Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters

to the editor should be sent to either of

the editors-in-chief, Chandler Davis or

Marjorie Senechal.

Nonegenarian Fibonacci Devotee

Please let me take this opportunity to make one more obeisance to the Fi­bonacci sequence.Fibonacci tended to take over my mathematical life from the time, many years ago, when I found that the occurrence of the numbers in leaf patterns needed more explaining. One thing led to another, decade after decade, paper after paper.1 I lived com­fortably among these numbers-until midnight of April 26, 2003. At that in­stant, I ceased to be 89 years old; and there seems little prospect of my ever again having a Fibonacci number as my age. To be sure, my rural route address is now Box 532, Route 1, a concatena­tion of Fibonacci numbers in reverse order, but that is small consolation.

Something more is needed to re­affirm my allegiance. Here is my offer­ing.

I will prove that the Fibonacci num­bers with odd index can be generated iteratively from the quadratic equation

(la) x2 + y2 = 3xy - 1

in the following way. Put x equal to any Fibonacci number with odd index;:::: 1, and solve (1a) for y; the larger root will be the Fibonacci number with the next larger odd index. The Fibonacci num­bers with even index are generated by

exactly the same procedure from the equation

(lb) x2+y2=3xy+l.

To prove these, I will use an imme­diate consequence of the defining iter­

ation Fn+l = Fn + Fn-t:

(2) Fn-2 + Fn+2 = 3F,.

I will also use the identity

(3) Fn-2Fn+2 = Fn2 + ( -1)n+l,

which is a special case of an identity in Hoggatt.2

Now I set x = Fn (n odd) in (la)

(4) Fr/ + y2 = 3FnY- 1,

and I am able to show that the larger root for y is F11+2. Substituting (3) on the left and (2) on the right of ( 4) re­duces it to

which does indeed have F,+2 as its larger root. Similarly for the assertion for even n.

Irving Adler

297 Cold Spring Road

North Bennington, VT 05257

USA

e-mail: iadler@sover.net

'For instance. my articles in J. Theor. Bioi. 45 (1g74), 1-7g: and J. Algebra 205 (1ggs), 227-243.

2V. E. Hoggatt, Jr. Fibonacci and Lucas Numbers (Houghton Mifflin, 1969). See p. 59.

4 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Bus1ness Med1a, Inc

BARRY KOREN

Computationa F uid Dynamics· Science and Too

The year 2003 marked the 1 OOth anniversary of both the birth of John von Neumann

and the first manned flight with a powered plane-both events of great importance for computational fluid dynamics.

he science of flows of gases and liquids is fluid dynamics, a subdiscipline of physics.

No courses in fluid dynamics are given in high school, as it requires too much math-

ematical background. Fluid dynamics is taught at university and at engineering col-

leges, for one cannot ignore fluid dynamics if one wants to design an aircraft, a rocket,

a combustion engine, or an artificial heart.

Particularly for aircraft design, knowledge and under­

standing of fluid dynamics-aerodynamics in this case-is

of major importance. Except for the dangerous gravity, all

forces acting on a flying plane are forces exerted by air. To

fly an aircraft safely (tanked up with fuel and with pas­

sengers on board), a precise knowledge, understanding,

and control of these aerodynamic forces is a matter of life

and death. Moreover, flying must not only be safe but also

fuel-efficient and quiet. For aerospace engineering, aero­

dynamics is indispensable.

Nowadays, both experimental and theoretical means are

available for investigating fluid flows. Wind tunnels are the

canonical tool for experimental aerodynamics. The Wright

brothers, who made the first manned flight with a powered

plane (Fig. 1), had at their disposal a wind tunnel, one they

themselves had made. Wind-tunnel testing has many dis­

advantages, but it is deemed trustworthy because real air

is used and not the virtual air of theoretical aerodynamics.

A Brief History of Computational Fluid Dynamics

Nowadays, the technological relevance of theoretical aero­

dynamics, of theoretical fluid dynamics in general, is widely

appreciated. However, in the past it was mainly an acade-

mic activity, with results that strongly differed from ex­

perimental observations. The technical applications of fluid

dynamics developed independent of theory. Theoretical

and technological breakthroughs have since closed the gap

between theory and practice, and today we see a fruitful

interaction between the two. The airplane has played a very

stimulating role in this development.

I proceed by highlighting some key developments from

the history of theoretical fluid dynamics with an eye toward

computational fluid dynamics.

Revolutionary innovations

Theoretical fluid dynamics has an illustrious history [1, 2,

3]. In the course of centuries, many great names have con­

tributed to the understanding of fluid flow and have helped

in building up theoretical fluid dynamics, step by step. The­

oretical fluid dynamics goes back to Aristotle (384-322 BC),

who introduced the concept of a continuous medium. In

my opinion, though, it actually began 2000 years later, when

Leonhard Euler published his equations of motion for the

flow of liquids and gases, on the basis of Newton's second

law of motion [4, 5].

Euler's idea to describe the motion of liquids and gases

© 2006 Springer Science+Business Media, Inc., Volume 28. Number 1, 2006 5

Figure 1 . Glass-plate photo of the first manned flight with a powered

plane (flight distance: 37 meters, flight time: 12 seconds), Kitty Hawk,

North Carolina, 1 0.35 h., December 17, 1 903. Prone on the lower wing:

Orville Wright. Running along with the plane to balance it by hand if

necessary: Wilbur Wright. Visible in the foreground: the rail from

which the plane took off and the bench on which the wing rested.

The very same day, the Wright brothers made a flight of 260 meters

and 59 seconds!

in the form of partial differential equations was a revolu­

tionary innovation. However, his equations, known today

as the Euler equations, were still unsuited for practical ap­

plications, because they neglect friction forces: only pres­

sure forces were taken into account. It was almost a cen­

tury later, in 1845, that George Stokes proposed fluid-flow

equations which also consider friction [6]; equations which,

for an incompressible flow, had already been found by

Claude Navier [7] and are now known as the Navier-Stokes

equations. With the introduction of the Navier-Stokes equa­

tions, the problem of understanding and controlling a large

class of fluid flows seemed to be within reach, as it had

been reduced to the integration of a handful of fundamen­

tal differential equations.

Although formulating the Navier-Stokes equations con­

stituted great progress, the analytical solution of the com­

plete equations was not feasible. (It remains one of the out­

standing open mathematical problems of the 21st century.)

One developed instead a large number of simplified equa­

tions, derived from the Navier-Stokes equations for special

cases, equations that could be handled analytically. More­

over, a gap continued between experimental and theoreti­

cal fluid dynamics. The former developed greatly during the

Industrial Revolution, independent of the latter. It was para­

doxical that the introduction of the Navier-Stokes equa­

tions led to a further fragmentation into different flow mod­

els, all of which described the flow of the same fluid (air

in our case)-a theoretically highly undesirable situation.

Theoretical fluid dynamics stagnated along a front of non­

linear problems. This barrier was finally broken in the sec­

ond half of the 20th century, with numerical mathematics,

at the expense of much-often very much-computational

work A key role was played in this by a Hungarian-born

mathematician, John von Neumann.

6 THE MATHEMATICAL INTELLIGENCER

Trail-blazing ideas from Princeton

In the tens and twenties of the past century, Budapest was a

fruitful breeding ground for scientific talent. It saw in 1903

the birth of John von Neumann (Fig. 2). In his early years von

Neumann received a private education; at the age of 10 he

went to school for the first time, directly to high school.

There, his great talent for mathematics was discovered. He

received extra lessons from mathematicians of the Univer­

sity of Budapest, among them Michael Fekete, with whom

von Neumann wrote his first mathematics paper, at the age

of 18. By then he was already a professional mathematician.

Von Neumann studied at the ETH ZUrich and the University

of Budapest; he obtained his PhD degree at the age of 22.

Next he moved to Gem1any, where he lectured at universi­

ties in Berlin and Hamburg. There he was particularly active

in pure mathematics: in set theory, algebra, measure theory,

topology, and group theory. He contributed to existing theo­

ries: the sure way to quick recognition. From the mid 1930s,

von Neumann chose a riskier way of working: breaking new

ground. He turned to applied mathematics in the sense of

mathematicians like Hilbert and Courant, i.e., not mathe­

matics applied to all kinds of ad hoc problems, but the sys­

tematic application of mathematics to other sciences, in par­

ticular to physics, with subdisciplines like aerodynamics. The

rapidly deteriorating political situation in Europe, from which

von Neun1ann had already emigrated to Princeton, 1 played a

·-------- -------

1Von Neumann was one of the many scientists who left Europe in the early 1 930s.

For a description of the fall of Gottingen under Nazi pressures. see Richard

Courant's biography [8].

Figure 2. John von Neumann, 1 903-1 957.

role in this decision. War was looming and brought in­

creasing demands for answers to questions related to mil­

itary engineering.

Whereas von Neumann had worked on a mathematical

basis for the equations of quantum mechanics before the

war, during the war he "lowered" himself to developing nu­

merical solution methods for the Euler equations. His idea

to compute possible discontinuities in solutions of the

Euler equations without explicitly imposing jump relations

was very original. Instead, von Neumann proposed the in­

troduction of artificial (numerical) diffusion, in such a way

that the discontinuities automatically appear in a physically

correct way: shock capturing, nowadays a standard tech­

nique. He also came up with an original method for ana­

lyzing the stability of numerical calculations: a Fourier

method, now a standard technique as well.

In 1944, the urgent need arose to apply von Neumann's

numerical methods on automatic calculators, computers,

beyond the scope of the machines of that time. This moti­

vated von Neumann to start working also on the develop­

ment of the computer. In 1944 and 1945 he did trail-blazing

work, writing his numerical methods for computing a fluid­

flow problem in a set of instructions for a still non-existent

computer. These instructions were not to be put into the

computer by changing its hardware or its wiring. Instead,

von Neumann proposed to equip computers with hardware

as general as possible, and to store the computing instruc­

tions in the computer, together with the other data involved

(input data, intermediate results, and output data). In 1949,

the first computer was realized which completely fulfilled

von Neumann's internal programming and memory princi­

ples: the EDSAC (Electronic Delay Storage Automatic Cal­

culator), by M. V. Wilkes, at Cambridge University. Today,

the two principles are still generally applied.

L. F. Richardson and Richard Courant and colleagues had

combined theoretical fluid dynamics and numerical mathe­

matics before von Neumann [9, 10], but still without clear

ideas about computers-without computer science. Compu­

tational fluid dynamics (abbreviated CFD) is a combination

of three disciplines: theoretical fluid dynamics, numerical

mathematics, and computer science. Because von Neumann

brought in this last discipline, he can be considered the found­

ing father of CFD. A detailed description of von Neumann's

contributions to scientific computing is given by Aspray [11).

A good overview of his other pioneering work can be found

in the scientific biography written by Ulam [12].

Traveling from place to place as an honored mathe­

matician with many social and political obligations, von

Neumann must have had very little time to write down his

scientific ideas. He published only one paper about both

shock capturing and the aforementioned stability analysis,

and that not until 1950 [13].

On his many travels, von Neumann visited the Nether­

lands. In 1954, he was an invited speaker during the Inter­

national Congress of Mathematicians held in Amsterdam.

A tea party with Queen Juliana of the Netherlands was

arranged for a select group of participants, among them

John von Neumann (Fig. 3).

Figure 3. John von Neumann and colleagues at Soestdijk Palace. Above: all together, John von Neumann front row, far left. Queen Juliana,

with white handbag, is flanked by the two new recipients of the Fields Medal: Jean-Pierre Serre (with Herman Weyl's hands on his shoulders)

and Kunihiko Kodaira. At the right of von Neumann: Mary Cartwright.

© 2CXJ6 Springer Science+ Business Media, lnc., Volume 28, Number 1, 2006 7

Pioneering work in Amsterdam

In the third quarter of the 20th century, computer science

was a new and growing discipline. Initially, the Netherlands

played no significant role in the development of computer

science, but the country was quickly moving forward. In

1946, the Mathematisch Centrum (MC) was founded in Am­

sterdam. The mission of this new institute was to do pure

and applied mathematics research in order to increase "the

level of prosperity and culture in the Netherlands and the

contributions of the Netherlands to international culture."

(Not at all the pure ivory tower.) The foundation of the MC

did not proceed without struggle. The most prominent

Dutch mathematician of the day, L. E. J. Brouwer of the

University of Amsterdam, was of the opinion that mathe­

matics should be indifferent towards the physical sciences

and even rejecting of technology; an odd point of view, con­

sidering the work of mathematicians like Hilbert, Courant,

and von Neumann. With the MC, Brouwer wanted to tum

Amsterdam into the new Gottingen of pure mathematics.

It did not work out that way. His biographer feels that

Brouwer was sacrificed to the foundation of the MC ([14],

p. 479).

The founders of the MC had heard about von Neumann's

ideas about machines which should be able to perform a

series of calculations as independently as possible. They

wanted the MC to have a computing department in order

to develop a computer and to execute advanced comput­

ing work. Aad van Wijngaarden (Fig. 4), former student of

J. M. Burgers of the Delft University of Technology, was

appointed as the first staff member of the MC, in 194 7. That

year he made a study tour to visit von Neumann in Prince­

ton. Van Wijngaarden and his co-workers designed and con­

structed the first Dutch computer: the ARRA I (Automa­

tische Relais Rekenmachine Amsterdam I, Fig. 5). New

Figure 4. Aad van Wijngaarden, 1 916-1987 (photo courtesy of CWI).

8 THE MATHEMATICAL INTELLIGENCER

computers were designed and built (one per design only),

in 1955 exclusively for the Fokker aircraft industries:

the FERT A (Fokker Elektronische Rekenmachine Type

ARRA). Much human labor was required to perform com­

putations on these early computers. At the MC, this was

done by young women (Fig. 6), schooled in mathematics

by Van Wijngaarden. A highlight was the project for the de­

velopment of the Fokker Friendship airplane, a numerical

project on which Van Wijngaarden and his "computing

girls" worked from 1949 until 1951. The computations con­

cerned oscillations of the airplane's wing in subsonic flow:

flutter. The first computing work was still very "external"

and machine-dependent; for each computation, cables had

to be plugged into the computer. With the accomplishment

of internal programming as proposed by von Neumann, at­

tention shifted entirely to the invention of algorithms and

their coding as computer programs. Edsger Dijkstra, a later

Turing Award recipient, was appointed at the MC as the

first Dutch computer programmer. Van Wijngaarden and

Dijkstra left an international mark on computer science

with their contributions to the development of the pro­

gramming language Algol 60 [15], and Van Wijngaarden

later added to that reputation with Algol 68 [16]. In 1979,

Van Wijngaarden was awarded an honorary doctorate for

his pioneering work by the Delft University of Technology,

and the MC grew into the present CWI (Centrum voor

Wiskunde en Informatica), which celebrates its 60th an­

niversary in 2006.

Computational fluid dynamics research on the basis of

the Euler or Navier-Stokes equations, of the same funda­

mental character as that established by von Neumann, was

not done in the Netherlands of 1945-1960. For this funda­

mental work, we have to go to the United States and the

Soviet Union of the 1950s.

Figure 5. The first computer at the MC and in the Netherlands, the ARRA I in its final set-up. From left to right: power frame and the three

arithmetic registers. On the table in the middle: the punch-tape reader. Some 1200 relays are at the back of the machine {photo courtesy of

CWI).

A continuous flow of CFD from New York

In early December 1941, a passenger ship carried a 15-year­

old Hungarian boy from Europe to the United States. The

boy, along with his parents, was escaping the tragic fate

threatening European Jews. (It was to be the last passen­

ger ship from Europe's mainland to the United States for

years to come. During the voyage, the United States was

drawn into the Second World War by the attack on Pearl

Harbor.) The young ship passenger carried with him two

letters of recommendation from his teachers. It seems

likely that von Neumann saw those letters brought by his

young fellow-countryman, for the boy, Peter Lax (Fig. 7),

had a meteroric rise to success. In 1945, while still a

teenager, he became involved in the Manhattan Project. In

1949, he received his PhD degree from New York Univer­

sity, with Richard Courant as his thesis advisor, and in 1951,

Figure 6. Female arithmeticians at the MC, the "girls of Van Wijngaarden." In the foreground: Ria Debets, later the spouse of Edsger Dijk­

stra. {photo courtesy of CWI).

© 2CX>6 Springer Science+ Business Media, Inc., Volume 28, Number 1, 2006 9

Figure 7. Peter D. Lax.

he became assistant professor there. His work in mathe­matics continues to this day and has led to many honors and awards, among them the 2005 Abel Prize in mathe­matics [ 17].

Like von Neumann, Lax is a homo universalis in math­ematics. He has performed ground-breaking research, and has been a productive and versatile author of mathematics books. His books deal with such diverse topics as partial differential equations, scattering theory, linear algebra, and functional analysis. Above all, he is known for his research on numerical methods for partial differential equations, in particular for hyperbolic systems of conservation laws, such as those arising in fluid dynamics. Lax's name has been given to several mathematical discoveries of impor­tance to CFD:

• the Lax equivalence theorem [18], stating that consis­tency and stability of a finite-difference discretization of a well-posed initial-boundary-value problem are neces­sary and sufficient for the convergence of that dis­cretization,

• the Lax-Friedrichs scheme [ 19], a stabilized central finite­difference scheme for hyperbolic partial differential equations,

• the Lax-Wendroff scheme [20], a more accurate but equally stable version of the Lax-Friedrichs scheme,

• the Lax entropy condition [21], a principle for selecting the unique physically correct shock-wave solution of nonlinear hyperbolic partial differential equations that al­low multiple shock-wave solutions, and

• the Harten-Lax-Van Leer scheme [22], a very efficient nu­merical method for solving the Riemann problem.

Like von Neumann, Lax was (and still is) a strong pro­ponent of the use of computers in mathematics. A quote

1 0 THE MATHEMATICAL INTELLIGENCER

from Lax: "The impact of computers on mathematics (both applied and pure) is comparable to the roles of telescopes in astronomy and microscopes in biology."

Despite the Second World War and the Cold War, Lax has always had very good connections with scientists worldwide. One such relation is with a famous Russian mathematician, mentioned in the next section.

A brilliant idea from Moscow

A substantial part of the Euler and Navier-Stokes software used worldwide is based on a single journal paper [23], dis­tilled by the then-young Russian mathematician Sergei Kon­stantinovich Godunov (Fig. 8) from his PhD thesis.

Godunov proposed the following. Suppose one has a tube and in it a membrane separating a gas on the left with uni­formly constant pressure, from a gas on the right with a like­wise uniformly constant but lower pressure (Fig. 9, top). If the membrane is instantaneously removed-the traffic light changes from red to green-then the yellow gas will push the blue gas to the right; the interface between the two gases, the contact discontinuity, runs to the right. At the same time, two pressure waves start running through the tube: a compression wave running ahead of the contact dis­continuity and an expansion wave running to the left (Fig. 9, bottom). In the 19th century, the Euler flow in this tube, a shock tube, had already been computed by Riemann, with "pencil and paper" [24]. (For this old work of Riemann, Duivesteijn has written a nice, interactive Java applet [25].) For the computation of the flow in a tube in the case of an initial condition which has more spatial variation, Godunov proposed to decompose the tube into virtual cells (Fig. 10, above), with a uniformly constant gas state in each cell, and with each individual cell wall to be considered as the afore­mentioned membrane (traffic light). To know the interac-

Figure 8. Sergei Konstantinovich Godunov.

rarefaction wave contact discontinuity shockwave .....

Figure 9. Shock tube. Top: condition of rest in left and right part: high and low pressure, respectively. Bottom: condition of motion with shock

wave and contact discontinuity running to the right and rarefaction wave running to the left. (drawing: Tobias Baanders, CWI).

tion between the gas states in two neighboring cells, one in­stantaneously 'removes' the cell wall separating the two cells, and computes the Riemann solution locally there, and hence the local mass, momentum, and energy flux (Fig. 10, bottom). This is done at all cell faces. With this, the net transport for each cell is known and a time step can be made. A plain method and a very simple flow problem, so it seems. If one can do this well, the flow around a com­plete aircraft or spacecraft can be computed. The remark­able property of the method is that at the lowest discrete level, that of cell faces, a lot of physics has been built into it, not just numerical mathematics.

The more cells, the better the accuracy, yet also the more expensive the computation. Godunov did not have ac­cess to computers, but to "computing girls," who called Godunov and his fellow PhD students "that science," and who received payment on the basis of the number of com­putations they performed, right or wrong. No real CFD there either!

In 1997, Godunov received an honorary doctorate from the University of Michigan, and a symposium was organized

for him at the university's Department of Aerospace Engi­neering. At that symposium, in a one-and-a-half-hour lec­ture, Godunov gave insight into his earlier research, whose strategic importance was not appreciated in the Soviet Union at the time. This historic lecture has since been pub­lished [26, 27].

A second important result in Godunov's classical paper from 1959 [23] is his proof that it is impossible to devise a linear method which is more than first-order accurate, with­out being plagued by physically incorrect oscillations in the solution: wiggles (Fig. 11). With a first-order-accurate method, the solution becomes twice as accurate and re­mains free of wiggles when the cells are taken twice as small. With a second-order-accurate method, the solution becomes four times more accurate then, but-unfortu­nately-possibly wiggle-ridden.

Wiggles can be very troublesome in practice. For ex­ample, a simple speed-of-sound calculation in a single cell only may break down the entire flow computation, because of a possibly negative pressure. The wiggle problem does not occur only with Godunov's method; it is a general prob-

Figure 10. Shock tube divided into small cells. Top: cells. Bottom: wave propagation over all cell faces. (drawing: Tobias Baanders, CWI).

© 2006 Springer Science+Bus1ness Media, Inc., Volume 28, Number 1, 2006 1 1

pressure pressure

1

0

Figure 1 1 . Right and wrong pressure distribution. Left: without wiggles. Right: with wiggles. (drawing: Tobias Baanders, CWI).

lem. A drawback of Godunov's method is that it is com­puting-intensive; at each cell face, the intricate Riemann problem is solved exactly.

Technology pushes from Lelden

It took about two decades before good remedies were found for the wiggles of higher-order methods and the high cost of the Godunov algorithm. The aid came from an as­tronomer. In space, large clouds of hydrogen are found. Simulation of the flow of this hydrogen provides models of the development of galaxies. The literally astronomical

Figure 12. Bram van Leer (photo: Michigan Engineering).

1 2 THE MATHEMATICAL INTELUGENCER

speeds and pressures which may arise in these computa­tions impose high demands on the accuracy, and particu­larly the robustness, of the computational methods to be applied. While still in Leiden, in the 1970s, the astronomer Bram van Leer (Fig. 12) published a series of papers in which he proposed methods which are second-order ac­curate and do not allow wiggles. The fifth and last paper in this series is [28). Furthermore, Van Leer introduced a computationally efficient alternative to the Godunov algo­rithm [29): two technology pushes, not only for astronomy but also for aerospace engineering, as well as for other dis-

ciplines. In 1990, Van Leer was awarded an honorary doc­torate for this work by the Free University of Brussels.

Efficient solution algorithms from Rehovot and

other places

Broadly speaking, how does an Euler- or Navier-Stokes­flow computation around an aircraft work? The airspace out to a large distance from the aircraft, may be divided into (say) small hexahedra, small 3D cells. Just as in the 1D shock tube example, one can then compute for each cell the net inflow of mass, momentum, and energy, using at each cell face the Godunov alternative ala Van Leer or other alternatives, like the Roe scheme [30] or the Osher scheme [31]. The finer the mesh of cells around the aircraft, the grid (Fig. 13), the more accurate the solution, but also the higher the computing cost. A grid of one million cells for an Euler- or Navier-Stokes-flow computation is not un­usual. Suppose that we want to simulate a steady flow. We then have to solve, per cell, five coupled nonlinear partial differential equations. The cells themselves are coupled as well: what flows out of a cell flows into a neighboring cell (or across a boundary of the computational domain). In Navier-Stokes-flow computations, the flow solution in a sin­gle cell may influence the flow solutions in all other cells. In our modest example, we may have to solve a system of five million coupled nonlinear algebraic equations. Effi­cient solution of these millions of equations is an art in it­self. Many efficient solution algorithms have been devel-

oped, the most efficient of which are the multigrid algo­rithms. Multigrid methods were invented at different loca­tions and by several people. A leading role has been played by Achi Brandt from the Weizmann Institute of Science in Rehovot, Israel [32]. Multigrid algorithms have a linear in­crease of the computing time with the number of cells. This may seem expensive-2, 3, or 4 times higher computing cost for a grid with 2, 3, or 4 times more cells, respectively­but it is not. In numerical mathematics, no bulk discount is given. For many solution algorithms the rule is 22,32,42, . .. times higher computing cost for a grid with 2,3,4, . .. times more cells! For the interested reader, a book on multi­grid methods is [33].

Present State of the Art in CFD

An example

A quick impression will now be given of what can be done with CFD by looking at a standard flow problem. It con­cerns the recent MSc work of Jeroen Wackers. From scratch, he developed 2D Euler software in which the grid is automatically adapted to the flow, and what follows de­scribes one of his results. Consider the channel depicted in Figure 14, and in it a uniformly constant supersonic air flow (from left to right) at three times the speed of sound. One may consider the channel to be a stylized engine inlet of a supersonic aircraft. In fact it is just a benchmark geometry [34, 35]. The red vertical valve at the bottom of the chan­nel instantaneously snaps up, so that, together with the red

Figure 13. Cross sections of a hexahedral grid around the Space SHuttle.

© 2006 Spnnger Science+ Business Med1a, Inc .. Volume 28, Number 1. 2006 1 3

' \ ' \ flow speed ' \ 7 _) ..J

speed of sound

I Figure 1 4. 20, parallel channel. In it, a parallel plate and a vertical valve which is still open.

horizontal plate, it forms a step which suddenly chokes part of the channel.

Figure 15 shows a computational result. We see how the uniformly constant initial solution and the grid have de­veloped after some time. The computational method highly satisfies the often conflicting requirements of numerical stability, accuracy, and monotonicity on the one hand, and computing and memory efficiency on the other [36].

Books, journals, and software

Twenty years ago, textbooks on CFD were rare, but sev­eral are available now (see, e.g., [37, 38, 39, 40]). There are

>

>

X

X

also scientific journals dedicated to CFD. Moreover, off­the-shelf CFD software can be purchased these days. Each issue of, e.g., the monthly Aerospace America contains col­orful, full-page advertisements for CFD software. A practi­cal overview of the CFD literature, software, and also va­cancies can be found on the Web site of CFD Online [41].

Today, CFD's role is about as important as that of ex­perimental fluid dynamics. And CFD continues to grow. It is fed by improvements in both computer science and nu­merical mathematics. In addition, CFD itself stimulates re­search in computer science and numerical mathematics: a fruitful interaction.

Figure 15. Computational result some time after instanteously closing the lower part of the channel. Top: iso-lines of density. When the ver­

tical valve is still in the open position, the density in the entire channel is constant (everywhere the same blue color as at the inlet). Bottom:

computational grid automatically adapted to flow solution.

14 THE MATHEMATICAL INTELLIGENCER

At present, CFD enters into full cooperation with other disciplines, such as structural mechanics (computational fluid-structure interactions) and electromagnetism (com­putational magnetohydrodynamics ).

Outlook

The fact that commercial CFD software is a success is proof of the practical importance of the theoretical fluid-dynam­ics work since Euler. The growing availability of CFD soft­ware may seem to be a threat for CFD research; CFD re­searchers seem to make themselves redundant by their own success. Yet, this growing software availability may also be considered a good development. Not everyone has to write his/her own Euler or Navier-Stokes code. Coding such soft­ware from scratch gives the best insight and is pleasing work, but it may easily take too much time.

Education

A new question arises: How to teach CFD, now that it has become more and more important as an easily available, au­tomatic tool? Not just factual knowledge but also under­standing of the mathematical and physical principles of CFD remains indispensable, not only when practicing it as a sci­ence, but also when using it as a tool. The CFD-tool user must know and understand these principles well in (1) pos­ing the computational problem, (2) choosing the numerical method to solve that problem, and (3) interpreting the com­putational results. The user must know the possibilities and limitations of computational methods and should be able to assess whether the computational results obtained fulfill the expectations or not. If not, it should be found out why. Thus CFD is not solving flow problems by blind numerical force. On the contrary, stimulated by the growing potential of CFD, still more complicated flow problems will be considered, problems which will require even more knowledge and un­derstanding of flow physics and numerical mathematics.

Research

As CFD becomes more and more mature, it also becomes more difficult to contribute fundamental research to it. In re­cent decades a PhD student can hardly do such fundamental work as Godunov did. Students will have to acquire an ever­growing knowledge and understanding of CFD before they can start working in it themselves. On the other hand, thanks to the availability of CFD tools, the possibilities for applica­tion of CFD are far greater now than in Godunov's era. Just how CFD will develop remains unpredictable, and this is part of what makes it an exciting and attractive discipline.

In CFD plenty of research questions remain. New fluid­flow problems will continue to arise, and there will cer­tainly be times when we may say with Orville Wright, "Isn't it astonishing that all these secrets have been preserved for so many years just so that we could discover them!"

REFERENCES

[1] T. von Karman, Aerodynamics, McGraw-Hi l l , New York, 1963.

[2] H . Rouse and S. lnce, History of Hydraulics, Dover, New York,

1963.

[3] J. D . Anderson, A History of Aerodynamics, Cambridge University

Press, Cambridge, 1997.

[4] L. Euler, Principes gf!meraux du mouvement des fluides, Memoires

de I 'Academie des Sciences de Berlin, 11 (1755), pp. 274-31 5 .

[5] M . D. Salas, Leonhard Euler and his contributions to fluid me­

chanics, AIM-paper 88-3564, AIM, Reston, VA, 1988.

[6] G. G. Stokes, On the theories of the internal friction of fluids in mo­

tion, and of the equilibrium and motion of elastic solids, Transac­

tions of the Cambridge Philosophical Society, 8 (1845), pp. 287.

[7] C. L. M . H. Navier, Memoire sur les lois du mouvernent des f/uides,

Memoires de I'Academie des Sciences, 6 (1822), pp. 389-440.

[8] C. Reid, Courant in Gottingen and New York. The Story of an Im­

probable Mathematician, Springer-Verlag, New York, 1976.

[9] L. F. Richardson, Weather Prediction by Numencal Process, Cam­

bridge University Press, Cambridge, 1922.

[1 0] R. Courant, K. 0. Friedrichs, and H. Lewy, Ober die partie/len Dif­

ferenzgleichungen der mathematischen Physik, Mathematische

Annalen, 1 00 (1 928), pp. 32-74.

[11] W. Aspray, John von Neumann and the Origins of Modern Com­

puting, MIT Press, Cambridge, Massachusetts, 1990.

[1 2] S. Ulam, John von Neumann, 1903-1957, Bulletin of the Ameri­

can Mathematical Society, 64 (1958), pp. 1-49.

[13] J. von Neumann and R. D. Richtmyer, A method for the numeri­

cal calculation of shocks, Journal of Applied Physics, 21 (1950),

pp. 232-237.

[14] D . van Dalen, L. E. J . Brouwer, 1 881-1966, Het Heldere Licht van

de Wiskunde, Bert Bakker, Amsterdam, 2002.

[15] P. Nauer (ed .) , Revised Report on the Algorithmic Language Algol

60 (available for download from http://www.masswerk.at/

algol60/report.htm).

[16] A. van Wijngaarden et al., Revised Report on the Algorithmic Lan­

guage Algol 68, Springer-Verlag, Berl in, 1 976.

[17] http://www.abelprisen.no/en/.

[18] P. D. Lax and R. D. Richtmyer, Survey of the stability of l inear fi­

nite difference equations, Communications on Pure and Applied

Mathematics, 9 (1 956), pp. 267-293.

[19] P. D. Lax, Weak solutions of nonlinear hyperbolic equations and

their numerical computation, Communications on Pure and Ap­

plied Mathematics, 7 (1954), pp. 159-1 93.

[20] P. D. Lax and B. Wendroff, Systems of conservation laws, Commu­

nications on Pure and Applied Mathematics, 13 (1960) , pp. 217-237.

[21] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Math­

ematical Theory of Shock Waves, SIAM, Philadelphia, 1973.

[22] A. Harten, P. D. Lax, and B. van Leer, On upstream differencing

and Godunov-type schemes for hyperbolic conservation laws,

SIAM Review, 25 (1 983), pp. 35-61.

[23] S. K. Godunov, Finite difference method for the numerical com­

putation of discontinuous solutions of the equations of fluid dy­

namics, Mathemat1cheskfi"Sborn1k, 47 (1959), pp. 271-306. Trans­

lated from Russian at the Cornell Aeronautical Laboratory.

[24] G. F. B. Riemann, Ober die Fortpflanzung ebener Luftwellen von

endlicher Schwingungsweite, in : Gesammelte Werke, Leipzig,

1876. Reprint: Dover, New York, 1953.

[25] G. F. Duivesteijn , Visual shock tube solver (to be downloaded from

http://www. piteon.ni/cfd/) .

[26] B. van Leer, An Introduction to the article "Reminiscences about

difference schemes", by S. K. Godunov, Journal of Computational

Physics, 153 (1999), pp. 1-5.

© 2006 Spnnger Science+ Business Media, Inc., Volume 28, Number 1, 2006 15

[27] S. K. Gudunov, Reminiscences about difference schemes, Jour­

nal of Computational Physics, 1 53 (1 999), pp. 6-25.

