Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters

to the editor should be sent to the

editor-in-chief, Chandler Davis.

Review by Harold Edwards of

3 Books

James Thurber's short story The Mac­beth Murder Mystery is a hilariously

misplaced analysis of Shakespeare's

Macbeth written as though the play

were a whodunnit. Now you have pub­

lished a Thurberesque review of three

popular mathematics books about the

Riemann Hypothesis (RH)-review by

Harold M. Edwards, Mathematical In­

telligencer, vol. 26, no. 1, 2004-written

as though they were academic tomes.

To make matters worse, Edwards

doesn't believe it possible to explain

RH to non-mathematicians: he bases

this opinion on his failure to teach lib­

eral arts students that V2 is irrational,

blithely ignoring the obvious alterna­

tive hypothesis about his own teaching

ability. Edwards understands the dif­

ference between books aimed at pro­

fessional mathematicians and books

aimed at a general readership, but de­

cides that "it is only as a mathemati­

cian that I can evaluate the books."

Why? Can't a mathematician be a nor­

mal human being too, or at least imag­

ine what one might be like? It is as

though the Thurber character, having

tried and failed to write a tragedy, has

decided that tragedies are impossible

to write, and is therefore reviewing one

as if it were a detective story.

When reviewing The Music of the Primes by Marcus du Sautoy, which I

have read, enjoyed, and thought rather

inspiring, Edwards grumbles, "as a

sometime historian of mathematics,"

about the lack of citations of historical

sources. But, Professor Edwards, it's

not a history of mathematics, it's a

book for the general reader and posi­

tively shouldn't be cluttered up with

footnotes.

Edwards complains about du

Sautoy's "habit of introducing a private

phrase to describe something and for­

ever calling it by its new name rather

than the one used by everyone else."

But why on earth shouldn't he? The re­

placement of tired cliches and techni-

cal jargon by new, striking metaphors

is a mark of good writing, isn't it?

Amazingly enough, not everyone else

uses the term "modular arithmetic": I

imagine the reader du Sautoy had in

mind has never heard of modular arith­

metic, so it seems laudable for du

Sautoy to try to come up with a fresher,

more insightful expression, and I think

his idea of "clock calculator" isn't bad

at all. Personally, I liked du Sautoy's

metaphorical image of a landscape in

which the zeroes of the zeta function

are the points at sea level. I don't see

any reason for complaint. As for "for­

ever calling it by its new name" ... well,

if du Sautoy had reverted to the old

name, Edwards would have criticized

him for inconsistency. Or if he hadn't,

I would. Edwards's struggle with du

Sautoy's reference to "ley lines," which

he eventually decides "is apparently a

term used in British surveying," sug­

gests that du Sautoy credits his read­

ers with a broader general knowledge

than is actually possessed by Edwards.

Edwards seems determined to tell

us that mathematicians are obsessed

with problems like RH entirely for their

own sake, without any interest at all in

their history or context. He says that to

believe that the fascination of RH

arises from the information it would

give mathematicians about prime num­

bers "is a profound misunderstanding

of our tribal culture, like believing

mountaineers want to climb Mount

Everest in order to get somewhere."

Well, who knows what the true motives

for climbing Mount Everest are? I do

know, from the time I lived in Malaysia,

that the first Malaysian to climb Ever­

est was given a handsome financial re­

ward by the company he worked for: I

rather imagine that, like the rest of us,

he had mixed motives.

Edwards tells us that the books un­

der review "grossly overstate the con­

nection of RH to prime numbers": in

support of this he points out that Rie­

mann himself switched his attention

from �to �. a transformed version of {

© 2004 Spnnger Sc1ence+Business Media, Inc., VOLUME 26, NUMBER 4, 2004 5

But the fact that Riemann found it

more convenient to study a function in

one form rather than another says ab­

solutely nothing about its connection

with prime numbers. It would be ec­

centric if not insane to write a popular

(or, I should think, any other) book

about RH without emphasizing its im­

portance in prime number theory. In­

deed, Edwards's own book Riemann's Zeta Function (which, by the way, we

should have been told about right from

the outset of his reviews of books on

much the same subject) starts with a

reference to Riemann's paper On the Number of Primes Less Than a Given Magnitude and finishes with a proof of

the prime number theorem. In his de­

scription of the Riemann hypothesis

for the Millennium prizes, Bombieri

(whom I suppose Edwards might ad­

mit as a member of the "tribe" of math­

ematicians) writes that "The failure of

Harold Edwards replies:

Du Sautoy's failure to give any indication

of the sources of his stories is a problem

because so many of those stories are so

questionable. I state my reasons for

doubting some of his statements, and I

doubt many others. Whether through

footnotes or otherwise, he should justify

his more surprising assertions. Writing

for a naive audience does not give him a

license to invent history.

When he gives the name "Riemann's

6 THE MATHEMATICAL INTELLIGENCER

the Riemann hypothesis would create

havoc in the distribution of prime num­

bers. This fact alone singles out the

Riemann hypothesis as the main open

question of prime number theory." Of

course people who work on RH be­

come wrapped up in it-otherwise

they'd have no chance of success-but

the reason that RH stands out among

all the other interesting problems that

obsess mathematicians is precisely its

history and its position in mathematics

as a whole, particularly its connection

with prime numbers.

I started to write this letter because

I felt irritated at what seemed to me to

be a sneering attitude toward a book I

had enjoyed reading. But, having started

to think more carefully about Edwards's

reviews, I fmd it just plain silly that they

are written from the viewpoint of some­

one for whom the books were not in­

tended. The writing of mathematical

magical ley line" to the critical line

Re s = 112, du Sautoy credits his read­

ers not only with a broader general

knowledge than I possess but also with

a broader knowledge than the Ameri­can Heritage Dictionary of the Eng­lish Language possesses.

To say that "it is only as a mathe­

matician that I can evaluate the books"

is not to say that I am evaluating them

in any way except as books written for

readers who are not mathematicians.

books for a general readership is an art

quite distinct from academic writing,

and such books deserve to be reviewed

on their own terms. In addition, Ed­

wards paints an unrealistically depress­

ing picture of mathematicians as people

even more inward-looking and obses­

sive about their little problems than any

group of technical experts is bound to

be: mathematicians aren't quite as un­

aware of the context of their work as he

seems to want us to think

Next time you want a reviewer for

an academic mathematical tome, I sug­

gest you ask a Shakespearean scholar,

or a thriller writer, or perhaps even an

author of popular mathematics books.

Eric Grunwald

1 87 Sheen Lane

London SW1 4 SLE

UK

e-mail: EricGrunwald@compuserve.com

As I believe the review makes clear, I

tried to decide whether they would

convey inspiration, enjoyment, and a

reasonably accurate picture of the sub­

ject to such readers. I don't deny Mr. Grunwald's right to an opinion, and

don't know why he would deny mine.

Courant Institute of Mathematical Sciences

New York University

New York, NY 1 00 1 2 USA

e-mail: edwards@cims.nyu.edu

Four Poems Philip Holmes

Celestial Mechanics

At dawn, when my appr nti brought m

bowl and pitch r, h ·aid the city was tir

with talk of one Kop mik, who would hav it

that th un is a ftx d tar.

My teaching, my word the c thirty y ru the un i ftx d; all lse · in tracks about her

which will not leav

Th bodi , God' , m asur the p

in each lap ru1d ·way of p riod,

mark th ir future in each pull again t anoU1 r. Th od we fear d

and it was mine, its pivot ur r

than my gl, could tell m ; and my own place

fix d for v r, though � w h ard me, <mel few r

et in ili years' p, ·ag .

I k pt ·iience for my chur<·h. I would be

rack d for tlti my know! dge, and must rack

my lf for holding it, iliough it b

truth and all els clark.

lei god whom I t acli d, teady me now

against th rumour; I t me not drift utman1ed,

who nanwd your path ·; my w rd cannot

be drean1S: the v ry \\·orl mo\·e on U1 m.

The day' duti fold about me. Heav n turn ,

and earth, and on it what we know of H av n.

We make our littl gai.I1S. Why hould I burn,

except from vanity, if honour go to him?

I was apart from that. Tho day , th movement

was all, to t it right. Then II av n'. hand

was h r or not and distant anyway as doubts

alas which now I know ar al o mine.

"Celestial Mechanics." "Clear Air Turbulence," and "Background Noise" are reprinted by permission from the

author's The Green Road, Anvil Press Poetry, 1986.

© 2004 Springer Science+ Business Media, Inc., VOLUME 26, NUMBER 4, 2004 7

Background Noise

The wind crambl s and thunders over hill

with a voic far b low what we can hear.

Whal ong, bird ongs boom and twin r.

Sea, air, v rything' a chao of ignals

and even tho w 'v nan1 d v r and fall

Clear Air Turbulence

The Dakotas and th n Wy ming Wlinkl tmder as th air wrap about u -

only the cale differs: tho fin grain

and peaks ar th land' flow, wh r y ars

ext nd to millennia ut lif£ bring up

the plateau' tr t h with a I ap

and up h r onds ·aunt as the wingtips dip

and boun , br aking ight of the Wlinkled fac

b low, th n w blown outhward off ridge .

Thi air w 'r tum d and bucked in

p · and fill. hug cell over tho e rang

h now hiUg again and pull straight.

'n n, the patterns stagger and br ak up;

what we would impose on them br a

How an the air' heated, turning cha e n

a fit end to its local order? And v n granting tlli ,

I till know that, in flight, volum and pre

far I properly described k ep u aliv .

Why wi h to xplain th m, if we can rely on

what's not underst od'? W an't. The plane drop

an instant. We're forced again to look

past the surface, th Iillis and knotted air, to the blank place, alway ju t ahead, where

if only for a moment, th h rut tops.

8 THE MATHEMATICAL INTELUGENCER

Tectonic Order

Banff lnt mational R arch tation

Alb rta, anada

pril 13, 2004

It took five hundr d million years to tmn

afloor into ki-nm. We annat hold

o larg a tiling in nlind, but can we learn

a mall U1ing from it? Thi w eping cene

was peaked and t rra d by th heat and cold

that took fiv hundr d million y ars to turn

Perhaps tl1e view' too p rf t to disc m

much beyond tl1e bald fa t we'v b n told:

it took fiv htmdred nlillion y ars to tum

tile world that brought u h r , that shape and arn

ur living ... \ ait! Is this th ' ay to mould

a ubtl r thing in nlind'? To h lp us I am

that liv go on but ours will n t return?

Th' m ting is the first and last we'll hold:

it took five hundred million y ars to tum

tlli morning out. Be mindful, think, and learn.

Department of Applied and Computational Mathematics

Princeton University

Princeton, NJ 08544-1 000

USA

e-mail: pholmes@math.princeton.edu

·�·ffli•i§rr6'h£119.1,1rrlll,iihfj Marjorie Senechal, Editor I

The Mysterious Mr. Ammann Marjorie Senechal

This column is a forum for discussion

of mathematical communities

throughout the world, and through all

time. Our definition of "mathematical

community" is the broadest. We include

"schools" of mathematics, circles of

correspondence, mathematical societies,

student organizations, and informal

communities of cardinality greater

than one. What we say about the

communities is just as unrestricted.

We welcome contributions from

mathematicians of all kinds and in

all places, and also from scientists,

historians, anthropologists, and others.

Please send all submissions to the

Mathematical Communities Editor,

Marjorie Senechal, Department

of Mathematics, Smith College,

Northampton, MA 01 063 USA

e-mail: senechal@smith .edu

Mathematics is an oral culture, passed

down from professors to students, gen­

eration after generation. In the math

lounge, late in the evening, when the

theorem-scribbling dwindles and e-mail

morphs into screen savers, someone

opens a bottle of wine, another brings

out the cake, and the stories begin. Kepler was mystical, Newton alchemi­

cal. Hotheaded Galois died in a duel.

Godel starved logically, to avoid being

poisoned. The stories roll on without

end. Stories of giants, their genius and

foibles: yesterday's giants, giants to­

day. Wiener, the father of feedback,

couldn't find his way home. The peri­

patetic Erdos woke his hosts at 4 in the

morning. You know, stories like that.

Robert Ammann too was a brilliant eccentric. Yes, I knew him. His

story isn't like that. 0 0 0 0

"Wait a minute," Jane interrupts me again. "Who was Robert Ammann?"

