the other, by de­

fault, Greek science, European science

from the Renaissance to nowadays as

well as science in the USA, would be

left as non-ethnic sciences (white sci­

ence versus colored science?). If we

continue to follow the unspoken logic

of this division, these sciences should

then show the opposite qualities and be

a contrario universal. We should not

be surprised then to find here and there

in some papers hasty judgments and

worn-out commonplaces on these "dif­

ferent" civilizations, which could be de­

fined as the eurocentrism the editors

intended to stigmatize: "The transfor­

mation of the word science as a dis­

tinct rationality valued above magic is

uniquely European" (H. Selin, p. vi) or

"the development of this concept of ra­

tionality (i.e., European's 17th century)

was not universal. For example, it was

not paralleled in Islamic society where

men were denied rational agency; they

were held to lack the capacity to

change nature or to understand it.

Knowledge was instead to be derived

from traditional authority" (D. Tum­

bull, Rationality and the disunity of the sciences, p. 47). One of the authors

(R. Eglash, Anthropological perspec­tives on ethnomathematics) is clearly

conscious of the difficulty of defining

what ethno-mathematics or non-West­

em mathematics actually are, and also

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005 85

aware of the disparity in level and con­

tents of what the editors group under

this heading. He therefore suggests

distinguishing between mathematics

of "non-Western empire civilisations"

(India, China, Japan, Arab world)­

which could be compared to "West­

em" mathematics-and mathematics

of "indigenous societies." However,

this leaves us to wonder what consti­

tutes an "indigenous society." Would it

not be more epistemologically conve­

nient to try to distinguish between

mathematics and what could be called

proto-mathematics, and to try to spec­

ify the criteria that would enable us to

define each of them (criteria of ratio­

nality, modes of proof, degree of ab­

straction, nature of the problems

solved . . . )?

We could try to define mathematics

as a human activity the purpose of

which is to solve abstract problems

dealing with number and magnitude or

with geometrical figures, with methods

having some degree of exactness and

generality; moreover, this activity is ex­

plicitly bound, by those who practise it

(geometers, mathematicians . . . ) and

name it (geometry, wasan . . . ), to some

criteria of rationality (even if they

could differ) and to the necessity of

some type of proof (whatever it is,

should it be the constraining form of

euclidean formalism, or a mere justifi­

cation of calculus procedures and al­

gorithms). Proto-mathematics would

then be a contrario a set of calculus

techniques and geometrical construc­

tions, directly born of practical neces­

sities or esthetical concerns, and aimed

at solving problems arising in a some­

how organized society (the knowledge

of these techniques and geometrical

constructions is not restricted to a dis­

tinct social class, and those who know

them and use them do not bother about

the necessity of proof). Let us note that

this implies no disrespect for proto­

mathematics; furthermore the opposi­

tion between mathematics and proto­

mathematics is not so clear-cut as this

brief account suggests, and more often

it will be possible to distinguish in

proto-mathematics the forms of more

sophisticated future mathematical the­

ories.

86 THE MATHEMATICAL INTELLIGENCER

Part One

Leigh N. Wood, Communicating

mathematics across culture and time

The author intends to stress the diffi­

culties inherent in communicating

mathematics across cultures and time

periods. Mostly relying on erratic and

superficial knowledge, the author

jumps haphazardly from Babylonian

tablets to the counting system in Ocea­

nia and Papua New Guinea, from

Greece to China . . . Historians of sci­

ence will certainly be pleased to learn

that "Leonardo Fibonacci . . . [solved]

cubic equations" (p. 10) (sic).

Ron Eglash, Anthropological

perspectives on ethnomathematics

This paper is the only one to mention

the difficulties in defining ethno- or

non-Western mathematics and to bring

up the ideological dangers of this en­

deavour together with its a-historiciz­

ing effect. In answer to some papers,

like Helen Veran's, he remarks judi­

ciously that "what goes under the name

of multicultural mathematics is too of­

ten a cheap shortcut that merely re­

places Dick and Jane counting marbles

with Tatuk and Esteban counting co­

conuts" (p. 20).

Edwin J. Van Kley, East and West

A brief survey of what European sci­

entific development owes to China.

Conjectures the author puts forward as

if they were facts, e.g. , "Simon Stevin

whose sailing chariot probably was in­

spired by description of similar Chi­

nese devices . . . also introduced deci­

mal fractions and a method of

calculating an equally tempered musi­

cal scale both of which might also have

been inspired by Chinese example" (p.

32). His conclusion is as condescend­

ing as it is naive: "Obviously Asians are

as able to master its intricacies [i.e., in­

tricacies of the rational scientific in­

dustrialized culture] as are Europeans

and Americans" (p. 34)!

David Turnbull, Rationality and the

disunity of the sciences

In this paper the author expounds the

thesis of "post modem" relativism: that

the concept of rationality is relative,

and hence science is but a social prod-

uct essentially dependent on commu­

nity and tradition.

Helen Veran, Logics and

mathematics: challenges arising

in working across cultures

This paper combines the author's per­

sonal experience teaching mathemat­

ics in Australian Aboriginal and Niger­

ian classrooms (see below) with a

criticism of colonialism and general

considerations on the relativity of logic

and rationality. It concludes that "the

foundationist mode of ontology/episte­

mology is a moral project of European

colonising" (p. 72) (sic).

Ubiratan d'Ambrosio, A historical

proposal for non Western

mathematics

This paper focuses on the social, politi­

cal, and cultural factors in the dynamics

of the transfer and production of math­

ematical knowledge in the colonies, as

well as on the recognition of non-Eu­

ropean forms of mathematics extant or

buried in the colonial process.

Part Two

The chapters in the second part describe

individual cultures and their mathemat­

ics. They vary greatly in the fields of

study and in the level of the discussion.

Eleanor Robson, The uses of

mathematics in ancient Iraq,

600o-600 BC

The author, who deplores the fact that

the history of mathematics is too often

written by mathematicians or mathe­

matics historians and intends to rem­

edy that, makes "a first approximation

to a description of Mesopotamian nu­

merate quantitative or patterned ap­

proaches to the past, present, and fu­

ture; to the built environment and the

agricultural landscape and to the nat­

ural and supernatural worlds" (p. 95).

James Ritter, Egyptian mathematics

A conscientious, well-documented study

of Egyptian mathematics by one of the

specialists in the field.

Jacques Sesiano, Islamic

mathematics

Following the inner logic of this book,

i.e., the division of mathematics into

two distinct fields, Western and non­

Western, the author stresses that

Mesopotamian mathematics would be

the ancestor to Arab mathematics.

However, he gives no evidence of how

it was transmitted across the centuries,

and carries on to suggest, still with no

proof, that algebra originated in India.