[28] B. van Leer, Towards the ultimate conseNative difference scheme.

V. A second-order sequel to Godunov's method, Journal of Com­

putational Physics, 32 (1 979), pp. 1 01 -1 36. Reprint: Journal of

Computational Physics, 135 (1997}, pp. 229-248.

[29] B. van Leer, Flux-vector splitting for the Euler equations, in Lec­

ture Notes in Physics, Vol. 170, Springer-Verlag , Berl in , 1 982, pp.

507-5 1 2 .

[30] P . L. Roe, Approximate Riemann solvers, parameter vectors, and

differences schemes, Journal of Computational Physics, 43 ( 1 981 ) .

pp. 357-372.

[31] S. Osher and F. Solomon, Upwind difference schemes for hyper­

bolic systems of conseNation laws, Mathematics of Computation,

38 (1 982), pp. 339-374

[32] A Brandt , Multi-level adaptive solutions to boundary-value prob­

lems, Mathematics of Computation, 31 (1 977), pp. 333-390.

[33] U. Trottenberg, C. W Oosterlee, and A Schuller, Multigrid, Aca­

demic Press, New York, 2001 .

[34] A F. Emery, An evaluation of several differencing methods tor in­

viscid fluid flow problems, Journal of Computational Physics, 2

(1 968), pp. 306--331 .

[35] P. R. Woodward and P. Colella, The numerical simulation of two­

dimensional fluid flow with strong shocks, Journal of Computa­

tional Physics, 54 (1 984), pp. 1 1 5-1 73.

[36] J. Wackers and B. Koren, A simple and efficient space-time adap­

tive grid technique for unsteady compressible flows, in Proceed­

ings 1 6th AIM CFD Conference (CD-ROM), AIM-paper 2003-

3825, American Institute of Aeronautics and Astronautics. Reston,

VA, 2003.

[37] P. Wesseling, Principles of Computational Fluid Dynamics.

Springer-Verlag, Berl in , 2001

[38] P. J. Roache, Fundamentals of Computational Fluid Dynamics,

Hermosa, Albuquerque. NM, 1 998.

[39] Ch. Hirsch, Numerical Computation of Internal and External Flows.

Vol. 1 Fundamentals of Numerical Discretization, Vol. 2 Computa-

AUTHOR

BARRY KOREN

CWI P.O. Box 94079

1 090 GB Amsterdam

The Netherlands

e-mail: barry.koren@cwi.ni

Barry Koren studied Aerospace Engineering at the Delft Insti­

tute of Technology, and Computational Fluid Dynamics at the

Von Karman Institute for Fluid Dynamics in Belgium. He is now

leader of the research group in Computing and Control at the

Dutch Centre for Mathematics and Computer Science (CWI)

in Amsterdam, and also professor of Computational Fluid Dy­

namics at the Delft Institute of Techno logy. More information

can be found at http://homepages.cwi.nV-barry/.

He is married and the father of three children.

tiona/ Methods for lnviscid and Viscous Flows, Wiley, Chichester,

1 988-1 990.

[40] C. A J. Fletcher, Computational Techniques for Fluid Dynamics,

Vol. 1 Fundamental and General Techniques, Vol. 2 Specific Tech­

niques for Different Flow Categories, Springer-Verlag, Berl in, 1 988.

[41] http://www cfd-online.com

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Ancient Egypt ian Mathemat ics : New Perspectives on O ld Sources Annette lmhausen

Send submissions to David E. Rowe,

Fachbereich 1 7 -Mathematik,

Johannes Gutenberg University,

055099 Mainz, Germany.

Pro captu lectoris

habent suafata libelli

(Terentianus Maurus)

I f books, in general, have their own special fates-which depend on their

readers-the same is true for the mathematical "books" from ancient Egypt. Indeed, modem editors and sub­sequent readers have strongly influ­enced the way we view them today. And even now, readers of the third mil­lennium can alter the fate of these early texts by their careful (or careless) reading. 1

Sources and Early Historiography

For the past fifty years, the reputation of Egyptian mathematics has been rather poor. This has been due in part to the very limited number of available primary sources, particularly when compared with the vast collections of cuneiform mathematical texts pro­duced in Mesopotamia. In ancient Egypt the production of mathematics (as well as literature) took place in cities. Then, as today, Egyptian cities were located along the Nile, and hence close to water. This circumstance has had significant consequences for con­temporary Egyptological research. On the one hand, papyrus, the main writ­ing material in this culture, was de­pendent on absolute dryness for its preservation, a condition found in the Egyptian desert where most papyrus finds were made. However, in ancient Egyptian cities, where writings con­cerned with the mundane affairs of daily life were discarded after use, this condition was usually not fulfilled. Therefore, most of the written evi­dence documenting the role of mathe­matics in Egyptian social, economic, and cultural life must be assumed lost forever. On the other hand, to the ex­tent that such sources may still be re­trievable some day, practical problems stand in the way. The locations of an­cient Egyptian cities often coincide with those of modem urban centers.

This makes it next to impossible to ex­cavate at a number of locations where extant remains might still be found.

Among the few known (excavated) cities, the Middle Kingdom town of Lahun (also known as lllahun or Kahun) is exceptional, having yielded the rich­est findings of Middle Kingdom papyri so far, among which incidentally are a number of mathematical fragments. 2

The two most significant sources, how­ever, the famous Rhind and Moscow mathematical papyri, were bought on the antiquities market, making their prove­nance uncertain. These and most of the other known mathematical sources were already published by 1930.

The achievements of the earliest re­searchers who studied these texts, es­pecially those who worked during the first half of the twentieth century, were enormous. As editors, they managed to penetrate a foreign vocabulary of tech­nical terms, which placed them in po­sition to make a first attempt at un­derstanding Egyptian mathematical methods. 3 As was common at that time, ancient sources and achieve­ments were viewed and evaluated by means of direct comparison with mod­em conventions and results. In many respects, it was found that Egyptian mathematics had little in common with the methods found in modem mathe­matical textbooks. Nevertheless, with some effort the mathematical content of the ancient texts could be "decoded" and "translated" into modem mathe­matics.

Unfortunately, this type of reading often entailed a loss of the most strik­ing characteristics of the original sources, a drawback that was little ap­preciated at the time. Not surprisingly, the "achievements" of Egyptian math­ematics, judged in terms of a different mathematical culture (from more than 3000 years later), looked rather crude and simple. One of the early leading au­thorities on ancient mathematics was Otto Neugebauer, who wrote his dis­sertation on Egyptian methods of cal-

© 2006 Springer Science+ Business Med1a. Inc., Volume 28, Number 1 , 2006 1 9

culating with fractions.4 Afterward, Neugebauer turned his attention away from Egyptian mathematics to study Mesopotamian mathematics and as­tronomy, which he believed was a higher level of scientific achievement. As he once expressed this:

Egypt provides us with the excep­tional case of a highly sophisticated civilization which flourished for many centuries without making a sin­gle contribution to the development of the exact sciences. [ . . . ] It is at this single center (Mesopotamia) that abstract mathematical thought first appeared, affecting, centuries later, neighbouring civilizations, and fi­nally spreading like a contagious disease.5

It was surely in part due to the out­standing quality of the early scholarly contributions that readers accepted so readily this kind of negative assess­ment of Egyptian mathematics. As in­dicated already, this situation was compounded by the lack of new source material which-had it been there­would have required those capable of reading Egyptian texts to reflect upon the assessments of their predecessors. Thus, in the case of Mesopotamian mathematics, where new source mate­rial is still being uncovered on an al­most regular basis, readers' opinions have changed significantly over time. H

Lacking this wealth of textual material, readers of the Egyptian texts seemed to have no basis for questioning the standard views of earlier experts like Neugebauer. Indeed, once the major Egyptian mathematical papyri became available in English or German trans­lation, various historians of mathemat­ics began contributing new ideas based on their own readings of these first translations. Often these involved mod­em mathematical symbolism, leading to results that had almost nothing in com­mon with the original source text. 7

This once common approach has now been recognized as both anachro­nistic and misleading. Indeed, for the last 20 years historians of mathematics have started to take up and to rework the subject of ancient mathematics.8 It is now generally accepted that histori-

20 THE MATHEMATICAL INTELUGENCER

ans of mathematics cannot work on a source text without knowing the lan­guage in which it is written or the cul­tural background it comes from. At the same time, it has become obvious that mathematical knowledge is not uni­versal. It is neither independent of the cultures in which it is produced and used, nor has it developed universally from basic beginnings to more and more advanced stages of knowledge. This dependence on cultural back­ground begins already with number systems and number concepts, as has been demonstrated by various scholars working on ethnomathematicsY More advanced mathematical techniques and concepts have also been shown to be dependent on the culture that cre­ated them. 10

Current Research

From this description of past research, it follows that the editions of Egyptian mathematical sources are by now outdated. It is to be hoped that new editions can be published before the current ones reach their centenary. Likewise, older studies of Egyptian mathematics, those written more than 30 years ago, must be read with cau­tion, bearing in mind the kind of ap­proach past researches typically took. For an up-to-date introduction to the subject, the reader should consult the articles by Jim Ritter. 11 In the follow­ing sketch, I will attempt to give an overview of the state of current re­search, illustrated with selected exam­ples from the source material.

Although there have been no spec­tacular new finds of mathematical pa­pyri, extant sources, including the much-studied Rhind and Moscow pa­pyri, still offer many clues about the role of mathematics in Egyptian life. Alongside these, the Lahun mathemat­ical fragments have just been re-edited, including several previously unpub­lished fragments. 1 2

Other texts are still awaiting proper publication, such as the mathematical fragments of Papyrus Berlin 6619. The earlier publications from 1900 and 1902 only contain facsimiles of the two largest fragments. Moreover, the inter­pretations of them then given are not without problems. n The Cairo wooden

boards are currently available in two very small and hardly legible photos with a discussion of some of their con­tent. While a number of demotic math­ematical texts have been published, no detailed study of Egyptian mathemat­ics in the Graeco-Roman period is available yet. 14

Evidence from the Predynastic Period

Apart from the extant mathematical texts, however, there are further sources available throughout Egyptian history which inform us about aspects and uses of mathematics as it evolved in ancient Egypt in periods from which no mathematical texts are extant. Writ­ten evidence exists from as early as around 3000 B.C., the oldest dating from shortly before the unification of Egypt. It comes from the tomb Uj at Abydos15 and consists of writing on pottery as well as on little tags of bone and ivory. These tags all reveal holes, suggesting they were probably once at­tached to some perishable goods from this grave, thus indicating their prove­nance and quantity. u; The quantities were rendered using elements and style familiar from the Egyptian num­ber system in later times, i.e., a deci­mal system without positional notation (see Figure 1). In this system, each power of 10 up to 1 million was repre­sented by a different sign. In order to write any number, the respective signs, written as often as needed, were jux­taposed in a symmetric way. Note that the hieroglyphic writing, which is what most people associated with ancient Egypt, was used mostly on stone mon­uments. For daily life purposes, Egypt­ian scribes wrote with a reed (dipped in ink) on papyrus or so-called ostraca (limestone or pottery shards). The

Figure 1 . Number representations on the

tags from tomb Uj.

script used in this writing is more cur­

sive and abbreviated than hieroglyphic script. Several signs can be combined to form ligatures, whereas the writing

itself can vary a great deal, depending

on the individual scribe Gust like mod­

em handwriting).

Mathematics in the Old Kingdom

After the unification of Egypt under a

single king (around 3000 B.C.), the Old

Kingdom (OK; 2686-2160 B.C.) brought

forth the first period of cultural bloom

in Egyptian history. Extant architec­

tural remains, like the pyramids, as

well as such artifacts as the scribal

statues, demonstrate a high level of

cultural attainment by this time. There

can be little doubt that mathematical

techniques lay at the heart of this de­

velopment as a significant tool for han­dling organizational and administrative

problems. To achieve something on the

scale of the pyramids, mathematics

was necessary not only for architec­

tural planning but also for the organi­

zation of labor. The scribal statues,

which depict high officials from this

period, demonstrate the importance of

the administrative system. Despite this,

there is practically no written evidence

for mathematical practices extant from

this time. Many of the monumental hi­

eroglyphic inscriptions are still ex­

tant-but these, of course, focus on

eternity and tell us little about Egypt­

ian daily life and the affairs in which mathematics played an important part. Only very few papyri from this period

have survived, some in a very frag­

mentary state.

Nevertheless, there is other direct

evidence of Egyptian mathematical

techniques, for example from the plan­

ning and execution of building projects

such as a mastaba from Meidum (see

Figure 2). Around the comers of this

mastaba, beneath the ground level,

four 1-shaped mud-brick walls had

been built. On these walls a series of

diagrams can be found, which indicate

the slope of the sides of the mastaba.

This method of handling sloped sur­faces points to the development of a

concept which is well documented in

the mathematical texts. 17 To express

sloped surfaces, such as the sides of a pyramid, the Egyptians used the so-

7 p a I

m s

'seqed'

Figure 2. Indication of a sloped surface at Meidum.

called sqd. This Egyptian term is de­

rived from the verb qd, meaning "to

build." The sqd was used to measure

the horizontal displacement of the

sloped face for each vertical drop of

one cubit, that is the length by which

the sloped side had "moved" from the

vertical at the height of one cubit. The

sqd was always indicated in palms, and

if necessary, digits. Although we have

textual evidence for this concept only

from the Middle Kingdom onward,

sketches from the Old Kingdom indi­

cate that it was in use during this ear­

lier period. Note that the parallel lines

drawn on the mud bricks are spaced at

a distance of one cubit or seven palms. Furthermore there is early evidence

for several metrological systems. While

these units can also be found in later

mathematical texts, their appearance

in administrative papyri as well as in the inscriptions and depictions from

tombs indicates that these systems go

back at least to the Old Kingdom. Some

of these systems changed over time, but the sources from the Old Kingdom

suffice to trace these changes.

Calculations with Unit Fractions

One of the most intriguing aspects of

Egyptian mathematics concerns spe­

cial methods for calculating fractions,

which were understood in ancient

Egypt as inverses of integers. 18 Hence,

the Egyptian notation for fractions did

not consist of a numerator and de­

nominator, but rather a special symbol was used alongside an integer to des­

ignate the corresponding fraction. An exception was the fraction %• which had a special sign. The fractions t. i• and ± were also written by using spe-

cial signs (indicating that these may be

older) rather than by using the general

Egyptian notation. 19 In modem stud­

ies, Egyptian fractions are usually

described as unit fractions, and it is

often suggested that the Egyptians

"restricted" themselves to calculations with fractions having a numerator of

one. 20 As explained in the paragraph

above, however, this is a rather anachro­

nistic view. Moreover, seen from a mod­

em perspective, the Egyptian system

inevitably appears awkward and un­

necessarily restrictive.

One of the first to study Egyptian

computations with fractions was Otto

Neugebauer, who devised a notational

system that parallels the Egyptian no­

tation. Fractions, as inverses of inte­

gers, are rendered by the value of

the integer with an overbar: thus, +. would be written as 5, i as 6, etc. Th� exceptional fraction % was rendered by Neugebauer as 3, whereas _1_, �. and _I_ - 2 ,J 4 appeared as 2, etc. This notational sys-

tem, which closely mirrors the Egyptian

concept of fraction, has become the

standard way of writing Egyptian frac­tions in modem textbooks.

Following this concept of fractions

as inverses of integers, the next step­

consequently-was to express those

parts that correspond to a multitude of

inverses. This was done by (additive)

juxtaposition of different inverses.

Thus, � was written in the Egyptian sys-4 - -

tern as 2 4, whereas a general fraction

was given as a sum of different in­

verses written in descending order ac­

cording to their size. (Note that this no­

tation enables one to be as accurate as necessary by considering only ele­

ments up to a certain size.)

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1, 2006 21

Egyptian techniques of multiplica­tion and division (see below for a more detailed description) frequently in­volved the doubling of a number. This could be done very easily if the num­ber to be doubled was an integer or the inverse of an even integer. However, to double an odd Egyptian fraction (when the result is supposed to be a series of different inverses only) can be quite difficult to accomplish. Consequently, it proved useful to prepare tables giv­ing the results for doubling the inverses of odd numbers. These can be found in the so-called 2 7 N tables still extant in two sources: at the beginning of the Rhind Mathematical Papyrus (for odd N = 1 - 101) and in the Lahun frag­ment UC 32159 (for odd N = 1 - 21).

Figure 3 shows the fragment UC 32159 in which the numbers are arranged in two columns. The first col­umn shows (what we call) the divisor N, except for the first entry which shows both the dividend 2 and the di­visor 3. This is followed by a second column that altematingly shows frac­tions of the divisor and their value (as a series of inverses). Thus, the second line starts with the divisor 5 in the first column: it is 2 7 5 that shall be ex­pressed as unit fractions. This � foJ lowed in the second column by 3, 1 3, - - -15, a.!ld 3. This has to be read � 3 of 5

- - - - -is 1 3, and 15 of 5 is 3. Since 1 3 and 3 added equal 2, the series of unit frac­tions needed to represent 2 7 5 is 3 15.

The 2 7 N table in the Rhind papyrus shows the same arrangement of num­bers; however, the solutions there are marked by the use of red ink.

Obviously, the representation of 2 7

N as a series of unit fractions is not unique. However, the Egyptian 2 7 N Table uses for each N only one of the theoretically possible representations. Those we find in the Lahun fragment, for example, are identical to the ones found in the table of the Rhind papyrus. And whenever an odd fraction is dou­bled within the mathematical texts, it is this same representation that we find used.

This circumstance has fascinated a number of experts on additive number theory. In fact, there have been several attempts to crack the puzzle posed by the 2 7 N Table by finding the criteria that led the Egyptians to employ just these particular representations. Yet, while it is possible to describe some of the general tendencies-e.g. , represen­tations with fewer elements are fa­vored as are also representations with larger inverses, etc.-it has not been possible to establish strict mathemati­cal rules that explain the choices the Egyptians mathematicians made. Rather than criticizing them for their lack of insight-or blaming them for not having followed strict rules that would comply with a different mathe­matical concept of fractions devised by another culture several thousand years

2 3

5

7

9

11

13

15

17

19

21

3 2

3 1 3 15 3 - - - -

4 1 2 4 28 4

6 1 2 18 2 - - - -

6 1 3 6 66 6 - - - - - -

8 1 2 8 52 4 104 8

10 1 2 30 2 - - - - - -

12 1 3 12 51 3 69 4 - - - - - - -12 1 2 12 75 4 114 [6]

14 1 2 42 2

later-it seems more appropriate to recognize that mathematics is, indeed, culturally dependent; our modem point of view may not afford us the best picture of past achievements. Thus, in­stead of trying to concoct an explana­tion of the Egyptian solutions by using modem mathematics, it may be more rewarding simply to "accept" the Egyptian table and examine its use and usefulness within the mathematical en­vironment that employed it.

Mathematical Problem Texts

from the Middle Kingdom

Apart from tables, the mathematical texts also include special procedures articulated within problem texts. As these names indicate, such texts set out a problem and then give instruc­tions showing how to solve it. Proce­dure texts derive from an educational setting. They may have been written by a teacher, who was compiling a hand­book, or perhaps by a student engaged in practicing mathematical techniques. An appreciation of this context is im­portant for understanding these texts, which were intended to prepare scribes for the mathematical tasks they would later have to execute as part of their daily work.21 Given that these texts were written for this type of mathematical education, it should not be expected that we can learn how the Egyptians developed their mathemati­cal knowledge from sources of this na­ture.

The extant hieratic mathematical texts contain roughly one hundred problems. Furthermore, in the largest of these texts, the Rhind Mathematical Papyrus (see Figure 4), we can discern an arrangement of these problems ac­cording to their rising level of diffi­culty. This is not to be judged by purely mathematical aspects alone but also by additional knowledge (often from a practical background) which is neces­sary to solve the problems. This can be seen, for example, in pRhind, problems 31-34 and those immediately follow­ing, problems 35-38. Mathematically, both groups teach a procedure for de­termining an "unknown" number if its

Figure 3. Fragment UC 32159: 2 .;- N table (Copyright Petrie Museum of Egyptian Archaeol- sum with fractions of itself is given. ogy, University College London). The procedure for solving the prob-

22 THE MATHEMATICAL INTELLIGENCER

lems in both groups is roughly the same. However, in the second group (pRhind, problems 35-38), the "un­known" number is not an abstract num­ber but a quantity of grain. Therefore the result, which is determined in the same way as in the preceding prob­lems, needs to be transformed after­wards into the respective metrological units.22

The style of Egyptian mathematical problem texts can best be appreciated by looking at an actual example, like problem 56 of the Rhind Mathematical Papyrus:

Method of calculating a pyramid, 360 is its base, 250 is its height. You shall let me know its inclination. You calculate half of 360. It results as 180. You divide 180 by 250. 2 5 50 of a cubit results. 1 cubit is 7 palms. 23 You multiply with 7.

7 \ 2 3 2 \ 5 1 3 15 \ 50 10 25 1ts inclination: 5 25

palms

Problem 56, like the other four ex­amples of pyramid problems found in the Rhind Papyrus (nos. 57, 58, 59, and 59b ), teaches the relation between the base, height, and inclination of the sides. This example complements the OK sketch found on the walls around the mastaba with sloping sides, which was discussed above. In fact, the tech­nical term sqd-the number of palms the slope of a slanted plane recedes per vertical difference of one cubit-is ex­plicitly indicated in the problem text. Thus, the base, height, and inclination of a pyramid are linked by the relation:

1 /2 base inclination = 7 palms X

height

The problem above presents a pyramid with base (360) and height (250); its in­clination is to be calculated. The pro­cedure calls for calculating half of the base and dividing this by the height. The result is then multiplied by 7 to ob­tain the inclination in palms. Having grasped "what is going on" in this prob­lem, let us now take a second, closer look at the Egyptian text and its means of structure.

The text begins-as is typical for mathematical problem texts-with a title "Method of calculating a pyra­mid." Note that the beginning of the ti­tle is written in red ink (rendered in my translation in bold). This use of red ink helps the reader recognize at a glance the beginnings of individual problems. The title of mathematical problems is very often given as "Method of . . . " fol­lowed by a key word which indicates the type of problem. In our example, the key word is the Egyptian mr, "pyramid."

After this title, the given data are in­troduced, and they are always specific numerical values. This statement of the data is generally followed by a question or command, outlining the problem that the scribe shall solve. In this ex­ample: "You shall let me know its in­clination." Next, we see a sequence of instructions, followed by intermediate results. This procedure then leads to the numerical solution of the problem. Each instruction usually consists of one arithmetic operation. The Egyptian mathematical language distinguishes addition, subtraction, multiplication, division, halving, inverting, squaring, and the extraction of square roots. These individual mathematical opera­tions are expressed without any use of mathematical symbols. The instruc­tions themselves are always given as complete sentences.

Furthermore, in this part of the text, a special verb form is used, the so­called sd.m.IJr=f. The name consists

of the Egyptian verb "to hear" (sd.m), which is used in Egyptian grammars to demonstrate different conjugations, its characteristic morphological element (IJr) and the suffix pronoun of the third person singular (f). Its function is to express a "general truth" which results as a necessary sequence from previ­ously stated conditions.24 In the math­ematical texts, the sd.m.IJr=f is used for both instructions and announcing intermediate results. As for the latter, the verb form expresses "mathematical facts"-if 2 and 2 are added, the result will necessarily be 4. The use of the sd.m.IJr=f in the instructions under­lines the specific procedural character of the text: the sequence of instruc­tions necessarily has to be followed to solve the problem. The last i�s.!_ruction given, the multiplication of (2 5 50) by 7 is followed by a scheme of numbers. This carries out the actual multiplica­tion in the Egyptian manner, which may now be described.

Multiplication (and division) are ex­ecuted following a scheme that uses two columns of numbers. 25 Each mul­tiplication begins with the initialization which is found in the first line of the scheme: a dot is placed in the first col­umn and the number to be multiplied in the second column. The multiplica­tion is carried out by subsequent oper­ations in both columns using a variety of techniques, depending on the nu­merical values of the numbers that shall be multiplied. The aim is to find the multiplier as a combination of en­tries in lines of the first column. The respective lines of the second column will then be the result of the multipli­cation.

Figure 4. Rhind Mathematical Papyrus, No. 56 Copyright The British Museum.

Problem 56 of the Rhind Papyrus shows the notation used to compute 7 X 2 5 50. The initialization is followed by three more lines, each of which in­dicates

_ o�e

.!!_f the required fractional

parts (2, 5, 50) of the multiplier. How the individual entires of the second col­umn were found is not obvious. It is possible that there may have been ta­bles for fractional parts of 7, as this was a number that leads to compli­cated calculations, but which came up frequently due to the metrological con­ventions.26

© 2006 Springer Science+Business Media, Inc., Volume 28, Number I, 2006 23

Finally, the result of the problem is announced. Next to the text of the problem there is a sketch indicating characteristic measurements for this problem, i.e., the values of base and height (see Figure 5). This step-by-step layout can be found in virtually all Egyptian problem texts. This being the case, one can easily see that the formal aspect of phrasing mathematics in the form of procedures will be completely lost if a problem is "translated" into a modem algebraic equation (in this case: inclination = (� base/height) X 7 palms). While this formula has the ad­vantage of informing a modem reader at a single glance how an ancient measure was defined, it conveys nothing what­soever about the procedural character of Egyptian mathematics. Moreover, al­gebraic formulae played no part in Egyptian mathematics so that the above formulation for the sqd is anachronistic, at best, as it is foreign to the methods actually found in Egyptian texts.

Analyzing Egyptian Problem Texts

As it happens, a closer analysis of the problem texts reveals many hitherto unnoticed methodological features of Egyptian mathematics. Indeed, the procedural format can be used as a key to analyze not only individual problems but also various types of problems as found in the mathematical papyri. To get beyond a superficial understanding of Egyptian mathematics, however, a method was needed that enabled a reader to analyze and compare the Egyptian procedures. Such an ap­proach was first proposed by Jim Rit­ter.27 In my dissertation I have adapted this method to analyze the various pro­cedures used in all hieratic mathemat­ical problems. 28

The analysis of a specific problem text can be carried out by rewriting it in two stages. In the first, one keeps the numerical values indicated in the source text but rewrites the instruc­tions by replacing the rhetoric for­mulations with modem symbols that indicate the respective arithmetic op­erations. The data are noted at the be­ginning of the scheme by their numer­ical values. Thus, for the example cited above (pRhind, problem 56), the text would be rewritten as follows:

24 THE MATHEMATICAL INTELLIGENCER

Figure 5. Sketch at the end of Rhind Mathe·

matical Papyrus, No. 56.

Method of calculating a pyramid, 360 is its base, 360 250 is its height. 250

You shall let me know its inclination. You calculate half of 360.

(1) 2 X 360 It results as 180. = 180 You divide 180 by 250.

(2) 180 -7- 250 2 5 50 of a cubit results.

1 cubit is 7 palms. You multiply with 7.

= 2 5 50

(3) 2 5 50 X 7

The result allows one to see at a glance whether the arithmetic operations to be carried out were simplified by the choice of data. For example, in prob­lem 43 of the Rhind papyrus, the cal­culation of the volume of a granary with circular base, the diameter of the granary is given as 9. This greatly fa­cilitates the calculational procedure, the first step of which is to determine � of the diameter. In the values of our 9 problem, the given data were 360 and 250. While the first step, halving 360, is fairly straightforward, the second, the division of the result of the first step by the second datum results in a frac­tion of three parts, which then has to be multiplied by 7. Thus, by compari­son, the data in problems 58 and 59 re­sult in easier calculations.

This first stage of rewriting is espe­cially helpful when dealing with a cor­rupt text, as the modem reader is forced to follow the source text and identify the procedure in a step-by-step fashion. It then becomes immediately apparent where specific difficulties arise in the source.

To further analyze the text so as to reveal how its procedures are related

to those used in other problems, it is necessary to distinguish between dif­ferent types of numbers that can ap­pear throughout the procedure. The first numbers a reader encounters are the data of the given problem. From the second instruction on, three types of numbers are possible: data, intermedi­ate results, and constants. To distin­guish these, and also to get a clearer view of the structure of the procedure, a second stage of rewriting is required. In this stage the data are indicated by symbols D;, whereas intermediate re­sults are specified by a number in paren­theses (x) which specifies the step in the procedure that leads to the given re­sult. The only actual numbers that now appear in the rewritten text are con­stants. Thus, for our example, the result of this second rewriting is as follows:

Method of calculating a pyramid, 360 is its base, D1

250 is its height. D2 You shall let me know its inclination. You calculate half of 360.

(1) 2 X Dl It results as 180. You divide 180 by 250.

(2) (1) -7- D2 2 5 50 of a cubit results. 1 cubit is 7 palms. You multiply with 7. (3) (2) X 7

In my dissertation I have analyzed the procedures of all hieratic math­ematical problems by rewriting the procedure in the form of a symbolic algorithm. This makes it possible to compare the various procedures used and analyze their respective complex­ity. The analysis of problems by means of their procedures or algorithms thus constitutes a powerful tool for com­paring the structure of individual math­ematical problem texts. From this, one can learn a great deal about Egyptian mathematical techniques. Within the Rhind Mathematical Papyrus, for exam­ple, one fmds groups of problems with similar procedures (pRhind, No. 24-27), as well as a progression within one group from basic procedures to more elaborate ones (pRhind, No. 69-78).

Identifying an unambiguous sym­bolic algorithm can sometimes be

straightforward, as in the example above. Unfortunately this is not the case with all problems. Individual instruc­tions may be missing-sometimes they are replaced by a written calculation, or several steps are summarized in one in­struction only. These types of difficul­ties can sometimes be overcome by tak­ing into account all of the available source material. If-as in the Rhind Pa­pyrus-several problems of the same kind are available and their procedures are identical insofar as they are explic­itly stated, then those problems which lack certain instructions can occasion­ally be reconstructed by means of the more detailed problems.

I would like to stress in this context that both types of rewriting are merely tools for analyzing specific aspects of the procedures found in the problem texts, whereas the source texts them­selves remain central and should never be neglected in any analysis. Taking the three versions of the procedure to­gether, however, enables one to form a more complete analysis that includes not only the various procedures but also technical mathematical vocabulary, as well as the relation of drawings and cal­culations carried out in writing con­nected with the procedure, and others.

Mathematics within the Context

of Egyptian Culture

Another integral part of the reassess­ment of Egyptian mathematics concerns its role within Egyptian culture. Mathe­matics was one of the key elements of scribal training in pharaonic Egypt. It provided the scribes with a crucial tool they needed to fulfil their administrative tasks as well as to plan and carry out construction projects. Consequently, many of the mathematical problems they dealt with were related to practical matters, e.g., the distribution of rations, the volume of granaries, or the amount of produce to be delivered by a worker. Our understanding of mathematical problems of this kind is at least partially dependent on our appreciation for these larger contexts.

This can be demonstrated with the so-called bread and beer problems, which appear against the background of economic activity, baking and brew­ing, under the control of a local au-

thority (state or temple). A quantity of grain is taken from a granary and then given to workers who produce bread and/or beer from it. Obviously it was necessary to ascertain the quantity-in loaves of bread or vessels of beer-of a given quality (in this case measured by grain content) that was equivalent to the amount of grain initially given to the workers. The mathematical side of this control is represented by the bread and beer problems. 29 The terminology used in these problems is taken from the respective technological language. Thus the bread and beer problems evolve around the psw, a unit which measures how many loaves of bread have been made from one 1}1;,3.t of grain. Apart from the psw, there are two additional standard phrases indicating the use of specific kinds of grain prod­ucts and their quantities. Obviously, this has further consequences for the re­spective calculations. Similar observa­tions can be made for other groups of practical problems as well. These gen­erally involve not only the "basic" math­ematical terminology but also further knowledge related to the technological or administrative background. This usu­ally makes them not only more difficult to understand but also less likely to be "mirrored" by a familiar problem in modem mathematics. Thus, early his­torical research often neglected this area of Egyptian mathematics.

However, as is obvious from the or­dering of the problems found in the Rhind Papyrus, it was precisely these practical problems that were consid­ered more advanced. After all, the aim of the mathematical handbooks was to prepare scribes for their daily admin­istrative work. So if we want to obtain insights into Egyptian mathematics, we must consider these problems and try to understand them. The setting of the individual problem may help to point to further sources (not only textual) which may be useful to understand the additional terminology and practice. Furthermore, it is this type of problem that indicates other possibilities of gaining knowledge about Egyptian mathematics apart from the restricted corpus of mathematical texts. The ac­tual output of the scribes in doing their daily work provides us with numerous

documents that prove the use of mathe­matical techniques. Thus Michel Guille­mot has used a ration text from Kahun to analyze mathematical practices. 30 These can be linked to techniques taught in mathematical papyri.31 It is to be hoped that this example can be followed for other texts as well.

The most promising sources still to be explored in this respect are the Reis­ner Papyri. This set of four papyrus rolls contains calculations for the building of a sanctuary, including ra­tion tables, actual building calcula­tions, as well as the administration of workshops. 32 They not only enlarge the meagre set of seven problems related to architecture which are known from the Moscow (problem 14) and Rhind (problems 56-60) papyri, but they also demonstrate that the amount of work done was linked to a specific number of workers (and rations) per day.