The denizens of the lounge sprawl in self-organized clumps. My clump in­cludes Jane, a first-year graduate stu­dent just learning the lore; Carl, in his third year of graduate work, who's just passed his orals; and Richard, a col­league from elsewhere. Jane and Carl sit on the rug, as befits their apprentice status. Richard relaxes on the black leather sofa, a 20-pound calculus text under his head: he gave the colloquium lecture this evening.* I slouch in an armchair that has seen better days.

"You've never heard of Ammann?" Carl plays incredulous. "Everyone knows about 'Ammann tiles,' and 'Am­mann bars.' In tiling theory, anyhow."

"He was a pioneer in the morphol­ogy of the amorphous," says Richard.

"The what of the what?" Jane asks. "Non-periodic tilings, chaotic fluids,

fractal coastlines, aperiodic crystals, that sort of thing," Richard explains.

"Toward the end of the twentieth cen­tury, scientists in many fields, includ­ing math, discovered that 'disorder' isn't random, it's a maze of subtle pat­terns."

"Ammann was one of the first to dis­cover non-periodic tiles and tilings. And he showed their amazing variety," I tell her. "He didn't prove much, but he had vivid insights into their nature. He settled open questions, posed new ones, and sparked imaginations."

"Artistic imaginations too," Carl says. "A painter in Berlin incorporates Ammann bars in his designs. And they're being used in a pavilion at the Beijing Olympics.''1

"A physicist I know laid an Ammann tiling, with real tiles, in the entrance hall in his home," Richard adds. "And a vice-president at Microsoft has in­corporated all of Ammann's two-di­mensional tilings in the new home he's building. On floors and walls and grilles."

"You're telling me what he did, not who he was," Jane reminds us.

"Robert Ammann, the person, re­mains almost unknown," I say. "This is his story, as I learned it."

0 0 0 0 I'll begin, not with his birth in Boston on October 1, 1946, but with an an­nouncement in Scientific American.

The August 1975 issue, to be exact. "For about a decade it has been known that there are tiles that together will not tile the plane periodically but will do so non-periodically. . . . Penrose later found a set of four and finally a set of just two," Martin Gardner wrote in his monthly column, "Mathematical Games." That's Penrose as in Roger Penrose, the famous mathematician and gravitation theorist, son of a psy­chologist of visual paradoxes. Father and son had sent impossible figures to M. C. Escher, who used them in his lith-

'Jane, Carl, and Richard are surrogates for you, the reader. Their questions-your questions, my questions­

guide us through the puzzles of Ammann's work and life.

1 Q THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+Business Media, Inc.

Figure 1. Left: Ammann's "octagonal" tiling in the en­

trance to Michael Baake's home; photo by Stan Sherer.

Right: Ammann grille in the home of Nathan Myhrvold;

courtesy of Nathan Myhrvold.

ographs "Ascending and Descending" and "Waterlall." Penrose's new discov­ery, to which Gardner alluded, seemed even more impossible.

Floor tiles-triangular tiles, paral­lelogram tiles, hexagonal tiles, oc­tagons with squares-repeat over and over, like ducks in a row and rows of ducks. Wall tiles do too, and tilings in art. Even Escher's wriggling lizards, plump fish, haughty horsemen, and winsome ghosts arrange themselves in regular, periodic arrays. Non-peri­odic tilings? What could they be? Gardner gave no details, drew no pic­tures: Penrose was waiting for a patent. "The subject of non-periodic tiling is one I hope to discuss in some future article," Gardner concluded his column.

For thirty years, from 1956 to 1986, Martin Gardner intrigued young and old, amateurs and scientists, unknown and famous, geniuses and cranks, with mathematical games, puzzles, diver­sions, challenges, problems. His read­ers deluged him with solutions, some of them valid, some of them pseudo. Scientific American hired assistants to help weed out the nonsense.

Another pair of planar non-periodic tiles? Could this be true? And the first set of non-periodic solids? Who was this Robert Ammann? Gardner knew just about everyone who knew any­thing about non-periodic tilings at that time: Roger Penrose, Raphael Robin­son, John Conway, Ron Graham, Benoit Mandelbrot, Branko Griinbaum, Geoffrey Shephard. He'd never heard of Ammann. Nor had they.

"I am excited by your discovery," Gardner replied on April 16. Ammann's tiles seemed quite different from the Penrose pair Gardner planned to write about later. "Would you object to my sending your tiles to Penrose for his comments? Are you planning to write a paper about them? . . . Tell me some­thing about yourself. How should you be identified. A mathematician? A student? An amateur mathemati­cian?"2

F ... ., �obnr A,..,...,,,. 1'301 11•���-··· ST l-1wt>ll 1 tf•<S Or\� J

"I would not mind your mentioning my tiles or sending them to Penrose, as I am not planning to write a paper about them," Ammann wrote back " . . . I consider myself an amateur doodler, with math background."

0 0 0 0 "Penrose tiles have been made into puzzles," Jane remembers. She crosses the lounge to the table and takes a box from a drawer. "A mystifying mixture of order and unexpected deviations from order," she reads from the label. "As these patterns expand, they seem to be always striving to repeat them­selves but instead become something new."

Jane dumps dozens of small, thin plastic tiles onto the table, four-sided polygons with notched edges. The black ones, dart-like, are all the same size; the white ones are identical kites.

She pulls up a chair and tries to put a kite and a dart together to make a par­allelogram. But the notches don't fit.

"That's the reason for the notches," I tell her. "If you could make a paral­lelogram with these tiles, then you could cover the plane with them, the way square ceramic tiles cover a floor."

"No, she couldn't," Carl interjects. "The plane is infinite, theoretically. She'd need infinitely many tiles. She only has a hundred or so."

"Of course. But you know what I meant. Don't be so picky, it's after 10 p.m." I tum to Jane. "The notches pre­vent you from making a parallelogram or a repeat unit of any kind. So every tiling with kites and darts is non-peri­odic. That's why they're called non-pe­riodic tiles. "3

Ammann's response to the August announcement reached Gardner's desk the following spring. "I am also inter­ested in nonperiodic tiling," Ammann wrote, "and have discovered both a set of two polygons which tile the plane only nonperiodically and a set of four solids which fill space only nonperiod­ically."

Figure 2. Ammann's two polygons-notched rhombs-which tile only non-periodically, and

his sketch of part of a tiling with these tiles. [Ammann to Gardner, undated, spring 1976.]

VOLUME 26, NUMBER 4, 2004 1 1

Figure 3. A kite and a dart.

Figure 4. The deuce with two possible ex­

tensions. For simplicity, the notches are not

shown.

Jane picks up some more tiles and fits four of them together; I recognize the configuration known as the "deuce."4 She starts to add another, then hesitates.

"Strange. A kite fits in this spot, but so does a dart."

"Penrose tilings aren't jigsaw puz­zles," I remind her. "In Penrose tilings you sometimes have choices."

"And different choices lead to dif­ferent tilings," Richard calls out from the sofa. I'd thought he'd fallen asleep. "Penrose tilings aren't individuals, they're species. Species with infinitely many members."

"What kind of infmity?" asks Jane. "Countable, or uncountable?"

"Un! Yet all the tilings look just alike-as far as the eye can see. Any fi­nite patch of tiles in one Penrose tiling turns up in all of them. Infinitely often."

"Borges! Escher! Where are you when we need you!" Carl gasps in mock horror.

0 0 0 0 "I am most intrigued-indeed, some­what startled-to see that someone has rediscovered one of my pairs of non-pe­riodic tiles so quickly!" Penrose wrote to Gardner, who'd sent Ammann's let-

12 THE MATHEMATICAL INTELLIGENCER

ter to several experts, with Ammann's permission. "It seems that his discovery was quite independent of mine!"

Penrose explained that he'd found not one, but two pairs of non-periodic tiles in 1974; the intriguing kite and dart that Gardner had in mind, but also a pair of rhombs, one thick and one thin. Penrose understood that though the tiles look very different, any tiling built with one pair can be converted into a tiling by the tiles of the other.

Start, for example, with a tiling by kites and darts. Bisect the tiles into tri­angles. Then recombine the triangles in situ into rhombs.

Figure 5. Left: a portion of a kite and dart

tiling, with the tiles bisected into triangles.

Right: the triangles are joined to form

rhombs.

Ammann, who'd seen neither set, had indeed rediscovered Penrose's rhombs and rhomb tilings, but by a very different route. And soon, in addition to the three-dimensional non-periodic tiles-I'll come back to those later-he found five new sets in the plane. He an­nounced his discoveries in a flurry of letters to Gardner, with hand-drawn fig­ures and hand-waved proofs. 5

Gardner sent the letters on to the ex­perts, who found Ammann's construc­tions ingenious and insightful. They grasped his ideas immediately, from his sketchy drawings.

Penrose's tilings are hierarchical. That is, they repeat not in rows, but in scale: the small tiles combine into larger ones, which combine into larger ones, which combine into larger ones . . . ad infinitum. Ammann's tilings are hierarchical too. And he had devised some intriguing variations. For exam­ple, the large tiles in most hierarchical tilings are larger copies of the smaller ones, but he found an example where they're not.

The experts who dissected Am-

mann's claims never found a mistake, though the jury's still out on a few of them. But the letters were odd. How had Ammann found his remarkable tiles? Why didn't he publish his results in mathematics journals, like everyone else? He had a droll sense of humor, they all could see that. But Ammann's "friend" Dr. Bitwhacker must have been a private joke for Gardner, chronicler of Dr. Matrix's mathematical adventures.6

0 0 0 0 "Why did anyone care about non-pe­

riodic tiles?" Carl wants to know. "It's deep stuff," I reply. "They're re­

lated to Turing machines and the de­cidability of the tiling problem."

"The tiling problem?" "It's an old, old problem. Imagine

you're a tile maker, back in deep an­tiquity. A rich patron hands you a fancy template and asks you to use it for thousands and thousands of tiles to cover her palace floor. Before you fire up your kiln, you'd better be sure the shape really is a tile. If copies don't fit together you'll be in big trouble."

"What's the problem? Why not make a dozen or so and test them?" asks Jane.

"Even if your dozen do fit together, how do you know you can add still more? In fact there are cases where you can't; Ammann found a tile that can be entirely surrounded by three rings of copies of itself, but not four."7

"So the tiling problem is: given a shape or set of shapes, is there a gen­eral procedure, one that works in every case, that determines whether you can cover the plane with it or them?"

"You mean, of course, the infinite plane, not just a palace floor," Carl re­minds us.

"Of course," I yawn. "I'd try to arrange a few tiles into

some sort of quadrilateral that I can re­peat in a periodic array," Jane contin­ues.

"That's the whole point!" I wake up. "Can you always do that? Hao Wang proved that a decision procedure ex­ists if and only if any set of shapes that tiles the plane in any manner can also be arranged in a periodic tiling. "8

"You mean, a decision procedure exists if and only if non-periodic tiles do not?"

Figure 6. Two kites and two half-darts make one bigger kite; one kite and

two half-darts make one bigger dart, and this can be repeated. Thus every

kite and dart tiling is at once a tiling on infinitely many scales.

" 1] , .. , ... rr. r.,.r"fo'oo•r.

I ,_.., •�f"ln<��l'\f"' • C"'....., l)f •v I• t•r tn I,., rtn" �n t� Cl • f'Yt' l•'""· i ..• ..,..,..,...11•, 1 "'""• ,.. ""'""'"' """"''""' "��'"'",.. -.v ""1"1 1'\lln•r� 'ln1 """• ,,.,..,� t.._n l'\1 "'""' -1 CAll .,..., c,....,v•r •<1 "" n�, .. �,.l"'"""'l,. �tlt..,"'"· F11r ._..r rl• .-11• w111 t.-,11..,, � ,.., ..-.v r•"lv t� ;t-.rt'l<��•'llli lllll•'\t l•"t•r (I '11 ,.f'lrt Y"\ 1 "' ci')('IV of tn. .. r•ol"J.

I 1

j l ..

l • J l i

·�t "''"h""'"· ; a�·-r""V �"'"""'rt A·,.·�"

Figure 7. Ammann's hierarchical tilings. [Ammann to Gardner, May 20,

1976.]

Figure 8. Another version of (a portion of) the

first tiling in Ammann's May 20, 1976, letter to

Gardner (see Fig. 7). Copies of the two small

tiles can be combined into larger ones, as

shown by the shaded tiles. All the tiles in the

infinite tiling can be combined into larger ones

in this way, again and again, so the tiling re­

peats on all scales. Look carefully: the shaded

tiles are not exact enlarged copies of the

smaller tiles of which they are composed.