The anachronistic use of the word al­

gebra-Indian algebra "dealing with

the practical needs of daily life and

trade," geometrical algebra of the an­

cient Greeks-leads the author to un­

derrate the radical novelty and impor­

tance of al-Khwarizmi's Algebra, which

for him is "probably not very original"

(p. 143). This preconceived idea con­

cerning the supposed lack of original­

ity of Arab mathematics, backed up by

some allusions to the so-called con­

formism of mathematicians ("a refer­

ence to the ancients had almost the

weight of a formal proof' (p. 139), is

belied by all the scientific literature.

This portrayal of Arab mathematics, in

addition to its many errors of perspec­

tive-importance given to minor au­

thors or to secondary problems and

words, and misunderstanding of the

novelty of others-leaves aside entire

chapters of these mathematics (infini­

tesimal mathematics, geometrical

transformations, spherical geometry

. . . ). No wonder that the bibliography

given at the end of the paper is quite

outdated; it ignores almost all the fun­

damental publications of the last 20

years, both specialized studies and en­

cyclopedic works.

Tzvi Langermann and Shai Simonson,

The Hebrew mathematical tradition

In this paper Tzvi Langermann investi­

gates numerical speculation and num­

ber theory (mainly based on the works

of Abraham ibn Ezra (1092-1 167), a

central figure in Hebrew mathematics)

and Hebrew contributions to geometry

(mainly translations from Arabic to He­

brew of Euclid's works), while Shai Si­

monson focuses on algebra.

As with Sesiano, the anachronistic

use of the word "algebra," applied to

Babylonian algorithms as well as to the

so-called geometrical algebra of the an­

cient Greeks, leads Shai Simonson to

paint a quick and rather summary pic-

ture of this chapter of the history of

mathematics, mixing up epochs and

traditions: "Throughout these 3000

years, the Greeks, Indians, Chinese,

Muslims, Hebrews and Christians seem

to have done no more than present their

own versions of solutions to linear and

quadratic equations, which were well

known to the Babylonians" (p. 174). Si­

monson then studies the contributions

of Abraham ibn Ezra and Levy ben Ger­

shon to algebra. In doing so, he de­

scribes Levy's method for square and

cube root extraction, in great detail.

However, this probably owes less to

Babylonian or Chinese origins (spanning

centuries and distances with no textual

evidence of such transmission), than to

the classical method set out by al­

Khwarizmi (IXth century) in his arith­

metrical treatise (translated several

times into Latin in Spain during the XI­

Ith century), or by al-Karaji in his al-Kafi fi al-hisab (end Xth beginning Xlth cen­

tury) for the square root, and by al­

Samaw' al (died 117 4) for the cube roots.

As for ibn Ezra, if he knew the approx­

imation formula V x2 ± A = x ± _:'\_ , � r + A . Z,r

(not v x- :c:: A = '--i-' as wrongly mdi-

cated by the author), it seems more

likely that he borrowed it from al­

Khwarizmi than from the Babylonians

or Greeks .

Thomas E. Gisldorf, Inca

mathematics

This paper, which ambitiously aims to

describe "the development of Inca

mathematics in order to give a cultural

perspective and to emphasize the in­

terdependence of mathematics and

non-mathematical factors" (p. 189),

shows how difficult it is to study a

dead civilization which has left no

other written traces than decorative

patterns bearing geometrical motives

or qui pus (devices formed by knotted

strings). If the needs of this central­

ized society, as presented by the au­

thor (water control, civil and agricul­

tural engineering, astronomy, and

time-keeping) seem to have been gen­

erally the same as in other civiliza­

tions and surely imply some mathe­

matical knowledge, the lack of written

testimonies or oral transmission

nonetheless makes it difficult to re-

constitute. This leads the author to

put forward as fact second-hand hy­

potheses which have been taken from

different authors and are quite often

contradictory.

Michael P. Closs, Mesoamerican

mathematics

A precise, well-documented paper in

which the author studies in detail the

numeration and calendar systems in

the Olmec, Zapotec, Epi-Olmec, Maya,

Mixtec, and Aztec civilizations.

Daniel Clark Orey, The ethno­

mathematics of the Sioux tipi

and cone, Walter S. Sizer, Traditional

mathematics in Pacific cultures,

Paulus Gerdes, On mathematical

ideas in cultural traditions of central

and southern Africa

Three papers in the same vein. Reha­

bilitating the culture of the Sioux Indi­

ans after they have been slaughtered is

certainly a worthy project, but I cannot

see how the fact that Clark Orey de­

termines the height of a tipi and the

center of gravity of its base, or gives a

(false) calculation of his own of its lat­

eral area, helps preserve the beauty of

this civilization. If the observation of

the straight spears of New Guinea war­

riors leads Walter S. Sizer to assert that

"most cultures display considerable

geometric understanding through their

manufactured products" (p. 260), the

discovery of ancient lunar calendars

makes Paulus Gerdes wonder "who but

a woman keeping track of her cycles

would need a lunar calendar?" (p. 3 14)

before concluding that "women were undoubtedly the first mathematicians"

(p. 314)!

Helen Veran, Aboriginal Australian

mathematics: disparate mathematics

of land ownership and Accounting

mathematics in West Africa: some

stories of Yoruba number

In the first paper, H. Veran studies the

subtle and unusual links the Aborig­

ines have forged from time imme­

morial with their native land, their

despoilment at the hands of the

colonizers, their continuing conflict

with the Federal Government, as well

as the kinship systems in their society.

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005 87

This study, combined with the axiom

"politics, logic and mathematics are in­

separable" and the constant use of

analogy, leads the author to somewhat

strange conclusions concerning the

mathematics of a society in which

there seems to be no use of numbers,

such as "land is a matrix of vectors"

(p. 308). The second paper is a lexical

study of the numeration system of

Yoruba (Nigeria).

Jean Claude Martzloff, Chinese

mathematical astronomy

A serious and documented study of

Chinese mathematical astronomy by

one of the specialists of the field.

T. K. Puttaswamy, The mathematical

accomplishments of ancient Indian

mathematicians

A brief survey of ancient Indian math­

ematics, marred both by numerous

mistakes in mathematical formulas

and geometrical figures, and by ety­

mologies of pure fantasy.

Jochi Shigeru, The dawn of Wasan

(Japanese mathematics}

This is an introduction to the history of

Japanese mathematics, with a very

comprehensive bibliography.

Kim Soo Hwan, Development of

materials for ethno-mathematlcs in

Korea

The author illustrates his interest in

school and everyday mathematics by

the description of some everyday tra­

ditional objects to which one can ap­

ply mathematics.