Evidence of Mathematics

in the New Kingdom

While the mathematical texts date al­most exclusively from the Middle King­dom, other sources are available from all periods of Egyptian history. The Wilbour Papyrus, a text from the New Kingdom, is an official record of mea­surements and assessments of fields over a distance of 90 miles along the Nile. The fields are given by localization and acreage, their assessments referring to taxes specified in amounts of grain.

Another major opportunity to find relevant sources of mathematics for the New Kingdom is provided by the excavation of Deir el Medina. Deir el Medina is the modem name of an an­cient Egyptian village on the West Bank of the Nile opposite Luxor. The village was inhabited by workmen who were responsible for the construction and decoration of the tombs in the Val­ley of the Kings. Deir el Medina has

yielded a huge quantity of artifacts and texts relating to daily life in the New Kingdom-similar to the findings at Lahun for the Middle Kingdom. Among the sources are ration texts, building plans, as well as texts for the educa­tion of scribes. The ostracon in Figure 6 shows a fragment of an exercise in the multiplicative writing of large numbers. It shows in the first column

© 2006 Springer Science+Bus1ness Media, Inc., Volume 28, Number I , 2006 25

Figure 6. Deir el Medina: Remains and Ostracon with Number Exercise.

(on the right) the numbers 600,000, 700,000, and 800,000 and in the second (middle) column the numbers 5,000,000, 6,000,000, and 7,000,000 written by the sign for the number 100,000 (or 1 ,000,000) with the respec­tive multiplicative factors (6, 7, and 8 and 5, 6, and 7) below. The third col­umn (left) shows again the sign for 1,000,000 and two illegible signs below.

Conclusions

Although Egyptian mathematics will probably never have the vast number of sources that still can be found in other cultures like India or Mesopotamia, there is more available than has been used so far. 33 The analysis of all the available mathematical texts, taken along with the additional material from administrative economic and literary contexts related to Egyptian mathe­matics, is certain to provide a better foundation for understanding its role within Egyptian culture. This inte­grated approach represents an impor­tant advance beyond the early studies that relied exclusively on an internal analysis of a small corpus of mathe­matical texts, which served for several decades as the sole basis for assessing nearly three millennia of mathematical life in ancient Egypt. By carefully rereading these classical mathematical texts while according the new sources a serious first reading, we may antici­pate that the fate of Egyptian mathe­matics faces an exciting future.

NOTES

1 . I thank David Rowe for his comments on

previous versions of this article and for his

corrections of my English. I also thank

Richard Parkinson of the British Museum

and Stephen Quirke of the Petrie Museum

26 THE MATHEMATICAL INTELLIGENCER

for permission to include photographs of

sources.

2. See Annette lmhausen and Jim Ritter,

"Mathematical Papyri , " in : Mark Collier and

Stephen Quirke (eds.), The UCL Lahun Pa­

pyri: Religious, Literary, Legal, Mathematical

and Medical, Oxford: Arcaheopress 2004.

3. Among the early editions, the most note­

worthy are still Thomas E. Peel, The Rhind

Mathematical Papyrus. British Museum

10057 and 10058, London: Hodder and

Stoughton 1 923, and Wasili W. Struve,

Mathematischer Papyrus des Staatlichen

Museums der Sch6nen Kunste in Moskau

(Quellen und Studien zur Geschichte der

Mathematik, Abteilung A: Quellen, Vol. 1 ) ,

Heidelberg: Springer 1 930.

4. Otto Neugebauer, Die Grundlagen der

agyptischen Bruchrechnung, Berl in: Julius

Springer 1 926.

5 . Otto Neugebauer, A History of Ancient

Mathematical Astronomy (Part Two). Berlin,

Heidelberg, New York: Springer 1 975: 559.

6. See for example the interpretations of

Plimpton 322, e.g. , compare Joran Friberg ,

"Methods and traditions of Babylonian

mathematics: Plimpton 322 , Pythagorean

triples and the Babylonian triangle param­

eter equations," Historia Mathematica 8

(1 98 1 ): 277-318 and the recent reassess­

ment by Eleanor Robson (Eleanor Robson,

"Neither Sherlock Holmes nor Babylon : a

reassessment of Plimpton 322 , " Historia

Mathematica 28 (200 1 ) ; 1 67-206 and

Eleanor Robson, "Words and pictures: new

light on Plimpton 322," American Mathe­

matical Monthly 1 09 (2002): 1 05-1 20).

7. An extreme example of this is Richard

Gill ings, "The Volume of a Truncated Pyra­

mid in Ancient Egyptian Papyri , " The Math­

ematics Teacher 57 (1964): 552-555.

8. For Egyptian mathematics, see for exam­

ple James Ritter, "Chacun sa verite: les

mathematiques en Egypte et en Me­

sopotamie , " in : Michel Serres (ed.) , Ele-

ments d'histoire des sciences: 39-61 ,

Paris: Bordas 1 989: James Ritter, "Egyp­

tian Mathematics," in: Helaine Selin (ed.),

Mathematics Across Cultures: The History

of Non-Western Mathematics : 1 1 5- 1 36,

Dordrecht, Boston, London: Kluwer 2000,

as well as Annette lmhausen, Agyptische

Algorithmen: Eine Untersuchung zu den

mittelagyptischen mathematischen Auf­

gabentexten, Wiesbaden: Otto Harras­

sowitz 2003. For Greek Mathematics, cf.

Serafina Cuomo, Ancient Mathematics,

London, New York: Routledge 2001 ,

Michael N. Fried and Sabetai Unguru,

Apollonius of Perga 's Conica. Text, Con­

text, Subtext, Leiden: Brill 2001 , as well as

David Fowler, The Mathematics of Plato's

Academy: A New Reconstruction (Second

Edition), Oxford: Clarendon Press 1 999,

and Reviel Netz, The Shaping of Deduc­

tion in Greek Mathematics: A Study of

Cognitive History (Ideas in Context 5 1 ) , Cambridge: Cambridge University Press

1999. For Mesopotamian mathematics,

see most recently Jens H0yrup, Lengths,

Widths, Surfaces. A Portrait of Old Baby­

lonian Algebra and its Kin, New York:

Springer 2002, and Eleanor Robson,

Mesopotamian Mathematics, 2 1 00-1 600

BC: Technical Constants in Bureaucracy

and Education (Oxford Editions of

Cuneiform Texts XIV), Oxford: Clarendon

Press 1 999.

9 . See Gary Urton, The Social Life of Numbers.

A Quechua Ontology of Numbers and Phi­

losophy of Arithmetic, Austin , Texas: Uni­

versity of Texas Press 1 997, and Marcia As­

cher, Mathematics Elsewhere. An

Exploration of Ideas across Cultures, Prince­

ton, N.J . : Princeton University Press 2002.

1 0. See, for example, for Mesopotamia Jens

H0yrup, Lengths, Widths, Surfaces. A Por­

trait of Old Babylonian Algebra and its Kin,

New York: Springer 2002.

1 1 See note 8.

1 2 . See Annette lmhausen and Jim Ritter,

"Mathematical Papyri , " in: Mark Collier and

Stephen Quirke (eds ) , The UCL Lahun Pa­

pyri: Religious, Literary, Legal, Mathematical

and Medical, Oxford: Arcaheopress 2004.

Another mathematical fragment will be pub­

lished in the next volume of that series.

1 3 . See Oleg Berlev, "Review of William Kelly

Simpson: Papyrus Reisner I l l : The Records

of a Building Project in the Early Twelfth

Dynasty, Boston : Museum of Fine Arts

1 969," Bibliotheca Orienta/is 28 (1 97 1 ):

324-326, esp. p. 325.

1 4 . Richard Parker, "A Demotic Mathematical

Papyrus Fragment," Journal of Near East­

ern Studies 1 8 (1 959): 275-279; Richard

Parker, Demotic Mathematical Papyri,

Providence, R . I . : Brown University Press

1 972; Richard Parker, "A Mathematical Ex­

ercise-P. Dem. Heidelberg 663 , " Journal

of Egyptian Archaeology 61 (1 975):

1 89-1 96. A list of Demotic mathematical

ostraca can be found in Jim R itter, "Egypt­

ian Mathematics , " in: Helaine Selin (ed.),

Mathematics across Cultures. The History

of Non-Western Mathematics , Dordrecht:

Kluwer 2000: 1 34, note 27.

1 5 . See Gunter Dreyer, Umm ei-Qaab I . Das

pradynastische Konigsgrab U-j und seine

fruhen Schriftzeugnisse, Mainz: Von Zabern

1 998

1 6 . For a discussion of the inscriptions on these

tags, see Gunter Dreyer, Umm ei-Qaab I.

Das pradynastische Konigsgrab U-j und

seine fruhen Schriftzeugnisse, Mainz: Von

Zabern 1 998, pp. 1 37-145, and John

Baines, "The Earliest Egyptian Writing: De­

velopment, Context, Purpose," in: Stephen

D. Houston, The First Writing. Script Inven­

tion as History and Process, Cambridge:

Cambridge University Press 2004: 1 50-1 89.

1 7 . See problems 56-60 of the Rhind Mathe­

matical Papyrus.

1 8. Jim Ritter, "Mathematics in Egypt, " in :

Helaine Selin (ed.), Encyclopedia of the

History of Science, Technology and Med­

icine in Non-Western Cultures, Dordrecht,

Boston, London: Kluwer 1 997, p. 631 .

1 9. For the prehistory of Egyptian fractions and

their development see Jim Ritter, "Metrol­

ogy and the Prehistory of Fractions," in:

Paul Benoit, Karine Chernla, Jim Ritter

(eds.) , Histoire de fractions, fractions d'his­

toire: 1 9-34, Basel, Boston, Berlin:

Birkhauser 1 992.

20. See, for example, the description of Cou­

choud: " . . . i l ne semble avoir connu que

les fractions unitaires, c'est a dire celles

dans lesquelles le numerateur est toujours

equivalent a ! 'unite, . . . " (Sylvia Couchoud,

Mathematiques Egyptiennes. Recherches

sur les connaissances mathematiques de

I 'Egypte pharaonique, Paris: Le Leopard

d'Or 1 993, p. 2 1 ) or that of Gill ings: "When

the Egyptian scribe needed to compute

with fractions he was confronted with

many difficulties arising from the restriction

of his notation. His method of writing num­

bers did not allow him to write such sim­

ple fractions as % or % because all fractions

had to have unity for their numerators (with

one exception)." (Richard J. Gill ings, Math­

ematics in the Time of the Pharaohs, Cam­

bridge, Mass . : MIT Press 1 972, p. 20).

2 1 . See Jim Ritter, "Egyptian Mathematics , " in:

Helaine Sel in (ed.), Mathematics across

Cultures. The History of Non-Western

Mathematics, Dordrecht, Boston, London:

Kluwer 2000, p. 1 20.

22. For a discussion of the use of an abstract

number system and conversions into

metrological systems, see Jim Ritter,

"Egyptian Mathematics," in : Helaine Selin

(ed.), Mathematics across Cultures. The

History of Non-Western Mathematics,

Dordrecht, Boston, London: Kluwer 2000,

pp, 1 2 1 - 1 22.

23. The cubit was the Egyptian standard mea­

sure of length. 1 cubit consisted of 7

palms; each palm, of 4 digits.

24. An English example for its use would be

the statement "If I have a stone in my hand,

and let it drop, then the stone falls to the

ground." The last part of this statement

"then the stone falls to the ground" is

where the sqm.l)r==f is used in an Egypt­

ian text.

25. Examples can be found in Annette

lrnhausen and Jim Ritter, "Mathematical

Fragments: UC321 1 4, UC321 1 8, UC321 34,

UC321 59-UC32 1 62, " in : Mark Collier and

Stephen Quirke, The UCL Lahun Papyri:

Religious, Literary, Legal, Mathematical

and Medical (British Archaeological Re­

ports International Series 1 209): 7 1 -96,

Oxford; Archaeopress 2004, esp. pp. 85-86.

26. The New Kingdom Ostracon Senmut 1 53

may be interpreted as a table of + in this

way, see David Fowler, The Mathematics

of Plato's Academy: A New Reconstruc­

tion (Second Edition), Oxford: Clarendon

Press 1 999, p. 269.

27. Jim Ritter, "Chacun sa verite: les rnathe­

rnathiques en Egypte et en Mesopotamie,"

in : Michel Serres (ed.), Elements d'histoire

des sciences: 39-61 , Paris: Bordas 1 989

(English edition: Jim Ritter, "Measure for

Measure: Mathematics in Egypt and

Mesopotamia," in: Michel Serres (ed.) , A

History of Scientific Thought. Elements of

a History of Science: 44-72, Oxford:

Blackwell 1 995).

28. Annette lrnhausen, Agyptische Algorith­

men. Eine Untersuchung zu den mittel­

agyptischen mathematischen Aufgaben­

texten (Agyptologische Abhandlungen 65).

Wiesbaden: Otto Harrassowitz 2003.

29. For a detailed discussion of these prob­

lems see Annette lmhausen, "Egyptian

Mathematical Texts and their Contexts, "

Science in Context 1 6, 2003: 367-389.

30. Michel Guillemot, "Les notations et les pra­

tiques operatoires permettent-elles de par­

ler de 'fractions egyptiennes'?", in: Paul

Benoit, Karina Chemla, Jim Ritter (eds.), His­

loire de fractions, fractions d'histoire, Basel,

Boston, Berlin: Birkhauser 1 992: 53-69.

31 . Annette lmhausen, "Calculating the Daily

Bread: Rations in Theory and Practice,"

Historia Mathernatica 30 (2003): 3-1 6.

32. A first attempt to analyze the mathemati­

cal content of some parts of the Reisner

Papyri has been made by Richard J .

Gil l ings, Mathematics in the Time of the

Pharaohs, Cambridge, Mass. MIT Press

1 972, pp. 2 1 8-231 .

33. A variety of architectural sources (with

mathematical implications) can be found in

Corinna Rossi, Architecture and Mathe­

matics in Ancient Egypt, Cambridge: Cam­

bridge University Press 2004.

AUTHOR

ANNETTE I MHAUSEN

Trinity Hall

Cambridge University

Trinity Lane, Cambridge CB2 1 T J UK

e-mail: ai226@com.ac.uk

Annette lmhausen studied mathe­

matics, history of mathematics, and

Egyptology and received her PhD

from Mainz University (Germany) un­

der David E. Rowe. She has held fel­

lowships at the Dibner Institute for the

History of Science and Technology

(Cambridge, Mass.) and at Trinity Hall

(Cambridge University, England). She

is currently completing a book that

outlines the historical development of

ancient Egyptian mathematics, situ­

ating it within the larger social, eco­

nomic, and cultural background. In

her spare time she enjoys running 1 /2

and 2/3 marathons.

© 2006 Springer Science t Business Media. Inc., Volume 28. Number 1, 2006 27

M?tffii•i§u6hl%1i@i§4fii.J .. t§.id M i c hael Kleber and Ravi Vaki l , Ed itors

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 94305-21 25, USA

e-mail: vakil@math.stanford .edu

The Locker Puzzle Eugene Curtin and Max Warshauer

Suppose I take the wallets from you and ninety-nine of your closest

friends. We play the following game with them: I randomly place the wal­lets inside one hundred lockers in a locker room, one wallet in each locker, and then I let you and your friends in­side, one at a time. Each of you is al­lowed to open and look inside of up to fifty of the lockers. You may inspect the wallets you find there, even check­ing the driver's license to see whose it is, in an attempt to find your wallet. Whether you succeed or not, you leave all hundred wallets exactly where you found them, and leave all hundred lockers closed, just as they were when you entered the room. You exit through a different door, and never communi­cate in any way with the other people waiting to enter the room. Your team of 100 players wins only if every team

member finds his or her own wallet. If you discuss your strategy beforehand, can you win with a probability that isn't vanishingly small?

We develop a more mathematical formulation to facilitate a precise dis­cussion of the problem. This consists of numbering our players, and replac­ing wallets by player numbers! Our game is played between a single Player A against a Team B with 100 members, B1, B2, . . . , B100. Player A places the numbers 1, 2, . . . , 100 randomly in lock­ers 1, 2, . . . , 100 with one number per locker. The members of Team B are ad­mitted to the locker room one at a time. Each team member is allowed to open and examine the contents of exactly 50 lockers. Team B wins if every team member discovers the locker contain­ing his own number. Team B is allowed

28 THE MATHEMATICAL INTELLIGENCER © 2006 Spnnger Sc1ence+Bus1ness Med1a. Inc.

an initial strategy meeting. No com­munication is allowed after the initial meeting, and each team member must leave the locker room exactly as he found it. It is important to realize that the solution does not involve some trick to pass information from one player to another. We could equally well make 100 copies of the room and make an identical distribution of num­bers into lockers for each room, then ask the members of Team B to perform their searches simultaneously, with one person per room.

Each individual will succeed in find­ing his own number with probability 1/2. If they act independently, they must get lucky 100 times in a row, and the team will win with probability only e/z)100. Team B needs some help! Amaz­ingly there is a strategy which gives sig­nificant probability of success for Team B. Even if we give the problem with 2n players on Team B each of whom can examine n out of 2n lock­ers, Team B can apply the strategy to succeed with probability over 30% regardless of how large a value we take for n. Your problem is to find this strategy.

Searching For Ideas

Let's play with some ideas using a more manageable number of players. To be as concrete as possible, let's switch to the case of 10 players on Team B, each of whom can examine 5 out of 10 lock­ers. Here random guessing by each player is already somewhat hopeless and succeeds with probability (l/z)10 = 1�24 . A first try to improve the probability of success is to search for a clever way to assign a set of lockers for each person to examine. Certainly we can improve over random guessing in this manner. For example if team members 1-5 ex­amined lockers 1-5, and team mem­bers 6-10 examined lockers 6-10, they would succeed provided numbers 1-5 are placed in lockers 1-5. Number 1 is placed somewhere in the first 5 lock­ers with probability 5/10, then given

that number 1 is so placed, number 2 is also in the first 5 with probability 4/9 and so on. Following this plan, Team B will succeed with probability

5 4 3 2 1 _ 1 10 9 8 7 6 - 242 "

While this is an improvement over ran­dom guessing, it still leaves Team B with slim chances. Although the scheme fails, it is worth noticing that if B1 finds his number in this scheme, then B6 will find his number with prob­ability 5/9 (as he will look in 5 lockers not including the one containing the number 1 ), but B2 will find his with probability only 4/9. The success or failure of B1 can influence the proba­bilities of success of the other mem­bers. This is the first clue!

An ideal strategy would be one where if B1 succeeds then everyone else does too. Note that this would al­low the whole team to succeed half the time even though each individual member fails half the time. This ideal is not attainable, but perhaps you can find a strategy where if B1 succeeds, then everyone else is more likely to succeed. No method of preassigning lockers will accomplish this, as if B1 finds his number in locker k anyone with locker k in their preassigned set has his chances reduced. This suggests that the locker choices will have to de­pend on information not available at the initial meeting. The only such in­formation available is the numbers a player finds inside the lockers he opens. With this further hint try one more time to find a good strategy be­fore we proceed to the solution!

Developing the Solution

Once we realize that the locker B1 opens at any stage can depend on what he has found inside the lockers he has already opened, the number of possi­ble strategies to consider is enormous, even in the 10-player case. The strategy must tell B1 which locker to open first (10 choices), which locker to open next if he is not lucky on the first try (9 choices for each of the possible 9 numbers he may see), which to open third if he is not lucky on his second attempt either (8 choices for each of the 9 X 8 possible sequences of 2 numbers he has seen so far), and so

on. So B1 alone has 10 X 99 X 89X8 X 79xsx7 x 69xsx7x6 possible strategies. To compute the number of strategy choices for the whole team, we raise this to the lOth power and get a num­ber 28,537 digits long! How are we to choose one?

In this section we will show that one very simple strategy lets the team win with remarkably high probability. The strategy for any one player is entirely un­remarkable; the magic arises from the fact that the chances of the different players winning are highly correlated. Moreover, in the next section, we will show that the strategy is in fact optimal.

Fortunately the good strategy is simple to implement and the choice of the next locker does not depend on the entire sequence of numbers seen but only on the most recent number. The good strategy has player Bi start by opening locker i. Then if he finds num­ber k at any stage and k =I= i, he opens locker k next. Notice that player Bi, never opens a locker (other than locker i) without first finding its number, so each time he opens a new locker he must find either his own number or the number of another unopened locker.

Again let's look at a particular case with 10 players and suppose, for exam­ple, that the numbers are distributed in the order 6,8,9,7,2,4, 1,5, 10,3. Player B1 first examines locker 1 and finds the number 6. So he looks in locker 6 fmd­ing the number 4, then locker 4 finding the number 7, then finally in locker 7 finding his number. When he finds his number, B1 will now know that B6, B4, and B7 will look in exactly the same lockers in the same cyclic order, each finding his number on the 4th try! He also knows that none of the other play­ers will waste any tries on these lockers.

We may represent any permutation of numbers into lockers by listing the cycles. The permutation 6,8,9, 7,2,4, 1 ,5, 10,3 gives the cycles (6, 4, 7, 1) (8, 5, 2) (9, 10, 3), and Team B succeeds be­cause there is no long cycle. To find the probability that Team B wins, we count the number of permutations of 10 num­bers with a cycle of length 6 or longer. First let's count how many have a 6-cy­cle. Choose which 6 elements go into the 6-cycle, arrange them in cyclic or­der, and then pick an arbitrary permu-

tation of the remaining 4 elements. The number of ways to do this is (10) ' ' -

10!5!4! -

10! 6

5.4. - 6!4! - 6

So 116 of the 10! permutations have a 6-cycle, and a random permutation has a 6-cycle with probability 1/6. The same argument can be used to find the probability of a permutation of 1-10 having a cycle of any length longer than 6. (We warn that the argument does not work for counting the number of per­mutations of 1-10 with a 5-cycle (or shorter) as the permutation could have two 5-cycles.) A permutation of 10 numbers has a 7-cycle with probability 117 and so on, and the probability of a cycle oflength 6 or larger is 1/6 + 117 + 1/8 + 1 /9 + 1 / 1 0 = 1 62 7/2 5 2 0 =

0.645635. This gives the probability that Team B will fail, so of course Team B wins with probability 1 - 1627/ 2520 = 893/2520 = 0.354365. Over 35% of the time, all 10 members of Team B find their own wallets!

Will this idea be good enough for the initial version with 100 players? We can do the analogous computation and see that this pointer-following strategy works with probability 1 - ( 1151 + 1152 + . . . + 1/100) = .31 1828.

Notice that while our strategy has still performed remarkably well for 100 players, the probability of success was still less than in the 10-player version. As we increase the number of players, does the success rate decrease to zero, or does it always stay above a certain positive number? With 2n players and 2n lockers, Team B will win provided that the permutation of numbers in lockers has no cycle of length n + 1 or longer. The probability of such a long cycle is IJ:�1 -1-. By viewing this ex-

n+k pression as an upper Riemann sum for

f2n -1- dx and a lower Riemann sum n x + 1 for f2n l dx we obtain n X ( 1 ) l2n 1 ln 2 - n + 1 = n x + 1

dx

n 1 12n 1 s L -- s - dx = ln 2.

k � l n + k n X So II:� 1 -1- ---+ ln 2 as n ---+ oo; moreover n+k the sum increases monotonically with n. So the expression 1 - II:� 1 -1- giv­n+k ing the probability of success is

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 2006 29

monotonically decreasing to 1 - ln 2 = 0.306853. Team B wins with the pointer­following strategy with probability ex­ceeding 30%, regardless of the number of players and lockers. Now that we have found a good strategy, we turn our attention to whether it provides the best possible solution.

Is Pointer-Following Optimal?

We establish the optimality of pointer­following by comparing the game con­sidered above (Game 1) with a new game (Game 2) between the same ad­versaries, Player A and Team B. For simplicity we give the argument in terms of the 10-player versions. Recall that in Game 1 we are allowing each player to examine 5 lockers. We first modify this rule and say that each player must continue examining lockers until he has opened the locker containing his number, and then he is not allowed to open any further lockers. Team B wins if no player opens more than 5 lockers. This change makes no difference to who wins in Game 1, but it will clarifY the comparison with Game 2.

In Game 2, Player A again distrib­utes the 10 numbers at random in the 10 lockers. Then all of team B is invited into the locker room together. Team member B1 is required by the rules to start opening lockers and continue un­til she reveals the number 1 . Once she has opened the locker containing the number 1 , she may not open any fur­ther lockers; then, the lowest-num­bered member of Team B whose num­ber has not yet been revealed is required to take over opening lockers until he finds his number and so on. Team B continues until all lockers are opened. Again Team B wins if no indi­vidual team member opens more than 5 lockers. Before proceeding, we invite you to consider the following ques­tions: With what probability can Team B win Game 2? What strategy should the team members employ? Does their choice of strategy even matter?

Let's sit in the locker room and ob­serve Team B in the process of playing Game 2. We record the progress, list­ing the numbers in the order in which they are revealed. Our list of numbers is sufficient to determine how many lockers were opened by each player.

30 THE MATHEMATICAL INTELLIGENCER

For example, if we record the list 2,6, 1 , 4,9,7, 10,8,3,5, we know that player B1 revealed the numbers 2, 6, and 1. Then player B3 was required to take over, and he opened the lockers containing the numbers 4, 9, 7, 10, 8, and 3, in that order. Then player BR opened the re­maining locker containing the number 5. In this example Team B lost, as player B3 opened 6 lockers. Notice that we will record any given ordering of the numbers 1-10 with probability 1110!. The first number revealed is 2 with probability 1110, no matter which locker is opened, given that the first is 2 the second will be 6 with probability 119, and so on. What strategy is Team B following here? It makes absolutely no difference! Team B can choose lock­ers at random or follow the most so­phisticated plan; we still get probabil­ity 1/10! for each of the 10! possible orders in which the numbers could be revealed. In Game 2 Team B's proba­bility of success is completely inde­pendent of strategy.

To find the probability that Team B wins, we must count how many of the 10! possible orders of the numbers 1-10 represent wins. We employ a ver­sion of the classical records-to-cycles bijection [6, p17] to assign a permuta­tion written in cycle notation to each ordering. The first cycle of our permu­tation consists of the numbers opened by B1 in order; the second cycle, the numbers opened by the second locker opener; and so on. So, for example, 2,6, 1 ,4,9, 7, 10,8,3,5 ----> (2,6, 1)( 4,9, 7, 10,8,3) (5). Furthermore we see that each per­mutation arises in this manner from a unique ordering of the numbers 1-10. We first write the permutation in cycle notation, rotate each cycle so that the lowest number in the cycle is written last, and then order the cycles so that their last numbers are in ascend­ing order. For example (9, 7,8)(1,3, 10,5) (2 ,4 ,6) = (3, 10 ,5 , 1 ) (4 ,6,2)(8 ,9 , 7) ----> 3, 10,5, 1,4,6,2,8,9,7. We have established a one-to-one correspondence between lists for which Team B wins and the permutations of 1-10 with no cycles of length greater than 5. Thus the proba­bility that Team B wins Game 2 is the probability that a random permutation of 1-10 has no cycle of length greater than 5, and we have already computed

this as 893/2520 = 0.354365. This is ex­actly the probability of success for Team B in Game 1 using pointer fol­lowing!

Our analysis has a significant con­sequence for Game 1. Team B can take any Game 1 strategy and adapt it to Game 2 as follows: If player Bi is open­ing lockers in Game 2, he can use his Game 1 strategy for choosing lockers to open, simply observing the contents without wasting a turn if the indicated locker is already open. Thus if a strat­egy succeeds in Game 1 for a particu­lar distribution of numbers into lock­ers it will also succeed in Game 2. If there were a better strategy for Game 1 we could apply it in Game 2 and get a better chance to win this game also. But this is impossible, as all strategies for Game 2 lead to the same probabil­ity of success.

We have one final small puzzle: Happy with their optimal strategy for Game 1 , Team B began a sequence of matches with Player A, but they soon found themselves down 10 to 0. What do you suspect Player A is doing? (It seems that Player A subscribes to the Intelligencer and has devised a plan to defeat Team B.) What can Team B do to counter Player A's plan?

History of The Locker Puzzle

Our problem was initially considered by Peter Bro Miltersen, and it appeared in his paper [ 4] with Anna Gil, which won a best paper award at the ICALP conference in 2003. Miltersen says of the problem, "I think it started spread­ing when I presented it to several peo­ple at Complexity 2003, which was held in Aarhus, where I was a local orga­nizer." In their version there is one numbered slip of paper for each player on the team. Player A then colors each slip either red or blue. Each member of Team B may examine up to half the lockers. He is then required to state or guess the color of the slip of paper with his number. Again every team member must state or guess his color correctly for the team to win. Initially Miltersen expected that Team B's probability of success would approach zero rapidly as the number of players increased. However, Sven Skyum, a colleague of Miltersen's at the University of Aarhus,

brought his attention to the beautiful pointer-following strategy. Finding this is left as an exercise in the paper.

Miltersen and G:il originally consid­ered the case where there are n team members and 2n lockers, half of them empty; each team member still gets to open up to half of the lockers. This is a more difficult problem. Clearly sim­ple pointer-following will not work as empty lockers do not point anywhere. It is an open question whether the win­ning probability must tend to zero for large n.

In [5] Navin Goyal and Michael Saks build on Skyum's pointer-following to devise a strategy for Team B in a more general setting, varying both the pro­portion of empty lockers and the frac­tion of lockers each team member may open. As the number of players in­creases, their probability of success for Team B approaches zero less rapidly than conjectured in [4]. And fixing the number of players and fraction of lock­ers each may open, their probability of

winning remains nonzero even as more empty lockers are added.

The problem also appeared in Joe Buhler and Elwyn Berlekamp's puzzle column in the Spring, 2004 issue of The Emissary [3] , with lockers replaced by ROM locations and colored numbers re­placed by signed numbers. Here it is pointed out that the team benefits from the members carefully planning their guessing strategy as well as their locker searching strategy. For example, if there are 2n lockers and the longest cy­cle has length n + 1 , the team members caught in the n + 1 cycle can guess in such a manner that they all guess cor­rectly or all guess incorrectly. The trick is the same as that employed in the hat problem of Todd Ebert [2]. Variations of the hat problem are described in Joe Buhler's article in this column [ 1 ] and in Peter Winkler's book [7, p66, p120]. The locker problem will be discussed in a fu­ture edition of Winkler's book also.

We thank Joel Spencer for intro­ducing us to the problem, and we thank

Ravi Vakil and Michael Kleber for en­couraging us to write this note and pro­viding many useful suggestions.

REFERENCES

[ 1 } Joe Buhler, Hat tricks. Math. lntelligencer

24 (2002), no. 4, 44-49.

[2} Todd Ebert, http://www.cecs.csulb.edu/

�ebert/

[3} The Emissary, http://www.msri .org/publi­

cations/emissary/

[4] Anna Gal and Peter Bro Miltersen. The Cell

Probe Complexity of Succinct Data Struc­

tures, Proceedings of 30th International

Colloquium on Automata, Languages and

Programming (ICALP) 2003, 332-344.

[5} Navin Goyal and Michael Saks, A Parallel

Search Game, to appear in Random Struc­

tures and Algorithms.

[6} Richard Stanley, Enumerative Combina­

torics Vol 1 . Cambridge Studies in Ad­

vanced Mathematics, 49, Cambridge Uni­

versity Press, Cambridge, 1 999

[7} Peter Winkler, Mathematical Puzzles: A

Connoisseur's Collection, A K Peters, Ltd . ,

Natick, MA, 2004.

MAX WARSHAUER Department of Mathematics

Texas State Un iversity-San Marcos

San Marcos, TX 78666-4945 e-mai l : mw07@academia.swt.edu

EUGENE CURTIN

Department of Mathematics

Texas State University-8an Marcos

San Marcos, TX 78666-4945 e-mail: ec01 @txstate.edu

Max Warshauer received his Ph.D. at LSU under Pierre Conner

and is now Professor in the Mathematics Department at Texas

State, where he founded and directs Mathworks" a center for Math­

ematics and Mathematics Education, one of five programs in Texas

to receive the 2001 Texas Higher Education Star Award for Clos­

ing the Gaps. Max was one of 10 individuals in the country to re­

ceive the 2001 Presidential Award for Excellence in Science, Math­

ematics, and Engineering Mentoring. His hobbies include playing

chess, go, Ping-Pong , and bicycling. His research interests have

included quadratic forms, analysis of algorithms, and mathematics

education.

Eugene Curtin, a native of Ireland , received a Ph .D. from Brown

University in 1 988 under Thomas F. Banchoff. He is current ly a

professor at Texas State University. His research interests have in­

cluded differential geometry, r ing theory, combinatorics , and graph

theory. Eugene has been a faculty member with the Honors Sum­

mer Math Camp under Max Warshauer's directorship every sum­

mer since 1 992. He enjoys mathematical games and puzzles, and

likes to play chess, go, and backgammon. He was chess cham­

pion of Ireland in 1 984 and 1 985, and chess champ ion of Texas

in 1 99 1 , 1 992, and 1 998

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 2006 31

RAJENDRA BHATIA AND JOHN HOLBROOK

Noncom m utat ive

Geometric Means

For, in fact, what is man in nature? A Nothing in comparison with the Infinite, an All in comparison with the Nothing, a mean between nothing and everything.