(i �cr r r G<·" � �I

f" .. t ,., ,A('" 1 t tT.:r �· ��..· �

�·�t .... f\kld cr , �er ,� ' " \0 'f' \"cl� � /e (. "(1i ,.

Figure 9. Ammann to Gardner, April 14, 1977.

11)

VOLUME 26, NUMBER 4, 2004 1 3

Figure 10. A tile that can be surrounded by

three rings of copies of itself but not four.

[Robert Ammann, 1991]

" Exactly. Back in the early 1960s, when Wang posed the question, he and everyone else assumed that a decision procedure would be found. They were wrong. Robert Berger found the first non-periodic tiles in 1966. But there were 20,426 different tiles, so it was only of theoretical interest."

" Well, he showed that the tiling problem is formally undecidable," Jane says. "That's enough!"

" If we had a jazzier name for non­periodic tiles no one would ask 'who cares,' " Richard observes. " No one asks who cares about chaos and frac­tals. Some of us tried calling tiles ape­riodic if all their tilings are non-peri­odic. But the name never caught on. And no one has come up with anything better."

() () () () I first heard of Anunann's work through the grapevine, but I didn't grasp its im­portance until I read Tilings and Pat­terns.9 I was one of the lucky readers of an early draft. The first few chapters arrived unannounced in the mail on the first day of the spring semester in 1978.

The authors had no idea how glad I'd be to see it. Tilings play a key role in the geometry of crystal structures, my research field at the time, so I had an­nounced a course on them, mainly to teach myself. No textbook existed; most mathematicians dismissed tilings as " recreational math" in those days. I would pull together articles from crys­tallography journals, Martin Gardner's columns, books on design, and a few

•q, = (1 + Vs)/2

14 THE MATHEMATICAL INTELLIGENCER

mathematics articles I knew of, and piece them together somehow.

That hubris sustained me through the fall and into winter. But now the clock ticked toward class. I dreaded facing the students: I had nothing to say. The tiling literature was incoher­ent, incomplete, inconsistent, and, worst of all, incomprehensible. To forestall the disaster, if only for a few minutes, I checked my mailbox on the way to the classroom. I opened the bulky package and ran to the phone.

The authors, Branko Griinbaum and Geoffrey Shephard, agreed to let me and my students work through it and send comments. Tilings and Patterns became the course.

The book galvanized research on tilings, including my own. Griinbaum and Shephard had gathered, sifted, re­viewed, and revised everything that had ever been written, in any language, living or dead, on tilings and patterns in the plane. Tiles of so many kinds! Polygonal tiles, star-shaped tiles, tiles with straight edges, curvey tiles, tiles symmetrically colored. These omni­scient authors filled in gaps, corrected mistakes, compared and synthesized different approaches, proposed new terminologies, and classified tilings with various properties.

Martin Gardner's article on the kites and darts and John Conway's account of their remarkable properties had just been published. 10 In chapters sent later, Griinbaum and Shephard de­scribed that and much more: Wang tiles, Robinson's tiles, and five Am­mann sets, A1 through A5, some marked with lines they called Ammann bars.

Yet except for his letters, no one knew a thing about Ammann. No one, not Gardner, Penrose, Griinbaum, nor Shephard, had met him.

() () () () " Are these lines the Ammann bars?" Jane asks, handing me a kite. She's no­ticed the thin lines etched on the tiles, each kind of tile etched alike.

" Right," I reply. " With Ammann bars, you don't need the notches. You can't make a parallelogram if you keep the bars straight."

" Ammann bars are a grid for the tiling,'' I continue. "As Anunann ex­plained it, the 'pattern is based on fill­ing the plane with five sets of equidis­tant parallel lines at 36- and 72-degree angles to each other, and placing a small tile wherever two lines intersect at a 36-degree angle, and a large tile wherever they intersect at a 72-degree angle.' If you look closely at the lines on the tiles you've laid, you'll see how it works."

" Some lines are closer than others,'' Jane points out. " I thought you said Ammann's lines were equally spaced."

"They were, in his first letter to Gardner, the one I just quoted. But equally spaced lines can't be drawn on the tiles so that each tile of each kind is marked the same way. Ammann modified the spacings later. The pat­tern of intersections is the same."

Jane adds tile after tile. The patch grows like a crazy quilt. The lines remind me of a children's game called pick-up sticks, but those fall any which way.

" Hmm,'' Carl says. "The long and short distances form a sequence, . . . L SL SLL SL . . . "

"Keep going," says Richad. " . . . L SLL SL SLL SLL SL SL

L S L S L L S L . . . ,'' he reads out. " Omigod!" Jane exclaims. " Fi­

bonacci rabbits! Where did they come from?"11

" From the hierarchical structure," I show her.

"Penrose tilings are riffs on Fi­bonacci numbers and the golden ratio," Richard pontificates. "4> crops up everywhere: it's the ratio of long to short tile edges, the ratio of kite to dart areas, and the ratio of the relative num­bers of darts to kites in the infinite tiling."*

" So the mysterious and ubiquitous key to ancient architecture, pine cones, and pentagrams is also the key to non­periodicity!" says Jane.

" No, it's not,'' Carl deflates her. " At first people thought it might be, but Ammann found pairs of non-periodic tiles where all those ratios are V2. The square and rhomb tiling Richard showed you-the one in the hallway­is the most famous example."

Figure 11. A kite and a dart marked with Am­

mann bars. The notches are not shown.

Jane returns to the tiling puzzle. A few minutes later she exclaims, "The tiles don't fit any more. I've hit a dead end."

"The deuce commands a far-flung empire," Richard explains unhelpfully. "He controls tiles far away, tiles not yet laid down."

"Cut the metaphor, just tell me why I'm stuck"

"Some choice you made, a few steps back, is incompatible with the Fibonacci sequence you're hatching here."

"Oh." "There's no way you could have

known that," I console her. "The choices seemed equally valid at the time."

"So very like life," she mutters. "Anyway, I'm not sure I can really tile the infinite plane with these things. I mean"-she glances at Carl-"in prin­ciple, if I had an endless supply of tiles."

"You can," I reply. "It's yet another consequence of the hierarchical struc­ture."

"You must have slept through my talk," says Richard. "I showed you how to get complete Penrose tilings by pro­jecting the tiles from higher dimen­sions. De Bruijn invented that method. Start with five sets of equally spaced parallel lines, just like Amman's origi­nal ones-de Bruijn calls them penta­grids. He showed that the criss-cross pattern of lines in the plane is a slice of a periodic tiling in five-dimensional space."

"But I'm stuck down here," Jane per­sists.

Richard ignores her. "Then de Bruijn does some abracadabra-more pre­cisely, he takes the pentagrid's dual­and projects it down to the plane.

Figure 12. Left: Ammann bars; right: Penrose tiles with Ammann bars superimposed.

Voila, the Penrose tilings! in the plural! You get them all if you shift the slice around. And the matching rules also fall out of the sky!"12

"Is there some abracadabra so I can continue?"

"Remove some pieces and try again." I pour a second glass of wine. "When you get stuck in a non-periodic tiling you can always repair it. Unlike life."

"How far should I backtrack?" "No one can say." 0 0 0 0

Over the next decade, assisted by Tilings and Patterns and spurred by the startling discovery of quasicrys­tals-crystals with atoms arranged in non-periodic patterns-tilings leaped from the game room into the solid state lab. 13 Mathematicians, physicists, chemists, x-ray crystallographers, and materials scientists found a common passion in non-periodic tiles. "We're all amorphologists now," a physicist told me. Penrose tilings and Amman tilings were buzzwords of the day. And still no one I knew had ever met Ammann.14

We, the growing community of tiling specialists, attended conference after conference, all over the world. In those days, before the Internet, keeping up in a hot field meant being there. Ammann was often invited but always declined, if he answered at all. In the spring of 1987, Branko Grtinbaum again pleaded with him, "Would you please recon­sider? Without exaggeration, I am con­vinced that you have shown more in­ventiveness than the whole rest of us taken together."15 Again Ammann said no. The mysterious Mr. Ammann," he'd signed a letter to Gardner. Mysterious he remained.

But the mystery man's most recent refusal was postmarked Billerica, MA­an hour and a half from my home in Northampton. So I sent him a note, invit­ing him to dinner at my home to meet Dick de Bruijn, who was visiting from the Netherlands. De Bruijn was strong bait-his powerful analysis had lifted Penrose's tilings from two dimensions to five and Ammann's work from doo­dle to theory. Even so, I was as sur­prised as delighted when he accepted.

November 19, 1987, a cold, rainy day. Our guest arrived after dark, three hours late. He was neatly dressed, short and a little stout, his very high forehead framed by black hair and black-rimmed glasses. I guessed his age about forty. He shook my hand limply, avoiding my gaze.

Bob didn't make small talk, not even hello. As I took his dripping raincoat, he pulled sheets of doodles from a brown paper bag: his latest discoveries, his newest results. Dick and I looked at them carefully, but couldn't decipher them. I asked what they meant. Bob's answers were vague. Dick explained his pentagrid theory. Bob showed no interest. This wasn't rudeness, I sensed. He seemed far away, and ineffably sad. Fortu­nately, dinner was waiting. Dick and I did most of the talking at dinner, but Bob seemed glad to be with us, and he answered our questions when asked.

"How did you discover your tilings?" we wanted to know.

"I'd been thinking about the lines of red, blue, and yellow dots used to re­produce color photographs in newspa­pers," Bob replied. "I drew lines criss­crossed at appropriate angles, and

VOLUME 26, NUMBER 4, 2004 15

Dear Mr. Gardner,

I got your latest letter, and am·enclosing a diagram showing two sets of "Ammann bars" (thanks for naming them after me) based on the a ratio 1 : V2 and the resulting forced tiles. Of course, there are actually four sets of solid lines at 45° angles and two sets of dotted lines at 90° angles crossing the figure, but the extra sets have been omitted for clarity. I believe it is possible to find a set of nonper­iodic heptagonal tiles, but such a set would be large (over 10 tiles) and not very esthetically appealing.

You wanted to know more about my friend Dr. Bitwhacker. He is the author of seveeal booksi, including the 1972 "Autcbbiography Of Clifford Irving". He recieved a rather large cash advance from the publishers for that book, but he k spent a few months in jail for fraud when the publishers discovered Clifford Irving had absolutely no connection with the book. (Clifford Irving1 as you may remember, wrote the ·

"Autobiography Of Howard Hughes'). Best,

(�� Ch-vw1U<#lV "The Mysterious Mr. Ammann"

Figure 13. "The Mysterious Mr. Amman." [Ammann to Gardner, February, 1977. (Exact date

unknown.)]

stared at them for awhile. The tiles just popped out at me."

In a letter to Gardner, Bob men­tioned he had some "math back­ground "; I asked what he meant. "A lit­tle calculus, and some programming languages," he said. He'd been a soft­ware engineer for twelve years, but now he worked in a post office all day, every day, sorting mail. Because, he told me, civil service jobs are secure.

Had he heard about quasicrystals? Yes; he'd been in touch with some physicists who were studying them. He'd even gone to Philadelphia to see them once.l6 They'd told him to call them collect if he had any new ideas, but so far he hadn't.

After dinner, as he was leaving, Bob gave me a typescript of an arti­cle he'd written, a revolutionary new theory of dinosaur extinction. 17 He hoped I could help get it published. I lent him a book on fractals he hadn't yet seen.

"Our conversation was very touch­ing, really," I wrote Griinbaum the next day. "Ammann is not in communication with this world, and knows it, and seems ambivalent about it. He's not a complete recluse, but I see now why he won't attend conferences."

Bob and I stayed loosely in touch, mainly at Christmas. I wrote once ask­ing to interview him about how he made his discoveries; I wasn't sur­prised that he didn't reply. 18

0 0 0 0 "Bob was autistic, or Asperger's maybe," says Jane, looking up from the

tion. In a world of his own, and the vi­sual thinking."

"Not so fast," I snap. "That describes most of us. Besides, thinking in images is one thing, but visual genius is another."

"We don't need vision anymore," says Richard. "Ammann bars and pentagrids are more important than the tiles them­selves. We have equations for them. We can feed parameters into computers, and the computers draw the tilings."