Centre d'Histoire des Sciences et des

Philosophies Arabes et Medievales

CNRS UMR 7062

7 rue Guy M6quet

94801 Villejuif Cedex

France

e-mail: h.bellosta@free.fr

After Math by Miriam Webster

WAYNE, PA, ZINKA PRESS, 1 997. 280 PP., US$1 2.95,

ISBN 0-96-47 1 7 1 1 -5

88 THE MATHEMATICAL INTELLIGENCER

The Parrot's Theorem by Denis Guedj

NEW YORK, THOMAS DUNNE, 2001 . 352 PP., US$24.95,

ISBN 0-31 -228955-6

The Fractal Murders by Mark Cohen

BOULDER CO, MUDDY GAP PRESS, 2002. 282 PP.,

US$25.00, ISBN 0-97 - 1 89860-X

The Da Vinci Code by Dan Brown

NEW YORK, DOUBLEDAY, 2003. 454 PP., US$24.95,

ISBN 0-38-550420-9

The Curious Incident of the Dog in the Night-Time by Mark Haddon

NEW YORK, DOUBLEDAY, 2003. 240 PP., US$22.95,

ISBN 0-38-550945-6

Leaning towards Infinity by Sue Woolfe

NEW YORK, FARRAR STRAUS & GIROUX, 1 997. 393 PP.,

US$24.95, ISBN 0-57-1 1 9905-4

REVIEWED BY MARY W. GRAY

The past few years have seen a pro­

liferation of portrayals of real

mathematicians in popular media.

Sylvia Nasar's A Beautiful Mind; Hugh

Whitemore's Breaking the Code; Paul

Hoffman's charming rendition of the

life of Erdos, The Man Who Loved Only Numbers; and David Auburn's Proof come to mind as contributing to the

caricature of mathematicians as pecu­

liar, to say the least-especially if one

thinks of what Hollywood did to

Nasar's story. Rumors have circulated

that a film about the Unabomber is in

the works, and perhaps a version of

Ahmed Chalabi's adventures will in­

clude his brief fling as a mathemati­

cian. His subsequent shady political ca­

reer may have some echoes of the

tactics of Newton in Carl Djerassi's re­

cent play Calculus. Television has

been somewhat kinder, producing Si-

man Singh's The Proof, with the superb

opening scene of Wiles's reflections. In

a more esoteric vein, an opera about

the eleventh-century Islamic mathe­

matician Ibn Sina (Avicenna) recently

premiered in Qatar. Fictional mathe­

maticians have also enjoyed some at­

tention, in, for example, Tom Stop­

pard's Arcadia or his screen play for

Enigma; Matt Damon and Ben Af­

fleck's Good Will Hunting; or Daniel

Aronofsky's chaotic Pi. Mathematicians and mathematics

have also invaded the mystery genre.

Not that this is new. Sherlock Holmes

dabbled in mathematics from time to

time, and his nemesis Moriarty was a

mathematician as well as a master

crook. Michael Innes's Weight of the Ev­idence is a classic, Desmond Cory's Pro­

fessor Dobie stories date back a few

years, and Scott Turow's first huge suc­

cess, Presumed Innocent, featured a

mathematician. Not all mathematicians

in mysteries are fictitious. A very un­

pleasant John Wallis figures promi­

nently and Newton makes a cameo ap­

pearance in lain Pears's wonderfully

atmospheric An Instance of the Fin­gerpost. In Philip Kerr's Dark Matter Newton himself is the detective, albeit

in his role as warden of the Royal Mint.

And Escher's mathematics, if not the

artist himself, appears in Jane Langton's

The Escher Twist. A cadre of more con­

temporary mathematicians is thinly dis­

guised in Maths d mort, written by Mar­

got Bruyere, a long-time administrator

at IHES (lnstitut des Hautes Etudes Sci­

entifiques de Bures-sur-Yvette).

In fiction the representation of

women in mathematics appears higher

than in real life; of the mysteries re­

viewed here, only The Curious Inci­dent of the Dog in the Night-Time has

no woman mathematician involved in

the plot. (Yes, the parrot is female!).

Presumed Innocent is in this tradition,

as are the Laurie King books featuring

Sherlock Holmes's apprentice and

later wife, the mathematics student

Mary Russell; P. M. Carlson's series

with statistician Maggie Ryan; Arca­dia; and Proof

Various theories for the increased

interest in and popularization of math­

ematics in the media have been put

forward. Perhaps for many authors it

is the strangeness of mathematics that

makes mathematicians attractive sub­

jects, although the writers may feel

compelled to exaggerate our peculiar­

ities. For others-and perhaps for their

readers-the notion that mathemati­

cians are, after all, very weird if not ac­

tually deranged, may justify their math

phobia. And yet often the books, plays,

and films about mathematics and

mathematicians written by non-mathe­

maticians are wonderfully sensitive to

us and our work.

The heightened attention to mathe­

matics in the media has done nothing

for the recruitment of students, at least

in the U.S. and the U.K. It has been con­

jectured that the spurt in applications

to law schools in the U.S. can be traced

back to television's LA Law or to the

0. J. Simpson trial. Unfortunately the

appearance of the competing statisti­

cians in the latter coincided, it is re­

ported, with peak usage of water in the

homes tuned in to the trial. Maybe an

episode of Law and Order in which the

outcome of the trial depends on statis­

tical evidence would be a more effec­

tive recruitment device. Although we

do not want our field overcrowded, it

would be nice to have more students in

undergraduate courses because they

love mathematics and not because it is

a requirement for graduation. Are there

so many more lawyers than mathe­

maticians because we are so litigious,

or is it the other way around? Or can it

be because faking it is so much easier

in law than in mathematics? Of course,

lawyers from Fermat through Cayley

and Sylvester did produce significant

mathematics, but the day of the ama­

teur mathematician may be past, al­

though several of the books reviewed

here do feature such anomalies.

Magic realism in the hands of a mas­

ter like Gabriel Garcia Marquez is, well,

magical. The same cannot be said for

the supernatural After Math adven­

tures of the dead mathematician Ray

Bellweather and equally dead graduate

student Glen Vesper, as they combine

forces with two colleagues still among

the living to expose their murderer.

Not that there was ever much doubt

about who was the murderer. Author

Amy Babich (Miriam Webster is a pseu­

donym), a mathematician known in the

Austin environmental community as an

advocate for the use of bicycles and as

a city council candidate, claims to fol­

low the revelatory maxim of Anthony

Trollope, "The author scorns to con­

ceal from the reader any secret which

is known to himself." However well

this may have worked for Trollope, in

her case it takes out the mystery, leav­

ing very little except for the atmo­

sphere of a university mathematics de­

partment.

The author generally has a good feel

for this life, but contrary to her char­

acterization, it is NOT the case that all

statisticians prefer their chocolate in

the form of M&Ms when it comes to

eating, no matter what else they may

use them for. My colleagues have a

strong preference for Godiva, but then

. lett ing nearly

al l the princ ipal

guard ians of the

sacred doctrine

over the centuries

be male seems a

bit negative .

none of us regularly wear Birkenstocks

nor find it necessary to run our hands

over our bodies to determine whether

or not we are wearing clothes, other

characteristics of mathematicians, ac­

cording to Babich. Nor can I agree with

her assertion that "mathematics is

about rules." Doesn't that just perpet­

uate another stereotype?

Ajler Math also suffers from an

overabundance of characters, few of

them well developed; from excessive

reliance on quotes from Shakespeare,

Trollope, Byron, Heine, Raymond

Chandler, Wittgenstein, Gogol, Goethe,

and others; and from coy asides from

the author to the reader. At least the

murdering mathematician is not crazy,

just venal. However, his complacency

as he contemplates a long imprison­

ment, remarking that good mathemat­

ics has been done in prison, seems to

overlook the propensity of the state of

Texas to impose the death penalty.