-Blaise Pascal

A veraging operations entered mathematics rather early. .rt.Fascinated as they were by geometric proportions, the ancient Greeks defined as many as eleven different means. The arithmetic, geometric, and harmonic means are the three best-known ones. If Pascal had one of these in mind when he composed his Pensees [P] , he would soon have realised that mixing zero and infinity is a source of as many problems as mixing mathematics and divinity.

For centuries, mathematicians perfom1ed their opera­tions either on numbers or on geometrical figures. Then in 1855 Arthur Cayley introduced new objects called ma tri­ces, and soon afterwards he gave the laws of their algebra. Seventy years later, W emer Heisenberg found that the non­commutativity of matrix multiplication offers just the right conceptual framework for describing the laws of atomic mechanics. Matrices were found to be useful in the de­scription of classical vibrating systems and electrical net­works as well. For mathematicians, analysis of linear op­erators was a subject of intense study throughout the twentieth century and into the twenty-first century.

Many quantities of basic interest such as states of quan­tum mechanical systems and impedances of electrical net­works are defined in terms of matrices. Mixing of the un­derlying systems in various ways leads to corresponding operations on the matrices representing the systems. Not surprisingly, some of these are averaging operations or means.

Of the three most familiar means, the geometric mean combines the operations of multiplication and square roots. When we replace positive numbers by positive definite ma­trices, both of these operations involve new subtleties. In this article we introduce the reader to some of them.

0 0 0

32 THE MATHEMATICAL INTELLIGENCt:R © 2006 Spnnger Sc1ence t- Bus1ness Media, Inc

Let IR+ be the set of all positive real numbers. Given a and b in IR + a mean m(a,b) could be defined in different ways. It is reasonable to expect that the binary operation m on IR+ has the following properties:

(i) m(a,b) = m(b,a).

(ii) min(a,b) :S m(a,b) :S max(a,b).

(iii) m( aa,ab) = am( a,b) for all a > 0. (iv) m(a,b) is an increasing function of a and b. (v) m(a,b) is a continuous function of a and b.

The three familiar means, arithmetic, geometric, and har­monic, satisfy all these requirements. Other examples of means include the binomial means, also called the power means, defined as ( aP + bJ! )lip

mp(a,b) = 2 , - oc :S p :s; x.

Here, it is understood that for the special values p = 0 and ±: oc we define mp(a,b) as the limits

mo(a,b) = limp __, omp(a,b) = v;;:b, mx( a,b) = limp --> xmp( a,b) = max( a,b ) ,

m-x(a,b) = limp __, -xmp(a,b) = min(a,b) .

The arithmetic and the harmonic means correspond to the cases p = 1 and - 1 , respectively. Inequalities between means have been studied for a long time. See the classic [HLP] , and the more recent [BMV]. A sample result here is that for fixed a and b, mp(a,b) is an increasing function of p. This includes, as a special case, the inequality between the three familiar means.

There exists a fairly well-developed theory of means for positive definite matrices. Let MnCO be the set of all n X

n complex matrices, §n the collection of all self-adjoint el­ements of Mn(IC), and iJ=Dn that of all positive definite ma­trices. The space §n is a real vector space and iJ=D n is an

open cone within it. This gives rise to a natural order on

§n. We say that A 2: B if A - B is positive definite or pos­

itive semidefinite. Two elements of §n are not always com­

parable in this order. Every element X of GLn (the group

of invertible matrices) has a natural action on iJ=D n· This is

given by the map r x(A) = X* AX. We say that A and B are

congruent if B = r x(A) for some X E GLn. In the special

case when X is unitary, we say that A and B are unitarily

equivalent. The group of unitary matrices is denoted by QJn· Now we have enough structure to lay down conditions

that a mean M(A,B) of two positive definite matrices A and

B should satisfy. Imitate the properties (i)-(v) for means

of numbers. This suggests the following natural conditions:

(I) M(A,B) = M(B,A).

(II) If A :::::: B, then A :::::: M(A,B) :::::: B.

(III) M(X*AX,X*BX) = X*M(A,B)X, for all X E GLn. (IV) M(A,B) is an increasing function of A and B; i.e., if

A1 2: A2 and B1 2: B2, then M(A1,B1) 2: M(A2,B2).

(V) M(A,B) is a continuous function of A and B.

The monotonicity condition (IV) is a source of many in­

triguing problems in constructing matrix means. This is be­

cause the order A 2: B is somewhat subtle. For example, if

A = [� �1 and

then A 2: B but A2 ;t B2.

What functions of positive numbers, when lifted to pos­

itive definite matrices, preserve order? This is the subject

of an elegant and richly applicable theory developed by

Charles Loewner. Letf be a real-valued function on IR+. If

A is a positive definite matrix and A = 'i.Aiuiui is its spec­

tral resolution, thenf(A) is the self-adjoint matrix defined as j{A) = 'i.J{Ai)uiui. We say that j is a matrix monotone junction if for all n = 1 , 2, . . . , the inequality A 2: B in IJ=Dn implies j{A) 2:j{B). One of the theorems of Loewner says

thatfis matrix monotone if and only if it has an analytic con­

tinuation to a mapping of the upper half-plane into itself. As a consequence, the functionj{x) = xP is matrix monotone if

and only if 0 :::::: p :::::: 1. The function f(x) = log x is matrix

monotone, but j{x) = exp x is not. We refer the reader to

Chapter V of [B] for an exposition of Loewner's theory.

Returning to means, the arithmetic and the harmonic

means of A and B are defined, in the obvious way, as

teA + B) and [ tCA - 1 + B- 1) ] - 1, respectively. It is easy to

see that they satisfy the conditions (I)-(V) above.

The notion of geometric mean in this context is more

elusive, even treacherous. Every positive definite matrix A

has a unique positive definite square root A 112. However, if

A and B are positive definite, then unless A and B com­

mute, the product A 112 B112 is not self-adjoint, let alone pos­

itive definite. This rules out using A112Bl12 as our geomet­

ric mean of A and B, except in the trivial case when AB =

EA. We should look for other good expressions in A and B

that reduce to A 112B112 when A and B commute. One plau­

sible choice is the quantity

(1) ( log A + log B ) _ . ( AP + BP )lip exp 2 - hmp __. o 2 .

The equality of the two sides of (1) was noted by Bhag­

wat and Subramanian [BS], who studied in detail the

"power means" occurring on the right-hand side. This

too is not monotone in A and B, as can be seen by choos­

ing positive defmite matrices X and Y, for which X 2: Y but exp X ;t exp Y, and then choosing A and B such that

X = t (log A + log B) and Y = t log B.

The condition (III), sometimes called the transformer

equation, is not innocuous either. Our failed candidates fail

on this count too.

The noncommutative analogue of v;;J; with all desirable

properties turns out to be the expression

(2) A#B = A 112 (A - 1/2 BA - 112)v2 Avz,

that was introduced by Pusz and Woronowicz [PW] in 1975. At the outset it does not appear to be symmetric in A and

B; but it is, as we will soon see. The monotonicity in B is

assured by the facts that congruence preserves order (B1 2:

B2 implies X*B1X 2: X*B,y() and the square root function is

matrix monotone.

Symmetry in A and B is apparent more easily from an al­

ternative characterisation ofA#B due to T. Ando [A]. We have

(3) A#B = max{x : X = X* and [� �1 2: o}. Among its other characterisations, one describes A#B as

the unique positive definite solution of the Riccati equation

(4) XA- 1X = B.

We call A#B the geometric mean of A and B. It has the de­

sired properties (I)-(V) expected of a mean M(A,B) : prop­erty (III) may be verified easily from (3) or ( 4). It satisfies the expected inequality

( A - 1 + B- 1 ) - 1 A + B (5) 2

:::::: A#B :::::: -2- ,

and has other pleasing properties. Many of these were de­

rived by Ando [A].

Two positive definite matrices A and B can be diago­

nalised simultaneously by a unitary conjugation r u if and

only if they commute. In the absence of commutativity, A

and B can be diagonalised simultaneously by a congruence

in two steps:

(A,B) rA-"' (I ,A - 112 BA - 112) � (I,D),

where U is a unitary such that U* (A - 112BA - 112) U is a di­

agonal matrix D. This takes some of the mystery out of the formula (2). In fact, any mean m(a,b) of positive numbers

leads to a mean M(A,B) of positive definite matrices by the

procedure M(A,B) = r AJt2(m(I,D)). To ensure that M is an

increasing function of A and B, we have to assume that the

functionf(x) = m(l,x) is matrix monotone. The formula (2) corresponds to the case when m(a,b) = (ab)112.

© 2006 Spnnger Science+ Business Media, Inc., Volume 28, Number 1, 2006 33

The indirect argument we have used to deduce the sym­metry of the geometric mean is not necessary. Let m( a,b) be any mean, letf(x) = m(l,x), and

(6) M(A,B) = A112j (A- 112BA- 112) A112.

Though this expression seems to be asymmetric in A and B, in fact M(A,B) = M(B,A). For this we need to prove

f(A - 112BA - 112) = A - 112B112j (B- 112AB- li2)B112A - 112. Using the polar decomposition A - 112B112 = PU, where P is positive definite and U unitary, this statement reduces to

This, in tum, is equivalent to saying that for every eigen­value A of P, we have

But that is a consequence of properties (i) and (iii) of the mean m. A similar argument verifies (III).

A simple corollary of this construction is the persistence of inequalities like (5) when one passes from positive num­bers to positive definite matrices. Kubo and Ando [KA] de­veloped a general theory of matrix means and established a correspondence between such means and matrix mono­tone functions.

What happens when we have three positive definite ma­trices instead of two? The arithmetic and the harmonic means present no problems. Plainly, they should be defined as �(A + B + C) and [�(A - 1 + B-1 + c-1)] - 1, respectively. The geometric mean, once again, raises interesting problems.

We would like to have a geometric mean G(A,B,C) that reduces to A 113 B113C113 when A, B, and C commute with each other. In addition it should have the following properties.

(a) G(A,B,C) = G(1r(A,B,C)) for any permutation 1r of the triple (A,B,C).

(/3) G(X*AX,X*BX,X*CX) = X*G(A,B,C)X for all X E GLn. ( y) G(A,B,C) is an increasing function of A, B, and C. (8) G(A,B,C) is a continuous function of A, B, and C.

None of the procedures presented above for two matrices ex­tends readily to three. The expressions (2), (3), and ( 4) have no obvious generalisations that work The idea of simulta­neous diagonalisation does not help either: while two posi­tive definite matrices can be diagonalised simultaneously by a congruence, generally three can not be. Defining a suitable geometric mean of three positive definite matrices has been a ticklish problem for many years. Recently some progress has been made in this direction, and we describe it now.

0 0 0

One geometry cannot be more true than another; it can only be more convenient.

-Henri Poincare [Po}

While the geometric mean A#B has been much studied in connection with problems of matrix analysis, mathemati­cal physics, and electrical engineering, a deeper under-

34 THE MATHEMATICAL INTELLIGENCER

standing of it is achieved by linking it with some standard constructions in Riemannian geometry.

The space Mn(C) has a natural inner product (A,B) = tr A*B. The associated norm I IAib = (tr A*A)112 is called the Frobenius, or the Hilbert-Schmidt, norm. If A is a matrix with eigenvalues A 1, . . . , An, we write A (A) for the vector (A 1, . . . , An) or for the diagonal matrix diag(A 1 , . . . , An)·

The set IP n is an open subset of §n and thus is a differ­entiable manifold. The exponential is a bijection from §n onto IP n· The Riemannian metric on the manifold IP n is con­structed as follows. The element of arc length is the dif­ferential

(7) ds = IIA - 112 dA A - 112l lz.

This gives the prescription for computing the length of a differentiable curve in IP n· If y : [ a,b] � IP n is such a curve, then its length, obtained by integrating the formula (7), is

(8) L(y) = r � �y- 112(t)y'(t)y- 112(t)lb dt. a

If A and B are two elements of IP n, then among all curves y joining A and B there is a unique one of minimum length. This is called the geodesic joining A and B. We write this curve as [A ,B], and denote its length, as defined by (8), by the symbol 82(A,B). This gives a metric on 1Pn called the Riemannian metric.

From the invariance of trace under similarities, it is easy to see that for every X in GLn the map r X : IP n � IP n is a bi­jective isometry on the metric space (IP n,82).

An important feature of this metric is the exponential met­

ric increasing property (EMI). This says that the map exp from the metric space (§11,l l · l lz) to (IPnh) increases distances. More precisely, if H and K are Hermitian matrices, then

(9)

To prove this, one uses the formula (8) and an infinitesi­mal version of (9):

(10)

for all H, K E §n· Here Defi (I() is the derivative of the map exp at the point H evaluated at K, i.e.,

efi+tK - elf (11) Defi(K) = limt--.o t

There is a well-known formula due to Daleckii and Krein (see [B], chapter V, for example) giving an expression for this derivative. Choose an orthonormal basis in which H = diag(A b . . . , An). Then [ fl'; - eAi J Defi(K) =

A · _ A · kij . '· J

(The notation here is that [Xij] stands for a matrix with en­tries Xij·) From this, one sees that the (iJ) entry of e-H12Defi(K)e-H12 is

( 12) sinh(Ai - Ai)/2

(A.; - Aj)/2 kij·

Since sinh X 2: 1, the inequality ( 10) follows from this. X

In the special case when H and K commute, a calcula­tion shows that there is equality in (9). In this case the func­tion exp maps the line segment [H,K] in the Euclidean space §n isometrically onto the geodesic segment [ef/,ef<] in !f1> n· If A = eH and B = eK, this says that the geodesic seg­ment joining A and B is the path

y(t) = eC1-t)H+tK = eC1-t)HetK = A1-t Bt, 0 :::; t :::; 1 .

Further, o2(A,y(t)) = to2(A,B) for each t in [0, 1 ] . The case of noncommuting A and B can b e reduced

to the commuting case using the fact that r A-lt2 is an isom­etry on the space (ifl>n,82). The geodesic segment [I ,A - 1!2 BA - 112) is parametrised by y0(t) = (A - 112BA - 112)1, by what we said about the commuting case. So, the geodesic (A,B) = !fA'"{J),fAv.{A- 112BA-112)) is parametrised by

(13) y(t) = A112 (A- 112 BA- 112)1 A112, 0 :::; t :::; 1.

This shows that the geometric mean A#B defined by the fonnula (2) is nothing but the midpoint of the geodesic join­ing A and B in the Riemannian manifold !f1> n· Thus while (2), (3), and (4) might have ap,Reared as over-imaginative non­commutative variants of �. very natural geometric con­siderations lead to the same notion of mean as is given by (2). Note that for each t, y(t) defmed by ( 13) is a mean of A and B corresponding to the functionf(x) = :xf in the for­mula (6). Those means are not symmetric, however: (I) fails unless t = 1/2.

This discussion also gives an explicit formula for the metric 82. We have o2(A,B) = 82 (I ,A - 112 BA - 112) =

2 2. 5 2 1 . 5 0. 5

lllog I - log (A - 112BA - 112) 1 12 = lllog(A - 112BA - 112) l lz. The ma­trices A - 112 BA - l/2 and A - 1 B have the same eigenvalues. So, this can be expressed as

The inequality (9) captures an essential feature of lfl>n : it is a manifold of nonpositive curvature. To understand this, consider a triangle with three vertices 0, H, and K in §n· Under the exponential map, this is mapped to a "triangle" with vertices I, exp H and exp K in !f1> n· The lengths of the two sides [O,H) and [O,.K] measured by the norm l l · l lz are equal to the lengths of their images [I, exp H) and [I, exp K] measured by the metric 82 . By the EMI (9), the length of the third side [ exp H, exp K] of the triangle in !f1> n is larger than (or equal to) IIH - Kllz. The general case of a geodesic triangle with vertices exp A, exp B, exp C in !f1> n may be re­duced to the special case by applying the congruence fexp(-A/2) to all points and thus changing one of the ver­tices to I. This is often described by saying that two geo­desics emanating from a point in !f1> n spread out faster than their pre-images (under the exponential map) in §n·

It is instructive here to compare the situation with that of IUn, a compact manifold of non-negative curvature (Fig­ure 1 ). In this case the real vector space i§n consisting of skew-Hermitian matrices is mapped by the exponential onto IUn. The map is not injective; it is a local diffeomor­phism.

Using the formula (11) with H and K in i§n, we reduce

exp(iA)

0 0.5 1 .5 2 2.5 Figure 1. Three curvatures, showing a comparison of a Euclidean (curvature zero) triangle in §2 with its images under exp(-) in P2 (nonposi­

tive curvature) and exp(i·) in Q.J2 (non-negative curvature). The colours indicate matching vertices. Note that the geodesics emanating from

exp(A) spread out faster than Euclidean ones (compare the straight lines at A), whereas those emanating from exp(iA) spread more slowly.

© 2006 Springer Science+Business Media, Inc., Volume 28. Number 1 , 2006 35

A#C

B c B#C

Figure 2. Geodesic distance from A#B to A#C is no more than half

that from B to C. Joining the midpoints of the sides of a geodesic

triangle in IP'n results in a triangle with sides no more than half as

long. Iterating this procedure leads to the construction of Ando, Li,

and Mathias, described in the text.

H to diag( iA 1 o • • • , iAn) with A1 real. Instead of (12) we have now

sin(Ai - AD/2 (Ai - AJ)/2 kii·

Since [ sin x [ :S 1, the inequality (10) is reversed in this case, X as is its consequence (9), provided elf and eK are close to each other.

expA

Returning to IP n and the geometric mean, it is not diffi­cult to derive from the information at our disposal the fact that given any three points A, B, and C in 1Pn we have

1 (15) o2(A#B,A#C) :S 2 o2(B,C).

This inequality says that in every geodesic triangle in IP n with vertices A, B, and C, the length of the geodesic join­ing the midpoints of two sides is at most half the length of the third side. (If the geometry were Euclidean, the two sides of (15) would have been equal.) Figure 2 illustrates (15).

We saw that the geometric mean A#B is the midpoint of the geodesic [A,B]. This suggests that we may possibly de­fine the geometric mean of three positive definite matrices A, B, and C as the "centroid" of the geodesic triangle Ll(A,B,C) in 1Pn.

In a Euclidean space �. the centroid x of a triangle with vertices x1, x2, X3 is the point x = �(x1 + Xz + x3). This is the arithmetic mean of the vectors x1, x2, and x3. This point may be characterised by several other properties. Three of them are:

(M1) x is the unique point of intersection of the three medians of the triangle ll(x1,x2,x3), as in Figure 3;

(M2) x is the unique point in � at which the function

attains its minimum; (M3) x is the unique point of intersection of the nested

sequence of triangles [lln} in which ll1 = Ll and ll1+ 1 is the triangle obtained by joining the mid-

�--__,. expC

A � c

Figure 3. In the hyperbolic geometry medians may not meet. While the medians of a Euclidean triangle intersect at the centroid, the corre­

sponding median geodesics of a triangle in IP'n may not intersect at all. A 3-D wire model would make it clear that, generically, the medians

do not even intersect in pairs.

36 THE MATHEMATICAL INTELLIGENCER

points of the three sides of t:.J (Figure 2 mimics this

construction in the non-Euclidean setting of IP' n).

To define a geometric mean of A, B, and C in IP' n we may

try to imitate one of these definitions, now modified to suit

the geometry of IP' n· Here fundamental differences between

Euclidean and hyperbolic geometry come to the fore, and

(Ml), (M2), and (M3) lead to three different results.

The first definition using (Ml) fails. The triangle

t:.(A,B,C) may be defined as the "convex set" generated by

A, B, and C. (It is clear what that should mean: replace line

segments in the definition of convexity by geodesic seg­

ments.) It turns out that this is not a 2-dimensional object

as in ordinary Euclidean geometry (see Figure 4). So, the medians of a triangle may not intersect at all in some cases

(again, see Figure 3).

With (M2) as our motivation, we may ask whether there

exists a point X0 in IP' n at which the function

J(X) = 8�(A,X) + 8�(B,X) + 8�(C,X)

attains a minimum. It was shown by Elie Cartan (see, for

example, section 6. 1 .5 of [Be]) that given A, B, and C in IP' n•

there is a unique point Xo at which f has a minimum. Let

G2(A,B,C) = X0, and think of it as a geometric mean of A,

B, and C. This mean has been studied in two recent papers

by Bhatia and Holbrook [BH] and Moakher [M].

In another recent paper [ALM], Ando, Li, and Mathias

define a geometric mean G3(A,B,C) by an iterative proce­

dure. This iterative procedure has a nice geometric inter­

pretation: it amounts to reaching the centroid of the geo­

desic triangle 11(A,B,C) in IP'n by a process akin to (M3).

0.25

0.2

0 . 15

0.1

0.05

0.2

0 0.2 0.3 0.4

Starting with /11 as the triangle 11(A,B,C) one defines /12 to

be 11(A#B,A#C,B#C), and then iterates this process. Figure 2 shows the beginning of this process. The inequality (15)

guarantees that the diameters of these nested triangles de­

scend to zero as 112n. It can then be seen that there is a

unique point in the intersection of this decreasing sequence

of triangles. This point, represented by G3(A,B,C), is the

geometric mean proposed by Ando, Li, and Mathias.

It turns out that the two objects G2(A,B,C) and G3(A,B,C) are not always equal (Figure 5 illustrates this phenome­

non). Thus we have (at least) two competing notions of the

centroid of 11(A,B,C). How do they do as geometric means?

The mean G3(A,B,C) has all of the four desirable properties (a)-(8) that we listed for a mean G(A,B,C). Properties (a), (/3), and (8) are almost obvious from the construction. Prop­

erty ( y)-monotonicity-is a consequence of the fact that

the geometric mean A#B is monotone in A and B. So mo­

notonicity is preserved at each iteration step. The mean

G2(A,B,C) does have the desirable properties (a), (/3), and

(8). Property (/3) follows from the fact that r X is an isome­

try of (IP' n,82) for every X in GLn. However, we have not been

able to prove that G2(A,B,C) is monotone in A, B, and C. We

have an unresolved question: Given positive definite matri­ces A, B, C, and A' with A 2: A', is G2(A,B,C) 2: G2(A' ,B,C)?

An answer to this question may lead to better under­standing of the geometry of IP'n, the best-known example

of a manifold of nonpositive curvature. Certainly this is of

interest in matrix analysis. Computer experiments suggest

an affirmative answer to the question.

Finally, we make a brief mention of two related matters.

The Frobenius norm is one of a large class of norms called

0.5 0.6 0.7 0.8 0.9

Figure 4. Conv (A,B,C) is not two-dimensional. In the hyperbolic (nonpositive curvature) geometry of l?m the convex hull of a triangle (formed

by successively adjoining the geodesics between points that are already in the object) is not a surface but rather a "fatter" object.

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 2006 37

unitarily invariant norms or Schatten-von Neumann

norms. These norms 1 1 · 1 1"' have the invariance property I IU A VII"' = I IAII"' for all unitary U and V. Each of these norms corresponds to a symmetric norm <P on IR11; that is, a norm <P that is invariant under permutations and sign changes of coordinates. The correspondence is given by I IA I I''' = <P (s,(A), . . . , Sn(A)), where s,(A) 2: · · · 2: sn(A) are the singular values of A. Common examples are the Holder norms <Pp(X) = o:lxj:V)11P and the corresponding Schatten

norms I IAIIv = (� s}(A))11P, 1 ::::: p ::::: oc. The Frobenius norm is the special case p = 2.

For each of these norms we may define a metric o,v on 1P'11 as in the formula (14). The EMI in the form (9) or (10) remains true (see [B2]). The import of this remark is that, with any of these metrics, IP'n is a Finsler manifold of non­positive curvature; the special Frobenius norm arises from an inner product and gives rise to a Riemannian structure. In recent years metric spaces of non positive curvature have been studied in great detail; see the comprehensive book by Bridson and Haefliger [BrHa] . The spaces IP' n with norms 1 1 · 11''' are interesting and natural examples of such spaces.

0 0 0 But the whole wondrous complications of interference, waves, and all, result from the little fact that :i:p - px is not quite zero.

-Richard Feynman [ FLS j

The generalised version of EMI has a fascinating connection with yet another subject: inequalities for

�he matrix exponential function discovered by physicists and mathematicians. Many such in­

equalities compare eigenvalues of the matri­ces ef!+K and eHeK, and are much used in

1uantum statistical mechanics and lately in quantum information theory. In [S] I.

Segal proved for any two Hermitian matrices H and K the inequality

Figure 5. The "Cartan surface" contains G2(A,8,C) but not G3(A,8,C).

The Cartan surface consists of points minimizing the convex combi­

nations all�(A.xJ + bll�(B,x) + cll�(C,x); here the colours of the points

shown are chosen to reflect the relative strengths of the weights a,b,c.

Thus G2(A,8,C) corresponds to 1 /3, 1/3, 1/3 (see yellow dot on sur­

face). The small black circle locates G3(A,8,C), which is not on the

surface in general. Thanks to J.-P. Shoch for computing this picture

of a Cartan surface.

38 THE MATHEMATICAL INTELLIGENCER

Here A 1 (X) is the largest eigenvalue of a matrix X with real eigenvalues. In a similar vein, we have the famous Golden­Thompson inequality

(17)

The matrices ef!+K and ef112eKeH12 are positive definite. So, the inequalities ( 16) and (17) say

llefl+K]IP ::::: l lefll2eKeH12I IP, for p = 1 ,oc.

The EMI (9) generalised to all unitarily invariant norms is the inequality

By well-known properties of the matrix exponential, this implies

(19)

This inequality, called the generalised Golden-Thompson inequality, includes in it the inequalities (16) and (17) . The origins of these inequalities and their connections with quantum statistical mechanics are explained in Simon [Si] (page 94). Still more general versions have been discovered by Lieb and Thirring, and by Araki, again in connection with problems of quantum physics. See Chapter IX of [B] . Gen­eralisations in a different direction were opened up by Kostant [K], where the matrix exponential is replaced by the exponential map in more abstract Lie groups.

A common thread running between matrix analysis, Rie­mannian and Finsler geometry, and physics! Pascal would have approved.

REFERENCES

We have included some articles that are related to our theme but not specifically mentioned in the text. [A] T. Ando, Topics on Operator Inequalities, Lecture Notes, Hokkaido

University, Sapporo, 1 978.

[ALM] T. Ando, C . -K. Li, and R. Mathias, Geometric means, Linear Al­

gebra Appl. 385(2004), 305-334.

[Be] M. Berger, A Panoramic View of Riemannian Geometry, Springer­

Verlag, 2003.

[B] R. Bhatia, Matrix Analysis , Springer-Verlag, 1 997.

[B2] R . Bhatia, On the exponential metric increasing property, Linear

Algebra Appl. 375(2003), 2 1 1 -220.

[BH] R . Bhatia and J . Holbrook, Riemannian geometry and

matrix geometric means, to appear in Linear Algebra Appl.

[BrHa] M . Bridson and A. Haefl iger, Metric Spaces of Non­

positive Curvature, Springer-Verlag, 1 999.

[BMV] P S. Bullen, D . S. Mitrinovic, and P. M . Vasic, Means and Their

Inequalities, D. Reidel, Dordrecht, 1 988.

[BS] K. V. Bhagwat and R. Subramanian, Inequalities between means of

positive operators, Math. Proc. Camb. Phil. Soc. 83(1 978), 393-401 .

[CPR] G. Corach, H . Porta, and L. Recht, Geodesics and operator

means in the space of positive operators, Int. J. Math. 4(1 993),

1 93-202.

[FLS] R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures

on Physics, volume 3, page 20-1 7, Addison-Wesley, 1 965.

A U T H OR S

JOHN HOLBROOK

Department of Mathematics and Statistics

University of Guelph

Guelph, Ontario N 1 G 2W1

Canada

e-mail: jholbroo@uoguelph.ca

RAJENDRA BHATIA

Indian Statistical Institute

7, S. J. S. Sansanwal Marg

New Delhi 1 1 0016

India

e-mail: rbh@isid.ac.in

Rajendra Bhatia did his doctoral studies at lSI Delhi with Kalyan Mukherjee. He has been based there most of the quarter-century since,

along with his wife lrpinder and their son Gautam.

John Holbrook is now Professor Emeritus at Guelph. He and his wife Catherine divide their time between Guelph and Fowke Lake

(farther north), generally in the company of children, grandchildren, and cats.

This photograph of the authors (courtesy of Peter Semrl) shows them in yet another continent, Europe: in the beautiful Alps of Slove­

nia. The photo may also serve as encouraging evidence that it is possible to collaborate on mathematical projects and remain on good

terms!

[HLP] G. H. Hardy, J. E. Littlewood, and G. P61ya, Inequalities, Cam­

bridge University Press, 1 934.

[K] B. Kostant, On convexity, the Weyl group and the lwasawa de­

composition, Ann. Sc. E. N. S. 6(1 973), 4 1 3-455.

[KA] F. Kubo and T. Ando, Means of positive linear operators, Math.

Ann. 246(1 980), 205-224.

[LL] J . D. Lawson and Y. Lim, The geometric mean, matrices, metrics,

and more, Amer. Math. Monthly 1 08(200 1 ), 797-81 2.

[M] M . Moakher, A differential geometric approach to the geometric

mean of symmetric positive-definite matrices, SIAM J. Matrix Anal.

App/. 26(2005) , 735-747.

[P] B. Pascal, Pensees, translation by W. F. Trotter, excerpt from item

72, Encyclopaedia Britannica, Great Books 33, 1 952.

[Po] H. Poincare, Science and Hypothesis , from page 50 of the Dover

reprint, Dover Publications, 1 952.

[PW) W. Pusz and S. L. Woronowicz, Functional calculus for sesquil in­

ear forms and the purification map, Reports Math. Phys. 8(1 975),

1 59-170

[S] I . Segal. Notes towards the construction of nonlinear relativistic quan­

tum fields I l l , Bull. Amer. Math. Soc. 75(1 969), 1 390-1 395.

[Si] B . Simon, Trace Ideals and Their Applications, Cambridge Univer­

sity Press, 1 979.

© 2006 Springer Science+ Business Media, I n c . , Volume 2 8 , Number 1 , 2006 39

Numerica l Properties of Rudolph M ichael Sch indler's Houses in the Los Angeles Area Jin-Ho Park

Does your hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the caje

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?

Jj 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

The architecture of Los Angeles mir­

rors the diversity of cultures rep­

resented within the city and includes

the work of many eminent contempo­

rary architects, such as Frank 0. Gehry. Any serious study of Los Ange­

les architecture invokes the name

of Rudolph Michael Schindler as a ma­

jor inspiration. Schindler's numerous

buildings in Los Angeles, built over

thirty years, from the 1920s to the

1950s, are recognized as icons of twen­

tieth-century design.

Mathematicians interested in ex­

ploring and discovering Los Angeles ar­

chitecture will find Schindler's build­

ings (Fig. 1) of particular interest. To

make the most of your visit, you should

understand the underlying principle of

Schindler's architecture: his propor­

tional system (he called it "reference

frames in space") and its unique nu­

merical properties.

Pacific Ocean

Schindler's primary concern in ar­

chitecture is the assembly and delin­

eation of spaces. He crystallized his

idea of space in his mind, rather than

by visualizing it through a physical

model, or bodily movement combined

with the perceptual process, or other

methods. As a result, once you enter a

Schindler house, you find yourself in a

space-form that provides a full array of

new spatial experiences, and, at times,

surprises that arise from the complex­

ity of the spatial flow.

It is hard to predict Schindler's

space-forms by examining his plans,

elevations, and sections. For Schindler,

these were just two-dimensional nota­

tional forms, like scales in music. His

approach was not one of figure and

ground. Rather, the interior and the ex­

terior spaces are intertwined. In the in­

terior, the spaces flow into each other

and, on the exterior, simple and recti-

Figure 1 . Map of Los Angeles, and Schindler's buildings discussed in the present paper.

40 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Sc>ence- Bus>ness Media, Inc

Figure 2. R. M. Schindler: The McAimon House, 1 935, 271 7-2721 Waverly Drive, Los Angeles; The Mackey Duplex, 1 939, 1 1 37 South Cochran

Avenue, Los Angeles.

linear space-forms are highly inter­

locked. Their union creates a rhythm

of complex combinations. Thus it is of­

ten hard to distinguish where the build­

ings start or end; frequently they lack

a clear main fa<;ade. Therefore, to ap­

preciate Schindler's space-forms, one

must visit the houses in person, and ex­

perience them through the movement

of the body in space.

Architecture, for Schindler, began

with thinking about and reasoning with

the building, shaping and feeling it as

a structure in the mind (Schindler,

1946). Here, the proportional system

plays a significant role in composing a

specific compositional sensibility with

the practicalities of physical construc­

tion.

The purposes of his system were,

first, to provide a mental structure for

conceptualization before pencil met pa­

per and second, to communicate to the

builders a map locating elements, al­

lowing them to easily scale off dimen­

sions. By doing so, the location of every

element of the building is identified ac­

curately in the convenience of composi­

tion and construction. Thus, no obscure

or arbitrarily unrelated measurements

are involved in the system.

Usually, Schindler recommended a

basic unit of 4 feet ( 48 inches), to be

used with simple multiples and with 1l2, 1h and 114 subdivisions. In many cases,

he used 114 and 1l2 subdivisions for

plans and 113 for vertical subdivision.