"Some of Bob's tilings don't have bars," I remind him. "The world of non­periodic tilings exceeds every theory yet devised, and always will. Visual imagination still has a role here."

Richard rolls his eyes: "The projec­tion method and hierarchical order cover the field."

I get up from the armchair and take a book from the shelf. "If you don't be­lieve me, listen to Penrose. 'The differ­ent kinds of tiling arrangements that are enforced by subtly constructed prototile shapes must, in a clear sense,

defy classification ... we have had our eyes opened to the vast additional pos­sibilities afforded by quasi-symmetry and hierarchical organization, yet even this cannot be the whole story.' "19

"Well, maybe so," Richard admits. Hierarchy is hierarchy.

"I still think Bob was autistic," Jane insists. "Most math geniuses were, Newton and Godel and Wiener and Erdos. Einstein too. The Mathematical Intelligencer ran an article about autism a few issues back. n20

"I read the article but I don't buy the argument," I reply. "Yesterday Freud, Asperger today, who knows what tomorrow. A Rorschach test of the times."

"What do you mean? The author said they had Asperger symptoms."

"No one-size-fits-all diagnosis can explain such complicated people. Take Norbert Wiener, for example. When he wrote his autobiography, in the early 1950s, the prevailing fashion was Freud. Everyone told Wiener that his emotional and social problems-he had lots of both-were due to his father. Wiener rejected that explicitly, even though his father was famously diffi­cult. He said it was too simplistic. "21

"Maybe Wiener was autistic and Bob's troubles were Freudian," cracks Richard.

"The press made Wiener's life even more miserable," I continue, ignoring him. "They made a huge fuss over prodigies back then. Another kid who entered Harvard at eleven cracked up in the spotlight."

"What did Bob say in that dinosaur paper?" Carl asks.

tiles. "It's obvious, from your descrip- Figure 14. Bob Ammann and N. G. de Bruijn, November 19, 1987; photograph by Stan Sherer.

16 THE MATHEMATICAL INTELLIGENCER

"They were killed off by nuclear weapons."

() () () () In March 1991, the moveable feast paused in Bielefeld, Germany.22 Birds twittered in the pine grove behind the university's new interdisciplinary re­search center. When I arrived, late in the evening, the lounge was filled with the usual suspects, thirty-three assorted sci­entists from nine different countries. No stories this time: the story was there. Bob sat quietly at the edge of the crowd, speaking when spoken to but not look­ing anyone straight in the eye.

Everyone was genuinely pleased to meet Bob at last and tried to put him at ease. (I guess my account of our meeting had spread.) As the days went on, he mingled more easily, almost naturally. Like the rest of us, he feigned interest in the lectures whether he un­derstood them or not. He joined us for meals. On the third day, nervous and halting, he gave his first-ever talk, on his three-dimensional non-periodic tilings and how they might model a par­ticular quasicrystal. I remember the talk as confusing and disorganized, but others insist it wasn't too bad. We agree on Bob's wistful conclusion: "That's all I have to say. I have no more ideas."

On the fourth and last evening, at the conference banquet, the organizers honored Bob with a special gift, a large three-dimensional puzzle of dark brown wood, and photographed him together with Penrose. In the picture, Bob looks off into space, with the faintest of smiles.

() () () () "I still don't see," says Carl, "why all those scientists cared about tiles that tile only nonperiodically. Wang's theo­rem was old hat by 1991. So why were you guys still talking about them at Bielefeld?"

Lots of reasons. What we used to call 'amorphous' turns out to be a vast largely unexplored territory, with reg­ular arrangements as one limiting case and randomness another. It's inhabited by non-periodic tilings, quasicrystals, fractals, strange attractors, and who knows what other constructs and crea­tures.

be a model for the quasicrystal?" Jane asks.

"I don't know. He intended to pub­lish something on it but never did. It might be possible."

"Ammann's 3-D tiles are the famous golden rhombohedra," Carl points out. "You can build anything with them, even periodic tilings. Bob must have found ways to prevent that. Did he notch them, or what?"

"He marked a comer of each facet of each rhombohedron with an x or an o, and claimed that if you match x's to a's you get a 3-D version of the rhom­bic Penrose tiles."

"And do you?" Carl asks. "Yes," I say. "Well, not exactly.

Socolar proved they force non-period­icity, but in a weaker sense than Pen­rose's. The whole question of matching rules deserves a fresh look They seem to come in various strengths: weak, strong, perfect. And the connections be­tween matching rules and hierarchical structure and projections is still murkY. And 3-D tiles are hard to visualize. So there's a thesis problem for you, if you need one."23

"Have any other 3-D non-periodic tiles been found?" asks Jane.

"Very few."24 () () () ()

I met Bob a third time six months later, in the fall of 1991. No longer so painfully shy, he accepted my invitation to speak at an AMS special session on tilings in Philadelphia-if I would pay his ex-

penses.25 He was eager to meet John Conway at last, and also Donald Cox­eter, with whom he had corresponded. 26

Bob's talk, mostly on his 3-D tiles, was more polished this time. Very pleased, I congratulated him. He beamed.

I never saw him again. Six years passed, with no word from

Bob. Then in 1997 my fractal book came back in the mail, without any note. At Christmas, I sent a card with a few words of thanks. Early in Janu­ary, I received a reply.

Dear Dr. Senechal, Your greeting card addressed to my

son Robert was received a few days ago. I am sorry to have to tell you that Robert died of a heart attack in May, 1994 . . .

Sincerely yours, Esther Ammann

She had sent me the book; she'd thought I'd known. Distressed, I con­tacted colleagues. None had heard from Bob since the Philadelphia meet­ing and none had known of his death.

In April, when the snow had melted, I drove the fifty miles to Brimfield to meet Bob's mother. Esther Ammann, a vigorous, intelligent widow of ninety­three, lived alone in a big house at the top of a hill with a panoramic view of the forest. The puzzle from Bielefeld sat proudly on her mantelpiece, sur­rounded by pictures of Bobby, her only child. Bobby in his cradle; Bobby at

"Did Bob's 3-D tiling tum out to Figure 1 5. Roger Penrose and Bob Ammann, March 21 , 1 991; photograph by Ludwig Danzer.

VOLUME 26, NUMBER 4, 2004 17

) 1�. ; _ l ' { tJ .J

t t •

_.. j

I

/ / /

5c. \ ' J_

/ <1- �. "-, ----->''

------6 ! 0 'i.

I ) / . .-----

'-}

10 � ... ,, ��� '

J

Figure 16. Ammann's drawing of the nets for his marked rhombohedra, in his first letter to

Gardner. The two on the left are obtuse, the two on the right acute. Cut them out and fold

them up!

three, with his favorite possession, a globe nearly his size; Bobby with his parents, Esther and August; Bobby with four or five cousins and gaily wrapped presents, in front of a Christ­mas tree; Bob with Roger Penrose in Germany. Over coffee she told me her story.*

Bob didn't mind sorting mail: he could let his mind wander. He'd always liked post offices. When he was little, Esther would hand him through the stamp window and leave him with the postal workers while she shopped. Bobby loved the maps on the walls, the routing books for foreign mail, the stamping machines, the sorting bins,

the trucks coming in and going out. The men enjoyed his arcane, intelligent questions. The tot knew more geogra­phy than anyone. One evening a dinner guest wondered whether the capital of Washington was Spokane or Seattle. "It's Olympia," chirped Bobby.

Bobby could read, add, and subtract by the time he was three. He tied sailors' knots, solved interlocking puz­zles, learned Indian sign language, oiled the sewing machine. He could ex­plain how bulbs grew into flowers, how frostbite turned into gangrene, how tis­sue healed in a bum, how teeth de­cayed, how caterpillars changed into butterflies.

But suddenly, before he was four, Bobby stopped talking. His doctors never knew why. For months, only Es­ther understood his mumbling; only she could explain him to his father, to his cousins, to the world. Gradually, with the help of a speech therapist, Bobby started speaking again, but slowly. He moved slowly, too. He never smiled. Nearsighted and absent­minded, he went in the "Out" doors and out the "In"s, and everybody laughed. Children bored him, so he wouldn't play with them, nor would they let him. He didn't like sports, but he loved jun­gle gyms, the high kind with criss­crossing bars.

"He was off the charts intellectually, but impossible emotionally," Esther continued. Bobby was happiest in the cocoon of his room, with his Scientific Americans and dozens of books, lots about math, some about dinosaurs.

Schoolwork bored him, so he didn't do it. Most of his teachers threw up their hands and gave him the low grades he'd technically earned. While his classmates struggled with fractions, Bobby computed the stresses on the ca­bles of the Golden Gate Bridge. He won the state math contests, and his SATs were near-perfect. MIT and Harvard in­vited him to apply, but turned him down after the interview. Brandeis Uni­versity accepted him. He enrolled, but he rarely left his dorm room and again got low grades. After three years, Bran­deis asked him to leave.

So Bob studied computer program­ming at a two-year business college and took a low-level job with the Hon­eywell Corporation near Boston. There he wrote and tested diagnostic rou­tines for minicomputer hardware com­ponents. Twelve years later, the com­pany let him go. He found another job but that company soon folded. In 1987, not long before I met him, Bob started sorting mail.

"The dinner at your house was a high point of his life," Esther sighed. "No one else reached out to him."

And the conference in Germany! "If only his father had lived to see his sue-

*I've incorporated recollections of members and friends of the Ammann family and, most extensively, Esther's brief, unpublished memoir of her son's first years into

my account of our meeting.

18 THE MATHEMATICAL INTELLIGENCER

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Figure 11. Esther Ammann's list of Bobby's books at age 12 (first of two pages).

cess! Bob died when his career was

only beginning. He'd have done so

much more if he'd been given the time."

Esther Ammann died in January,

2003.

0 0 0 0 The math lounge erupts in consterna­

tion.

"He'd have done so much more if

he'd been given more training!" Jane

exclaims.

"Didn't anyone at Brandeis notice

Bob was a math genius?" Carl protests.

"Lots of math students are socially

challenged; professors expect it."

"He didn't take math courses, ex­

cept a few in analysis," I reply. "Bob's

kind of math was out of fashion in the

1960s. It wasn't taught anywhere. My

course on tiling theory may have been

the first."

"Out of fashion? Math is timeless!"

Jane declares.

"Hardly, but let's leave Bourbaki out

of this. It's late, and I want to finish my

story."

"In any case, Bob wasn't trainable,

was he?" Richard says. "Pass the cake."

0 0 0 0 "He was a kind and gentle soul," John

Thomas, a childhood and close family

friend, told me. "I have encountered

many bright people during my studies

at Reed College and at MIT, but Bob is

the only person I have personally

known who had, without question, a

genius-level intellect."

They lost touch when John left for

college, but reconnected when he re­

turned to MIT. "Bob began weekly vis­

its, always on Wednesday nights. We

had supper, talked a little bit, watched

the original 'Charlie's Angels' show on

an old black-and-white TV that Esther

had given us, and talked some more."

Bob himself owned three TVs; he

watched them all at once.

Those were the Honeywell days.

"We shared a cubicle for a few years,

in the early 1970s," a co-worker, David

Wallace recalled. "Bob was very shy. It

was two years in the same office be­

fore I found out how he pronounced

his last name! Everyone in the depart­

ment pronounced it like the capital city

of Jordan: 'Ah-mahn.' He prounounced

it 'am-man. ' But he never corrected the

mispronunciation."

While we specialists played with

Bob's tilings, studying them, applying

them, extending them in new and sur­

prising directions, Bob's life kept hit­

ting dead ends. He backtracked and

tried again, over and over, but still

nothing fit.

John and his wife moved back to

Oregon. Esther didn't see much of her

son after that. Bob and his father had

never gotten along; August was strict

and impatient. And she hadn't known

how to conciliate.

Bob's co-workers kept their dis­

tance. "He had a weird sense of hu­

mor," David explained. "His stories

about the 'penguin conspiracy' and his

theory that Richard Nixon and Patty

Hearst were the same person were, to

put it mildly, enough to make one won­

der for his sanity. And he used his desk

to store food that really should have

been kept refrigerated-like ham­

burger! For up to a week! He was fond

of canned spaghetti but didn't like the

sauce so he'd wash the stuff in the sink

or the nearest drinking fountain."