It is fair to say that some readers

must have liked the book better than I

did. A recent site visit to Amazon.com

showed copies being offered for as

much as $296.

If Babich's primary goal was to

write a murder mystery, Denis Guedj's

in The Parrot 's Theorem seems to have

been to make the history of mathe­

matics palatable for the masses. Or, as

Simon Singh put it in a review in The Guardian: The Parrot's Theorem has

a "different objective from most other

fiction-to smuggle mathematical con­

cepts into the mind of the unsuspect­

ing reader by wrapping the maths in­

side an engaging plot." Guedj has

succeeded fairly well, although the

"mystery" is pretty transparent. We

have a real parrot bought in a Paris

market, a mathematical recluse in the

Amazon jungle, a delightful Parisian

bookseller, and an amazing deaf child

called Max, through whom the reader

is introduced to 5000 years of math his­

tory, from Thales to amicable numbers

to claimed proof of Fermat's Last The­

orem and the Goldbach Conjecture.

(Apostolos Doxiadis's Uncle Petros and the Goldbach Conjecture intro­

duces another recent fictional claimant

to the proof of the latter.) Perhaps the

frequent references to problems un­

solvable by use of a compass and ruler

(instead of straight edge) and a few

other inaccuracies are a result of mis­

translation. My main quarrel is with his

attribution of what I have always

thought of as the Problem of Dido to

Hassan Sabbah, a friend of Omar

Khayam. A nice touch is the link he

sees between the origins of mathemat­

ics and tragedy in ancient Greece.

The Parrot's Theorem would be an

excellent text for a general education

course in the history of mathematics,

although those with little mathematics

background might find some passages

tough going. Also, some of the more

colorful legends he describes cannot

be taken too seriously. Although the

book was a best-seller in France, it is

unlikely to repeat its success in the

United States. The mathematics is too

simple or too familiar to mathemati­

cians (of whom there are in any case

© 2005 Springer Sc1ence+ Business Media, Inc., Volume 27, Number 2, 2005 89

too few to confer best-seller status)

and too inaccessible and digressive for

others. Guedj's writing does combine

insight and appreciation with a joy he

would like his readers to share, but his

combination of whimsy, mathematics,

and mystery doesn't quite achieve the

proper balance.

My favorite among the books re­

viewed here is The Fractal Murders, where I think the author has achieved

a great balance between mathematics

and mystery. A Heidegger-obsessed for­

mer Judge Advocate General lawyer

turned private investigator and a Uni­

versity of Colorado mathematician

team up to solve the mystery, enlivened

by a fairly accurate lay description of

fractals. In fact, the book has Mandel­

brot's endorsement. The mysterious

deaths of three mathematicians who

were working on applications of Man­

delbrot's trading time theorem trigger

the investigation. The lives of academic

mathematicians are quite well cap­

tured, even though the investigator be­

trays a lack of understanding of the eco­

nomics of mathematics textbooks by

being shocked by a $44.95 price tag he

considers excessive.

The monetary value of fractals is

also central to the plot of Robert God­

dard's Out of the Sun, featuring one of

my favorite fictional creations, a female

mathematician who is described as be­

ing at the lAS at Princeton contempo­

raneously with Einstein, Godel, Man­

delbrot, and von Neumann. Both of

these books are helped by great loca­

tions, ranging from Nederland, Col­

orado, to a remote English village. (OK.

I confess that any book that uses Ne­

braska as even a minor locale has a lot

going for it as far as I am concerned;

we Nebraskans do not see much about

our home state in print, fact or fiction.)

They can be recommended as good dis­

tractions on the long plane rides and se­

curity delays faced by peripatetic math­

ematicians. Cohen is planning a series

featuring the same main characters, so

watch for their next appearance.

The Da Vinci Code has topped best­

seller lists and created a lot of contro­

versy with theologians, historians, and

mathematicians, among others. At

least eight books have been written de­

bunking various aspects.

90 THE MATHEMATICAL INTELLIGENCER

A symbologist and a cryptographer

set out to solve the murder in the Lou­

vre of its chief curator, and to halt at­

tempts to suppress entirely the preser­

vation of the "sacred feminine" in

religion. Although making the cryptog­

rapher a woman (and not a first for

Brown, as his earlier Digital Fortress introduced Susan Fletcher as the head

of code-breaking at the National Secu­

rity Agency) may be a gesture for fem­

inism, letting nearly all the principal

guardians of the sacred doctrine over

the centuries be male seems a bit neg­

ative. Granted, it's in keeping with

what we know about the leaders of

most religions, not to mention those in

influential positions of any kind.

It cannot be said that the reader's

understanding or appreciation of math­

ematics is enhanced by reading The Da Vinci Code, but for sheer escapism the

book is hard to beat. Yes, Brown im­

plies that the Golden Ratio is a ratio­

nal number, and the Da Vinci Code is

not much of a code, but this is a thriller,

not a textbook Criticisms seem to be

felt to be necessary by those who

probably not only secretly enjoy the

book but identify with the protago­

nists. Apparently these critics need to

display their superior knowledge, ig­

noring Brown's skill in using various

devices to make the book exciting

reading. They remind one of those in

the United States who felt obliged to

display their credentials by panning

Michael Moore's Fahrenheit 9/1 1 , as if

it were someone's dissertation, not a

skillful propaganda piece with some

basic truths and a few exaggerations.

The originality and factual basis of

Brown's theories may be in question,

but he has written an absorbing mys­

tery. As such, it is a page-turning suc­

cess.

Another best-seller is Mark Haddon's

The Curious Incident of the Dog in the Night-Time, borrowing from Conan

Doyle's "Silver Blaze." (Inspector Gre­

gory: "Is there any point to which you

would wish to draw my attention?"

Holmes: "To the curious incident of the

dog in the night-time." Inspector: "The

dog did nothing in the night-time."

Holmes: "That was the curious inci­

dent."] Originally classified as a book

for young people, The Curious Incident

has charming simplicity appealing to a

broader audience. The "detective," in

whose voice the book is written, is fif­

teen-year-old Christopher Boone, who

suffers from Asperger's syndrome, a

symptom of which is an inability to fig­

ure out what is going on in the minds

and emotions of others--not that every­

one does not have occasional problems

doing so. Christopher's curious incident

is less benign than Sherlock's, as the

dog is impaled on a garden fork Find­

ing out who did this leads to scary rev­

elations about Christopher's family,

and, crucial to Christopher himself, al­

most causes him to miss out on a math­

ematics exam to which he was looking

forward with great joy.

The narrative could have become

patronizing or pathos-filled, but Had­

don manages to achieve just the right

tone, making his hero endearing in

spite of his quirks. Some of these, like

his giving each chapter a prime num­

ber, are endearing, but others, such as

his refusal to be touched, are less so.