The 114 and 112 adjustments "humanize"

the coarseness of the space reference

frame itself. The architect must think

in 48-inch units, but after ± 12", ±24" re­

fmements, results always lie within a 12-

inch grid. A variety of L-shapes and other

arrangements are easily accommo­

dated. Such a unit is useful to measure

dimensional relationship of a build­

ing's heights, widths, and lengths. Verti­

cally, s11ch rectangles are elevated in 16-

inch steps to create spatial elements.

There are two reasons for his

choice of this unit. First, it had to be

related to the human figure to satisfy

all the necessary requirements for

rooms, doors, and ceiling heights; sec­

ond, for practical reasons, the 48-inch

module fit the standard dimensions of

materials and common construction

methods available in California at that

time.

Architectural drawings are the

strategic transfer of the architect's in­

tention and vision, accurately and pre­

cisely. Schindler utilized his space ref­

erence frame in a grid pattern in

three-dimensional space. Numbers and

letters are laid out on the grid on the

floor plans in sequence, and the verti­

cal module is identified with an eleva­

tion grade. This pattern was original to

Schindler. In his earlier designs, the

grid was present on drawings and in

the house; later, the grids disappeared

from the house and, at times, from the

drawings. This doesn't mean he aban­

doned his system; on the contrary, it

remains embedded in the designs as

the underlying principle.

Schindler's choice of 48-inch units

has two significant mathematical as­

pects. With the unit system, a space is

understood in the mind as a continu­

ous quantity in three dimensions. Aris­

totle bisects "quantity" into two parts:

magnitude and multitude. For Aristo­

tle, magnitude is that which can be

measured: it has indivisible, continu­

ous attributes. Multitude, which can be

counted with number, has divisible or

discrete attributes. Schindler's unit

system can be interpreted as measured

space replaced by counted space.

Thus, for Schindler, a space can be

measured numerically by the unit sys­

tem in the mind-with subdivisions

and multiples of the unit. It is a fully

rationalized method of creating a spa­

tial structure. He wrote:

[The space architect] must establish

a unit system which he can easily

carry in his mind and which gives

him the size values of his forms di­

rectly without having to resort to

mathematical computations. Mea­

surements in figures express ab­

stract size relations void of any hu­

man connotation and offering no

assistance for the imagination.

Thus, the unit modules replace mea-

surements. The designer can play with

them, in his mind, freely and accu­

rately, without any mechnical measur­

ing devices. Accordingly, space forms

are successively conceived from the

mental play of the unit system.

Divisibility is another essential fea­

ture of the unit system. The number 48

is divisible by 10 factors (1, 2, 3, 4, 6,

8, 12, 16, 24, 48), which form a sub­

module group and relate organically to

the unit system. This group forms a

© 2006 Springer Science+ Business Media. Inc., Volume 28, Number 1 , 2006 41

-----

-----

----

---..,.. -

Figure 3. Schindler's three-dimensional grid of space cubes.

whole numbers on drawings are com­

monly presented. These numbers are in­

crements of unit multiples with subdivi­

sions. At times, various rooms are not

rectangular in shape but interlocked,

overlapped, and even zigzagged. Despite

their shape, Schindler labeled room di­

mensions in terms of a X b on the draw­

ings. It is certain that he approximated

measurements of the room in a rational

manner with his space reference frame.

The notation would be the standard

room and component sizing notation:

a X b with a, the width, greater than the

length, b. Here there is no temptation to

reduce the terms by common factors or

ratios, since the repetition of the same

ratio throughout the parts is seen as pro­

ducing a confusion of scale. Schindler is

concerned about scale precisely be­

cause it would imply a change of the unit

of measurement within the same work

He wrote, " 'Scale' denotes a consistent

dimensional relationship of parts of a

structure to each other and to a basic

unit." Thus, with such a notation, scale

is always retained.

family, so that the measurements of the

whole building are in relationship to

one another. Thus, all the parts of the

dimensions harmonize through unifor­

mity of scale and rhythm. In other

words, unrelated dimensions in the

whole scale can be excluded. In

Schindler's words, " . . . only coarse­

ness allows him to break that rhythm

by introducing arbitrary unrelated di­

mensions into his layout." The 48-inch

unit is characterized as a "highly

composite number" by Lionel March

(1994a), who accounts for various

composite numbers, such as 36, 48, 60,

180, with a lattice diagram, showing a

hierarchical structure.

Through a process of abstraction,

Schindler began to construct and ex­

plore a variety of spaces and space­

forms in his mind. He defined spaces

as abstract geometric forms whose po­

sitions could be uniquely determined

by a given number of coordinates.

Here, Schindler's space reference

frame helps to measure the exact dis­

tance between points in space, and to

identify space-forms of a design in

precise locations, allowing for formal

and spatial coherence in his architec­

ture. It is easy to see how Schindler

worked with those unit dimensions in

his spatial computation. Because the

space architect can conceive of 4 X 4 x 4 foot "units" of space, it seems

quite reasonable to imagine such a

space element as being slightly less or

slightly more, :±:: 1/4, or simply halfway

42 THE MATHEMATICAL INTELLIGENCER

between, :±:: 1h in the mind. In the de­

sign process, Schindler is able to ad­

just the dimension bit by bit.

Schindler roughed in his designs us­

ing the 4 X 4 X 4 foot space reference

frame and located spatial elements rel­

ative to one another, introducing re­

finements between elements-taking a

little here, adding a little there. As he

explained, "It is not necessary that the

designer be completely enslaved by the

grid. I have found that occasionally a

space-form may be improved by devi­

ating slightly from the unit." Nor

was Schindler obsessed with particu­

lar numbers, such as the Fibonacci se­

quence or ratios. Design can be imag­

ined in the mind without recourse to

pencil and paper. This represents a ma­

jor distinction from the architect Le

Corbusier, who used regulating lines in

his early work, transforming them into

Le Modular, and was noted for the per­

sistence of the golden section in his

work after World War II.

Schindler also used the space refer­

ence frame to measure room sizes. In his

preliminary sketches, room sizes with

Among Schindler's 200 houses in

the Los Angeles area, three are dis­

cussed in this article-the Kings Road

House of 1920; the Lovell House

of 1923; and the How House of 1925.

They represent exquisite examples of

Schindler's finest work Most of his

houses are private residences and are

not accessible to the public. However,

the Kings Road house is open; since

1994, the house has been used as the

MAK Center for Art and Architecture

in Los Angeles, a satellite of the Mu­

seum for Applied Art in Vienna.

Kings Road House, 1 92 1 -22 835 North Kings Road,

West Hol lywood

The Kings Road house is the best

"representation of Schindler's "or­

ganic" type of dwelling.

180

Figure 4. The lattice diagram of highly composite numbers such as 36, 48, 60, and 180 are

shown (after Lionel March).

6 Bath

b B � B I Living R'M §

I I

Kilclk!=n l:>our Living R�l Living RM L.iving RM

0 0

8 ( o Patio r

J 0 L?gr,...._...._•r-- -.,..-

L.iving RM B<dR ·I B<dRM

IA IB K:: ID � IF IG

Patio Front Elevation

As opposed to imposing modem structures, the simplicity of its volume is in tune with its environment, creating a refined refuge from the bustle of Los Angeles. The house lies in a densely wooded grove. Three L-shape units are arranged in a pattern on a 200 X 100-

foot lot, forming a courtyard with en­closed patios and outdoor fireplaces. Each pair of units opens to outdoor liv­ing patios. The third one is formed by the kitchen, guest studio, and garage. The impression is of primitive boxes resting in their natural place. Schindler wrote, "The shape of rooms, their relation to the patios and the alternating roof levels, create an entirely new spatial interlock­ing between the interior and the garden." There is almost no difference in level be­tween the ground floor and the garden, suggesting an infinite extension to the open ground in accordance with the character of the land. Schindler's "or­ganic" building type is fully realized in this house; the house and the outdoors unite in perfect rapport to embrace their

Figure 5. Pueblo Ribera Court. Plan: elevations,

window, and furniture.

The structural components of the house are simple: concrete walls on one side and two wooden posts from the other side support all ceilings. "All parti­tions and patio walls are non-supporting screens composed of a wooden skeleton filled in with glass or with removable 'in­sulate' partition. These basic materials are used in a lucid way to form "the cave-

tent shelter of concrete, wood and can­vas" which relates the project to the cli­mate, the region, and the surroundings. Schindler appears to have erected a mod­ern hut, expressing the immemorial re­lationship between humanity and nature. It is a true sustainable aesthetic.

On the drawings, the dimensions and placements of various spatial

extraordinary surroundings. Figure 6. The Kings Road House, 1 921-22. 1/4 scale model constructed with basswood.

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 1 , 20C>6 43

Figure 7. The Kings Road House, 1 921-22. Courtyard view.

forms and details of the house are con­

trolled by Schindler's unit system. The

48-inch unit system is clearly identi­

fied in the plan. A 12-inch ( 1/4 unit

module) vertical module is employed.

Thus the heights of all window and

door mullions are based on a 12-inch

module. The rafters of the horizontal

wooden structures are regulated ac­

cording to a 24-inch distance-half the

unit module.

Most of the major rooms in the con­

struction drawings are approximately

measured in whole numbers, although

rooms are frequently not simple rec­

tangles. Unlike the complexity of the

space-forms, these involve surprisingly

few dimensions and corresponding ra­

tios. In the house there are six differ­

ent ratios of dimensions: 2 : 1 , 3:2, 4:3,

6:5, and 9:8. The noteworthy character

of the ratio is that most of the fractions

vary by adding 1 to both numerator and

denominator. This is equivalent to the

classical sequence of subsuperparticu­

lar numbers (March, 1998). Interest­

ingly, these room ratios are also found

in musical ratios within an octave: oc­

tave (2: 1 ), second (9:8), fourth ( 4:3), a

minor third (3:2), and a major third

(6:5). Schindler himself briefly referred

to musical ratios in his early discussion

of proportion in his unpublished 1917

Emma Church School lecture notes

(Park 2003; March 2003). In addition,

March (1994b) used a musical analogy

to examine the proportional design of

the Schindler house: in his paper "Dr.

How's Magical Musical Box" March re­

calls "architecture as frozen music."

44 THE MATHEMATICAL INTELLIGENCER

According to March, any analysis of ra­

tios and proportions of Schindler's

houses will inevitably correspond to

musical intervals. It does not mean the

musicality is intentional but it is a nec­

essary consequence of the relations of

small numbers.

Lovell House, 1 922-26 1 242 Ocean Avenue,

Newport Beach, California

Built for Dr. Philip Lovell, this house is

the most revealing project in Schind­

ler's oeuvre, displaying dramatic spa­

tial complexity as well as structural

resolution. Schindler uniquely devised

ingenious structures to control and

9 : 8

raise the upper-level sleeping quarters

above a flat beach site. In the house, he

used five reinforced concrete columns,

opening the ground level for play and

recreation. Liberating the ground floor

recalls one of Le Corbusier's five points,

plan libre. Piloti, or columns, in con­

crete and steel, carry the structural load,

lifting the box into space; interior walls

are then freely arranged according to cir­

culation or other functional require­

ments (Park and March 2003).

The 48-inch unit system is clearly

identified in the plan with numbers and

letters, and the 16-inch vertical module

in elevation with grades. The grid is

marked along the bottom with num-

2 : 1

6 : 5 4 : 3 3 : 2

2 : 1

2 : 1

Figure 8. The Kings Road House, 1 921-22. Ratios of each room.

bers from 1 to 16, and up the right-hand

side with letters A to J. Its vertical mod­

ule is based on a 16-inch unit system,

which controls not only the height of

the room but other elements as well,

including built-in furniture, chairs, ta­

bles, windows, doors, and clerestory,

providing uniformity of scale and pro­portional beauty. In Schindler's words,

"Floor plans and elevations were de­

signed by a scheme of unit lines to as­

sure uniformity of scale. All woodwork

including concrete forms and furni­

ture, was built of eight inch boards

with wide joints."

Sarnitz (1986) provides a simple but

interesting proportional study of the

house. His analysis of the house plans

and elevations is based on the belief

that subdivisions of square and double

square determine the overall propor­

tional system of the house.

How House 1 925 931 North Gainsborough Drive,

South Pasadena

The How House stands out from

Schindler's other works of this period

in its conspicuous and transparent play

about a diagonal axis that overlays a

48-inch unit system. The use of both

the modular and symmetrical systems

in the How House is subtle. Schindler

increases the significance of the diag­

onal axis by setting the orthogonal

lines of the ground plan at a 45° angle

to the boundaries of the lot and the

road frontage (Park 2000). The lower

portion of the building's volume is built

in concrete with Schindler's own "slab­cast" construction system, while the structure of the upper portion is red­

wood. The horizontal stratification of

the continuous concrete course as well as the battened boards on the wall of

the house were laid to coincide with a

16-inch vertical module. The module

incorporates the heights of all ele­

ments of the main structure, as well as

built-in furniture, chairs, windows, and

doors.

The How House is mainly deter­

mined by reflective symmetry about a diagonal axis. As Schindler writes "the rooms form a series of right angle

shapes placed above each other and facing alternately north and south."

Figure 9. The Lovell House, 1 922-26. Five-layered vertical structures of the Lovell house in­

tegrate all layered stairs and corridors.

bile-the living room, the dining room,

and Dr How's study-where the spatial

and structural setting of the whole

composition is based on the diagonal

axis. However, the floor plan of the pi­ano nobile does not conform to sym­

metry along the diagonal axis in a strict

manner: the symmetry is broken by ad­ditional spaces such as the kitchen and

the entrance hall, and also in certain

details such as the principal fireplace

and the stairway to the lower, bedroom

floor. This asymmetry does not depend

on an arbitrariness of personal taste or a rejection of the principles of symme­

try in the cause of modernism. Instead,

it is generated from a disciplined un­

derstanding of symmetry. The final de­

sign displays an abundance of symme­tries within the parts while negating

the strict symmetry of the whole.

The locations of the rooms are

clearly arranged according to his sub­

divisions and multiplications of the

unit module. The floor plan of the house

demonstrates an interesting propor­

tional relation with regard to his system.

The house is set within a 48-inch unit

module, numbered from 10-24 (a 14-

unit module) and lettered from H-W

(again a 14-unit module). All architec­

tural elements are disposed within this

14-module square along the diagonal

axis. In the house, Schindler employs a

16-inch (1/3 unit) vertical module. In RM Schindler-Composition and

Construction, Lionel March provided an

in-depth analysis of this house. Accord­

ing to March, its principal features fol­low the classical arrangement known as

ad quadratum. First of all, the layout of the house is placed within a 14-module

The analysis focuses on the piano no- Figure 1 0. R. M. Schindler, the Lovell House, 1 922-26. Facade.

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 1, 2006 45

Figure 1 1 . R. M. Schindler, the Lovell House, 1 922-26. Corner view.

square. Another square covers the prin­

cipal volumetric elements of the house:

the living room, dining room, the study,

and the roof terrace. This square with

comers at V-24 and L-14 is a 10 X 10

module. From the external comer of the

living room a 7 X 7 square defines the

edge of the dropped ceilings in the din­

ing room and study. This is clearly visi­

ble at the gallery level. The major space

divisions generate a series of propor­

tions, such as 7:5, 10:7, and 14:10, which

suggests that the layout of the house

may be ordered from a series of ad quadratum arrangements.

It also turns out that three pairs of

concentric squares define conceptually

the principal "blocks" of the house. First

of all, the living room of the piano no­bile is located on the center point of the

14 X 14 square. It lies in 5 X 5 modules.

In the plan of the piano nobile, the main

volume of the house is planned within a

91/4 X 91/4 module square. The center of

Figure 1 2. The Lovell House, 1 922-26. 1/4 scale study model.

this square is the center of the 60-inch

by 60-inch open shaft which provides

light and ventilation for the bedroom

floor from the roof terrace above. The

large 51/2 X 51/2 module square is the

terrace, measured to the outer edge of

the planters. The small % X % module

square locates a built-in light fixture in

the cantilevered porch for the terrace.

The three diagrams, taken together and

superimposed, show a sequence of cen­

tering points along the diagonal axis.

There are two structural layers at the

gallery level % module apart; the lower

layer, which has the same herringbone

pattern as the ceiling of the porch ex­

tending over the terrace as the pattern

on the living room ceiling, and the upper

layer, which splits into two wings over

the dining room and the study, respec­

tively. Both L-shaped layers are set along

the diagonal axis facing each other in op­

posing south and north orientations. In particular, the lower part demonstrates

the ingenuity of Schindler's method, al­

lowing two partially cantilevered beams

to cross above the open shaft area with­

out any support in the center.

The exposed ceiling structure of the

living room is particularly worth men­

tioning. It fills a square plan with a span

of 20 feet, or five 48-inch modules. Two

chimney stacks on adjacent sides em-

Figure 1 3. The Lovell House, 1 922-26. Analysis of plan and elevation of the Lovell house (after August Sarnitz).

46 THE MATHEMATICAL INTELLIGENCER

Figure 15. R. M. Schindler, The How House, 1 925. Interior and furniture designs.

Figure 16. R. M. Schindler, The How House, 1 925. Exterior view.

Figure 1 4. R. M. Schindler, The How House,

1 925. Living room roof structure.

© 2006 Springer Science+ Business Media, Inc . . Volume 28, Number 1 , 2006 47

H

\Q 11 12 13 14 15 16 11 18 18 20 21 22 2'� /

0 p 0 R

u

Figure 1 7. The How House, 1 925. Plan analysis of the How House showing the ad quadratum

arrangement.

phasize the diagonal axis (Figure 18). The structure of this ceiling is a system

of redwood beams at 1/2 module (24-

inch) intervals: each joist is secured

in a wall at one end while the other

end is nailed to a longer joist in a her-

Figure 1 8. The How House, 1 925. Axonometric.

48 THE MATHEMATICAL INTELLIGENCER

ringbone pattern along the diagonal

axis. With this arrangement, only one

beam is required to span the whole

space. The structure of the living room

ceiling provides an interesting series of

proportional relations. According to

The ceiling st ructm·c

of the Jh·ing room

The upper structure

of tlw gaUery floor

The lower structure

ur the gallery floor

The piano nobile

March (1994b), the ceiling of the How

House living room is set out in the man­

ner of the Greek gnomon: the sequence

of squares (1 x 1, 2 x 2, 3 x 3, . . . , 9 x 9) and the sequence of oblongs (2 X 1 ,

3 X 2, 4 X 3 , . . . , 9 X 8) (Figure 19).

REFERENCES

March, Lionel and Judith Sheine, eds. 1 994.

RM Schindler-composition and construction.

London: Academy Editions.

March, Lionel. 1 994a. "Proportion is an alive

and expressive tool. . . . " pp. 88-1 01 in RM

Schindler: Composition and Construction, L

March and J Sheine, eds. London, Academy

Editions.

March, Lionel. 1 994b. "Dr How's Magical Mu­

sic Box." Pp. 1 24-1 45 in RM Schindler:

Composition and Construction, L March and

J Sheine, eds. London, Academy Editions.

March, Lionel. 1 998. Architectonics of Human­

ism: Essays on Number in Architecture. Lon­

don: Academy Editions, John Wiley & Sons.

lliJ I= ill i= I==

n JL I oo I a

hl b

Il l c

Figure 1 9. The How House, 1925. Ceiling de­

sign.

March, Lionel. 2003. "Rudolph M. Schindler:

Space Reference Frame, Modular Coordina­

tion, and the 'Row' ," Nexus Network Jour­

nal 5, 2: 48-59.

Park, Jin-Ho. 2003. "Rudolph M. Schindler:

Proportion, Scale and the 'Row'," Nexus

Network Journal 5, 2 : 60-72.

Park, Jin-Ho. 2000. "Subsymmetry Analysis of

Architectural Designs: Some Examples," En­

vironment and Planning 8- Planning & De­

sign 27, 1 : 1 2 1 -1 36.

Park, Jin-Ho and Lionel March. 2003. "Space

Architecture: Schindler's 1 930 Braxton­

Shore project," Architectural Research Quar­

terly 7 , 1 : 51 -62.

Park, Jin-Ho. 2004. "Symmetry and Subsym­

metry of Form-Making: The Schindler Shel­

ters of 1 933-42," Journal of Architectural

and Planning Research 2 1 : 24-37.

Sarnitz, August. 1 986. R M Schindler: Architekt

1887-1953. Vienna: Edition Christian Brand­

statler.

Schindler, R. M. 1 946. "Reference Frames in

Space." Architect and Engineer 1 65: 1 0 , 40,

44-45.

AUT H OR

JIN-HO PARK

Department of Architecture

lnha University

Nam-gu, lncheon

Korea

e-mail: j inhopark@inha.ac.kr

Jin-Ho Park earned his BS in architecture from lnha University, Korea, and his MA and

PhD Degrees. also in architecture, from the University of California at Los Angeles. He

has taught at the University of Hawaii (Manoa), where he received the University of Hawaii

Board of Regent's Medal for Excellence in Teaching in 2002 and the ACSA/AIAS New

Faculty Teaching Award in 2003. He is presently associate professor at lnha University.

His articles have appeared in numerous journals, and he currently serves as corre­

sponding editor of the online Nexus Network Journal: Architecture and Mathematics.

N E W I N P A P E R B A C K

With an introduction by Thomas Banchoff

F LAT LA N D A Romance of Many Dimensions

Edwin Abbott Abbott Flatland has fascinated generations of readers, becoming a peren nial science-fiction favorile. A first-rate ficlional guide to the concept of multiple dimensions of space, the book wil l also appeal to those who are interested in computer graph­ics. In his inlroduction, Thomas Banchoff points out that there is no beHer way to begin exploring the problem of understand ing higher-di mensional slicing phenomena than readi ng lhis classic novel of the Victorian era.

Praise for Princeton 's previous edition: "One of the most imaginative, delightful and, yes, touching works of mathematics, this slender 1 884 book purports to be the memoir of A. Square, a citizen of an entirely two­dimensional world."- Washington Post Book World Princeton Science librory Paper $9.95 0-69 1 - 1 2366-7

Celebrating 100 Years of Excellence PRINCETON (0800) 243407 U.K.

' ' 800-777-4726 u.s. Umverstty Press math.pupress.princeton.edu

© 2006 Springer Science+ Business Media, Inc . . Volume 28, Number 1 , 2006 49

The Wedding Reuben Hersh

Philosophy: The bride and groom are here, and have

agreed to be wed today, in the presence of several wit­

ness s. Let the wedding begin. Logic, speak to Geometry.

Logic: I come to you, Geometry, my beautiful bride-to­

be, as your guide and your elder, your counselor and

corrector, your only lord and master, to have and to hold,

for good or for ill, in siclmess and in health, till death

do us part.

Geometry: You come to me, Logic, my groom and my

husband, as my guide and my junior, my counselor and

intem1pter, my self-named lord and master, to have and

to hold, for good or for ill, in siclmess and in health, till the unlmown and unknowable, whatever is to come at

the end of our joint dominion.

L: Do you, Geometry, take me as your guide and your

elder, your counselor and corrector, your only lord and

master, to have and to hold, for good or for ill, in sick­

ness and in health, till death do us prut?

G: I do, with a rese1vation.

Philosophy: We will hear your explanation.

G: I am the elder, you are the younger, Lord Logic.

P: Let it be so recorded, Geometry is the elder. Is that

your only reservation?

G: It could be. nless you wish to hear more.

L: You may go on.

G: You bring me strength and control. You bring me your

darling and precious offspring, 1, 2, 3, and many other

beautiful nun1bers. I an1 happy to be yow· bride.

L: This is well explained.

G: I bring you shape, fom1, and continuity. The possi­

bility of fruitful intercourse, and offspring uncountabl .

L: You lmow I love you, Geometry.

G: I know you love me, Logic. I do not know if I can love

you. You ru·e hru·d. You are unforgiving. You think you

can be my lord and master.

L: Look, is this a wedding, or what'?

G: It was announced and scheduled as a wedding. TI1e

guests ar here, with gifts. L: Yes, beloved friends and frunily have come. Mechan­

ics and Statistics, of course. Even the Stock Market, the

Census Bmeau, and the Atomic Energy Commission are

here.

G: Don't worry about them. They don't understand any­

thing, anyhow.

L: Easy for you to say. Who will pay the bills?

G: Oh, Logic, you are hopeless. The bills! Is this love or

business?

L: It' the business of love and the love of business.

G: Another paradox! I hate your silly little paradoxes! I hate them! Why did I ever agree to do this?

L: Because you need me.

G: Yes. It's true. I ne d you. But why and for what?

L: Without me to look after you, what would become of

you?

G: I don't know. What would become of me?

L: Look at your wretched, lo t cousins. umerology.

Weather and Financial Prediction. Old Pythagoras's

great-grand-children. \\lhere ar they now?

G: In the Gutter.

L: Yes. In the Gutter.

G: Why are they all in the Gutter?

L: Because they tried to live without Logic.

G: Will you be good to me? Will you be kind? Will you

ever be kind? L: I will be good for you. I will be as kind as I am able

to be.

P: Let it be ordered and written that on this day, in this

place, in the presence of witnesses both honorable and

dishonorable, Logic and Geometry were lawfully wed­

ded, to have and to hold, for good or for ill, in sickness

and in health, till death do them part. And may God some

day forgive us for what we do here today.

� believe � was Hermann Weyl who once said that the angel Algebra and the

devil Topology were struggling for the soul of Mathematics.)

Department of Mathematics and Statistics

University of New Mexico

Albuquerque, NM 871 3 1 USA

e-mail : rhersh@math.unm.edu

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 2006 51

lil§iit§'MJ Osmo Pekonen , Ed itor 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, P.O. Box 35, University of Jyvaskyla,

Fl-4001 4 Finland

e-mail: pekonen@mit.jyu.fi

Erich Kahler. Mathematische Werke. Mathematica l Works edited by Rolf Berndt and Oswald

Riemenschneider

BERLIN, WALTER DE GRUYTER, 2003. 971 PP. €228.00

ISBN 3- 1 1 -0171 1 8-X

REVIEWED BY ERNST KUNZ

This volume of almost one thousand

pages aims at describing the life,

activities, and convictions of Erich

Kahler (1906-2000), one of the great

and many-sided mathematicians of the

20th century, and a strong and fasci­

nating personality. It also deals with

the developments that have grown out

of his work

Nearly 700 pages contain most of

Kahler's original publications in math­

ematics and mathematical physics and

three of his philosophical essays. The

remaining 300 pages describe his life

and contain surveys by experts in the

fields in which Kahler was active, de­

scribing the impact of his work on

present-day mathematics, physics, eco­

nomics, and other areas of intellectual

life. That this part of the book is so

large shows the great influence that

Kahler's work still has. The editors de­

serve high appreciation for having col­

lected the pieces to compose a rich pic­

ture of modem science.

Of Kahler's 399-page book Geome­tria aritmetica, volume 45 of Annali di Matematica, which was considered

by many of his students, by many math­

ematicians, and maybe by Kahler him­

self as his main work in mathematics,

only the introduction is reproduced.

But its content is described in the long

article "Kahler Differentials and Some

Applications in Arithmetic Geometry"

by R. Berndt, who has also translated

much of the text from Kahler's unus­

ual philosophically motivated language

into modem mathematical terminal-

52 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Scrence+Business Media, Inc

ogy. Kahler had anticipated the publi­

cation of the book by surveys given at

conferences in various languages. The

book shows Kahler as a forerunner of

the theory of schemes, as was recog­

nized by Grothendieck in his Elements

de geometrie algebrique ("parmi les

travaux qui se rapprochent de notre

point de vue en Geometrie algebrique,

signalons l 'important travail de E.

Kahler"). At a memorial colloquium in

honor of Kahler at the University of

Leipzig, Kahler's former collaborator

H. Schumann reported that Kahler con­

sidered the Geometria aritmetica a

first step towards a more comprehen­

sive theory but abandoned the project

when the first volumes of Groth­

endieck's work appeared. A. Weil ex­

pressed concern in Mathematical Re­views that the unconventional style

and the unusual language would pre­

vent the book from receiving the at­

tention it deserved ("the author seems

to have done everything in his power

to discourage prospective readers and

is only too likely to have succeeded"),

and so it turned out. History has fol­

lowed the French school.

Kahler's main objective in Geome­

tria aritmetica was to unite number

theory and geometry. He emphasized

that for this purpose it was necessary

to study arbitrary local rings, not only

those containing a field, as was done

in classical algebraic geometry. Con­

trary to Grothendieck's more general

point of view, Kahler's schemes (called

''varieties") are contained in "algebraic"

fields, that is, fields which are finitely

generated over their prime field. He also

tried to embed both disciplines into a

philosophical system which was con­

ceived by him but not well understood

by others, and in fact hardly under­

standable for the reader without further

instruction. An explanation of his

system, a mathematical model of the

world, the universe of all existing things,

is given by him in the essay "Wesen

und Erscheinung als mathematische

Prinzipien der Philosophie" (Essence

and Appearance as Mathematical Prin­

ciples of Philosophy) and in other pub­

lications which are not contained in

the collected works. In his paper "In­

fmitesimal-Arithmetik" which appeared

four years after Geometria aritmetica,

maybe as a reaction to previous critics,

Kahler gave an outline of its contents

in ordinary mathematical language (ex­

cept that the local rings having a given

quotient field are called the faces (Seiten) of the field and valuation rings

the perfect faces). The mathematical

discipline to which the book belongs

and in which some of the most spec­

tacular successes were achieved in re­

cent years has adopted the name of the

book: Arithmetic Geometry. R. Berndt,

and in another essay J. B. Bost ("A Ne­

glected Aspect of Kahler's Work in

Arithmetic Geometry: Birational Invari­

ants of Algebraic Varieties over Number

Fields") describe some developments in

connection with the mathematical con­

cepts which were introduced by Kahler

in what was called by some his opus magnum. In another article by Berndt

there is a discussion of "Kahler's Zeta­

Function," and A Deitmar ("A Panorama

of Zeta-Functions") gives an overview of

results and conjectures about zeta func­

tions in general.

Much better known and much more

influential than Geometria aritmetica was Kahler's note "Ober eine be­

merkenswerte Hermitesche Metrik"

(On a remarkable Hermitian metric)

from 1932, which developed into a

large and important discipline-Kah­

lerian geometry, with "Kahler mani­

fold" as its main concept-which

had strong influence on many areas of

mathematics (differential geometry,

complex analysis, algebraic geometry)

and mathematical physics (relativity,

field theory). Here many notions have

their origin in this paper of 14 pages,

written by him at the age of 27, and

bear his name as in Kahler metric, Kah­

ler-Einstein metric, Kahler potential,

Kahler class, etc. In his report "The Un­

abated Vitality of Kahlerian Geometry"

J. P. Bourguignon describes Kahler's

paper and gives a detailed survey of all

developments since the paper was

written. He confesses that he and many

colleagues for a long time considered

Kahler as a figure of the first third of

the 20th century and that they were not

aware that he was still alive and active

in the 1980s. In a "Tribute to Berm

Erich Kahler" S. S. Chern appreciates

the influence that Kahler had on his

own work

In recent decades Kahler manifolds

have entered theoretical physics again

in the construction of models to unify

the fundamental laws of physics. In his

article "Supersymmetry, Kahler Geom­

etry and Beyond" H. Nicolai reports on

these developments.

Since early in his career Kahler's

work made extensive use of Cartan's

theory of exterior differential forms,

which he refined and recreated and

generalized in purely algebraic terms.

Many of his publications on the so­

called Kahler differential forms are de­

voted to this goal. His book Ein­

fiihrung in die Theorie der Systeme

von Differentialgleichungen (Introduc­

tion to the theory of systems of differ­

ential equations) of 1934, which is not

reproduced in the collected works,

uses differential forms. It became the

origin of what is now called Cartan­

Kahler theory, with the theorem of Car­

tan-Kahler as its fundamental result.

This part of Kahler's work is thor­

oughly discussed in a report by R.

Berndt and 0. Riemenschneider, which

contains a much more detailed survey

of all of Kahler's mathematical results

than can be given here. In another es­

say by I. Ekeland a problem of eco­

nomics is explained which leads to a

system of partial differential equations

of first order and which can at present

be solved only by means of the Cartan­

Kahler theorem.

Kahler wrote influential papers on

the singularities of complex functions

in two variables. W. D. Neumann re­

views these publications and puts them

in the perspective of the topological

and algebraic classification first of

plane curve singularities and then of

isolated hypersurface singularities.

Kahler's work in mathematical

physics started with three papers on

the 3-body and n-body problem and

one on fluid dynamics. Later work is

devoted to expressing fundamental

equations of physics such as the

Maxwell equations and the Dirac equa­

tion and some of their solutions in

terms of differential forms.

As a rule, the sciences select or de­

velop the tools from mathematics they

need to describe and to solve their

problems. Kahler's thinking went also

in the other direction. He was con­

vinced that the powers of highly de­

veloped disciplines such as arithmetic

and algebra were not sufficiently used

in physics ("Number theory is deter­

mined to become the leading power of

natural sciences"; "What is the mean­

ing for natural sciences of the re­

sources collected in algebra and num­

ber theory?"). Without being very

specific, he supported his views some­

times with bold conjectures and

speculations. It seems that theoretical

physicists have adopted few of these,

an exception being Kahler's (quater­

nionic) "new Poincare group" which

according to him should replace the

Lorentz group of classical relativity

theory. It is mentioned in a footnote of

Nicolai's contribution that the new

Poincare group is identical with the de

Sitter group from theoretical physics.