Bob's apartment resembled his desk

at the office. In 1976 the health in­

spector condemned it-though it was­

n't that bad-and Bob was evicted. He

stored all his furniture, except his TVs,

and moved into a motel on Rte. 4, mid­

way, it turned out conveniently, be­

tween Honeywell and the post office.

"Honeywell laid him off in one of their

quarterly staff reductions, regular as

clockwork for almost ten years!" David

told me. "Bob kept coming to work any­

way. They eventually put him back on

the payroll. When he was laid off a sec­

ond time a few years later, the security

guards were given his picture and told

not to let him back in the building."

Bob phoned John often over the

years. The invitation to Germany terri­

fied him. They talked about it end­

lessly. Somehow Bob found the courage to go. Esther didn't know he'd

gone until he came back.

The motel was Bob's home for the

rest of his life. He ate at the fast food

joint next door. One day the cleaning

woman found him dead in his room. A

heart attack, the coroner said. He was

46 years old.

Steve Tague, another Honeywell co­

worker and the executor of Bob's es­

tate, salvaged loose sheets of doodles

from the swirl of junk mail, old phone

books and TV guides, uncashed pay­

checks, and faded magazines Bob had

stuffed in the back of his car. Steve

found smaller items too, which he

placed in a white cardboard box. He

VOLUME 26, NUMBER 4, 2004 1 g

Figure 18. Request form for leave from the United States Post Office.

stored the box and the doodles in the

attic of his home in northern Massa­

chusetts, near the New Hampshire bor­

der. Ten years later, when I drove there

to talk with him, he showed them to

me.

I looked through the doodles. They

seemed just that.

In the white box I found the shards

of Bob's shattered childhood: two

cheap plastic puzzles; a little mechani­

cal toy; a half-dozen birthday cards, all

from Mom and Dad; school report

cards, from first grade through eighth;

a tiny plastic case with a baby tooth

and a dime; a tom towel stamped with

faded elephants and a single word, BOBBY. And some letters and clip­

pings and drawings, among them a

front-page news article, dated 1949.27

A little boy who is probably one of the smartest three-year-olds in the coun­try. . . . With a special love for geog­r-aphy, he can quickly name the capi­tal of any state or can point out on a globe such hard-to-find places as Mozambique and Madagascar . . . . He is now delving into the mysteries of arithmetic. He startled both his par­ents the other day by telling them that 'jour and two is six and three and three is six and five and one is six. "

In the picture little Bobby, looking

earnestly at the photographer, sits with

his globe.

20 THE MATHEMATICAL INTELLIGENCER

I almost missed the poem tucked in­

side a folded sheet of green construc­

tion paper. "I hope you'll write more

like this one!" Bobby's fifth-grade

teacher had written on the back.

I'm going to Mars Among the stars The trip is, of all things, On gossamer wings.

Figure 19. Undated (1949) clipping from The

Herald (Richland, Washington).

Acknowledgments

I am very grateful to members of

Robert Ammann's family, Esther Am­

man, Berk Meitzler, Grant Meitzler,

Russell Newsome, and Robert St. Clair,

for sharing memories, letters, pho­

tographs, and other family documents

with me; to friends of the Ammann

family, Jean Acerra, Eleanor Boylan,

Dixie Del Frate, Louise Rice, and Fred­

erick Riggs, for their anecdotes and in­

sights; and to Robert's friends and co­

workers Steven Tague, John Thomas,

and David Wallace. Berk Meitzler put

me in touch with all the others; Steven

Tague made invaluable material from

Robert's estate available to me.

Martin Gardner and Branko Grtin­

baum generously gave me access to

their large files of Ammann correspon­

dence. I am also grateful to Michael

Baake, Ludwig Danzer, Oliver Sacks,

Doris Schattschneider, Joshua Socolar,

and Einar Thorsteinn for advice and as­

sistance.

Michael Baake, Doug Bauer, Eleanor

Boylan, David Cohen, N. G. de Bruijn,

Dixie Del Frate, Frederick Riggs, Doris

Schattschneider, Marilyn Schwinn

Smith, Steven Tague, John Thomas,

and Jeanne Wikler read early versions

of this manuscript and made thought­

ful suggestions, most of which I have

adopted. I am also grateful for the en­

couragement and constructive criti­

cisms from fellow participants in two

workshops in creative writing in math­

ematics at the Banff International Re­

search Station at the Banff Centre,

Canada.

NOTES AND REFERENCES

1 . The artist is Olafur Eliasson. Einar Thorsteinn

supplied this information.

2. All letters to and from Martin Gardner

quoted in this article, except Ammann's

first, belong to the Martin Gardner Papers,

Stanford University Archives, and are used

here with kind permission.

3. Grunbaum and Shephard preferred the

term "aperiodic" for such tiles. Most au­

thors use the terms "aperiodic" and "non­

periodic" interchangeably.

4. John Conway's fanciful names-sun, star,

king, queen, jack, deuce, and ace-for the

seven vertex configurations allowed by

Penrose's rules seem permanent.

5. Grunbaum and Shephard proved many of

Ammann's assertions about his tiles; he

joined them as co-author of Ammann, R . ,

Grunbaum, B . , and Shephard, G. C . , "Ape­

riodic Til ings," Discrete and Computational

Geometry, 1 992, vol. 8, no. 1 , 1 -25.

6. Martin Gardner's chronicles of "Dr. Matrix"

include The Incredible Dr. Matrix; The

Magic Numbers of Dr. Matrix; and Trap­

doors, Ciphers, Penrose Tiles, and the Re­

turn of Or. Matrix.

7. Heesch's problem asks whether, for each

positive integer k, there exists a tile that can

be surrounded by copies of itself in k rings,

but not k + 1 . Such a tile has Heesch num­

ber k. Robert Ammann was the first to find

a tile with Heesch number 3 . Today tiles

with Heesch numbers 4 and 5 are known,

but the general problem is still unsolved .

8. Hao Wang, "Proving theorems by pattern

recognition. I I , " Bell System Tech. J. 40,

1 961 , 1 -42.

9. Branko Grunbaum and Geoffrey Shep­

hard, Tilings and Patterns, W. H. Freeman,

New York, 1 987.

1 0. Martin Gardner, "Extraordinary nonperiodic

tiling that enriches the theory of tiles , "

Mathematical Games, Scientific American,

January, 1 977, 1 1 0-1 21 .

1 1 . See Tilings and Patterns, Chapter 1 0.6,

"Ammann bars, musical sequences and

forced tiles , " pp. 571 -580.

1 2 . See N. G. de Bruijn , "Algebraic theory of

Penrose's non-periodic tilings of the

plane," Proceedings of the Koninglike Ned­

erlandse Akadernie van Wetenschappen

Series A, Vol. 84 (lndagationes Mathernat­

icae, Vol. 43), 1 981 , 38-66. De Bruijn

showed that the construction is really very

general. Using n-grids and n-dimensional

cubes, for any positive integer n, one gets

non-periodic tilings of non-Penrose types.

In general, the construction gives tilings

with many different tiles whose matching

rules, if they exist, remain a mystery, but a

few very interesting tilings have been found

in this way. See, e .g . , J .E.S. Socolar,

"Simple octagonal and dodecagonal qua­

sicrystals," Physical Review B, vol 39, no.

1 5 , May 1 5, 1 989, 1 05 1 9-51 .

1 3. See M. Senechal and J. Taylor, "Qua­

sicrystals: the view from Les Houches, "

The Mathematical lntelligencer, vol. 1 2 , no.

2, 1 990, 54-64.

1 4 . Gardner's files show that Benoit Mandel­

brot met Ammann once in 1 980. I had not

met Mandelbrot then.

1 5 . All letters to and from Branko Grunbaum,

except my letter after meeting Ammann ,

are used with Grunbaum's kind permis­

sion.

1 6 . Ammann visited and corresponded with

Paul Steinhardt and his students, Dov

Levine and Joshua Socolar.

1 7 . Robert Ammann, "Another Explanation of

the Cretaceos-Tertiary Boundary Event, "

unpublished.

1 8 . For the journal Structural Topology. The

editor, Henry Crapo, also wrote to Am­

mann about this but also received no

reply.

1 9. Roger Penrose, "Remarks on Tiling , " in R .

Moody (ed.) , The Mathematics of Long­

Range Aperiodic Order, Kluwer, 1 995, p .

468.

20. loan James, "Autism in Mathematics," The

Mathematical lntelligencer, vol. 25, no. 4 ,

Fall 2003, 62-65.

21 . Norbert Wiener, Ex-Prodigy, pp. 3-7,

1 25-1 42.

22. Conference, "Geometry of Quasicrystals,"

March 1 8-22, 1 99 1 , ZIF (Center for Inter­

disciplinary Research), Bielefeld University,

Bielefeld, Germany.

23. Joshua Socolar, "Weak Matching Rules for

Quasicrystals," Communications in Math­

ematical Physics, vol 1 29, 1 990, 599-6 1 9 .

I t should b e noted that Michael Longuet­

Higgins's "Nested Triacontahedral Shells,

or how to grow a quasicrystal , " The Math­

ematical lntelligencer, vol. 25, no. 2, Spring

2003, bears no relation to Ammann's con­

struction.

24. See, e .g . , P. Kramer and R. Neri, "On Pe­

riodic and Non-periodic Space Fillings of Em

Obtained by Projection, " Acta Crystallo­

graphica (1 984), A40, 580-587; L. Danzer,

"Three dimensional analogues of the planar

Penrose tilings and quasicrystals," Discrete

Mathematics, vol. 76, 1 989, 1 -7; and L.

Danzer, "Full equivalence between Soco­

lar's tilings and the (A,B,C,K)-tilings leading

to a rather natural decoration, " International

Journal of Modern Physics B, vol. 7 , nos. 6

& 7, 1 993, 1 379-1 386.

25. Special Session on Tilings, 868th meeting

of the American Mathematical Society,

Philadelphia, Pennsylvania, October 1 2-

1 3, 1 991 . The American Mathematical

Society does not pay honoraria or travel

expenses.

26. At the last minute Coxeter couldn't come.

They never met.

27. H. Williams, "Richland Lad, 3 , is Wizard at

Geography," The Herald (Richland, Wash­

ington), 1 949 (undated clipping). The Am­

mann family had moved from Massachu­

setts to Washington while August Ammann,

an engineer, worked on a nuclear power

construction project there.

VOLUME 26, NUMBER 4, 2004 21

M a them a tic a l l y Bent

The proof is in the pudding.

Opening a copy of The Mathematical

Intelligencer you may ask yourself

uneasily, "lthat is this anyway-a

mathematical journal, or what?" Or you may ask, "�there am !?" Or even

"ltho am !?" This sense of disorienta­

tion is at its most acute when you

open to Colin Adams's column.

Relax. Breathe regularly. It's

mathematical, it's a humor column,

and it may even be harmless.

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

Science Center, Williams College,

Williamstown, MA 01 267 USA

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

Colin Adams , Editor

Mangum, P. l . Colin Adams

The name's Mangum. Dirk Mangum, P.l. Yeah, that's right. I am a Prin­

cipal Investigator. On a National Sci­ence Foundation grant. Didn't start out that way, though. You don't just decide to be a P.I. No, you have to earn the right. For me, it wasn't anything I ex­pected. Just a fortuitous set of circum­stances, although it didn't seem fortu­itous at the time. Quite the contrary.

I was working as a snotnosed post­doc out of a sleazy hole-in-the-wall of­fice in LA. Actually, UCLA to be spe­cific. It was my third year of a three-year appointment, and I didn't have anything to show for the first two years except a stuffed wastebasket, a pile of empty Orangina bottles, and a whole lot of self-doubt.

My story begins on one of those days you get in LA. The sun was shin­ing, a slight breeze was ruffling the palm trees, and it was an even 70 de­grees. Actually, I just described every day in LA. It's enough to make you

want to scream. Just give me a cloud, or some fog. Or god forbid, a hailstorm. But no, there is the sun, day in, day out, beating a drum beat on your brain, banging out its sunny sun dance until you want to do things that would get you into serious trouble with Accounts Payable.

I was hunkered down in my office, feet up on the desk, sucking on my sec­ond bottle of Orangina for the day. I had been wrestling with the proof of a lemma all afternoon, but it had me in a double overhook headlock and the chances I would end up anywhere but on the mat were slim indeed. The constant drone of the air-conditioner sounded like a UPS truck tackling the Continental Divide. There was a knock at my door.