Readers are left to interpret for them­

selves words and actions that are a

mystery to Christopher, whose view of

the world is totally literal. Although

Christopher loves mathematics, there

is not a lot of it in the book (except for

some interesting bits about prime num­

bers and the Monty Hall problem). For

that matter, there is very little mystery,

but the book is brilliantly written.

Whether it is an accurate portrayal of

autism, I cannot say, but it feels au­

thentic. Unfortunately, the book may

feed the myth that anyone with As­

perger's syndrome is a mathematical

genius. On the other hand, it certainly

has led to an increased interest in, and

one would hope understanding of, the

condition, a recent indication of which

is a wonderful compendium, Ian Stu­

art-Hamilton's An Asperger Dictio­

nary of Everyday Expressions, ex­

plaining why "taking the bull by the

horns" is not just the tactic of a des­

perate matador. Maybe Christopher's

enthusiasm will even inspire others

with an eagerness to take mathematics

exams!

Finally, there is Leaning towards In­finity by Sue Woolfe, who has also writ­

ten a book, The Secret Cure, about

Asperger's. There is not a lot of mathe-

matics in Leaning towards Infinity, and

the only interesting mystecy is how it

won an award in its author's homeland

of Australia. No doubt the prize com­

mittee had few mathematically inclined

members. A complicated plot begins

with Ramant\ian's 1913 letter to Hardy

and involves three generations of

women: Juanita Montrose, the amateur

mathematician who is supposed to be

another Ramanujan; Frances her daugh­

ter, who was advised not to study math­

ematics as it might be injurious to her

health, and so ended up teaching litera­

ture, more suitable for a woman; and

Frances's daughter Hypatia, who is or­

ganizing the book her mother wrote. N ei­

ther Hypatia nor Woolfe admits to know­

ing anything about mathematics, an

assertion the reader can easily believe.

Juanita's creativity began, we are

told, with the understanding of Zeno's

paradoxes at age nine, and Frances fell

in love with the stocy that Einstein's wife

was responsible for e = mc2, giving

some idea of the level of sophistication

Woolfe employs. Borrowed from the life

of Sonya Kovalevskaia is the idea of

mathematical notes used as wall paper,

although obviously not to the same good

effect. After many unsatisfactory years

in her chosen field, Frances responds to

an ad soliciting entries for a context for

"radical" new ideas about mathematics,

the reward to be an invitation to present

the ideas at a conference in Athens. Her

winning contribution is the further de­

velopment of her mother's work on

Montrose numbers, apparently con­

ceived by Woolfe as something like

transfinite numbers that will turn the

mathematical world upside down.

Frances's alleged geometrical intuition

pushes the concept beyond the first of

these new numbers; she sees herself em­

ulating Tartaglia in winning the contest

and changing the world.

There are two themes in the book

that strike a realistic chord. The first is

the difficulty of combining mathemati­

cal research with primacy responsibility

for child care. The second is the atmos­

phere of an international mathematics

conference. With regard to the first, the

Montrose women are more or less fail­

ures, although their frustration is well

portrayed. Much about the second rings

true, but I doubt that any woman math-

ematician would go so far as to bare her

breasts to get attention when presenting

her results to an indifferent male-domi­

nated audience. (Maybe I have yet to see

true desperation!) In any case, both

Juanita and Frances go crazy and the

revolutionary mathematics never gets a

chance. That the "mathematics" is vague

and improbable is understandable, but

in general the book is difficult to wade

through, with the different narratives be­

coming confused.

It seems on the evidence here that

compelling mysteries are best written

by someone other than a mathemati­

cian, although Leaning towards Infin­ity shows that ignorance does not

guarantee success. Ultimately, it would

be interesting to know how many read­

ers previously uninterested in mathe­

matics are motivated to explore more

deeply concepts from the best of these

books such as prime numbers, growth

equations, probability, the golden ratio,

cryptography, and fractals.

REFERENCES

Daniel Aronofsky, Pi, film, Artisan Entertain­

ment, 1 998.

David Auburn, Proof, London: Faber & Faber,

2001 .

Dan Brown, Digital Fortress , New York: St.

Martin's Press, 1 998.

Margot Bruyere, Maths a mort, Paris: Aleas­

Editeur 2002 (originally published as Dis-mol

qui tu aimes, je te dirais tu ha1s, 1 990).

P. M. Carlson, Audition for Murder, New York:

Avon Books, 1 985.

-- , Murder is Academic, New York: Avon

Books, 1 985.

-- , Murder is Pathological, New York:

Avon Books, 1 986.

-- , Murder Unrenovated, New York: Ban­

tam Books, 1 987.

-- , Rehearsal for Murder, New York: Ban­

tam Books, 1 988.

-- , Murder in the Dog Days, New York:

Crimeline, 1 990.

-- , Murder Misread, New York, Double­

day Books, 1 990.

-- , Bad Blood, New York: Doubleday

Books, 1 991 .

Desmond Cory, The Catalyst, New York: St.

Martins Press, 1 991 .

-- , The Strange Attractor, New York:

MacMillan, 1 992.

-- , The Mask ofZeus, New York: St. Mar­

tins Press, 1 993.

--, The Dobie Paradox, New York: St.

Martins Press, 1 994.

Matt Damon and Ben Affleck, Good Will Hunt­

ing, film, Miramax Films, 1 997.

Carl Djerassi and David Pinner, Newton 's Dark­

ness: Two Dramatic Views, London: Imper­

ial College Press, 2003.

Apostolos Doxiadis, Uncle Petros & Gold­

bach's Conjecture, New York: Bloomsbury

USA, 2000.

Arthur Conan Doyle, Silver Blaze, Strand Mag­

azine, 1 892.

Robert Goddard, Out of the Sun, New York:

Henry Holt & Company, 1 997.

Paul Hoffman, The Man Who Loved Only Num­

bers, Westport CT: Hyperion Press, 1 998.

Michael Innes, The Weight of the Evidence,

New York: Dodd Mead, 1 943.

Philip Kerr, Dark Matter, New York: Crown Pub­

lishers, 2002.

Laurie R. King, The Beekeeper's Apprentice,

New York: St. Martins Minotaur, 1 994.

-- , A Monstrous Regiment of Women,

New York: St. Martins Press, 1 995.

-- , A Letter of Mary, New York: St. Mar­

tins Press, 1 997.

-- , The Moor, New York: St. Martins

Press, 1 998.

-- , 0 Jerusalem, New York: Bantam,

1 999.

-- , Justice Hall, New York: Bantam, 2002.

-- , The Game, New York: Bantam, 2004.

Jane Langton, The Escher Twist, Rollinsford

NH: Thomas T. Beeler Publisher, 2003.

Michael Moore, Fahrenheit 911 1 , film, Lions

Gate Films, 2004.

Sylvia Nasar, A Beautiful Mind, New York: Si­

mon & Schuster, 1 998.

lain Pears, An Instance of the Fingerpost, New

York: Putnam Publishing Group, 1 998.