The mathematical properties of the

new Poincare group are discussed in

an article by A Krieg.

Kahler, who was influenced by

Plato's concept of "ideas," Leibniz's

monadology, Hegel, and Nietzsche, gave

university courses in philiosophy. His

friends report that he considered his

texts combining mathematics with phi­

losophy and theology as more impor­

tant than his mathematical achieve­

ments, and that he suffered from the

fact that they did not receive the ap­

preciation he had hoped for. For him, at

least in his later years, mathematics was

a language designed to formulate and to

solve not only problems of the sciences

but much more ("mathematics as a

thread of Ariadne which guides us out

of the labyrinth of modem sciences"). I

quote also from the article "The Life of

Erich Kahler" by R. Berndt and A Bohm: "The ingenious mathematician

became a mathematical dreamer who

thought he could solve all problems of

this world by mathematical methods."

I do not feel competent to comment

© 2006 Spnnger Science+Bus1ness Media, Inc., Volume 28, Number 1 , 2006 53

on the philosophy of Kahler. He calls mathematics "an infinite refmement of language." In the collected works there are several attempts to guide the reader to his philosophical thinking. First there are three essays by Kahler himself, two in Italian, and the one which was men­tioned above. References to his other philosophical texts are given. Moreover two articles "Kahler's Vision of Mathe­matics as a Universal Language" by R. Berndt and "An Approach to the Phi­losophy of Erich Kahler" by K. Maurin serve the same purpose.

Kahler, who had a strong sense of mission, started his independent sci­entific production at the age of 17 and remained active until shortly before his death at 94. The principal stages of his academic career were:

1928 Doctor's degree at the Uni­versity of Leipzig under Leon Lichtenstein

1930 Habilitation in Hamburg 1931- As a Rockefeller fellow in 1932 Rome, he met Enriques, Cas-

1936

1948

1958

1964

telnuovo, Levi-Civita, Severi. B. Segre, and A. W eil Ordinary professor in Konigs­berg Successor to Koebe in Leipzig After serious political ten-sions with the East German administration Kahler ac­cepted a position at the Technical University of West Berlin Successor to Emil Artin in Hamburg

1974- Professor Emeritus, work-2000 ing mainly on his Mathe­

matical Philosophy and occasionally lecturing at physics and mathematics conferences.

The book under review does not con­tain a list of Kahler's former PhD-stu­dents. Here is one, maybe incomplete: Walter Thimm (Konigsberg 1939), Gtin­ter Hauslein (Leipzig 1955), Gerhard Lustig (Leipzig 1955), Armin Uhlmann (Leipzig 1957), Rolf Berndt (Hamburg 1969). Kahler also initiated the theses of Gtinther Eisenreich (Leipzig 1963) and Horst Schumann (Leipzig 1968), but

54 THE MATHEMATICAL INTELLIGENCER

since he had left East Germany he could not act formally as advisor.

It seems that Kahler's philosophical texts have been widely ignored. As mathematicians, we admire his numer­ous contributions to our science which have come to bear so much fruit inside and outside mathematics. With his col­lected works, the editors have given him a worthy monument.

Fakultat fUr Mathematik

U niversitat Regensburg

D-93040 Regensburg

Germany

e-mail:

ernst.kunz@mathematik.uni-regensburg.de

History and Science of Knots edited by J. C. Turner and

P. van de Griend

SINGAPORE, WORLD SCIENTIFIC, 1 996. 464 PP. US$78,

ISBN 981 -02-2469-9

REVIEWED BY KISHORE B. MARATHE

The book History and Science

of Knots consists of 18 chapters grouped into 5 parts: I. Prehistory and antiquity, II. Non-European traditions, III. Working knots, IV. Towards a sci­ence of knots?, V. Decorative knots and other aspects. The editors, Turner and van de Griend, have chosen a wide va­riety of experts as authors for the in­dividual chapters; more information about them may be found in "About the Authors" on page 419. The con­struction and use of knots, links, and braids from string-like objects pre­dates known human history. They oc­cur in diverse areas of human activity, from magic tricks and decorative arts to shipping, fishing, and religious and medical practice. Such structures also occur in nature, for example, in poly­mer chemistry and biology. In spite of the vast range of topics and the time frame, the book gives a representative sampling of knots and knot applica­tions. The historical aspects of each topic are also discussed. Knots are dis-

cussed from this broad perspective in all but one chapter. Only chapter 1 1 is devoted to knots as defined in mathe­matics. We will discuss the contents and highlights of each part and give a more detailed look at the part dealing with the mathematical theory of knots.

Part I deals with evidence for the use of knots from prehistoric times to the Egyptian civilization. The chronol­ogy of the knot technology over this vast period is summarized in Chapter 1 by studying various aspects of human and animal life which may have re­quired the use of knots. Chapter 2 is devoted to some speculations regard­ing the earliest forms of knots and their origin. Chapters 3 and 4 are written by archaeologists who examine the knot­ted structures found at many excava­tion sites in Europe and Egypt. The ear­liest such structure is a Mesolithic fish-net fragment found in 1913 on the Karelian isthmus (formerly in Finland) which uses a knot type used in Estonia and in the Finnish settlement. It is now called the "Antrea Russian Knot." The evidence of rock art and artifacts using knots in all major ancient civilizations, extends over thousands of years. This and other finds lead us to conclude that our prehistoric ancestors had both the materials and the skills for making complicated knots.

Part II has three chapters which give a sample of knot history in non-Euro­pean civilizations. The use of knotted cords or qui pus by the Incas is detailed in Chapter 5. There is an interesting discussion here of the construction of knotted cords and their use in repre­senting numbers and storing numerical data. Chapter 6 traces the rich history of knots in China covering a period of some eighteen thousand years. The main theme in this chapter is the de­scription of the decorative use of knots which began in ancient China and con­tinues to this day. Beautiful examples of such knots were much in evidence during ICM 2002 held in Beijing. Knots used by Inuit Eskimos are the subject of Chapter 7. I would like to note that knots and braids made by using strings and organic materials have been (and are now) used in religious functions in India since before the Vedic period. It

would be very interesting to study this, as excellent written sources are read­ily available.

Part III presents a historical account of two fields where knots have been extensively used. Chapter 8 deals with knots and ropes used by seamen and fishermen. The knowledge and use of knots was substantially affected in the transition of man from a land-dweller to a mariner. There is vast literature dealing with the use of knots and ropes at sea. The author has managed to give a very good presentation of this sub­ject in just a few pages. Chapter 9 dis­cusses what the author calls life-sup­port knots. It covers the use of knots in rock climbing, rescue work etc., with particular attention to their prop­erties in life-support tasks.

Part N is the longest, with five chap­ters. Behaviour, under load, of single stranded knots tied in fiber rope is studied in Chapter 10. Chapter 12 is de­voted to classification of knots by methods different from those used in topology. Various encyclopedias of knots are also briefly described. The work of Mandeville on trambling (i.e. producing sequences of knots by alter­ing one tuck at a time) is dealt with in Chapter 13. The last chapter is devoted to the history and techniques of cro­chet work. I found the discussion of the CADD (Computer Aided Doily Design) system and its applications developed by the author and her coworkers at the University of Waikato, New Zealand, quite interesting.

Chapter 1 1 should be of greatest in­terest to the readers of this magazine. To a mathematician, a knot is an em­bedding of a circle in the three-di­mensional Euclidean space R3 or its compactification, the 3-sphere S3. This definition is easily modified to obtain knots in any manifold. In particular, embedding of the standard unknotted circle is called the unknot. A system­atic study of knots was begun in the second half of the 19th century by Tait and his followers. They were moti­vated by Kelvin's theory of atoms mod­eled on knotted vortex tubes of ether. It was expected that physical and chemical properties of various atoms could be expressed in terms of prop-

erties of knots, such as the knot in­variants.

Though Kelvin's theory did not work, the theory of knots grew as a subfield of combinatorial topology. Tait classified the knots in terms of the minimal crossing number of a regular projection. Recall that a r·egular pro­

jection of a knot on a plane is an or­thogonal projection of the knot such that at any crossing in the projection exactly two strands intersect trans­versely. Tait made a number of obser­vations about some general properties of knots which have come to be known as the "Tait conjectures." In its sim­plest form the classification problem for knots can be stated as follows: Given a projection of a knot, is it pos­sible to decide in infinitely many steps if it is equivalent to an unknot. This question was answered affirmatively by W. Haken in 1961 , who proposed an algorithm which could decide if a given projection corresponds to an unknot.

However, because of its complexity it has not been implemented on a com­puter even after 40 years.

The simplest combinatorial invariant of a knot K is the crossing number c(K),

defmed as the minimum number of crossings in any regular projection of the knot K. The classification of knots up to crossing number 16 is now known [2]. The crossing numbers for some spe­cial families of knots are known, but the question of fmding the crossing number of an arbitrary knot is still unanswered. Another combinatorial invariant of a knot K that is easy to define is the un­

knotting number u(K), the minimum number of crossing changes in any pro­jection of the knot K which makes it into a projection of the unknot. Upper and lower bounds for u( K) are known for any knot K. An explicit formula for u( K) for a family of knots called torus knots, conjectured by Milnor nearly 40 years ago, has been proved recently by a num­ber of different methods.

One of the earliest investigations in combinatorial knot theory is contained in several unpublished notes written by Gauss between 1825 and 1844 and pub­lished posthumously as part of his estate. They deal mostly with his attempts to classify Tracifiguren or

plane closed curves with a finite num­ber of transverse self-intersections. Such figures arise as regular plane pro­jections of knots in R3. However, one fragment deals with a pair of linked knots. In this fragment of a note dated January 22, 1833, Gauss gives an ana­lytic formula for the linking number of a pair of knots. This number is a com­binatorial topological invariant. As is quite common in Gauss's work, there is no indication of how he obtained this formula. The title of the note "Zur Elec­trodynamik" ("On Electrodynamics") and his continuing work with Weber on the properties of electric and magnetic fields leads us to guess that it origi­nated in the study of the magnetic field generated by an electric current flow­ing in a curved wire. Maxwell knew Gauss's formula for the linking number and its topological significance and its origin in electromagnetic theory. In ob­taining a topological invariant by using a physical field theory, Gauss had an­ticipated Topological Field Theory by almost 150 years. Even the term topol­

ogy was not used in his era. It was in­troduced in 184 7 by J. B. Listing, a stu­dent and protege of Gauss, in his essay "Vorstudien zur Topologie" ("Preliminary Studies on Topology"). Gauss's linking number formula can also be interpreted as the equality of topological and analytic degree of a suitable function. Thus it can be con­sidered as an example of an index the­orem. Starting with this, a far-reaching generalization of the Gauss integral to higher self-linking integrals can be ob­tained. This forms a small part of the program initiated by Kontsevich [3] to relate topology of low-dimensional manifolds, homotopical algebras, and non-commutative geometry with topo­logical field theories and Feynman di­agrams in physics.

The Alexander polynomial provided a new type of knot invariant. There was an interval of nearly 60 years between the discovery of the Alexander polyno­mial and the Jones polynomial. Since then, a number of polynomial and other invariants of knots and links have been found. A particularly interesting one is the two-variable polynomial generaliz­ing both the Alexander polynomial and

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number � , 2006 55

the Jones polynomial. This polynomial is called the HOMFLY polynomial (a name formed from the initials of au­thors of the article [ 1 ]). The author of this chapter gives an excellent account of the history of topological knot theory and related theory of braids, bringing the readers up to the mid-1990s.

Some important recent develop­ments are not included in this chapter. Jones's work in the 1980s was a major advance in knot theory, leading to the resolution of several of the longstand­ing Tait conjectures. However, it did not resolve the chirality conjecture: If

the crossing number of a knot is odd,

then it is chiral (i.e. , not equivalent

to it,s mirror image). A 15-crossing knot which provides a counter-exam­ple to the chirality conjecture is given in [2]. Jones did not provide a geomet­rical or topological interpretation of his polynomial invariants. Such an in­terpretation was provided by Witten [6], who applied ideas from Quantum Field Theory (QFT) to the Chem­Simons Lagrangian. In fact, Witten's model allows us to consider the knot and link invariants in any compact 3-manifold M. Witten's ideas have led to the creation of a new area called Topo­logical Quantum Field Theory (TQFT) which, at least formally, allows us to express topological invariants of man­ifolds by considering a QFT with a suit­able Lagrangian. An account of several aspects of the geometry and physics of knots may be found in [4] and [5] .

Recently, several topological and geometric invariants of knots and links have been used in polymer chemistry and in studying the mathematical struc­ture of DNA. These early results have led molecular biologists to believe that knot theory may play an increasingly signifi­cant role in understanding the geomet­ric and topological properties of DNA and that these in turn may help in re­solving some of the riddles encoded in these basic building blocks of life. Un­derstanding the structure and dynamics of DNA, RNA, and proteins, in general, may very well require the forging of new mathematical tools.

The last part, V, deals with decora­tive knots and their use as symbols in heraldry and love. The history of

two widely practiced crafts, namely macrame and lace is given in Chapters 15 and 16, respectively. Chapter 17 de­scribes the various ways knots have been used in European Heraldry. The final chapter deals with the concept of love knot, its occurrence in literary works and the various forms that it has taken over the last five centuries or so. Indeed the most widely used synonym for marriage is "tying the knot."

I enjoyed browsing through all the chapters. They contain material that a mathematician would not normally come across in his work There is a well-known story about Alexander the Great unraveling the Gordian knot with his sword. Today's problems in knot theory, mathematical or otherwise, will require tools far more sophisti­cated than a sword.

BIBLIOGRAPHY

[ 1 ] R. Freyd, et a/., A new polynomial in­

variant of knots and links. Bulletin of Ameri­

can Mathematical Society (New Series,

1 2 :239-246, 1 985.

[2] J . Haste, M . Thistlethwaite, and J . Weeks.

The First 1 ,701 ,936 Knots. Mathematical ln­

telligencer, 20(4):33-48, 1 998.

[3] M . Kontsevich. Feynman Diagrams and

Low-Dimensional Topology. In First European

Congress of Mathematics, vol. I I Progress

in Mathematics, 1 20, pages 97-1 2 1 , Berlin,

1 994. Birkhauser. [4] K. B. Marathe, G. Martucci, and M . Fran­

caviglia. Gauge Theory, Geometry and

Topology. Seminario di Matematica deii'Uni­

versita di Bari, 262 : 1 -90.

[5] Kishore B. Marathe. A Chapter in Physical

Mathematics: Theory of Knots in the Sci­

ences. Mathematics Un/imited-2001 and

Beyond, pages 873-888. Springer-Verlag,

Berlin, 2001 .

[6] E. Witten. Quantum Field Theory and the

Jones Polynomial. Communications in Math­

ematical Physics, 1 21 :359-399, 1 989.

Max Planck Institute for Mathematics in

the Sciences, Leipzig, and

Department of Mathematics

1 3 1 7b Ingersoll Hall

City University of New York

Brooklyn College

2900 Bedford Ave.

Brooklyn, NY 1 1 21 0-2889, USA

e-mail: kbm@sci.brooklyn.cuny.edu

Codebreakers: Arne Beurling and the Swedish Crypto Program during World War I I by Bengt Beckman

PROVIDENCE, Rl, AMERICAN MATHEMATICAL SOCIETY,

2003. 259 PP , US $39, ISBN 0·821 8-2889-4

REVIEWED BY HAKAN HEDENMALM

The author of this book, Bengt Beck­man, is one of early members of

the Swedish cipher bureau FRA of Forsvarsstaben (Defense Staff Head­quarters), which was operational in 1941 but officially founded a year later. By the time Beckman came to FRA as a conscript in 1946, there was much discussion of Arne Beurling (1905-1986), the mathematics professor who had made himself famous at FRA by break­ing the German code of the Geheim­

schreiber (developed by Siemens in the 1930s) at a time pivotal for Sweden during World War II. This feat is of the same order of magnitude as the British effort to break the Enigma code dur­ing the same war. In England as well as in Sweden, mathematicians played a vital role for the intelligence deci­phering part of the war effort; perhaps the most famous mathematician work­ing for the British at Bletchley Park was Alan Turing. The Swedish effort to keep a low profile with regard to this intelligence gathering was quite suc­cessful; indeed, the importance of Beurling's contribution to Sweden's ability to keep out of the war was, un­til recently, known only to narrow cir­cles inside Sweden. By now, more than sixty years have passed, and the veil of secrecy has been lifted; Beckman, who stayed with FRA until 1991 , is able to tell the story as he remembers it, from what he picked up a long time ago, as well as from recent in-depth interviews with people more closely involved.

This story first aired in a 1993 Swedish Television documentary G sam i hemlig (G as in secret), pro­duced by Beckman and Olle Hager. In the book, Beckman is able to tell a

© 2006 Springer Sc1ence+Business Media, Inc., Volume 28, Number 1, 2006 57

much more detailed story, of course. With the translation to English, com­missioned by the American Mathemat­ical Society, the material is now avail­able to a considerably wider audience. The translator, Kjell-Ove Widman, who himself has been involved in cryptol­ogy at the Swiss company Krypto AG, has done a very thorough job and pro­duced a very readable text. However, the style differs a little from the origi­nal: Beckman tells a story by the camp­fire, but Widman's translation sets higher literary standards. Having for the moment assumed a slightly critical stance, let me also mention that the map of Stockholm on the page follow­ing xviii places Karlaplan a bit off, somewhere in the forest area north of the Royal Institute of Technology (KTH); the mark should be placed a lit­tle bit further southeast. It is also un­fortunate that the suggested explana­tion of Beurling's analysis is marred by cipher typos (on pp. 80, 82).

The book takes a historical per­spective on ciphers and begins with an exercise to decipher a coded message from the 18th century. This is quite en­joyable; the cipher is of simple substi­tution type, and it is just a matter of running a frequency analysis to guess the most common letters of the cipher key. Then the perspective changes a lit­tle, and automatic ciphering machines enter the picture, along with the Swedish names Damm, Hagelin, and Glyden. Then, a description of radio signal interception and cryptanalysis before 1939 follows.

The Kingdom of Sweden was poorly prepared for the war that broke out on the European continent on September 1, 1939. The situation became particu­larly dire on April 9, 1940, when Ger­many invaded neighboring Denmark and Norway. As Sweden could not af­ford a massive military build-up, it be­came imperative to be able to second­guess the German intentions regarding Sweden. Then Sweden's foreign policy could be modified to be more palatable for the Germans, and hence avoid ac­tual invasion. This was done quite suc­cessfully-German shipments of mate­rials and supplies as well as of troops were permitted in sealed transit trains through Sweden-and it is commonly

58 THE MATHEMATICAL INTELLIGENCER

believed that this is the reason Sweden was able to remain outside the war.

But this is not the whole story. The invasion of Norway offered Sweden the chance not to second-guess but to actually read the potential enemy's cards. The diplomatic traffic between occupied Oslo and Berlin was trans­mitted along Swedish telegraph lines, and the Swedes were able to tap the messages. The only problem was that the messages were not in plain text; moreover, the encryption was not of simple substitution type, as could be seen from a simple frequency analysis. Given the sheer volume of encrypted traffic, it was suspected that a machine was doing the encryption automati­cally. One day in 1940 Beurling, who had already been involved with some simpler cryptanalysis tasks for the Swedish Defense Department, col­lected the tapped telegraph traffic at the Karlaplan office dated May 25 and May 27, 1940, which he believed to be essentially free of transcription errors. After a couple of weeks, he had more or less cracked the code. This was an impressive feat, especially compared with the British Enigma effort, which was based on the physical capture of an encryption machine from the Ger­mans. If we think of the Geheim­

schreiber encryption as a kind of sub­stitution cipher, then the cipher key apparently was changing with each new letter of the message. Also, the ini­tial key settings were altered every few days. The way the Geheimschreiber

was made, it would not begin cyclically repeating its cipher on any given mes­sage, for the number of possible en­codings was much much bigger than the total quantity of information ex­changed over the entire war.

Beurling never revealed how he per­formed his feat; he would say that a magician never reveals his tricks. Nev­ertheless, Beckman offers a possible explanation, based on a reconstruction attributed to Carl-Gosta Borelius. The encryption may have been perfect in

theory, but in practice telegraph lines were not 100 percent reliable in those days, so the German telegraphers would frequently rerun parts of the message, using the same code. This al­lowed Beurling to get a foot in the

door, and using some sound hypothe­ses regarding the nature of possible codes on teleprinters (where each let­ter corresponds to a sequence of five Os and Is )-essentially combinations of permutations and transpositions­he was able to complete his task It should be noted that Beurling did this with a rather small data sample, and without actually having seen a Geheimschreiber. Today a Geheim­schreiber is on display in the Beurling library of the Mathematics Department at Uppsala University in Sweden, where Beurling worked in the 1940s.

At first, the Swedes carried out the deciphering manually, in accordance with Beurling's instructions, but later, and certainly by 1942, machines­called Apps-were doing the job. The value of having cracked the Geheim­schreiber depreciated toward the end of the war. The Germans sensed that their transmissions were being read, and reacted to it. By that time, how­ever, the risk that Sweden would get dragged into the war was much re­duced.

Beurling was a deep mathematician equipped with a difficult temperament. The stories about his disagreements, rows, or even outright fights with col­leagues are widely known in mathe­matical circles in Scandinavia. Some of these stories are retold in this book Beurling was apparently quite charm­ing to the ladies, and this aspect of his life, based on interviews with Anne­Marie Yxkull Gyllenband, takes up a chapter. He married twice. His first wife is not mentioned by name in the book, but it is known that she worked as a physician, and Beurling had two children with her. Later, in 1950, he met his second wife, Karin Lindblad, at the party his student Lennart Carleson hosted to celebrate his thesis defense at Varmlands nation, one of the student clubs in Uppsala. As far as I know, Karin was a friend of Carleson's, and was Forste Kurator at Viirmlands at the time, the highest post a student could assume at a student club in Uppsala. Karin and Arne remained together for the rest of their lives.

Beurling worked in three areas of mathematical analysis: potential the­ory, harmonic analysis, and complex

analysis. His collaborators were few

but well chosen: Ahlfors, Deny, Helson,

and Malliavin. Probably it was his im­

pressive collaboration and deep friend­

ship with Lars Ahlfors that landed him

the position of permanent member at

the Institute for Advanced Study in

Princeton, New Jersey, in 1954. In

Princeton, he took over the office pre­

viously occupied by Albert Einstein. It

is unfortunate that Beurling was not

able to continue on American soil the

extremely productive period he en­

joyed in Uppsala in the 1940s, with

students such as Goran Borg, Lennart

Carleson, Yngve Domar, Carl-Gustav

Esseen, and Bo I\jellberg. Had he been

able to wield more influence in Prince­

ton, the period of abstraction in math­

ematical analysis, which was such a

dominant theme in the 1950s and

1960s, might have been balanced by a

deep and elegant approach that fo­

cused not on form but on content.

Bengt Beckman has produced a fas­

cinating book that acquaints us with

some of the stars of 20th-century Scan­

dinavian mathematics. In addition, we

gain some insights into basic cryptol­

ogy.

Department of Mathematics

The Royal Institute of Technology

S-1 0044 Stockholm, Sweden

e-mail: haakanh@math.kth.se

From Newspeak to Cyberspeak. A History of Soviet Cybernetics by Slava Gerovitch

CAMBRIDGE, MA, THE MIT PRESS, 2002. xiv + 369 PP

US $42 ISBN 0-262-07232-7

REVIEWED BY PAUL JOSEPHSON

Most readers know of the impact

of ideological interference in the

practice of Soviet scientists. In the case

of biology, the peasant agronomist

Trofun Lysenko rose to the top of the

scientific establishment with Stalin's

personal endorsement. He advanced

Lamarckian notions of the influence

of acquired characteristics that sup-

planted genetics. After a 1948 national

conference that proclaimed Lysen­

koism the only true Soviet biology,

dozens of geneticists lost their jobs

and were exiled from the laboratories

to sheep-breeding farms at the end of

the empire; and genetics was purged

from university curricula. In physics,

too, several scientists joined ideo­

logues in an attack on quantum me­

chanics and relativity theory. The at­

tack nearly had disastrous results for

physicists, many of whom were forced

to abandon research. In both of these

cases, questions of epistemology, class

struggle, and other issues of impor­

tance in the official Soviet philosophy

of science, dialectical materialism,

played a role, as did Cold War pres­

sures to promote a new, Soviet science

that was different from, and better than

that in the West, particularly in the

United States.

In a superb contribution to the his­

tory and philosophy of science, Slava

Gerovitch considers the place of cy­

bernetics in Soviet philosophical dis­

putes, and the development of what

he calls "cyberspeak" in the postwar

USSR. Gerovitch, a research associate

at the Dibner Institute, has studied the

history of Soviet computing and cy­

bernetics for some time, and beyond

From Newspeak to Cyberspeak, is now

focusing on human-machine interaction

in the Soviet space program. Cyberspeak he defines as a universal language of man-machine metaphors described in

such terms as information and feedback

and control, e.g., the organism as an en­

tropy-reducing machine, the computer

as a brain, the brain as a computer.

Gerovitch joins several other scholars in

rejecting the notion that cybernetics was

severely damaged by ideological in­

terference in the form of such official

pronouncements that cybernetics was

a "reactionary pseudo-science." That in­

terference was relatively short-lived,

and scientists learned how to manage it.

Yet the fact that attacks were short­

lived and ignorant should not lessen

our appreciation of the way in which

they reflected the dangers of doing sci­

ence in the USSR generally. Those who

carried on the anti-cybernetics campaign

employed accepted ways of discourse

and dispute to rescue their careers and

EMI N ENT MATH EMATICIAN S

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ARTH U R CAYLEY Mathematician Laureate of the Victorian Age Tony Cri l ly Arthur Cayley ( 1 82 1 - 1 895) was one of the most prolific and important mathemati­cians of the Victorian era. His influence

sti I I pervades modern mathematics, in group theory (Cayley's theorem), matrix algebra (the Cayley-Hamilton theorem), and invariant theory, where he mode his most significant contribu­tions. Tony Cril ly, the world's leading a uthority on Cayley, provides the first definitive account of his life. $69.95 hardcover

S I R WI LLIAM ROWAN HAMILTON Thomas L. Hankins "This is an interesting, well-written biography of the great nineteenth­century mathematician." -Mathematical Reviews $22.50 paperback

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© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 1, 2006 59

grind their axes. These activities surely are seen in the West, but not with the fury and lasting costs to people and fields of research. Perhaps, as Gerovitch implies, because cybernet­ics was needed for radar and rocketry, the Cold War saved it from further at­tacks-in the way that research on the atomic and hydrogen bombs saved rel­ativity theory and quantum mechanics from ideological interference.

Gerovitch provides a detailed dis­cussion of what made attacks on cy­bernetics possible in the first place in a discourse that took place between the language of cybernetics and the language of Soviet ideology. He asserts that the ideological language was much more flexible than many historians have gathered. Soviet scholars learned to play by the rules of the establish­ment-the government bureaucracy and Party apparatus-in their rhetori­cal styles, modes of thought, and argu­ments to defend cybernetics. Like their opponents, who saw in cybernetics idealism, "kow-towing" to the West, among other dangers, they turned to quotation-mongering and label sticking (what others have called the "quote and club" method) to put them on the defensive. Gerovitch shows that the postwar ideological campaigns lacked coordination and coherence; they were rarely orchestrated from the top down.

In fact, Soviet cybernetics devel­oped along many of the same lines along which cybernetics developed in the West. Such scientists as Andrei Kol­mogorov and later Alexei Liapunov contributed to the foundations of cybernetics. Gerovitch discusses the work of these scientists against the background of comprehensive analysis of the contributions to the field of Nor­bert Wiener, Arturo Rosenblueth, and Claude Shannon.

Although cyberneticians were adept at disputation, there were significant pitfalls that awaited them until after Stalin's death. Their contributions were measured against the standards of the West, but they were required to avoid showing contamination with Western ideas. Soviet scholars had to avoid falling behind the West in com­puting and simultaneously following

60 THE MATHEMATICAL INTELLIGENCER

Western trends too closely. At the same time, a Cold War ideological battle against the West led to the cutting off of international contacts. Intellectuals were forced to toe the Party line. In this environment, the boundaries between academic and political disputes disap­peared.

An important facet of the history of computer science and the first Soviet computing machines was the connec­tion between cyberneticians and their powerful military and Communist Party patrons. One such example is Mikhail Lavrent'ev, who, as director of the In­stitute of Computer Technology, re­ceived facilities and protection from the new Moscow City Party Chief, Nikita Khrushchev. Later, as Party leader, Khrushchev enabled Lavrent' ev to found the Siberian city of science, Akadem­gorodok, with its own new Computer Center in the late 1950s. When it came to military purposes, computers were a technology without ideological devia­tions.

After the death of Stalin, in the mid-1950s Soviet computers were deified and entered the public realm, no longer to be held under wraps of military se­crecy. Cyberneticians touted comput­ers as paragons of objectivity based on quantitative knowledge, precise lan­guage, and precise concepts. They also commenced an attack against past ide­ological interferences of the philoso­phers and their allies among scientists and Party officials. Gerovitch's discus­sion of the battle against dogmatism and calcification of philosophical dis­course under Stalin at this time is en­gaging. Ultimately, Gerovitch writes, cyberneticians "overturned earlier ideological criticisms of mathematical methods in various disciplines and put forward the goal of the 'cybernetiza­tion' of the entire science enterprise" in search of objectivity in the life sci­ences and social sciences alike. (p. 199) This conclusion reinforces the sense that an excessive scientism de­veloped in the USSR.

In the 1960s cybernetics became a full-fledged science in the Soviet es­tablishment. It found an institutional home in a council of the Academy of Sciences of the USSR, experienced

rapid institutional growth, and saw a new publication, Problemy kiberni­

tiki, which was one of the most influ­ential such publications in the world. Its promoters claimed that cybernetics had become a universal, objective lan­guage, and as such would break inter­disciplinary barriers and legitimize the use of mathematical methods in other sciences. Its power was evident in the fact that even before Lysenko was de­posed, cyberneticians promoted ge­netic research in physics and chem­istry institutes, speaking about genes as units of hereditary information. Sup­porters thought cybernetics might be a panacea for reforming the Soviet economy through the creation of "opti­mal" models for planning and manage­ment. But scientists ultimately realized that central mainframes in large-scale systems to centralize input and output calculations for such a huge economy were simply unfeasible.

Once institutionalized, Gerovitch concludes, cybernetics became part of the Soviet establishment in service of the nation's management, administra­tion, direction, and government pur­poses. The goal was to control the en­tire national economy, technological processes, and so on, to ensure opti­mal governance. An alliance between cybernetics and dialectical materialism followed in the early 1960s. In the end, cyberspeak became so much a part of establishment thinking, so much the mode of dominant discourse, that its supporters grew disillusioned with ef­forts to apply it willy-nilly. Fissures in the cybernetics community as in Soviet society itself created new disputes. Some scientists had grown increasingly conservative and anti-Semitic, while others joined the dissident movement to protest increasing violations of human rights under Leonid Brezhnev. This sug­gests that scientism or not, cybernetics, like other sciences in other countries, could not avoid reflecting the social, po­litical, and cultural norms of the nation in which it developed.

Gerovitch only touches on reasons why the USSR failed to embrace the computer revolution, some of which have roots in the debates over "think­ing machines" that occurred from 1950

to 1965. He does not consider the de­cision to build computers based on reverse engineering after so many decades of success in building indige­nous machines. In addition, because his focus is on philosophy and intel­lectual history, some readers will need to seek other sources to gather the im­pact of the broader context of Soviet history and politics on cybernetics.

Gerovitch's study is based on a thor­ough use of archival materials and unpublished memoirs and interviews. This book will be of interest to ad­vanced undergraduate students, gradu­ate students, and teachers, as well as to computer scientists, historians, and philosophers. I recommend it highly.

Program in Science, Technology and Society

Colby College

5320 Mayflower Hil l

Waterville, ME 04901 -8853

USA

e-mail: prjoseph@colby.edu

Traditiona l Japanese Mathematics Problems of the 1 8th and 1 9th Centuries by Hidetoshi Fukagawa and

John F. Rigby

SINGAPORE, SCT PUBLISHING, 2002. 1 91 PP. US$50.00

ISBN 981 -04-2759-X

Japanese Temple Geometry Problems San Gaku by Hidetoshi Fukagawa and

Dan Pedoe

WINNIPEG, THE CHARLES BABBAGE RESEARCH

CENTRE, 1 989. 206 PP. US$40.00 ISBN 0-91961 1 -2 1 -4

REVIEWED BY CLARK KIMBERLING

The best thing about these books is their content, which is based on

problem proposals carved and drawn on Japanese wooden tablets dating from a span of isolation from the West. During that time Japanese mathemati­cians developed their own "traditional

mathematics," which, in the 1850s, be­gan giving way to Western methods. There were also changes in the script in which mathematics was written, and as a result, few people now living know how to interpret the historic tablets. One of these is Hidetoshi Fukagawa, the Japanese author of the two books. The 1989 book opens with these words:

A selection from the hundreds of problems in Euclidean geometry displayed on devotional mathemati­cal tablets (SANGAKU) which were hung under the roofs of shrines or temples in Japan during two cen­turies of schism from the west, with solutions and answers. Implicit in this description is the def­

inition of sangaku (often written San gaku and Sangaku ) . The phrase "with solutions and answers" applies to the books, not the sangaku. Dan Pedoe, co-author of the 1989 book, explains in the Preface:

There were few colleges or univer­sities in Japan during the period of separation from the west, but there were many private schools, and obviously many skilled geometers who wished to thank the god or gods for the discovery of a particularly lovely theorem, and also, it may be guessed, who were not averse to dis­playing their discoveries to other geometers . . . with the implicit chal­lenge: "See if you can prove this!" The 2002 book continues the col-

lection with additional problems and solutions. For both books, many of the solutions use modem methods. With admirably little overlap in content, the two books give historical descriptions, photographs, figures, calligraphy, solu­tions, and references, all well focused on sangaku. For a broader context, one may cite Chapter 22 of Yoshio Mikani's The Development of Mathe­

matics in China and Japan, second edition, Chelsea, 1974 (originally pub­lished in German, 1913).