"I'm not in," I yelled. There was a pause; then a second

22 THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+ Business Media, Inc.

knock I sighed, lifting my feet off the desk

"If you won't go away, you might as well come in."

The door swung open, and I just about swallowed my bottle of Orang­ina whole. Standing in the doorway was none other than Walter P. Parsnip, chair of the Berkeley Math Depart­ment. He was dressed suggestively, in a white buttondown, top button un­done to expose his clavicle, and slacks so worn you could almost seen through them at the knee. His shirt clung to his chest, the outline of his bulging stom­ach obvious for all to see.

I found it hard to believe he was here before me. I used to drool over this guy's articles when I was an un­dergraduate. He had a career built like a brick shipyard. And talk about legs. He published his first article in 1932,

and he was still going strong. Half the functions in Wang Doodle theory were named after Parsnip, and the other half were named after his dog.

I gave him a long look up and down and then said as smoothly as I could, "Well come on in here and take a load off."

He took his time coming in, giving

my eyeballs a chance to run over his body at will. I took full advantage of the opportunity. He slid into the over­stuffed leather chair that sat in front of my desk and stretched his legs out be­fore him.

I noticed a single bead of sweat work its tortuous way down his nose and then drop off, only to land on his extruding lower lip. I gulped.

"I'm . . . ," he started to say. "Oh," I said, cutting him off, "I know

who you are. What I don't know is what someone as hot as you wants with someone as cold as me."

"I'm in trouble," he said. "Who isn't?" I retorted. ''I'm in deep trouble," he said. He

fixed me with a look that would have made me swallow my tongue if I hadn't been chewing on it at the time.

He leaned forward conspiratorially, giving me a nice view down the inside of his well-used pocket protector. "I've got a theorem. It's a big one."

"I bet it is," I said, trying to sound casual.

But I knew that if Parsnip thought it was big, it would make Riemann Roch look like Zorn's Lemma.

"It implies Canooby." The Canooby Conjecture, perhaps

the biggest open problem in all of Pinched Rumanian Monofield Theory. You solve Canooby, and they deliver the presidency of the American Math Society to your doorstep.

"Doesn't sound like a problem to me," I said.

"It's joint work with Kazdan." I lifted an eyebrow. Kazdan was the

current darling of the math community. Twenty-six years old, Belgian, and bril­liant. So hot that if he were a waffle iron, you could pour batter and get fully cooked waffles in an instant. Bel­gian waffles.

I watched as Parsnip crossed his legs, his pant cuff riding up enough to expose some hairy leg just above his sheer black socks. He caught me tak­ing a gander.

"So, what's the problem with work­ing with Kazdan?" I asked.

"Kazdan isn't working with me any more. He dumped me for Vichy." Shwase Vichy was the youngest faculty member ever to get tenure a Chicago; he was still packing a lunch box. This must be hard on Parsnip.

"How can I help?" I asked, looking deep into his milky brown eyes. They were eyes you could spend a lot of time looking into. Why you would want to do that, I don't know, but people pick strange hobbies.

"It is a lemma," he said. "Just one lemma I need. With the lemma, I will have my proof."

"What makes you think I can help you with your lemma?" I asked, lean­ing back in my chair, trying to appear disinterested.

"They tell me you are the best when it comes to the theory of semiupper­pseudohypermultitudinal fluxions."

"Well, that was the title of my Ph.D. thesis. But you're the first person who ever pronounced it correctly."

"It is exactly what is needed to solve

my dilemma. What will it take to get you to help me, Dirk?"

He placed his hand on mine. I felt the warmth of his gnarled knuckles.

I smiled my most captivating smile. "Who in his right mind would tum down a chance to publish with you?"

He smiled back.

Over the next eight months I de­voted myself to the problem. I should have been writing papers based on my thesis, getting published to ensure a follow-up job. Instead, I thought of nothing but the lemma. I worked on it in the shower. I worked on it in the tub. I even worked on it at the office. It be­came an obsession.

I started to dream about it. There was one dream in which Parsnip and I were dancing the rhumba. Shwase Vichy danced over, laughed in that falsetto laugh of his, and said, "Oh, no, you are not doing math here." I woke up in a cold sweat.

And still, the lemma wouldn't budge. Parsnip notwithstanding, I was ready to give up. It seemed hopeless. But then, one day, as I was stepping off the bus, it hit me. I had an epiphany. Suddenly realizing what I had been missing, I couldn't believe my stupid­ity. All this time I had been working on semiupperpseudohypermultitudinal fluxions. When I should have been thinking about multihyperpseudoup­persemitudinal fluxions. I had been looking at it exactly backward. With this realization, I knew that I had not only solved the problem, but I had cre­ated a whole new field of mathematics.

The other passengers waiting to get off the bus began to push, but I didn't care. I knew I was right.

I rushed to my office, overwhelmed with excitement. I would have Parsnip's undying gratitude. A tenured position at Berkeley might be in my future. Parsnip picked up his phone on the first ring. "Hello, Parsnip? I solved your problem."

"You solved it?" he shouted into the phone. "That's amazing."

"Yes, it is," I said. "Why don't you come on down from Berkeley, and I'll show it to you. Then you can tell me how great I am."

"No, I can't wait," he said. "Please fax it to me now. I'll come down Mon­

day."

I should have smelled a double-deal­ing rat, but they have yet to perfect an odor-producing phone. So I faxed it to him.

The next morning, when I opened the LA Times, I saw the huge bold headline splashed across the page. "PARSNIP AND KAZDAN SOLVE CANOOBY." This time I did swallow my tongue, but luck­ily I quickly coughed it up. There was a huge picture of the two of them shaking hands with the governor. I had been played for a fool.

Figuring out what had happened took me less time than it takes a bam fly to find sustenance. Parsnip and Kaz­dan were working on Canooby the en­tire time, but they got stuck. They needed help, but they weren't about to let a pissant postdoc like me get my name on a theorem as big as this. So they devised their ruse: Parsnip comes to see me, acting the jilted collaborator, desperate for my aid. Sucker that I am, I fall head over heels. They figure I can't resist his charms, and they're right.

Once they have the fax, I'm history. Nobody will believe a postdoc without a single publication to his name, and with a job disappearing faster than the woolly mammoth. In a year, I would be pumping Slurpees at the local Seven Eleven.

For the first three days, I sat in my office and cried into my Orangina. Al­though diluted, the salt in the tears added zest. For the following three days I tried to figure out how to fran­chise salted Orangina.

On the seventh day, I received a grant proposal for review from the Na­tional Science Foundation. And won­ders of wonders, it was from Kazdan and Parsnip. They wanted five million dollars to study multihyperpseudoup­persemitudinal fluxions. Now, why the National Science Foundation sent the proposal to me for review, I'll never know. They certainly didn't know I in­vented the field. And it's unlikely they realized there was a connection be­tween multihyperpseudouppersemitu­dinal fluxions and semiupperpseudohy­permultitudinal fluxions. But for whatever reason, the osprey of oppor-

VOLUME 26, NUMBER 4, 2004 23

tunity had come to roost in my lap, and

I have to tell you, it felt good having it

there.

For the next two weeks, I worked

on multihyperpseudouppersemitudinal

fluxions. I saw vistas never before

glimpsed by man or beast. I wandered

the high plateaus of human thought,

breathing the rarefied air. To protect

myself from the elements, I built little

Quonset lemmas, small rounded pup

tents made out of words and symbols.

I thought I might need them if it rained.

And it did rain. First a little bit. And

then a lot. It poured as if the high

plateau of human thought lay beneath

a huge shower head, and somebody­

! don't know who-had turned it on

full. There was a deluge. For, you see,

I realized that multihyperpseudoup­

persemitudinal fluxions have ab­

solutely nothing to do with pinched Ru­

manian monofields or the Canooby

Conjecture. Yes, I had been mistaken.

Oops! My bad.

So I wrote a one-hundred-page re­

view of the grant proposal, pointing out

the error, and explaining how the field

of multihyperpseudouppersemitudinal

fluxions, although useless for the pur­

pose outlined in the proposal, was in

fact, just what is needed to model ap­

propriate salt content in carbonated

beverages.

Then I drove up to Berkeley, arriving

at the height of a lecture being given

by Parsnip on Canooby. Although he

saw me enter the lecture hall, it didn't

seem to shake him in the least. No, he

seemed to relish the opportunity to

show me how carefully he had con­

structed his deception. I sat down in

the front, right next to Kazdan.

Parsnip was going on about functor

24 THE MATHEMATICAL INTELLIGENCER

this and functor that, when I raised

my hand. He paused. I stood up and

said, "Cut to the chase. Who invented

multihyperpseudouppersemitudinal

fluxions?"

He actually smiled. "As everyone

knows, it was Kazdan and I. Don't you

read the papers?"

"Oh, yes, I read the papers," I said.

"But you know what they say. Don't be­

lieve everything you read."

"Young man, I'm not sure I under­

stand what you are getting at. Should I

know you? Are you a graduate student

visiting from out of town? Perhaps you

are looking for the cookies. They are

in the Math Lounge."

"The name's Mangum, Dirk Mangum,"

I said calmly. "But you know that."

There must have been something in

the way I said my name that made him

uncomfortable. The self-assured smile

fell from his face for just a second.

Then I fired. "If multihyperpseudo­

uppersemitudinal fluxions play such an

important role in the solution of the

Canooby Conjecture, then why is it that

they aren't connected? Canooby as­sumes that the fluxions are connected."

Parsnip's expression went from un­

sure to shocked in a split second.

Clearly, I had hit my mark. He gripped

the lectern for support as the blood

fled from his face. He was clearly in

pain.

"What do you mean they aren't con­

nected?" he croaked. Kazdan leaped up

from his chair, but there was nothing

he could do. The audience sat in

stunned silence as they watched the

tableau unfold. I fired again.

"I mean they aren't connected. Not at­

tached to one another. Capice? There is space in between them. Here's one and

here's another and you can't get from the

one to the other. Comprende? THEY COME IN MORE THAN ONE PIECE. So they don't apply to Canooby!"

Parsnip fell to one knee. A shudder

went through the audience. Kazdan

grabbed my sleeve, for what purpose I

don't know, but I shrugged him off, and

he fell back into his chair, stricken.

I smiled, then, at Parsnip. He

reached a trembling hand in my direc­

tion. "Dirk," he said. "Help me, Dirk."

For a moment, I almost felt sorry for

him. But I got over it.

"See you around", I said. "Actually,

I kind of doubt I will." I walked out the

door as he crumpled to the floor.

When I got back to LA, I submitted

the grant review. To quote from the let­

ter I received,

Never before have we received a review

that so clearly demonstrates the genius

of the reviewer, while also demon­

strating the entire paucity of ideas in

the original proposal. Not only do we

reject the proposal, but we would like

to give you a grant. How does a mil­

lion dollars sound? And that's just for

the first year. Any time you want ad­ditional funds, day or night, just call

the director of NSF Her home phone

number appears at the bottom.

Parsnip and Kazdan were so em­

barrassed that they dropped out of

Pinched Rumanian Monofield theory

entirely. Now they work in probability,

mostly taking turns pulling colored golf

balls out of bins. I ended up staying at

UCLA. After a while, you get used to the weather. And I have been a P.l. ever

since. If you need a P.l., give me a call. My number's in the book.

HINKE M. OSINGA AND BERND KRAUSKOPF

Crochet i ng the Lorenz Man ifo d

ou have probably seen a picture of the famous butterfly-shaped Lorenz attractor-

on a book cover, a conference poster, a coffee mug, or a friend 's T-shirt. The Lorenz

attractor is the best-known image of a chaotic or strange attractor. We are con­

cerned here with its close cousin, the two-dimensional stable manifold of the

origin of the Lorenz system, which we call the Lorenz man­ifold for short. This surface organizes the dynamics in the

three-dimensional phase space of the Lorenz system. It is

invariant under the flow (meaning that trajectories cannot

cross it) and essentially determines how trajectories visit

the two wings of the Lorenz attractor.

We have been working for quite a while on the devel­

opment of algorithms to compute global manifolds in vec­

tor fields, and we have computed the Lorenz manifold up

to considerable size. Its geometry is intriguing, and we ex­

plored different ways of visualizing it on the computer [6,9].

However, a real model of this surface was still lacking.

During the Christmas break 2002/2003 Rinke was relax­

ing by crocheting hexagonal lace motifs when Bernd sug­

gested, "Why don't you crochet something useful?"