Simon Singh, The Proof, video, WGBH, 2003.

Tom Stoppard, Arcadia, New York: Doubleday,

1 995.

Tom Stoppard, Enigma , film, Miramax Films,

2001 .

lan Stuart-Hamilton, An Asperger Dictionary of

Everyday Expressions, London: Jessica

Kingsley Publishers, 2004.

Scott Turow, Presumed Innocent, New York:

Farrar Straus & Giroux, 1 987.

Sue Woolfe, The Secret Cure, Sydney: Pica­

dor, 2003.

Department of Mathematics and Statistics

American University

Washington DC 2001 6-8050

USA

e-mail: mgray@american.edu

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005 91

Adventures in Group Theory: Rubik' s Cube, Merl in's Machine & Other Mathematical Toys by David Joyner

BALTIMORE AND LONDON, THE JOHNS HOPKINS

UNIVERSITY PRESS, 2002. 280 PP. US $69.95

HARDCOVER ISBN 0-801 8-6945-0

US $22.95 PAPERBACK ISBN 0-8018-6947-1

REVIEWED BY GERHARD BETSCH

Erno Rubik was born in Budapest.

He studied architecture and design

and is at present a professor of interior

design at the Academy of Applied Arts

and Design in his hometown, Bu­

dapest.

In the mid-seventies he patented a

cube-shaped mechanical puzzle that

has since captured the imagination of

millions of people worldwide. By 1982,

"Rubik's cube" was a household term,

and an entry in the Oxford English Dictionary. More than 100 million

cubes have been sold world-wide. The

more senior readers of The Intelli­gencer will remember the time around

1980, when playing with a Rubik's cube

was almost endemic.

It was clear from the beginning that

this toy could and should be consid­

ered in a systematic, or rather mathe­

matical way, with the tools of permu­

tation group theory. And very soon

papers and books on "cubic" problems

appeared in print. On the other hand,

Rubik's cube and similar games or toys

seemed to be a good opportunity, a

good starting point for an introduction

to group theory.

The book under review is another

attempt at an introduction to permuta­

tion group theory starting from fasci­

nating applications in "everyday life."

To quote the author: "All the abstract

algebra needed to understand the

mathematics behind Rubik's cube,

Lights out, and many other games, is

developed here . . . . This book began

as some lecture notes designed to

teach discrete mathematics and group

theory to students who, though cer­

tainly capable of learning the material,

92 THE MATHEMATICAL INTELLIGENCER

had more immediate pressures in their

lives than the long-term discipline re­

quired to struggle with the abstract

concepts involved."

Joyner's book owes much to the

monographs by Chr. Bandelow [ 1 ] and

D. Singmaster [2] .

A sketch of the contents may be in­

structive.

Chapters 1 and 2 deal with elemen­

tary mathematical concepts: Some

logic, sets, functions, relations; matri-

these chapters: finite projective linear

groups, Mathieu groups, Golay codes.

The "cubic" enthusiast will be quite

happy with the final presentation, Chap­

ter 15, which provides concrete solution

strategies for Rubik's cube, the Master­

ball, the Skewb, and a few others.

Most theorems in this book are

given with proof. The author provides

stimulating exercises, denoted as "pon­

derables."

ces and determinants; some elemen- REFERENCES

tary facts from combinatorics. Chapter [ 1 ] Christoph Bandelow: Inside Rubik's cube

3 is devoted to permutations, with a

fine introduction to "permutations and

bell ringing," or campanology, which is

the art and study of ringing cathedral

bells by permuting "rounds" of bells.

Cf. the murder mystery solved by Lord

Peter Wimsey, in Dorothy Sayers's

novel The Nine Tailors. This is fol­

lowed by a chapter on permutation

puzzles, which also provides a first

contact with the 2 X 2 and the 3 X 3

Rubik's cube.

Chapter 5 presents some elemen­

tary group theory: the quaternion

group, the finite cyclic groups, the di­

hedral and symmetric groups; the con­

cepts of subgroups, cosets, conjuga­

tion, actions and orbits. Chapter 6

expounds the mathematical structure

in some mathematical models of some

of the "Merlin's Magic Family" of

games. The following chapter is de­

voted to graphs, and graphical inter­

pretation of permutation groups (Cay­

ley graphs).

Chapters 8-10 provide more group

theory: the Platonic solids and their

symmetry groups; factor groups, direct

and semidirect products, wreath prod­

ucts; and the machinery of generators

and relations, as well as the presenta­

tion problem.

Chapter 1 1 is a highlight of the book:

the mathematical description of (legal)

moves of the 3 X 3 cube, the cube

group, and moves of order two.

To this point the book has ad­

dressed the "general reader," or the

"ordinary student." Chapters 12-14 are

more advanced and address readers

who have some knowledge of abstract

algebra (say, undergraduates majoring

in mathematics). A few key words from

and beyond, Birkhauser, Boston, 1 980.

[2] D. Singmaster: Notes on Rubik's magic

cube, Enslow, Berkeley Heights, N.J . , 1 981 .

Furtbrunnen 1 7

D 71 093 Wei! im Schonbuch

Germany

e-mail: Gerhard.Betsch@t-online.de

Origami Design Secrets: Mathematical Methods for an Ancient Art by Robert J. Lang

WELLESLEY, MA, A K. PETERS. LTD. 2003, ISBN

1 -5688 1 - 1 94-2 594 PP. US $48.00

REVIEWED BY THOMAS C. HULL

During the past 35 years or so the art of origami has been in a ren­

aissance. The word origami is Japan­

ese and reflects their historical prac­

tice, some say dating back to the 1600s,

of folding paper into representational

and abstract shapes. But modern

origami, in the sense of what one finds

when looking at a typical origami in­

struction book, only dates back to the

1940s and 1950s. That was when a

handful of individuals, namely the ma­

gician Robert Harbin of England, Lil­

lian Oppenheimer (the "grandmother"

of origami in the USA) of New York,

and paper-folding master Akira

Yoshizawa of Japan, took it upon them­

selves to communicate and popularize

this art to the rest of the world. As a

result, the first mass-marketed origami

instruction books began to appear, and

interest in the art slowly began to grow.

Technical advances in folding tech­

nique were created. Most notably,

Uchiyama from Japan attempted to

classify origami "bases" from which

many models can be derived, and the

method of "box pleating" was invented

by folders like Neal Elias and Max

Hulme. This led to more complex

origami model design, but nonetheless

there were respected paper folders in

the 1970s making statements like, "It is

impossible to make a grasshopper us­

ing one sheet of square paper" ([5], p.

131). The general impression seemed

to be that there were definite limits on

what one could do with origami.

This all changed when American

John Montroll published his first book,

Origami for the Enthusiast, in 1979

[6]. There it was, at the end of the book,

the most complex origami model pub­

lished up to that time: a grasshopper

with six legs, abdomen, thorax, wings,

head, and two antennae, all folded

from a single square of paper with no

cuts. Furthermore, his methods used

no box pleating, which was the most

complex technique devised at the time.