Mikani places sangaku in the per­spective of Seki Kowa, who has been called the Japanese Newton and father of Japanese mathematics. Although Seki's lifetime (1642-1708) preceded sangaku, his influence in algebraic and analytic methods set the stage for san-

gaku. Mikani writes, "The highest de­velopment of the Japanese mathemat­ics must of course be looked upon as the invention of . . . 'circular theory' "­and it is precisely the enchantment of circle-problems that pervades san­

gaku. Indeed, a majority of the prob­lems in the 1989 and 2002 books in­volve circles.

One of the foremost mathemati­cians represented in sangaku was Ajima Chokuen (1732-1798). (The fam­ily name is Ajima. The given name Chokuen is used in the 1989 book, but the more formal given name Naonobu is used in the 2002 book) The two books contain a number of spinoffs from Ajima's famous problem about three pairwise tangent circles in­scribed in a triangle. A view of Ajima's place in the books provides insights into the organization and mathematical tone of the two books and also gives insights into the work of one of the leading representatives of Japanese "traditional mathematics" (as it is called in the 2002 book).

On pages 28-30 of the 1989 book, Ex­ample 2.3 and Problems 2.3. 1 to 2.3. 7 are presented under the heading "Three Cir­cles and Triangles," followed by Exam­ples 2.4(1) and 2.4(2) and Problems 2.4.1 to 2.4. 7 under "Four Circles and Trian­gles." The presentation is in two-column format with figures in the right column. Each problem proposal is labeled "Ex­ample" or "Problem." Here are three items from the 1989 book:

Example 2.3: The three circles, 01Cr1), Oz(rz), and 03(r3) have ex­ternal contact with each other. The triangle ABC is formed by the com­mon tangents to the circles. Find the radius of the incircle of triangle ABC in terms of r1. r2, and r3.

Example 2.4( 1 ): I(r) is the incircle of triangle ABC, and the circles 01Cr1) , Oz(rz), and 03(r3) respec­tively touch AB and AC, BA and BC,

and CA and CB, and all touch I(r)

externally. Show that

r = Vr)r; + y:;;; + v;;;:;-. Problem 2.4.1: ABC is a triangle, I(r) its incircle. The circle 01(r1) touches AB and AC produced and

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 2006 61

also I(r) externally, and 02(r2), and 03(r3) are defined similarly. Show that

1 1 1 1 - = -- + -- + --

r v;v; y,;;; � . A footnote refers to page 106, in Part

I, titled "Solutions to Selected Problems and Answers." There, under the heading "The Malfatti Problem," solutions and historical comments are given. The con­struction of the circles as in Example 2.3 is often attributed to Malfatti, but the historical comments note that Ajima posed and solved the problem about 30 years before Malfatti did.

In modem times, this Ajima-Malfatti construction has received consider­able attention. For example, let A' be the touchpoint of circles 02(r2), and 03(r3), and cyclically, let B' = 03(r3) n 01(r1) and C' = 01Cr1) n 02Cr2). Then M'B'C' is perspective to MEG, and the perspector (John Conway's im­provement over "center of perspec­tive") is a point known as the Ajima­Malfatti point. For a discussion, visit the Encyclopedia of Triangle Cen­

ters-ETC:

http://facu1ty.evansvil1e.edu/ck6/ encyclopedia/ETC.htm1

and scroll down to X(179). Peter Yff found remarkable trilinear coordinates for the Ajima-Malfatti point:

(These are respectively proportional to the distances from the point to the sidelines BC, CA, AB.)

The 2002 book states "Ajima's Theo­rem" at the beginning of Chapter 4 and "Ajima's second theorem" at the begin­ning of Chapter 5. The second theorem is elsewhere cited in the book on sev­eral pages, as is a third theorem attrib­uted to Ajima, labeled "Boushajutu".

Another traditional mathematician represented in both books was Shoto Kenmotu (1790-1871). His configura­tion in Problem 3.2. 1 of the 2002 book is the earliest known (1840) construc­tion of a point now known as the Ken­motu point. The configuration, involv­ing three congruent isosceles right triangles, extends easily to the one de­picted here.

62 THE MATHEMATICAL INTELLIGENCER

A

The three congruent squares meet in the Kenmotu point, indexed as X(371) in ETC. Trilinears were found by John Rigby:

cos(A - 7T/4) : cos(B - 7T/4) : cos(C - 7T/4).

Both books are partitioned into Part 1 (problems) and Part 2 (essentially, solutions and comments). The 1989 book has chapter headings (1) Circles, (2) Circles and Triangles, (3) Circles and Polygons, (4) Polygons, (5) El­lipses (and One Hyperbola), (6) El­lipses and Circles, (7) Ellipses and Polygons, (8) Ellipses, Circles and Quadrilaterals, and (9) Spheres. The to­tal number of problems is 249.

The 2002 book chapter headings are (1) Number theory, (2) Numerical Analysis, (3) Geometry ofPolygons, (4) Geometry of Circles, (5) Geometry of Circles and Triangles, (6) Geometry of Ellipses, (7) Solid Geometry, and (8) Maxima and Minima. There are 287 problems.

In both books, the organization of material is indeed wonderful, when you stop to think that consecutive closely related problems started out in temples scattered across Japan. For example, in the 1989 book, for the nicely sequenced Problems 2.3. 1 to 2.3.5, we read in Part 2 that these orig­inated in various prefectures at various times: Fukusima in 1891, Iwate un­dated, Iwate in 1842, Miyagi in 1857, and Fukusima in 1901. Regarding such places and dates, Dr. Hiroshi Kotera of­fers a very attractive website:

http://www.wasan.jp/english/

(There is also a Japanese version.) There you will find a Clickable SAN­GAKU Map of Japan, showing 24 la­beled prefectures in which sangaku are

found. By clicking any of them, you will be able to examine individual tablets on which problems are posed. In par­ticular, you can click Tokyo, then se­lect English, and see not only a very well preserved tablet, but also markers indicating Ohkunitama Shrine, a date of March 1885, the text of the problem, and so on. Another website is also rec­ommended: Tony Rothman's (with the cooperation of Hidetoshi Fukagawa) Japanese Temple Geometry,

http://www2.gol.com/users/ coynerhm/0598rothman.html

Both books have extensive bibli­ographies. The 1989 book gives 78 items, and the 2002 book gives 109. The 2002 book has an Index.

Fukagawa's coauthor of the 1989 book, Dan Pedoe, is well known as the author of Circles (Pergamon, 1957), in which he wrote, regarding the nine­point circle, "This circle is the first really exciting one to appear in any course on elementary geometry." (This famous line is quoted at the beginning of a section in Coxeter's Introduction

to Geometry.) The coauthor of the 2002 book, John

F. Rigby, is well-known in geomet­ric circles. For example, Ross Hons­berger's Episodes in Nineteenth and

Twentieth Century Euclidean Geometry (Mathematical Association of America, 1995) reserves a page on which special acknowledgment is given to John Rigby for his contributions to the book. Pages 132-136 introduce a Rigby point which serves as a seed for families of points in ETC beginning at X(2677). Two other Rigby points are indexed in ETC as X(1371) and X(1372). For more on both kinds of Rigby points, visit Eric Weisstein's Math World:

http://mathworld. wolfram. com/ RigbyPoints.html.

Physicist Freeman Dyson's Fore­word (or "Forward", as it not inappro­priately appears) to the 2002 book is a noteworthy piece, reminiscent of his Imagined Worlds (Harvard University Press, 1997). The Forward opens with these words: "One of the most impor­tant scientific enterprises of the twen­tieth century is the search for ex­traterrestrial intelligent species." Once

contact is made, there will be the prob­lem of communication. Dyson writes, "Eighteenth-century Japan is stranger to me, in language and in historical tra­dition, than any other past or present culture on this planet. To my delight, I see in Fukagawa's books a collection of mathematical messages that are profoundly strange but none-the-less intelligible. Fukagawa has collected and arranged these messages so that their strange beauty is now accessible to everyone, eastern and western alike."

The 1989 book can be ordered di­rectly from The Charles Babbage Re­search Centre, P. 0. Box 272, St. Nor­bert Postal Station, Winnipeg, Canada R3V 1L6. To order the 2002 book, send a check payable to Mathematics

and Informatics to Susan Wildstrom, 10300 Parkwood Drive, Kensington, MD 20895-4040, USA. Include a letter telling whether you wish to receive the book (from Singapore) by surface mail (total $50.00) or by air mail (total $60.00).

Department of Mathematics

University of Evansville

1 899 Lincoln Ave

Evansville, IN 47722-0001 USA

e-mail: ck6@evansville.edu

Mathematics: A Very Short Introduction by Timothy Gowers

NEW YORK, OXFORD UNIVERSITY PRESS, 1 56 PP.

US $9.95, ISBN 0·1 9·285361 ·9

REVIEWED BY JEAN-MICHEL KANTOR

This is a very short review of Math­ematics: A Very Short Introduc­

tion, by Timothy Gowers, a very smart mathematician and an excellent com­municator of mathematics (the two concepts are distinct). Gowers's 1998 Fields Medal might be considered a proof of the first statement; this book, together with his already famous Clay Lecture on the importance of mathe­matics [ 1 ] , an argument for the second. He has done a wonderlul job in pro­ducing this rich little book on the es­sentials of mathematics.

The main part consists of three chapters:

• Models (turning practical problems into mathematical ones).

• The abstract method (or axiomatic method), for which he gives good ar­guments showing its power and sug­gesting its pedagogical usefulness (I will not open here the Pandora's box of discussing didactics).

• Proofs, which some people consider to be at the heart of mathematics. Gowers gives examples of proofs, and of seemingly obvious statements that need proofs.

The rest of this charming book is made up of stories, illustrations, ex­amples, and well-illustrated concepts such as limits, estimates, dimension, infinity . . . . Gowers has chosen a list of simple concepts of great mathemat­ical importance, which he presents to the curious reader. Finally, the chapter "Some frequently asked questions" re­sponds to what people generally ask about mathematicians (or so mathe­maticians imagine). Among the ques­tions, "Why do so many people posi­tively dislike mathematics?"

When people asked Henri Poincare why they never understood mathemat­ics at school, he answered them, "What I don't understand is that people don't understand mathematics!". Gowers sug­gests using the abstract approach. We wish him all the best!

I would strongly recommend this book to first-year undergraduates, al­though professional mathematicians will also find it a useful introduction to many beautiful examples on Gowers's Web site, such as the astonishing re­sults that won him the Fields Medal [2]. The general audience too will fmd much of interest on the Web site, in­cluding an article about what is defin­able in mathematics, and the text ver­sion [3] of the Clay Lecture, in which Gowers shows how mathematics is a subject where importance and beauty are connected.

REFERENCES

[1 ] The Importance of Mathematics. A Lecture

by Timothy Gowers, Springer VideoMATH,

2002. VHS/NTSC video tape. ISBN 3-540-

92652·6.

[2] http ://www . dpmms.cam .ac. uk/�wtg1 0

/vsipage.html

[3] http://www. dpmms.cam . ac . uk/�wtg 1 0/

importance.pdf

ADDED IN PROOF

See also The Princeton Companion to Mathe­

matics, under the supervision of T. Gowers,

to appear in 2006, Princeton University Press.

Universite de Paris VII

75251 Paris Cedex 75005

France

e-mail: kantor@math.jussieu.fr

Basebal l's Al l-Time Best H itters by Michael Schell

PRINCETON, NJ: PRINCETON UNIVERSITY PRESS, 1 999.

xxi + 295 PP. US $1 7.95 ISBN 0691- 1 2343·8 (PAPER)

REVIEWED BY JIM ALBERT

Baseball is one of the most popular tean1 games in the United States.

Professional baseball started near the end of the 19th century. Currently in the United States and Canada, there are 30 professional teams in the Amer­ican and National Leagues, and mil­lions of people watch games in ball­parks and on television.

Baseball is a game between two teams of nine players each, played on an enclosed field. A game consists of nine innings. Each inning is divided into two halves; in the top half of the inning, one team plays defense in the field and the second team plays of­fense, and in the bottom half, the teams reverse roles. The team that is batting during a particular half-inning, the of­fensive team, is trying to score runs. A player from the offensive team begins by batting at home base. A run is the score made by this player who ad­vances from batter to runner and touches first, second, third, and home bases in that order. A team wins a game by scoring more runs than its opponent at the end of nine innings.

A basic play in baseball consists of a player on the defensive team, called a pitcher, throwing a spherical ball (called a pitch) toward the batter. This

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 1 , 2006 63

Batting Career At-

Rank Player Average bats Career

Ty Cobb 0.3664 11434 1905-1928

2 Rogers Hornsby 0.3585 8173 1915-1937

3 Joe Jackson 0.3558 4981 1908-1920

4 Lefty O 'Doul 0.3493 3264 1919-1934

5 Ed Delahanty 0.3459 7505 1888-1 903

6 Tris Speaker 0.3447 1 0195 1907-1928

7 Ted Williams 0.3444 7706 1939-1960

8 Billy Hamilton 0.3443 6268 1888-1901

9 Dan Brouthers 0.3421 6711 1879-1904

10 Babe Ruth 0.3421 8399 191 4-1935

confrontation is called the batter's at­bat. The batter is attempting to make contact with the pitch using a smooth round stick called a bat. A strike is a pitch that is struck at by the batter and missed, or is not struck by the batter and passes through a region called the strike zone. A ball is a pitch that is not struck at by the batter and does not en­ter the strike zone in flight.

After a number of thrown pitches, the batter will either be put out or be­come a runner on one of the bases. The batter may be put out in several ways: (1) he hits a fly ball (a ball in the air) that is caught by one of the field­ers, (2) he hits a ball in "fair" territory, and first base is tagged before the bat­ter reaches first base, (3) a third strike is caught by the catcher.

A hitter can advance to become a runner and reach base safely by: (1) Receiving four pitches that are balls (outside of the strike zone). In this case, the batter receives a walk or base-on­balls and can advance to first base. (2) Hitting a ball in fair territory that is not caught by a fielder or thrown to first base before the runner reaches first base. There are different types of hits depending on the advancement of the runner on the play. A single is a hit when the runner reaches first base, a double is a hit when the runner reaches second base, a triple is a hit when a runner reaches third base, and a home

run is a "big" hit (usually over the out­field fence) when the runner advances around all bases safely.

Rating Players

Baseball players are rated with respect to their ability to hit and their ability to field. The classical measure of a

64 THE MATHEMATICAL INTELLIGENCER

player's ability to hit is the batting average (AVG), defined as the propor­tion of "official at-bats" (essentially a player's at-bats that are not walks) when the player gets a base hit. Above is a table listing the players since the beginning of professional baseball (1876) that have the ten highest batting averages. From this table, it appears that Ty Cobb is the best baseball hitter of all time. But was he really the best hitter? This book provides a serious at­tempt to make the proper adjustments to batting average so that one can make a fair comparison of players who played during different time periods.

Why Consider Batting Average?

It should be clarified what this book is about and what this book is not about. Although batting average (AVG) has been the standard way of measuring a player's batting ability since the begin­ning of professional baseball in 1876, it is well known that AVG is a relatively poor measure. The objective of a hitter is to help create runs for his team, and there are alternative measures of a player's hitting success that are much more positively associated with runs than AVG. Albert and Bennett (2003) give an overview of the many superior alternatives to AVG.

The book under review has been criticized by many people for compar­ing hitters with respect to batting average instead of alternative "good" measures of batting performance. I think this criticism is unfair. Schell makes it clear in the introduction that his analysis is not finding the best "bat­ters," but rather the best "hitters"-the ones who have the best chance of get­ting a hit. Given the current popularity

of the AVG as a measure of hitting per-formance, I believe that this is a rea-sonable aim. The definition of bat-ting average is carefully explained, and Schell can focus his efforts on the proper adjustment of this measure to make comparisons of players. (Schell is currently completing a second book, Premier Batsmen, that makes similar adjustments to other measures of hit-ting performance.)

What Adjustments Should

Be Made?

To illustrate some of the issues dis­cussed in this book, let's compare two great baseball hitters, Ty Cobb and Ted Williams. On the surface, Cobb appears to be the better hitter, for his career batting average was 0.366 compared to Williams's 0.344. But this superfi­cial comparison ignores some relevant concerns. First, we question whether 0.366 and 0.344 are representative mea­sures of the ability of the two players to get hits. Ty Cobb played baseball for 24 seasons. Figure 1 plots his season batting averages against his age.

In it, a smooth-fitting (loess) curve is placed on top of the plot. We see a general pattern in this plot that is typ­ical for many players. Baseball players, like many other athletes, mature and become more proficient in performance until a particular age, after which their performance deteriorates until retire­ment. A player's career batting average includes the periods of maturation and deterioration during which the player has a low batting average. A better measure of hitting performance might be the batting average for a player dur­ing the middle years of his career.

A second general concern in com­paring Cobb and Williams is that they played baseball during different eras and their careers did not overlap. The basic rules of baseball have remained the same over the years, but the play­ing conditions that affect the effective­ness of the pitchers have changed, and these changes have had a significant ef­fect on the batting averages of players. It is difficult to say that Cobb was a bet­ter hitter than Williams, because they played against different pitchers and the game was different in the two eras. For these reasons, a reported batting

• •

0.4 •

0.35 (!) �

0.3

0.25 •

20 25 AGE 30 35 40

careers. In the "Second Base" chapter,

Schell makes an adjustment for seasons

during which the batters had general ad­

vantages or disadvantages versus the

pitchers-he calls these periods hitting

feasts and famines. A mean adjusted bat­

ting average is computed, which is the

ratio of the player's batting average to

the average league batting average for

that particular season. This adjustment

diminishes the achievement of players

that had high batting averages during pe­

riods where hitting was relatively easy.

More subtle adjustments to batting

average are made in the "Third Base"

and "Home" chapters. The third ad­

justment accounts for differences in

the talent pool during different eras.

Schell claims that the standard devia­

tion of players' batting averages is a

useful measure of the talent pool of

players, and small standard deviation

values indicate more talent. A new ad­

justed batting average is proposed that

takes into account the difference in tal­

ent pools. This method essentially as­

sumes a percentile ranking for each

player for the era in which he played,

and then combines the percentile rank­

ings into a single list. In the "Home

Base" chapter, Schell makes the final

adjustment. A player's batting average

is adjusted for the ballpark where he

played half of his games. This is a te­

dious calculation, for adjustments have

to be made for each of the ballparks

where Major League baseball has been

played over the past 130 years. This

method multiplies a batting average by

a specific "ballpark effect."

Figure 1

average of, say, .320 is not very mean­

ingful unless we understand when this

average was achieved. Schell makes

the plausible assumption that the num­

ber of great hitters has stayed constant

over the years, and it is not meaning­

ful to look at a player's absolute bat­

ting average. Rather, he believes, a

player's batting average should be

judged relative to the batting averages

of other players who played during the

same seasons.

A third concern is related to the pool

of talented players during different

eras of baseball. Schell describes how

the recruiting pool of potential ball­

players has changed over the years.

Most of the players in the early years

of baseball were whites from the North­

east United States. In later years, base­

ball included players from the South,

later still, African-Americans, and now

players from Latin America and Asia.

The change in U.S. population and the

source of eligible players has had a sig­

nificant impact on the general talent of

the group of professional players. Gen­

erally, it can be said that the range of

talent has decreased over the years

(players are currently more homoge­

nous in terms of ability), and this

change should affect the comparison

of batting averages from different eras.

A final concern is that all ballplay­

ers play half of their games in their

home ballpark and half of their games

in ballparks away from their home

town. There are significant differences

in the dimensions of the ballparks, the

weather, the type of grass, and the con­

dition of the pitcher's mound (the hill

from which the pitcher throws), and

these ballpark characteristics can have

a significant effect on hitting statistics.

If one player hits for a 0.300 batting av­

erage in a "hitter's ballpark" and a sec­

ond player hits for an average of 0.300

in a "pitcher's ballpark," then one

would rate the second hitting perfor­

mance as superior because it was more

difficult to get hits in his ballpark

Thus, one needs to adjust a player's

batting average for his ballpark

Making the Adjustments

Schell carefully describes how to make

the four types of adjustments de­

scribed above to find the 100 best base­

ball hitters of all time. He starts by only

considering "qualifying" players-the

retired players who had at least 4000

at-bats and the current players who

have had at least 8000 at-bats. Note that

one of the players in the above list,

Lefty O'Doul, would be excluded from

the top list because he didn't have the

required 4000 at-bats. With this exclu­

sion, Schell displays the "traditional"

list of 100 best hitters, and in the four

chapters that follow (called "First

Base," "Second Base," "Third Base,"

and "Home"), adjusts the traditional

list using the four concerns.

In the "First Base" chapter, Schell

adjusts a player's batting average for

late career declines. Essentially, this

method adjusts a career A VG by con­

sidering only the proportion of hits in

the first 8000 at-bats. This adjustment

has the effect of raising the batting av­

erages of players whose averages de­

teriorated in the closing years of their

And the Best Hitter Is?

After performing all of these adjust­

ments to batting average, who are the

best hitters of all time? As expected,

many of the players who had high bat­

ting averages during the feast period of

the 1920s and 1930s get devalued, and

many of the current players rise in the

ratings. For example, Roger Hornsby,

who played in the 1920s, drops from

second on the traditional list to fifth on

Schell's list, and Rod Carew, a 1970s

player, rose from 28th (traditional) to

3rd (Schell). Surprisingly, the best hit­

ter of all time is Tony Gwynn, who re­

tired in the 2001 season. Schell's re­

sults received much media attention,

© 2006 Springer Science+Business Media, Inc., Volume 28, Number I , 2006 65

and the author had an opportunity to

meet with Gwynn at the San Diego ball­

park to discuss his findings.

General Assessment

This book provides a nice historical

view of baseball from a statistical

perspective. There have been notable

changes in the game over the years,

and one can view these changes by use

of time series plots of the appropri­

ate statistics. Through this historical

study, one learns a lot about the great

hitters of all time. Also, although many

baseball fans may disagree about the

final ranking of best hitters, the author

uses scientifically sound methods to

make the necessary adjustments in bat­

ting average. Although I might choose

different methods to make my adjust­

ments, I suspect that I would reach

similar conclusions. My only concern

in reading this book is the relatively

high technical level of the presentation;

the book can be read only by baseball

fans with a quantitative background,

yet most fans would be interested in

the conclusions. I also believe that the

adjustment procedures described here

would be helpful in constructing simi­

lar types of adjustments to measures of

performance in other sports.

REFERENCE

Albert J. and Bennett, J . , Curve Ball, Springer

(2003).

Department of Mathematics and Statistics

Bowling Green State University

Bowling Green, OH 43403

USA

e-mail: albert@math.bgsu.edu

Teaching Statistics Using Basebal l by Jim Albert ---.. -- ·-----

WASHINGTON. DC, MATHEMATICAL ASSOCIATION OF

AMERICA, 2003. 304 PP US $46.95. ISBN 0·88385-727-8

(PAPER)

REVIEWED BY MICHAEL J. SCHELL

Teaching Statistics Using Baseball (TSUB hereafter) addresses vari­

ous issues in statistics for which base-

66 THE MATHEMATICAL INTELLIGENCER

ball can provide insights. The author,

Jim Albert, suggests that TSUB "can be

used as the framework for a one-se­

mester introductory statistics class

that is focused on baseball" or "as a re­

source for instructors who wish to in­

fuse their present course in probability

or statistics with applications from

baseball." Regarding the first sugges­

tion, I would call such a course "sta­

tistics appreciation," for TSUB does

not cover the traditional statistics

material contained in introductory

courses, and very little formal infer­

ence is done.

The topics of TSUB are closely

linked with those of [2] . The book is or­

ganized into 9 chapters. The first chap­

ter is an introduction to the book; the

remaining chapters are composed of

an average of 5 case studies and 19 ex­

ercises, including "leadoff exercises"

featuring Rickey Henderson, probably

baseball's greatest leadoff hitter.

Chapter 2 introduces stemplots

(also called stem-and-leaf diagrams),

the "five-number summary" of a dis­

tribution (minimum, maximum, and

three quartiles ) , histograms, and graph­

ing of time series data. Chapter 3 com­

pares batches of data, principally using

parallel stemplots, boxplots, or stan­

dardization. Chapter 4 examines rela­

tionships between variables using scat­

terplots, correlation coefficients, and

regression, using the RMSE to evaluate

how well alternative measures (like

batting average and slugging average)

predict run-scoring. In Chapter 5, prob­

ability concepts are introduced and

applied to the design of baseball table­

top games. Plots of cumulative rates of

some baseball statistics (like the

earned run average of a pitcher over

the course of a season) are also fea­

tured. In Chapter 6, the binomial dis­

tribution and goodness-of-fit Pearson

residuals are presented in the case

studies, and the Poisson and geomet­

ric distributions are introduced in the

exercises. Chapters 7 and 8 focus on

statistical inference. Typically, data are

simulated according to various distri­

Obutional assumptions, and inference

among alternatives is conducted using

Bayes's rule, probability intervals, and

visual inspection of graphs. Chapter 9

models a baseball game as a Markov

Chain and uses transition probability

matrices to derive conclusions.

The case studies and exercises pose

interesting baseball questions, and the

book and accompanying Web site pro­

vide easy access to much baseball data.

Here are a couple of my favorites:

Case Study 2-5 presents an interest­

ing look at managerial strategy, by

studying the use of the sacrifice bunt.

When the dotplot of sacrifice attempts

is displayed in parallel (separately) for

the American and National Leagues,

there are noticeably fewer bunts in the

AL. The designated hitter rule provides

a "simple explanation for this discrep­

ancy."

Case Study 9-4 is built on previous

case studies in Chapter 9 and uses

Markov Chain concepts for its devel­

opment. Within an inning, there are 24

possible "states," based on the number

of outs (0, 1, or 2) and the runner con­

figurations (e.g., bases loaded). From

this model, Albert estimates the average

value of a home run to be 1.40 runs to

the team. This corroborates similar esti­

mates obtained by others by regressing

wins on various team batting statistics.

Albert has adroitly kept the statistical

development work to the minimum re­

quired to solve the question.

In other case studies, though, the

author simplifies the analyses ( un­

doubtedly for pedagogical reasons).

Students should know that these analy­

ses lack the rigor needed for more de­

finitive answers to the questions posed.

I now provide several such cases as an

aid to the potential instructor.

Case Study 3-5 is close to my heart,

for it involves comparison of four of the

best single-season batting averages in

baseball history. Albert standardizes the

raw averages by using the means and

standard deviations of "regular players"

(those with 2::400 at bats) from the four

seasons. For Rod Carew's 1977 and Tony

Gwynn's 1994 seasons, Albert obtains

batting average z-scores of 4.07 and 3.15,

respectively. Given that Gwynn's season

ranks first in my book, [3] (Carew's sea­

son ranks fourth), I was surprised that

the z-score differential in TSUB was so

disparate. Two simplifications of the

problem led to a substantial shift in the

findings. First, batting averages were

not adjusted for the players' home ball-

park in TSUB. Second, the definition of

a "regular player" biased the standard­

ization, because a players' strike short­

ened the 1994 season. Consequently,

only 63 players (2.2 players per team)

from 1994 were "regular players" com­

pared to 168 ( 6. 5 players per team) from

1977. These simplifications gave Carew

13 and 16 net batting points over

Gwynn, respectively, compared to the

more detailed analysis given in my

book, which yielded z-scores of 4.24 for

Gwynn and 4.03 for Carew.

For Case Study 7-3, Albert shows re­

sults from 150 simulations but notes

that they "aren't too precise," so he re­

simulates the example 1000 times.

Then, in actually solving the problem,

he uses about 333 simulations. If 150

isn't precise, and 1000 is, what about

333? I would have preferred that 10,000

simulations be used.

Sometimes TSUB relies heavily on

the reader's baseball knowledge. Having

discussed leadoff hitter Rickey Hender­

son's distribution of plate appearances

in the 24 runner and out situations for

1990 (p. 260), Albert asks the reader

whether Barry Bonds's 1990 profile

should be similar. The question seems to

spin on whether Bonds was a leadoff hit­

ter or not. Well . . . early in his career,

Bonds WAS a leadoff hitter. Bonds led

off only 13 games in 1990, but in 1989 he

led off 106 of the 159 games he played.

The casual fan might not know anything

about Bonds's lineup position at all. The

intermediate fan might think of him as a

cleanup hitter. The advanced fan wor­

ries about when he made the transition

from being a leadoff hitter.

In Chapter 8, TSUB proposes as­

sessing the "streakiness" of players by

visual comparison of player moving

average plots to simulated ones. How­

ever, most of the plots based on simu­

lated data are of different size, render­

ing comparison very difficult.

In summary, Jim Albert is to be ap­

plauded for providing a host of baseball

data and examples for use in learning

statistics. Sports examples provide one

of the best avenues for helping stu­

dents develop an interest in statistical

concepts. In 1997, Frederick Mosteller

[ 1] sounded the call that "statisticians

should do more about getting statistical

material intended for the public written

in a more digestible fashion" and relate

the interests of sports enthusiasts "to the

more general problems of statistics."

Through Curve Ball and Teaching Sta­tistics Using Baseball, Jim Albert has

responded well to that call.

REFERENCES

1 . Mosteller, Frederick. Lessons from sports

statistics. American Statistician, 51 :305-

3 1 0, 1 997.

2 . Albert, Jim and Bennett, Jay, Curve Ball:

Baseball Statistics and the Role of Chance

in the Game, Copernicus Books. New York,

2001 .

3. Schell, Michael J. Baseball's All-Time Best

Hitters: How Statistics Can Level the Play­

ing Field, Princeton University Press, Prince­

ton, 1 999.

Department of Biostatistics

University of North Carolina

Chapel Hi l l , NC 27599

USA

e-mail: michael_schell@unc.edu

Machine Learning: Discriminative and Generative by Tony Jebara

BOSTON, KLUWER,THE KLUWER INTERNATIONAL

SERIES IN ENGINEERING AND COMPUTER SCIENCE,

VOL. 755. 2004. 224 PP. HARDCOVER US$82,

ISBN 1 -400-7647-9

REVIEWED BY MARINA MElLA

I f "probability and statistics [ . . . ]

have separated, then divorced" [2],

machine learning is the place where

they coexist happily and occasionally

get up to interesting things together.

This book is the refreshing result of

just such an encounter.

To pursue another metaphor of the

same flavor, machine learning (ML) is

the estranged child of Artificial Intelli­

gence, which gave up on high-level,

rule-based representations of knowl­

edge for low-level, often stochastic

models estimated from data. It em­

braced probability from its inception,

discovered statistics in the 1990s, all

the while maintaining its roots in com­

putation. This was fortunate, as some

of machine learning's most famous

successes include an algorithmic in­

gredient.

One of the oldest and most studied

problems in ML is classification. Auto­

mated classification is often desired in

practical applications of computers.

For example, credit card companies

detect when a card is stolen from the

pattern of activity in the account on the

past few days; banks give credit ratings

"good = will pay = + 1" or "bad = will

default = - 1" to potential customers

based on a set of a dozen or so ob­

served characteristics (usually kept se­

cret by the banks). Recently, with the

advent of the massive scientific data

bases, automatic classification is in

demand in the sciences as well. In

biological sciences, automatic protein

classifiers are being developed that

predict the functional class of a protein

from its chemical formula. Astronomy

has introduced massive sky surveys,

with a telescope continually photo­

graphing the night sky for as long as a

few years. The number of celestial ob­

jects, from asteroids to remote galax­

ies, recorded this way is in the billions.

Hence, classifying each of them must

be done automatically.

Mathematically speaking, a classi­

fier is a function that takes some inputs

(the characteristics of the customer in

the credit rating problem, or the posi­

tion and image of the object if one

wants to classify astronomic observa­

tions) and outputs a label, e.g. ::!:: 1.

For example, a linear classifier is de­

fined by

f(x) = sign(81x1 + 8zxz + . . . 8mxm),

where x1, x2, . . . Xm are the inputs and

el, 8z, . . . em are the parameters of the

classifier. This classifier defines a hy­

perplane in m-dimensional space. All

examples (x1, Xz, . . . Xm) falling on one

side of the hyperplane are labeled + 1

(e.g., "good credit rating"), while the

points on its other side are labeled - 1

("bad rating"). The hyperplanef(x) = 0

itself is called the decision surface. The

values of the parameters, i.e., the

choice of the particular hyperplane, is

made based on a set of hand-labeled

examples called the training data.