The algorithm we developed "grows" a manifold in steps.

We start from a small disc in the stable eigenspace of the

origin and add at each step a band of a fixed width. In other

words, at any time of the calculation the computed part of

the Lorenz manifold is a topological disc whose outer rim is

(approximately) a level set of the geodesic distance from the

origin. What we realized is that the mesh generated by our

algorithm can be interpreted directly as crochet instructions!

After some initial experimentation, Rinke crocheted the

first model of the Lorenz manifold, which Bernd then

mounted with garden wire. It was shown for the first time

at the 6th SIAM Conference on Applications of Dynamical

Systems in Snowbird, Utah, in May 2003, and it made a sec­

ond public appearance at the Equadiff 2003 conference in

Hasselt, Belgium, in July 2003 [7] . The model is quite large,

about 0.9 m in diameter, and has to be flattened and folded

for transportation.

In this article we explain the mathematics behind the

crocheted Lorenz manifold and provide complete instruc­

tions that allow you to crochet your own. The images

shown here are of a second model that was crocheted in

the Summer of 2003. We took photos at different stages,

and it was finally mounted with great care and then pho­

tographed professionally. This second model stays

mounted permanently, while we use the first model for

touring.

We would be thrilled to hear from anybody who pro­

duces another crocheted model of the Lorenz manifold. As

an incentive we offer a bottle of champagne to the person

who produces model number three. So do get in touch

when you are done with the needle work!

The Lorenz System

The Lorenz attractor illustrates the chaotic nature of the

equations that were derived and studied by the meteorolo­

gist E. N. Lorenz in 1963 as a much-simplified model for

the dynamics of the weather [8]. Now generally referred to

as the Lorenz system, it is given as the three ordinary dif­

ferential equations: {:t = u(y - x), y = px - y - xz, z = xy - {3z.

(1)

© 2004 Spnnger Sc1ence+ Bus1ness Media, Inc., VOLUME 26, NUMBER 4, 2004 25

We consider here only the classic choice of parameters, namely u = 10, p = 28, and f3 = 2l The Lorenz system has the symmetry

(x,y,z) � ( -x, -y,z), (2)

that is, rotation by 7T about the z-axis, which is invariant under the flow of (1).

A simple numerical simulation of the Lorenz system (1) on your computer, starting from almost any initial condi­tion, will quickly produce an image of the Lorenz attractor. However, if you pick two points arbitrarily close to each other, they will move apart after only a short time, result­ing in two very different time series. This was accidentally discovered by Lorenz when he restarted a computation from printed data rounded to three decimal digits of accu­racy, while his computer internally used six decimal digits; see, for example, the book by Gleick [ 1 ] .

While the Lorenz system has been widely accepted as a classic example of a chaotic system, it was proven by Tucker only in 1998 [ 12] that the Lorenz attractor is actu­ally a chaotic attractor. For an account of the mathemat­ics involved see the lntelligencer article by Viana [ 13].

Stable and Unstable Manifolds

The origin is always an equilibrium of (1) . The eigenvalues of the linearization at the origin are

-f3 and - _u_+_1 + .!. v--:-c u-+....,.1""'?'+---:-4<T(---::-p---1-:7") 2 - 2

.

For the standard parameter values they are numerically

- 22.828, - 2.667, and 1 1.828

in increasing order. This means that the origin is a saddle with two attracting directions and one repelling direction. According to the Stable Manifold Theorem [2, 1 1], there ex­ists a one-dimensional unstable manifold wu(O) and a two­dimensional stable manifold W5(0), defined as

wu(O) = {x E �3 l lim q/(x) = OJ, t�-oc

W5(0) = {x E �3l lim q/(x) = 0}, t_.x

where cjJ is the flow of ( 1). The manifolds wu(O) and W5(0) are tangent at the origin to the unstable and stable eigen­space, respectively. We call W5(0) simply the Lorenz man­ifold. While most trajectories end up at the Lorenz attrac­tor, those on W5(0) converge to the origin instead.

The z-axis, the axis of symmetry, is part of the Lorenz manifold W5(0), which is itself invariant under rotation by 7T around this axis. Furthermore, there are two special tra­jectories that are tangent to the eigenvector of -22.828, which is perpendicular to the z-axis. They form the two branches of the one-dimensional strong stable manifold W55(0). All other trajectories on W5(0) are tangent to the z­axis.

Apart from the origin, the Lorenz system (1) has two other equilibria, namely

c�Vf3CP - 1), � Yf3(p - 1), P - 1) = ( �8.485, �8.485, 27),

26 THE MATHEMATICAL INTELLIGENCER

Figure 1. The two branches of the unstable manifold, one red and

one brown, accumulate on the Lorenz attractor. The little blue disc

is in the stable eigenspace and separates the two branches.

which are also saddles. They sit in the centres of the "wings" of the Lorenz attractor and are each other's image under the symmetry (2).

Figure 1 shows an image of the Lorenz attractor that was not obtained by simply integrating from an arbitrary start­ing condition, but by computing the one-dimensional un­stable manifold wu(O). Since the origin is in the Lorenz at­tractor, plotting wu(O) gives a good picture of the attractor. We computed both branches, one in red and one in brown, of the unstable manifold of the origin by integration from two points on either side of W5(0) at distance 10-7 away from the origin along the unstable eigendirection.

It is clear from Figure 1 that each of the branches of the unstable manifold visits both wings of the attractor, as is to be expected. In fact, because of the symmetry of equa­tions (1), the red branch is the symmetric image of the brown branch. Locally near the origin, each branch starts on a different side of the two-dimensional stable manifold W5(0). In Figure 1 we show a small local piece of WS(O) as a small blue disc.

The main question is: what does the global Lorenz man­ifold W5(0) look like, as it "wiggles" between the red and brown curves of Figure 1? Remember that W'(O) cannot cross W5(0).

Geodesic Level Sets

The Lorenz manifold, like any global two-dimensional in­variant manifold of a vector field, cannot be found analyt­ically but must be computed numerically. The knowledge of global stable and unstable manifolds of equilibria and periodic orbits is important for understanding the overall dynamics of a dynamical system, which we take here to be given by a finite number of ordinary differential equations. In fact, there has been quite some work since the early 1990s on the development of algorithms for the computa­tion of global manifolds. We do not give details here but

refer to [5] for a recent overview of the literature. The key

idea of several of these methods is to start with a uniform

mesh on a small circle around the origin in the stable eigen­

space and then use the dynamics to "grow" this circle out­

ward. The main problem one needs to deal with is that the

flow does not evolve the initial circle uniformly, so that the

mesh quality generally deteriorates very quickly.

The goal of our algorithm is to compute "nice circles"

on the Lorenz manifold to obtain a uniform mesh. Nice cir­

cles on the manifold are those that consist of points that

lie at (approximately) the same distance away from the ori­

gin. In other words, we want to evolve or grow the initial

circle radially outward (away from the origin) and with the

same "speed" everywhere. To formalize this, we consider

the geodesic distance between two points on the manifold,

which is defmed as the length of the shortest path on the

manifold connecting the two points. The geometrically

nicest circle is then a geodesic level set, which is a smooth

closed curve whose points all lie at the same geodesic dis­

tance from the origin.

The algorithm that we developed computes a manifold

as a sequence of approximate geodesic level sets; see

[4, 5] for the details. We start from a small disc in the sta­

ble eigenspace of the origin which we represent by a cir­

cular list of equidistant points around its boundary. This is

our first approximate geodesic level set. The algorithm then

adds at each step a new approximate geodesic level set,

again given as a circular list of points. To this end we com­

pute for every known point on the present geodesic level

set the closest point that lies on the new geodesic level set.

(This can be achieved by solving a boundary-value prob­

lem). When the distance between neighbouring points on

the new level set becomes too large, we add a new point

between them by starting from a point on the present level

set. Similarly, we remove a point when two neighbouring

points become too close. In this way, we ensure that the

distribution of mesh points along the new level set is close

to uniform. At the end of a step we add an entire band of

a particular fixed width to the manifold. The width of the

band that is added depends on the (local) curvature of ge­

odesics on W8(0).

Global Information Encoded Locally

We used our algorithm to compute the Lorenz manifold up

to considerable size, where we made use of the parametri­

sation in terms of geodesic distance; illustrations and ac­

companying movies of how the Lorenz manifold is grown

were published in [6, 9] and are not repeated here. Instead,

we show a crocheted model of the Lorenz manifold (see

Figures 4-6). The key observation is that, while the algo­

rithm computes each new mesh point as a point in IR3, the

essential information on the shape of the manifold is actu­

ally encoded locally!

This is illustrated in Figure 2a, which shows an en­

largement of a part of the Lorenz manifold with the trian­

gular mesh that was computed. Consecutive bands are

shown altematingly in light and darker blue; the mesh

points in the bottom right comer are closest to the origin.

a

b

Figure 2. A close-up of the mesh generated by our algorithm, show­

ing bands of alternating colour and the edges of the triangulation (a),

and (practically) the same close-up of the crocheted manifold (b).

New crochet stitches are added exactly where new mesh points are

added.

The mesh is formed from the mesh points on the geodesic

level sets. The diagonal mesh lines from bottom right to top

left are approximations of geodesics; they are perpendicu­

lar to the level sets. Whenever two such geodesics move

too far apart, a new one is started between them where a

new point is added.

The image is from a part of the manifold that is almost

flat. Because the circumference of a planar disc is linearly

related to its diameter, the number of new points being

added to the level sets depends linearly on the geodesic

distance covered. If the level sets are all at the same dis­

tance from each other, as in Figure 2, then a fixed number

VOLUME 26, NUMBER 4, 2004 27

of new points is added at each step. If the manifold is

cmved, however, then the number of points added during

the steps varies with the geodesic distance covered. For

positive local curvature, fewer points are added, whereas

for negative local curvature more points are added.

The crucial point is that the curvature of the manifold

is given locally on the level of the computed mesh simply

by the information where we added or removed points dur­

ing the computation.

Interpretation as Crochet Instructions

The observation detailed above allows us to interpret the

result of a computation by our algorithm directly as a cro­

chet pattern. Starting from a small crocheted circle, each

new band is created by making one or more crochet

stitches of a fixed length (translated from the width of the

respective band) in each stitch of the previous round. Ex­

tra stitches are added or removed where points were added

or removed during the computation; this information was

written to a file.

Figure 2b, the crocheted object, shows practically the

same part of the manifold as is shown in Figure 2a. You are

encouraged to look closely for the points in the crocheted

manifold where an extra crochet stitch was added and iden­

tify the same points in the computed mesh.

To preserve the geometry of the manifold one needs to

ensure that the horizontal width of the stitch used and its

length are in the same ratio as the average distance be­

tween mesh points on a level set and the width of the re­

spective band. The crocheting reader will be relieved to

hear that these considerations were translated into the cro­

chet instructions given below-simply following them slav­

ishly will give a good result.

We assume that the reader is familiar with the basic cro­

chet stitches, as they can be found in any book on chro­

cheting. Throughout we use the British naming convention

of stitches (ch, de, tr, dtr), which differs from the Ameri­can one (Table 1). The Lorenz manifold is crocheted in

rounds. Stitches in each round are counted with respect to

the previous round, starting from the number 0.

The first stitch of a round is 1 ch, 3 ch, or 4 ch, depending

on whether the round is done in de, tr, or dtr, respectively.

Each round is closed with a slip stich in the last ch of the

first stitch. From one round to the next, the colour alter­

nates between light blue and dark blue, which helps iden­

tify the different bands in the finished model. We found that

the end result is much better if the threads are cut after

each round, rather than carrying strands up the rounds.

Table 1 . Abbreviations of the crochet stitches used for the

Lorenz manifold.

Abbreviation British name American name

ch chain stitch chain stitch

de double crochet single crochet

tr treble crochet double crochet

dtr double treble crochet treble crochet

28 THE MATHEMATICAL INTELLIGENCER

Getting Started

To help with the interpretation of the instructions, we ex­

plain in more detail how to get started. We used a 2.50 mm

crochet hook with 4-ply mercerized cotton yam. Note that

the crochet hook is slightly smaller than recommended for

the weight of the yam; this is done to obtain a tighter gauge.