Rather, Montroll invented methods

that were merely natural generaliza­

tions of the classic bases (fish base,

bird base, and frog base) used by the

ancient Japanese.

John Montroll's grasshopper

Needless to say, Montroll's work

amazed everyone. Soon numerous pa­

per-folders were mimicking Montroll's

techniques and causing an explosion in

the level of complexity seen in origami.

This was when Robert Lang, then a

CalTech grad student, emerged with

ever-more-challenging models, like a

cuckoo clock (complete with cuckoo

that pops out when the pendulum is pulled) from a rectangular piece of

paper. And in Japan physicist Jun

Maekawa was creating models like his

devil (complete with eyes, nose, tongue,

ears, horns, tail, legs, and arms with five

fmgers on each hand), which took Man­

troll's innovations even further. Such

models were breaking away from the

concept of general "bases" and instead

developing ways to collapse the paper

that were individual for the subject at

hand. Furthermore, some of these new

models (Maekawa's devil among them)

were so elegant and exhibited such eco­

nomic use of the paper that one sus-

pected, as with the classic Japanese

crane and frog models, they were some­

how meant to be folded.

Concurrent with this was the emer­

gence of an increased interest in the

mathematics of origami. In fact, many

origami enthusiasts at the time (Lang,

Montroll, Maekawa, and Kawasaki, just

to name a few) were either mathemati­

cians, physicists, or engineers by day. It

is perhaps not surprising that people

with scientific interests would be drawn

to the challenge of origami design,

where the rules of economy and the

physical restraints of the paper must be

overcome, making each model design a

puzzle. Some of these same origamists

began observing mathematical patterns

among the creases in the models they

were folding. Thus emerged the so­

called Maekawa's Theorem, which

states that the difference between the

number of mountain (convex) and val­

ley (concave) creases at a flat-foldable

vertex in an origami crease pattern is al­

ways two, and Kawasaki's Theorem,

which states that the alternating sum of

the consecutive angles between creases

in a flat-foldable vertex is always zero

(see [1 ] and [2] for more details). The

French mathematician Jacques Justin,

who was also an origamist, discovered

these same basic results at the same

time, the 1980s. Mathematicians were

Illustration of the tree algorithm using one of Meguro's bugs. (Not all creases are shown.)

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2. 2005 93

paying attention to the on-going origami

revolution and initiating origami-math

research.

As origami entered the 1990s, paper­

folders were beginning to approach

complex design with more sophisti­

cated techniques. In fact, Fumiaki

Kawahata, Toshiyuki Meguro, and Jun

Maekawa in Japan and Robert Lang in

the United States independently dis­

covered a connection between origami

design and circle-packing. The idea is

that when designing something like an

origami insect, one needs to extract

many appendages, like legs, wings, and

antennae, from the square piece of pa­

per. Sometimes these appendages can

be made from the comers or edges of

the square. For example, the classic

Japanese crane makes appendages for

the head and tail from two opposite

comers of the square and the wings

from the other pair of comers. But a

more complex subject might require

appendages to be created from the in­

terior of the paper. In all these cases,

however, we can think of the ap­

pendage as being like an umbrella, that

is, a stick made from pleats radiating

from a point. Folded up, we would

have our appendage. Unfolded, we

would see that the area of paper de­

voted to this appendage would ap­

proximate a disk. Thus when planning

where appendages should be extracted

from the paper, we can try arranging

circles, possibly of different sizes for

different-sized appendages, on the

square. Our only requirement would be

that the circles not overlap and that the

center of each circle be contained in

the interior or the boundary of the

square. Finding an optimal arrange­

ment of such circles, where they might

be arranged symmetrically for ease of

folding or be as large as possible for ef­

ficiency of the design, would be the

first step toward developing a crease

pattern for a base that would have

all the appendages we need for our

origami insect.

This is, of course, easier said than

done. But it does suggest that an actual

algorithm for origami design might be

possible. Further, mathematical tools

like Maekawa and Kawasaki's Theo­

rems among others, had emerged by

the 1990s to help form crease patterns

94 THE MATHEMATICAL INTELLIGENCER

that would actually fold up. By the mid-

90s origamists were writing about such

algorithms in detail. The Japanese de­

signers called their algorithm the bun­ski, or molecule method, while Robert

Lang called his the tree method of

origami design. But their approaches

were similar: start by drawing a stick

figure, a tree graph, of your subject

with weights on the edges to indicate

their relative sizes. Then each leaf node

with its edge weight l determines a cir­

cle of radius l that must be packed on

the square. Find a way to arrange the

circles, making them reasonably large

and, for convenience, positioning them

along axes of symmetry of the square,

if possible. Then connect some of the

centers of neighboring circles with

creases to begin an outline of the crease

pattern. This will result in various poly­

gons drawn on the square with comers

at the centers of the circles. These

polygons can then be collapsed by in­

serting more creases-the molecules,

in the Japanese terminology-into each

polygon. For example, a triangle can

easily be collapsed by making creases

on the angle bisectors. Collapsing each

polygon should result in the paper

transforming into an object whose out­

line resembles the tree graph with

which we started. An origamist can

then, with skill, make this look like the

original subject.

But there were many gaps in this al­

gorithm. How does one find an optimal

arrangement of circles? What ifthe tree

graph has lots of vertices which are not

leaf nodes? And origamists had not

then discovered ways of collapsing all

possible polygons properly.

Robert Lang spent a lot of time in the

1990s and beginning of the 21st century

working on these details. The fruits of

his labor were compiled into the in­

creasingly complex versions of his

TreeMaker computer program, which al­

lowed a user to draw a stick figure,

weight the edges, and then watch as the

program used a constrained optimiza­

tion algorithm to fmd an optimal circle

packing. After Lang discovered his uni­

versal molecule for collapsing any poly­

gon, his program was able to go on to

produce a full crease pattern, which can

then be printed out and folded.

The book under review, Origami

Design Secrets, is Lang's magnum

opus, collecting all this work in printed

form. But this is not a mathematics

text, nor is it a strict technical manual.

Lang chose to strike a balance between

a book that describes origami design

algorithmically and one that appeals to

the origami community. Thus, 151 of

the 585 pages in the book are devoted

to folding instructions for a variety of

origami models. But even for non-ex­

perts at folding, these diagrams serve

to illustrate the sheer complexity that

can be found in modem origami.

Nearly 30 of the diagram pages are de­

voted to one model: Lang's tour-de­

force Black Forest Cuckoo Clock (an

insanely more elaborate version of his

earlier Cuckoo Clock), which has leg­

endary status among origamists. And

there are simpler models, like his Koi

Fish and Snail, which illustrate the de­

sign techniques on a smaller scale.

The book is written in a simple style

that will seem elementary to most math­

ematicians, and will only disappoint

people who are already deeply familiar

with Lang's techniques. For mathemati­

cians and origamists alike, Lang's ex­

pository approach introduces the reader

to technical aspects of folding and the

mathematical models with clarity and

good humor. Details of the underlying

mathematics are sprinkled throughout,

and those wishing for full details of the

optimization side of the tree algorithm

will fmd them in the last chapter.