Hence, training a classifier consists

of picking a function! and a set of val-

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 1 . 2006 67

ues for its parameters e. More ab­

stractly, one chooses a family '!fo of

functions parametrized by 81, . . . Om, a

criterion for selecting the values of the

parameters J(O; training data), and

some algorithm for obtaining the opti­

mal O's according to J. The first task,

choosing '!fo, is sometimes left to the

"application expert," but the second

and third are at the core of ML.

One way of practically approaching

classification was pioneered by Sir

Ronald Fisher and, to no surprise, typ­

ifies the point of view of statistics: As­

sume that one knows that the good

credit applicants have a distribution

p+ (x) (for example a multivariate nor­

mal) and the bad credit applicants are

distributed according to p-(x). Then,

to decide whether xnew deserves a

good or a bad credit rating we compare

the probabilities that the two mod­

els assign to xnew. For instance, if

p+(xnew)IP-(xnew) > 1, xnew receives

a good rating, otherwise it receives a

bad one. This approach can also be put

in aj(x) form, with

f(x) = sign[log p+(x) - log p-(x)] (1)

Because p+, p- are referred to as the

generative models of the two classes,

this approach is called the generative

approach to classification. By contrast,

directly choosing a decision surface

f(x) without resorting to generative

models is called discriminative classi­

fication.

The statistical approach induces

one to think of how the examples of

each class were generated, which is

more intuitive than directly selecting a

class '!fo of decision surfaces, especially

in high dimensions and when one deals

with more than 2 classes. Statistics

also offers principles-like Fisher's

Maximum Likelihood-and algorithms

for estimating the parameters of each

model from data. It was also proved

that, when the true model classes for

p± are known and the number of train­

ing examples tends to infinity, the gen­

erative classifier is optimal.

The problems appear in real condi­

tions, when one knows little about

the true distributions p±. What P to

choose? One can seemingly bypass this

68 THE MATHEMATICAL INTELLIGENCER

question by choosing a family of gen­

erative models rich enough to approx­

imate well any distribution the data

may have. But this way one may fall

prey to overfitting, which is exaggerat­

ing the importance of the inputs in pre­

dicting the data, when some of them

may be noisy or irrelevant. For exam­

ple, if a student sees a cat on the day

he fails an exam, and concludes that

seeing a cat on future exam days will

predict failure, then he is probably

overfitting the data.

The discriminative way to look at

classification is to pragmatically pick a

classifier f E '!fo that classifies the train­

ing set best, doing away with the gen­

erative model, i.e., with the assumption

that we know something about the

form of the "true" sources that gener­

ated the data. A now classic theory [ 1 ]

allows one to predict the classification

error rate on future data based on just

three numbers: the number of training

examples, the number of errors off on

these examples, and the capacity of '!fo the class of functions we are optimiz­

ing over. To understand the im­

portance of the last factor, think of over­

fitting. If the class '!fo is richer, something

measured by its capacity, then it is more

likely that the chosenfis overfitting the

observed examples but will do poorly

on new examples Gust like not seeing a

cat the next exam day will be a poor

predictor of success).

Discriminative methods have the

advantage of directly optimizing the

quantity of interest (the classification

error) instead of the description of the

classes separately through p+ , p-. In

practice they have been shown to be

more robust to overfitting, especially

when few training examples are avail­

able. However, their output cannot be

converted easily into a confidence

measure, and incorporating other

knowledge about the domain, some­

thing easily done with generative mod­

els, was only achieved on a case-by­

case basis.

How to combine the advantages of

generative models-ideal for incorpo­

rating knowledge about the domain

and returning a probabilistically mean­

ingful answer-with the principles of

discriminative classification that seem

to govern problems with finite data?

"Machine Learning: Discriminative and

Generative" offers a way of doing just

that: it is Maximum Entropy Discrimi­

nation (MED for short).

While introducing the core of the

MED method to a machine learning ed­

ucated audience can be done in one re­

search paper, this monograph has a

more ambitious goal. It aims to situate

MED in the context of its sources, to

motivate it, and to exhibit its many ties

with other old or recent advances in

machine learning.

The book opens with an overview of

machine learning, stressing the fun­

damental trade-off between choosing

a model expressive enough to solve

the problem and avoiding overfitting

the observed data. The next chapter

weaves in the precursor ideas for MED.

The generative and discriminative ap­

proaches are introduced. Each is illus­

trated by a notable success story in­

teresting in itself, but which also

supports the understanding of the rest

ofthe book For generative models, the

jewel of the crown are the "graphical

models," featured on the cover of the

book itself. Graphical models have be­

come a language for expressing prob­

abilistic dependencies between many

variables by way of graphs. Like any

good language, the concise represen­

tations are precise in a probabilistic

sense, and translate into very efficient

algorithms both for making inferences

from the model and for estimating the

models themselves.

The discriminative methods' reso­

nant success is the Support Vector Ma­

chine (SVM), also symbolized on the

book cover by a decision surface. SVM

demonstrates that important inventions

need more than one good idea; in the

present case three ideas converged from

domains as different as convex opti­

mization, reproducing kernel Hilbert

spaces, and statistical learning theory.

Chapter 3 introduces the MED core

ideas: Similarly to SVM, a family '!fo of

discriminative functions is chosen and

one explicitly enforces good classifica­

tion of the training examples. But, in­

stead of choosing just one f E '!fo, one

hedges the bets by averaging over all

the options. It is a weighted average, in

which an f gets a higher weight if it

makes fewer mistakes on the training

data. Beautifully and surprisingly, it

turns out that for many classes 21', solv­

ing this apparently more complex prob­

lem (an average over an infinite set of

functions) is no harder than choosing

one optimal! Even better, the task can

be cast into a convex optimization

problem with unique solution, that can

be solved by a generic algorithm.

One nice consequence of learning a

distribution instead of a single f is that

now one can naturally achieve most of

the goals listed above: One can construct

thefs as in (1) from generative models,

yet be explicitly optimizing the classifi­

cation error. One can interpret the clas­

sifier's output both discriminatively and

probabilistically. Prior knowledge of var­

ious kinds can be introduced.

Extensions to the main method are

presented in chapters 4 and 5. The for­

mer extends the MED approach to a

wider range of problems, like classifi­

cation with 3 or more classes (multi­

way classification), predicting a real­

valued "label" (regression), selecting

which inputs are relevant for classifi­

cation and which are not (feature se­

lection). Experimental results also

highlight the behavior of MED com­

pared with other methods. Chapter 5 discusses the case when some relevant

inputs are not observed (these are

called latent variables). For instance,

two stars that are close together in an

image may be part of a binary system

or may appear close only seen from the

Earth (much less likely!). Which is the

case is something that may perhaps be

inferred given other data, but cannot

be observed directly; it is thus a latent

variable. Finally, the discussion in

chapter 6 summarizes the ideas in the

book and outlines some directions for

further research.

Experiments punctuate most sec­

tions and the illustrations are very help­

ful (one wishes though that the Matlab

plots reproduced better). As the book

spans domains so varied as statistics,

optimization and functional analysis,

each with its own jargon, it is a challenge

for any author to fuse these into a co­

herent and precise language. The text

does a good job of it, but readers must

expect to do their part too. Those who

prefer a fully rigorous mathematical pre­

sentation will sometimes have to follow

the references. This is especially so in

the background chapters where not

everything is formalized, partly to keep

the book a manageable size.

For the ML researcher, understand­

ing MED and its sources of inspiration

is highly recommended, as the method

is at the center of a rapidly expanding

area of research. Chances are that,

even after reading this monograph, one

will see possible ways of expanding the

MED framework to new situations. For

someone wanting to apply ML solu­

tions to engineering or science, the

book offers a very flexible one-a so­

lution that can be adapted to a variety

of tasks and of problem characteris­

tics. And for the reader interested in

mathematical ideas at work in another

field, the topics in this book rate five

stars. The fundamental issues of ma­

chine learning are the mathematical

expression of problems that every sci­

entist has to deal with. The ideas MED

is based on (like maximum entropy,

exponential models, VC theory [ 1 ]) are

each on its own very powerful statisti­

cal and mathematical tools, lying at the

core of most notable successes of ma­

chine learning in the last decade. MED

itself is the result of combining them in

an elegant and surprising way.

Overall, the text is as an enthralling

presentation of ideas that move machine

learning these days. Let us remember

that ML is ultimately a field with practi­

cal significance, which currently is ex­

periencing substantial growth. It's excit­

ing to be there, and this book takes you

to the heart of the matter.

Department of Statistics

University of Washington

Seattle, Washington 981 95-4322

USA

e-mail: mmp@stat.washington.edu

REFERENCES

[ 1 ] V. Vapnik. Statistical Learning Theory. Wi­

ley, 1 998.

[2] Marc Yor. Review of "Weighting the Odds:

A Course in Probability and Statistics" by

David Williams. The Mathematical lntelli­

gencer, 25(2):77-78, 2003.

The Pol itics of Large Numbers. A History of Statistical Reasoning by Alain Desrosieres

Translated by Camille Naish

CAMBRIDGE, MA, HARVARD UNIVERSITY PRESS,

PAPERBACK, 2002. 384 PP. US $20.50

ISBN 0-674-00969-X

REVIEWED BY PATTI W. HUNTER

Today, the term statistics carries a

double meaning. On the one hand,

it refers to collections of numerical

facts about (among other things) polit­

ical, medical, or agricultural conditions.

On the other, statistics is a scientific dis­

cipline, a method of analyzing data. For

most of the nineteenth century and be­

fore, only the first meaning had wide­

spread use-a statistician was a col­

lector of numerical data. Not until the

twentieth century did any statisticians

think of themselves as mathematical

scientists.

How administrative and mathemat­

ical statistics came together is the sub­

ject of Alain Desrosieres's book, The Politics of Large Numbers. Drawing on

historical research of the last few

decades, Desrosieres examines the in­

teraction of these two domains, ana­

lyzing their development from the

point of view of the sociology of sci­

ence. He is particularly interested in

how the evolution of statistics ( admin­

istrative and scientific) sheds light on

the process by which such phenomena

as unemployment, poverty, and infla­

tion acquire objective status.

Covering the seventeenth through the

early twentieth centuries, Desrosieres

alternately considers the two faces of

statistics. Three chapters (1, 5, 6) ex­

amine the origins and development of

administrative statistics and their role

in the state, comparing Germany, Eng­

land, France, and the United States.

Chapters 2, 3, and 4 deal with subjects

more closely connected to the mathe­

matical face of statistics: seventeenth­

and eighteenth-century probability,

averages, correlation, and regression.

© 2006 Springer Science+ Business Med1a, Inc., Volume 28, Number I , 2006 69

Chapter 7 examines the social condi­tions in which sampling techniques originated. Chapter 8 considers prob­lems associated with choosing cate­gories into which to classify people and things. Moving into the twentieth cen­tury, chapter 9 traces the history of modem econometrics.

The book has a number of interest­ing and informative sections. Chapter 1 includes a detailed discussion of how the French Revolution shaped people's understanding of which aspects of social and demographic information were important and of what sets of categories most effectively classified that information. The chapter explor­ing Adolphe Quetelet's "average man" (chapter 3) provides an engaging glimpse into nineteenth-century debates among French medical scientists over the value of data about public health, par­ticularly in the context of efforts to identify the causes of the cholera epi­demic of the 1830s. In the second half of chapter 6, Desrosieres traces the professionalization of what became the United States Census Bureau. Here we see the influence of antebellum de­bates about the productivity of North­em versus Southern states, of dis­agreements about immigration policy, and of Franklin D. Roosevelt's New Deal.

Readers not familiar with the his­torical or mathematical details of the topics treated by Desrosieres may find his sociological analysis more accessi­ble after first reading for themselves the sources from which the author draws his material (for example, [ 1 , 2, 3] discuss the history of probability be­fore the twentieth century). In these works, they will find the writings of the historical figures analyzed carefully; they will learn something of the bi­ographies of people who play impor­tant roles in Desrosieres's discussion and of the professional and scientific contexts in which they worked. Math­ematicians will need no introduction to Blaise Pascal or Simeon Poisson. But what about Wesley Mitchell? The au­thor introduces him only as a "former census statistician" (p. 198), whose work took on new importance during World War I when Woodrow Wilson centralized national data collection.

70 THE MATHEMATICAL INTELLIGENCER

Some 80 pages further in, we learn that Mitchell founded the National Bureau of Economic Research in 1920, but the first discussion of this agency oc­curs thirty pages later. About Tjalling Koopmans, whose ideas figure promi­nently in the history of econometrics covered in chapter 9, some readers may find it helpful to be reminded of something more than the fact that he had a degree in physics. From where? What impact did this and his other training have on his economic thought? Desrosieres notes in his introduction that he is addressing readers from di­verse cultural backgrounds. He seems to treat the household names of each culture (and sometimes their ideas) as familiar to all.

This lack of attention to readers' di­verse knowledge of history would not alone make The Politics of Large

Numbers difficult for scholars new to Desrosieres's interests. Particularly in discussions of the broader questions about the sociological implications of the history of statistics, the exposi­tion relies on specialized vocabulary and rather complicated sentences. Set­ting out the purpose of the book, Desrosieres explains that he wants to link the technical history of statistics with its social history. "The thread that binds them," he writes, "is the develop­ment-through a costly investment­

of technical and social forms. This en­ables us to make disparate things hold together, thus generating things of an­other order" (p. 9, italics in the origi­nal). Desrosieres thus seeks to under­stand the development of these "forms," as well as how social phenomena like unemployment, and technical phenom­ena such as correlation managed to achieve objective status. As he puts it,

The amplitude of the investment 'in forms realized in the past is what con­ditions the solidity, durability, and space of validity of objects thus con­structed: this idea is interesting pre­cisely in that it connects the two di­mensions, economic and cognitive, of the construction of a system of equivalences. The stability and per­manence of the cognitive forms are in relation to the breadth of invest­ment (in a general sense) that pro­duced them. This relationship is of

prime importance in following the creation of a statistical system (p. 1 1 ). The introductory and concluding

chapters are most densely populated with such statements, but they can be found throughout the text. From a comparison with the original 1993 French edition, this style does not seem to be a consequence of the trans­lation.

Desrosieres's exploration of the re­lationship between statistics and the public sphere raises some intriguing questions about the mutual impact of mathematical ideas and the functions of the state. Readers with an interest in those questions and the willingness to fill in some historical detail and work their way through the exposition may find some thought-provoking an­swers.

Department of Mathematics

Westmont College

Santa Barbara, CA 931 08 USA

e-mail: pwhunter@juno.com

REFERENCES

[ I ] Lorraine Daston, Classical Probability in the

Enlightenment, Princeton University Press,

1 988.

[2] Theodore Porter, The Rise of Statistical

Thinking: 1820- 1900, Princeton University

Press, 1 986.

[3] Stephen M. Stigler, The History of Statis­

tics: The Measurement of Uncertainty be­

fore 1 900, Harvard University Press, 1 986.

Mathematical Circles, vol. I , II, I l l by Howard Eves

WASHINGTON, DC, THE MATHEMATICAL ASSOCIATION

OF AMERICA, 2003. US $98.00. ISBN 0-88385-542-9, 0-

88385-543-7, and 0-88385-544-5

REVIEWED BY STEVEN G. KRANTZ

G ian-Carlo Rota observed that we mathematicians are more likely to

be remembered for our expository work than for our research. (The ex­ceptions are figures like Gauss, Rie­mann, and Cauchy.) While only a hand­ful of research mathematicians from the 1950s and 1960s are worth even a

mention, the name of Howard Eves (191 1-2004) stands tall. Everyone has heard of Howard Eves. Why? Eves did little research, but he wrote texts and articles on a vast array of subjects, ranging from combinatorial topology to geometry to complex variables to number theory. He is best remembered for his many wonderful expository books. Notable among these is the six­volume Mathematical Circles collec­tion.

Using Google, I found no fewer than 2400 hits for Howard Eves. He is spoken of in glowing terms-"eminent mathematician and educator Howard Eves." Sounds perhaps like Saunders MacLane or Steve Smale. But it is Howard Eves, because Eves spoke to all of us with authority, with credibil­ity, and with sincerity, in language that we can all understand. What was his secret?

Eves was a considerable authority on mathematical history. That is a rather recondite subject area, and had he limited his publications to math his­tory journals then he would probably languish in well-deserved obscurity. But he chose to share his erudition by way of anecdotes and aphorisms about our collective heroes. He wrote well, he wrote accurately, and he wrote with conviction. Eves's Mathematical Cir­cles volumes constitute one of our col­lective treasures. The original Prindle, Weber, and Schmidt editions are now

Calculus: The Elements MICHAEL COMEN ETZ

537 pp $46 softcover (981 -02-4904-7)

$82 hardcover (981 -02-4903-9)

Both editions have sewn bindings

out of print, and we are fortunate that the Mathematical Association of Amer­ica has seen fit to republish the six books in three elegant volumes.

What is so special about these books? These days, with the great vogue in popular mathematical writing, we are beset with volumes about An­drew Wiles, about the Riemann hy­pothesis, about the history of zero, about chaos and fractals, and about any other quasi-mathematical topic that the public has a hope in hell of un­derstanding. It should be noted that Eves was one of the pioneers of math­ematical popularization and of mathe­matical story-telling. And his books are serious. He does not tell-at least in his first five volumes-of the digestive quirks of Descartes or the romantic peccadilloes of Galois. He really wants to tell us about the mathematical en­terprise, about the people who do mathematics and why they do it. Eves's purpose is serious, and his result-his written record-is substantive and valu­able. We are fortunate to have these col­lections of stories.

Among the more charming anec­dotes are • The story of why Thales never mar­

ried. • Four versions of the death of Archi­

medes. We have all heard the story of how Archimedes told one of Mar­cellus's troops to get out of his light-the soldier was disturbing his

circles. The irate soldier ran the great scholar through with his lance. Eves offers at least three other pos­sible versions of the story.

• The story of how Napier identified his servant who was stealing.

• The story of how l'Hopital obtained the rule named after him from Jo­hann Bernoulli.

• A determination of who were the second and third most prolific math­ematicians in history (after Leon­hard Euler).

• The story of how the Indiana legis­lature passed a law to declare it pos­sible to square the circle (this may be the first and primary source for the story).

• The story of how Walter Koppelman (in 1970) at the University of Penn­sylvania was shot and killed by his graduate student Robert H. Cantor.

Thus we see a range of events, from the historical to the current. The stories are told with a compelling accuracy and authority, rendered in concise and lively prose. Reading these stories is like eating Fritos: you cannot stop with just one.

There is always a danger with se­quels: You have told all your best stuff in the first volume. When your public or your publisher comes to you with demands for more, then you must cook something up. And it may not be up to the standard of the "stories of a life­time" that you set forth in Volume I.

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

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

to calculus, but also more logically rigorous in its discussion of the theoretical side than is usual. . . This style of explanation is well chosen to guide the serious

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 publ isher's website)

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

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

A selection of the Scientific American Book Club World Scientific Publishing Company

http://www. worldscientific.com 1 -800-227-7562

© 2006 Spnnger Sc1ence+Business Media, Inc., Volume 28, Number 1, 2006 71

Howard Eves stood up pretty well to

this challenge. But around Volume 5 one can perceive some dissipation.

Whereas Volume 1 contained stories

like the invention of the slide rule and

the mathematical treasures of the

Rhind papyrus, Volume 5 contains trite

stories about the addition of vectors

and the formulation of IOU notes. Vol­

ume 6 degenerates to the boringly

anecdotal:

A student once threw a book at a

mathematics instructor.

"I wouldn't have done that," re­

marked another student.

"Why not?" asked the culprit.

"Because that's no way to treat a

book," was the reply.

"Must we first take all these prelim­

inary courses?" asked a mathemat­

ics student.

"There's only one endeavor in which

one can start at the top, and that's dig­

ging a hole," replied the instructor.

The trouble with humor is that it can

rapidly become dated. I imagine that

someone thought these last two stories

were amusing at some time or another,

but they seem pretty pointless at the

moment.

Still and all, Howard Eves's magnum opus is a noble one, and an important

part of our literature. It is arguably the

first source for many important and

popular mathematical stories. A great

many of us cut our teeth on these books,

and we re-read them with pleasure.

Eves has a great spirit, and a great sense

of mathematical culture. He makes us

feel good about ourselves and what we

do. He shows us as erudite and human

and charming all at the same time. His

stories are never mean-spirited or criti­

cal. They show mathematicians for

what they are, in the strongest and most

positive sense of the word. They make

us feel good about our profession and

our enterprise. They leave us thirsting

for more.

When I read modem inept volumes

about the history of 7T and the proof of

Fermat's last theorem and the lore of

wavelets-mostly written by people

who do not know what they are talk­

ing about, just anxious for a quick

buck-I wish that Howard Eves were

still writing. I wish that mathematical

72 THE MATHEMATICAL INTELLIGENCER

exposition and popularization were ex­

ecuted by those who are qualified to do

it. Of course the cognoscenti are all too

busy proving theorems and going to

conferences. You will not catch them

writing books like Mathematical Cir­cles. And we are all the poorer for it.

Howard Eves was a virtuoso at his

craft. He knew exactly what he knew,

and he knew his limitations. He wrote

about things that he had mastered and

understood. He set a marvelous exam­

ple, to which we could all aspire. And

I hope that some of us will.

Department of Mathematics

Washington University in St. Louis

St. Louis, MO 631 30

e-mail: sk@math.wustl.edu

http://www.math.wustl.edu/�sk

Encyclopedia of Genera l Topology edited by K. P. Hart,

Jun-iti Nagata, and J. E. Vaughan ----- · --------- -- ---- . -----

AMSTERDAM, ELSEVIER, 2004. 526 PP € 1 45, ISBN 0-444-50355-2.

REVIEWED BY HANS-PETER A. KUNZI

As a student you may have been

taught that normality is neither

hereditary nor finitely productive.

Later on in your career you may have

encountered intricate problems in var­

ious contexts that could be reduced to

seemingly plain topological questions

which, nevertheless, you could not im­

mediately solve. For instance, in my

case, I could not find answers to some

questions about the behaviour of nor­

mality in box products, or to the prob­

lem of whether each compact topology

is coarser than a compact topology in

which all compact subsets are closed.

It is difficult to be a mathematician

and not use some basic concepts and

methods from General Topology every

now and then. Over the past decades a

few polished techniques and some

fundamental basic terminology of this

field have become so well known and

efficient that they now belong to the

folklore of today's university mathe­

matics. Less known to the mathemati-

cal community are many modem-and

often much more sophisticated and

complex-developments that thus far

have not had many applications be­

yond their area of origin.

The encyclopedia under review tries

to cover the basic as well as many of

the more specialized and new develop­

ments in the field of topology. It ad­

dresses both established mathemati­

cians and students, independent of their

area of specialization, and it is designed

to lead them quickly to the terminology

and results that might be useful to their

own investigations in other areas.

It explains terms like "Hedgehog",

"Weak P-Point", "Cliquish Function",

"Talagrand Compactness", "Thick Cov­

ers", and "Resolutions". Similarly, it

discusses concepts like "the Sous­

lin Hypothesis", "Hit-and-Miss Topolo­

gies", "the Normal Moore Space Con­

jecture", "the Blumberg Property", "the

Class MOBI", or "Bing's Example G". It

also deals with difficult open problems:

Is the class of meta-Lindelof spaces

preserved under perfect maps, or is

there a Dowker space with a u-disjoint

base? Finally, a wealth of known re­

sults and methods are treated: What

is the Dugundji Extension Theorem?

When is the completion of a topologi­

cal ring a field? Does the Sorgenfrey

line have a connected Hausdorff com­

pactification? How can one prove with

the method of elementary submodels

that the cardinality of a first-countable

compact Hausdorff space is at most

that of the continuum?

The encyclopedia includes about 120 articles contributed by a similar number

of topologists from all over the world.

Mostly written by experts in the spe­

cialized fields, these articles outline, in

a short sequence of definitions and re­

sults, many basic aspects of the treated

topics. The average length of a contri­

bution is about four pages. Very few

proofs are given. The articles are col­

lected under ten headings that imitate

Section 54 of the 2000 Mathematics Sub­

ject Classification as used by Mathe­

matical Reviews and Zentralblatt MATH.

The key words listed below will give

the reader an idea about the contents of

the work. They roughly follow the titles

of the articles as they are listed in the

table of contents of the book. Some

readers might want to skip the list be­low. I decided to include it in spite of a frowning referee, because it gives the reader a thorough and comprehensive first impression of the book In light of the high concentration of the presented results in the volume, the full flavour of the book cannot be grasped by merely glancing through a few examples of the­orems and techniques, as they are dis­cussed in the present review.

GENERALITIES: Topological Spaces, Modified Open and Closed Sets, Cardinal Functions, Conver­gence, Several Topologies on One Set (Minimal and Maximal Topologies).

BASIC CONSTRUCTIONS: Sub­spaces, Relative Properties, Product (Quotient, Adjunction, and Cleav­able) Spaces, Hyperspaces, Inverse and Direct Systems, Covering Prop­erties, Locally (P)- and Rim(P)­Spaces, Categorical Topology, and Special Spaces.

MAPS AND GENERAL TYPES OF SPACES DEFINED BY MAPS: Continuous (Open, Closed, Perfect, and Cell-Like) Maps, Extensions of Maps, Topological Embeddings (Universal Spaces), Continuous Selections, Multivalued Functions, The Baire Category Theorem, Ab­solute Retracts, Extensors, Gener­alized Continuities, Spaces of Func­tions in Pointwise Convergence, Radon-Nikod:Ym (Corson, Rosenthal, and Eberlein) Compacta, Topologi­cal Entropy, and Function Spaces.

FAIRLY GENERAL PROPERTIES: Separation Axioms, Frechet (Se­quential, and Pseudoradial) Spaces, Compactness (Local Compactness, Sigma-Compactness, Countable Com­pactness, and Pseudocompactness ), The LindelOf Property, Realcom­pactness, k-Spaces, Dyadic Com­pacta, Paracompact Spaces (Gener­alizations and Countable Variants), Extensions of Topological Spaces, Remainders, The Cech-Stone Com­pactification (in Particular of N and �), Wallman-Shanin Compactifica­tion, H-Closed Spaces, Connected­ness, Connectifications, and Special Constructions.

SPACES WITH RICHER STRUC­TURES: Metric Spaces, Metriza­tion, Special Metrics, Completeness,

Baire Spaces, Uniform and Quasi­Uniform Spaces, Proximity Spaces, Generalized Metric Spaces, Mono­tone Normality, Probabilistic Metric Spaces, and Approach Spaces.

SPECIAL PROPERTIES: Contin­uum Theory, Dimension Theory (General, and of Metrizable Spaces), Infinite Dimension, Dimension Zero, Linearly Ordered and Gener­alized Ordered Spaces, Unicoher­ence and Multicoherence, Topolog­ical Characterizations of Spaces, and Higher-Dimensional Local Con­nectedness.

SPECIAL SPACES: Extremally Disconnected (Scattered, and Dow­ker) Spaces.

CONNECTIONS WITH OTHER STRUCTURES: Topological Groups (Rings, Division Rings, Fields, and Lattices), Free Topological Groups, Homogeneous Spaces, Transforma­tion Groups and Semigroups, Topo­logical Discrete Dynamical Systems, Fixed Point Theorems, and Topo­logical Representations of Algebraic Systems.

INFLUENCES OF OTHER FIELDS: Descriptive Set Theory, Consistency Results in Topology (Quotable Principles, Forcing, and Large Cardinals), Digital Topology, Computer Science, Non Standard Topology, Topological Games, and Fuzzy Topological Spaces.

CONNECTIONS WITH OTHER FIELDS: Banach Spaces, Measure Theory, Polyhedra and Complexes, Homology, Homotopy, Shape The­ory, Manifolds, and Infinite-Dimen­sional Topology. Throughout the book it is assumed

that the reader has some basic knowl­edge of set theory, algebra, and analysis.

It is always easy to criticize various shortcomings of a book of this kind: Without doubt, each expert in the area will find some subject that in his or her opinion is lacking or is dealt with in­sufficiently. Similarly, it is easy to feel that some topics treated are of minor importance for the field and could have been less stressed or even completely neglected. Thus some readers of this encyclopedia might miss sections on Constructive Topology, Topological Ordered Spaces, or Frame Theory,

while others might wonder whether the concept of Cleavability or the the­ory of Approach Spaces are already so well established that they deserve their own sections in this volume.

Also, in light of the large number of authors, the nature of the articles is un­even in many respects, even after some unifying work was done by the editors. In particular the articles about the more specialized topics often and un­avoidably have to assume a certain fa­miliarity of the reader with basic con­cepts. This familiarity cannot be gained by reading a few introductory articles in the encyclopedia. Thus while the en­cyclopedia will certainly look impres­sive on the shelves of a library or on the desk of a mathematician, the ques­tion remains whether it is of practical use.

As can be guessed from the short de­scription above, the book will be very helpful to those who want to get a first overview of specific areas in the field and are looking for some references. However the work surely cannot re­place the usual text books, mono­graphs, or original research papers. In some sense, the value of encyclopedias in mathematics is quite limited: Essen­tially they can only provide quick ori­entation for informed readers. But they can hardly be relied on to teach com­plete novices, and they are generally too superficial for the working spe­cialist.

Those who want to study some of the sketched theories in any depth will necessarily have to learn more from the references at the end of each arti­cle, and from a general list of basic ref­erences that is used throughout the book Persons who are simply inter­ested in the solution to an isolated problem, for instance, need to know at once the basic facts of "the theory of bomological convergences", might prefer finding the latest literature deal­ing with their question with the help of the Internet. Indeed, the book will be most useful to those who already know a lot about topology and now still want to deepen their understanding of vari­ous areas closely related to their own field of interest.

Certainly some discussions in topol­ogy seminars could be based on parts

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 1 , 2006 73

of the more elementary articles. The students would be asked to fill in the missing arguments and appropriately expand the presentation of the material by consulting the pertinent literature.

Nevertheless the Encyclopedia of General Topology is a remarkable book, and one to which the editors have contributed a huge amount of work Such efforts are important in a time when the value of specialized original research is often overvalued at the cost of a systematization of the natural historic developments of mathematical knowledge.

The book represents a significant attempt to classify and order much of the work done in general topology and related areas in recent decades. It will inform future generations of mathematicians about the research already conducted and may thus avoid unnecessary du­plications and subsequent disappointments. Without such books, complex theories cannot develop properly, and even excellent mathe­matical concepts and results are soon forgotten and do not survive their discoverers or inven­tors.

Let us finally mention that the historical background of many of the theories presented is discussed in some detail in the three volumes thus far published by C. E. Aull and R. Lowen ( eds.) under the title "Handbook of the History of General Topology", Kluwer Academic Pub­lishers, 1997-2001.

Department of Mathematics and Applied Mathematics

University of Cape Town

Rondebosch 7701

South Africa

e-mail: kunzi@maths.uct.ac.za

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kj@ij,j.iq.U.I§i R o b i n W i l s o n I

The Ph i lamath' s Alphabet-L

Lagrange: Joseph-Louis Lagrange (1736-1813) wrote the first 'theory of functions', using the idea of a power se­ries to make the calculus more rigor­ous, and his mechanics text Mechani­

que analytique was highly influential. In number theory he proved that every positive integer can be written as the sum of four perfect squares. Laplace: Pierre-Simon Laplace (1749-1827) wrote a fundamental text on the analytical theory of probability and is also remembered for the

Lagrange

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

'Laplace transform' of a function. His monumental five-volume work on ce­lestial mechanics, Traite de mechani­que celeste, earned him the title of 'the Newton of France'. Leibniz: Although Newton could claim priority for the calculus, Gottfried Wil­helm Leibniz (1646-1716), who devel­oped it independently, was the first to publish it. His notation, including dyldx and the integral sign, was more versatile than Newton's and is still used. Leibniz's calculus was different from Newton's, being based on sums and differences rather than velocity and motion. Liu Hui: An ancient Chinese work, the Jiuzhang suanshu (Nine chapters on the mathematical art), contains the cal­culation of areas and volumes, the eval­uation of roots, and the systematic so­lution of simultaneous equations. Around 260 AD, while revising the Jiu­zhang suanshu, Liu Hui calculated the

Laplace

areas of regular polygons with 96 and 192 sides and deduced that 7T lies be­tween 3.1410 and 3. 1427. Logarithmic spiral: The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, . . . occurs throughout nature, and the ratios of successive terms 1/1, 2/1, 3/z, 5/3, . . . tend to the 'golden ratio' 1/2(1 + Y5) =

1.618. . . . A 'golden rectangle' with sides in this ratio has the prope1ty that the removal of a square from one end leaves another golden rectangle; this process is shown on a Swiss stamp, which also features the closely related logarithmic spiral, found on snail shells and ammonites. Lyapunov: The Russian mathemati­cian Aleksandr Lyapunov (1857-1918) was much influenced by Chebyshev. He worked on the stability of rotating liquids and the theory of probability. In 1918 his wife died of tuberculosis; Lya­punov shot himself the same day and died shortly after.

100

leibniz

t·<:� . . 'HELVETIA If '· ! l / I

Liu Hui

S;' , �- .:· . // I �\ I !a ' \ . / - ! . ' /><: �l �· _ 80

Logarithmic spiral

Lyapunov

76 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Business Media. Inc.