Two other features make this book

especially valuable. One is the sheer

abundance of gorgeous figures. This

alone makes the book suitable for un­

dergraduates to understand (although

Lang's clear writing will suit them too)

and is a boon for readers struggling to

visualize the tricks of paper-folding.

The other is that Lang was not sat­

isfied with only describing his tree al­

gorithm. He also includes, with histor­

ical references, numerous other design

techniques that are part of an origami

creator's toolbox. These include graft­

ing multiple crease patterns together,

splitting crease patterns to insert more

details, box pleating, and even finer el­

ements of the tree theory like rivers

and stub points. Being as complete as

possible in this regard makes Origami Design Secrets a literal encyclopedia of

paper-folding methods. It also offers

never-before-seen continuity and clas­

sification of all that has been discov­

ered by origami designers over the past

30 years. Further, Lang had the foresight

to describe box pleating, which for over

a decade has been considered by many

as an old-school design technique from

the 1970s, in a more modem design con­

text. This is especially helpful for those

who want to understand the models of

Satoshi Kamiya, a young Japanese

folder who for the past several years has

been redefining people's conception of

ultra-complex origami (see [4]). Kamiya

uses box pleating in ways no one

thought was possible. Lang's book is

current with such developments.

The book also points out open prob­

lems in this field. The biggest is that the

tree algorithm only produces uniaxial bases, that is, bases whose flaps (ap­

pendages) all lie along a single axis, or

line. There are a number of origami

bases without this property, and in­

corporating them into Lang's algorithm

is a wide-open area for study.

In fact, the problems dealt with in

Origami Design Secrets have grown

over the past decade into the emerging

Acknowledgment

field of computational origami. One

of the now-classic results in this field is

the fold-and-cut theorem, which states

that by folding a piece of paper one can

remove with a single cut of a scissors

any planar embedding of a graph (with

its interior). (See [3]) The proof of this

result actually uses a method similar in

concept to Lang's universal molecule.

Furthermore, one of the authors of this

result is MacArthur Award winner Erik

Demaine of MIT, who has done more

than anyone to galvanize this field. It is

not an exaggeration to say that

Origami Design Secrets is required

reading for mathematicans interested

in computational origami.

Those wishing to learn about math­

ematical methods in paper-folding will

have a frustrating time finding re­

sources. Aside from a proceedings

book from a 2001 conference (see the

reference for [ 1]) on origami in math,

science, and education, there are no

detailed texts or monographs in the

field. Origami Design Secrets fills a

huge void in that regard. It is highly rec­

ommended for mathematicians and

students alike who want to view, ex­

plore, wrestle with open problems in,

or even try their own hand at the com­

plexity of origami model design.

REFERENCES [ 1 ] T. Hull , The combinatorics of flat folds: a

survey, Origam1'J: Proceedings of the Third

International Meeting of Origami Science,

Mathematics, and Education, T. Hull ed. ,

A . K . Peters (2002), 29-38.

[2] K. Kasahara and T. Takahama, Origami for

the Connoisseur, Japan Publications, 1 987.

[3] E. Demaine, M . Demaine, and A. Lubiw,

"Folding and One Straight Cut Suffice, " Pro­

ceedings of the 1 Oth Annual ACM-SIAM

Symposium on Discrete Algorithms (SODA

'99}, Baltimore, Maryland, January 1 7-1 9,

1 999, 891-892.

[4] S. Kamiya, Web page: http://www.asahi­

net.or.jp/�qr7s-kmy/

[5] T. Kawai, Origami, Hoikusha Publishing

Co. , 1 970.

[6] J . Montroll, Origami for the Enthusiast,

Dover, 1 979.

Thomas Hull

Department of Mathematics

Merrimack College

North Andover, MA 01 845

USA

e-mail: thull@merrimack.edu

We acknowledge with thanks that Lyon P. Robinson of Sydney contributed the

photographs in the article about the Sydney Opera House in vol. 26 (2004), no.

4, 48-52.

© 2005 Springer SCience+ Business Media, Inc., Volume 27, Number 2, 2005 95

'"-J@ij,j.Mg.h.i§i Robin Wilson ]

The Philamath' s Alpha bet-H

Halley: Edmond Halley ( 1656-17 42) is

primarily remembered for the comet

whose return he predicted and which

is named after him. Having observed

the comet in 1682, he predicted its re­

turn in late 1758 or early 1 759, and its

appearance did much to vindicate

Isaac Newton's theory of gravitation.

Earlier, Halley had been influential in

persuading Newton to develop this the­

ory and publish its conclusions in the

Principia Mathematica. Hamilton: William Rowan Hamilton

(1805-1865) was a child prodigy who

discovered an error in Laplace's trea­

tise on celestial mechanics while still a

teenager and became Astronomer Royal

of Ireland while a student. He did im­

portant theoretical work in mechanics

Halley

Please send all submissions to

the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes,

MK7 6AA, England

e-mail: r.j.wilson@open.ac.uk

and geometrical optics, and revolu­

tionised algebra with his investigations

into non-commutative systems. This

stamp commemorates his discovery of

quaternions.

Heisenberg: Werner Heisenberg (1901-

1976) took an algebraic approach to

quantum mechanics, realising that quan­

tities such as position, momentum, and

energy can be represented by infinite

matrices. Using the fact that matrix mul­

tiplication is non-commutative, Heisen­

berg deduced his 'uncertainty principle',

that it is theoretically impossible to de­

termine the position and the momentum

of an electron at the same time.

Hipparchus: The first trigonometrical

approach to astronomy was provided

by Hipparchus of Bithynia (190-120

BC), possibly the greatest astronomi­

cal observer of antiquity. Sometimes

called 'the father of trigonometry', he

discovered the precession of the

equinoxes, produced the first known

star catalogue, and constructed a 'table

of chords' yielding the sines of angles.

Hua Loo-Keng: Hua Loo-Keng [Hua

Luogeng] (1910-1985) is most well

Hamilton's quatemions

�p · �q � h

Heisenberg

Htiun�hz Uns<lo4rftrtbltion

96 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Science+Bus1ness Media, Inc

known for his important contributions

to number theory-particularly, trigono­

metric sums and Waring's problem­

and to the study of several complex

variables. Throughout the Cultural

Revolution of 1966-1976, he travelled

widely through China, lecturing on in­

dustrial mathematics to audiences of

up to 100,000 factory workers.

Huygens: The scientific contributions

of Christian Huygens (1629-1695) were

many and varied. He expounded the

wave theory of light, hypothesised that

Saturn has a ring, and invented the pen­

dulum clock and spiral watch spring. In

mathematics he wrote the first formal

probability text, introducing the con­

cept of ' expectation', and analysed such

curves as the cycloid and the catenary.

Hipparchus

Hua Loo-Keng

Huygens