The finished model Lorenz manifold up to geodesic dis­

tance 1 10.75 is then about 0.9 m in diameter, and required

four 100 g balls of yam. Obviously, using a thicker crochet

hook and yam will lead to an even bigger manifold. The

complete crochet instructions are given in the Appendix;

here we explain briefly how to read the compact crochet

notation.

Begin (rndl) with a foundation chain in light blue of 5

ch stitches that are closed with a slip stitch to form a ring.

The first round consists of 10 de. This means that one starts

with 1 ch followed by 9 de in the loop, after which the ring

is closed with a slip stitch. The small disc obtained so far

represents the Lorenz manifold up to geodesic distance

(gd) 2.75. The next round (rnd2) is done in dark blue us­

ing a tr stitch. The total number of stitches doubles to 20

in this round, which means that 2 tr stitches are made in

each de. (Recall that the 10 de are numbered from 0 to 9.)

The crocheted disc has now grown to represent the Lorenz

manifold up to gd 4.75. In the next round (rnd3) the geo­

desic distance grows to gd 8. 75 with dtr crochet stitches.

As in rnd2, there are 2 dtr in each tr. Starting from rnd4,

crochet stitches are no longer doubled at each previous

stitch.

Notice that 20 new crochet stitches are added in each

round from rnd2 to rnd7; then the number of stitches starts

to vary from round to round, but essentially remains con­

stant when counted over two consecutive rounds. This

means that roughly up to rndlO of gd 36.75 the Lorenz man­

ifold is a flat disc, allowing the algorithm to take large steps,

which is translated to dtr crochet stitches. From gd 36.75

onward, all rounds are worked in tr crochet stitches.

In rnd37, that is, at gd 90. 75, stitches are deleted for the

first time. The notation - 515 means that the treble crochet

stitch at position 515 merges with the one at position 514.

This is done as follows: treble crochet stitch 5 14 is not fin­

ished completely; that is, one does not bring the yam

around the hook and pull it through the last two loops on

the hook. Similarly, treble crochet stitch 515 is then cro­

cheted except for this last step. The two stitches are cro­

cheted together by pulling a loop of yam through all three

loops at once. Note that - 515 is followed by 515 so that a

second treble crochet stitch is made in position 515, which

effectively undoes the deletion of the stitch. This corre­

sponds to an adjustment of the mesh points by the algo­

rithm, and we kept the instructions to be faithful to the

computed mesh. In later rounds, for example, in rnd39,

crochet stitches are truly deleted.

Comparison with Crocheting the Hyperbolic Plane

The idea to crochet a model of the Lorenz manifold was

born quite suddenly in December 2002, but was indirectly

influenced by our knowledge of the Intelligencer article

"Crocheting the Hyperbolic Plane" by Henderson and

Taimina [3]. Indeed, when their article came out in 2001

we had already developed our algorithm for manifold

computations, but somehow the idea of crocheting did not

click. As soon as we decided to crochet a mathematical

object ourselves, we of course had another look at their

paper.

Their idea is to crochet a model of hyperbolic space by

starting from a row (or a round) of a fixed number of chain

stitches and then adding rows (rounds), all of the same ba­

sic crochet stitch. The trick is to add one extra crochet

stitch every N stitches. In other words, the number of

stitches increases per row (round) and this leads to nega­

tive local curvature as was explained earlier. The smaller

N, the more extra crochet stitches are added and the larger

the negative curvature of the resulting object. This curva­

ture is constant as the procedure is repeated the same

everywhere.

From a crocheting point of view, crocheting a model of

hyperbolic space is quite simple as it involves the same cro­

chet stitch and counting to N. An expert needle worker will

be able to do this "on the side" while having a nice con­

versation or watching TV. Crocheting the Lorenz manifold,

a

b

c

Figure 3. The Lorenz manifold in the process of being crocheted, shown "as flat as possible" (left column) and doubled-up along the line of

symmetry (right column); from (a) to (c) the manifold is shown up to rnd26 (gd 68.75), up to rnd39 (gd 94. 75), and up to rnd47 (gd 1 10.75).

Where the manifold is rippled, the curvature is most negative.

VOLUME 26, NUMBER 4, 2004 29

on the other hand, requires continuous attention to the in­structions in order not to miss when to add or indeed re­move an extra crochet stitch. This involves much counting and checking of each round. In fact, Hinke crocheted the Lorenz manifold in the course of two months in an esti­mated 85 hours, which corresponds to about 300 stitches per hour for the total of 25,511 stitches. (To translate this time estimate to your own crocheting skills, be warned that Hinke is an expert at crochet and counting alike!)

A Shapeless Crocheted Topological Disc

Initially, up to a geodesic distance of about 36. 75, the Lorenz manifold is virtually flat as a pancake. It then starts picking up a lot of negative curvature near the positive z-axis, around

which it spirals. The lower part of the manifold with z < 0

has almost zero curvature. It is impossible to flatten out the crocheted manifold on a table, as the region of strong neg­ative curvature forms more and more folds.

Figure 3 shows the crocheted manifold at three differ­ent stages of progress and flattened out as much as possi­ble. The images on the left show the manifold as a rippling disc; the z-axis corresponds to the vertical line through the center of each panel. In the images on the right the cro­cheted manifold has been folded double along the z-axis. To "absorb" some of the curvature, the z-axis is then no longer a straight line in the upper part of the images, but even this is not enough to avoid the increasing (with di­ameter) rippling of the object.

Figure 4. The crocheted Lorenz manifold in front of a white background, which brings out the mesh and the symmetry.

30 THE MATHEMATICAL INTELLIGENCER

Mounting the Crocheted Lorenz Manifold

When we first saw the crocheted but yet unmounted Lorenz

manifold shown in Figure 3c we had some doubts whether

we could get it into the required final shape. However, as

explained above, the crocheted manifold "knows" its shape

in three-space because of the locally encoded curvature in­

formation. When mounting the manifold, it (almost) auto­

matically falls into its proper shape. To achieve this we

found that only three ingredients are required:

1. fixing the z-axis with an unbendable rod;

2. supporting the outer rim with a bendable wire of the cor­

rect length;

3. Supporting the manifold in the radial direction with a

single bendable wire that runs from rim to rim and

through the origin.

For the third task one could choose the geodesics that are

locally perpendicular to the z-axis, but we prefer to use the

strong stable manifold W88(0), which is basically the orbit of

(1) that starts off at the origin in the direction perpendicu­

lar to the z-axis. Because it is an orbit, it is not a geodesic

of the Lorenz manifold, but rather illustrates the difference

between the geometry of the manifold and the dynamics on

it. We computed W88(0) with the software from [10].

The next step was to identify the sequence of holes from

Figure 5. The crocheted Lorenz manifold in front of a black background, which gives a good impression of the manifold as a two-dimen­

sional surface.

VOLUME 26, NUMBER 4, 2004 31

a

b

Figure 6. Two more views of the crocheted Lorenz manifold in front of a white and a black background. The view in panel (b) differs by a ro­

tation of about 15 degrees from the view in panel (a), which in turn differs by about 15 degrees from the view in Figures 4 and 5.

one crocheted round to the next through which the posi­tive and negative z-axis and both branches of W88(0) go.

This information is collected in the weaving instructions in the Appendix.

To mount the Lorenz manifold we wove an unbendable thin kiting rod through the z-axis and bendable wires through the last round and the location of W88(0); details of this pro­cedure can be found in the mounting instructions in the Ap­pendix. Modulo rotations and translations in IR3, there are only

32 THE MATHEMATICAL INTELLIGENCER

two results of mounting the manifold, leading to the Lorenz manifold itself with a right-handed spiral around the z-axis, or its mirror image with a left-handed spiral around the z-axis. By giving the rim wire the right twist one can ensure that one obtains the former solution. The final step is to bend the sup­porting wires so that they are nice and smooth and the cro­cheted model indeed resembles the Lorenz manifold.

The final result is shown in Figures 4-7. The carbon fibre rod fixing the z-axis is vertical and in the centre of the images.

The image in Figure 4 shows the Lorenz manifold pho­

tographed with a white background, so that the crocheted

mesh is clearly visible. Furthermore, one can see through the

manifold and get an impression of the part that is hidden. This

emphasizes the rotational symmetry of the Lorenz manifold.

Figure 5, on the other hand, shows the Lorenz manifold pho-

tographed with a black background. One cannot see through

the mesh any longer, and the manifold appears as a two-di­

mensional surface. Notice the wire in the position of the strong

stable manifold W88(0) that supports the Lorenz manifold. Fig­

ure 6 shows two different views taken from different angles,

again against a white and a black background to emphasize

Figure 7. A close-up view of the crocheted Lorenz manifold in front of a white background. The vertical rod is the z-axis, and the wire emerg­

ing from the origin is the strong unstable manifold W55(0); notice also the wire supporting the outer rim of the manifold.

VOLUME 26, NUMBER 4, 2004 33

the mesh and the surface, respectively. Finally, Figure 7 is an

enlargement of the Lorenz manifold that shows the strong sta­

ble manifold W58(0) :running from the origin until it meets the

rim. Notice that W88(0) is perpendicular to the bands (ap­

proximately geodesic level set) only near the origin and cer­

tainly not close to the rim. In other words, it is clearly not a

geodesic.

It is our experience that the crocheted model of the Lorenz

manifold in Figures 4-7 is a very helpful tool for under­

standing and explaining the dynamics of the Lorenz system.

While the model is not identical to the computer-generated

Lorenz manifold, all its geometrical features are truthfully

represented, so that it is possible to convey the intricate struc­

ture of this surface in a "hands-on" fashion. This article tries

to communicate this, but for the real experience you will have

to get out your own yam and crochet hook!

Acknowledgments

The unmounted Lorenz manifold in Figure 3 was digitally

photographed by B.K. at the Design Office of the Faculty

HINKE M. OSINGA

of Engineering. The mounted Lorenz manifold was carefully

set up by the authors at the photo studio of the Photographic

Unit of the University of Bristol, and then digitally pho­

tographed by Perry Robbins. Electronic postprocessing was

expertly done by Greg Jones at the Design Office of the

Faculty of Engineering. We are very grateful to both Perry

and Greg for their patience and attention to detail, and for

dealing so well with our enthusiasm and perfectionism.

REFERENCES

[1 ] J. Gleick. Chaos, the Making of a New Science, William Heine­

mann, London, 1 988.

[2] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynami­

cal Systems, and Bifurcations of Vector Fields. Springer-Verlag,

Second edition, 1 986.

[3] D. W. Henderson and D. Taimina. Crocheting the hyperbolic plane.

The Mathematical lntelligencer, 23(2) : 1 7-28, 2001 .

[4] B. Krauskopf and H. M. Osinga. Two-dimensional global manifolds

of vector fields. CHAOS, 9(3):768-774, 1 999.

[5] B. Krauskopf and H. M. Osinga. Computing geodesic level sets

BERND KRAUSKOPF

Bristol Centre for Applied Nonlinear MathematiCS

Department of Eng1neenng Mathematics

Queen's BUikJong

UniverSity of Bnstol

Bristol BS8 1 TR

Un�tad Kingdom

e-mad: H.M.Osinga bnstol.ac.uk

Hinke Osinga learnt crocheting, and other handcraft techniques,

from her mother around the age of seven. A b1t later she got a

Ph.D. 1n Mathematics from the University of Groningen in 1 996 un­

der the d1rect10n of Henk Broer and Gert Vegter. Her research was

on the computation of normally hyperbolic 1nvariant man1folds, and the ideas developed then have been generalized (and disguised)

throughout her work. She held postdoctoral pos�tlons at the Geom­

etry Center, Univ9rSity of M1nnesota, and at the Cal1fomia Institute

of Technology in Pasadena. She moved to England 1n 2000, first

to the UniverSity of Exeter. She JOined the Department of Eng1neenng

Mathematics at the UniverSity of Bristol in 2001 The Lorenz man­

ifold IS her first project that comb1nes handcraft with mathematics.

34 THE MATHEMATICAL INTELLIGENCER

e-ma1l: B.Krauskopf bnstol.ac.uk

Bernd Krauskopf got h is Ph.D. 1n Mathematics from the UniVersity

of Gron1ngen 1n 1 995 under the d1rect1on of Floris Takens and Henk

Broer. After a year as v1sit1ng professor at Cornell Un1vers1ty, and

a two-year Postdoctoral pos1tion at Vrije Universiteit Amsterdam,

he jo1ned the Department of Eng1neering Mathematics at the Uni­

versity of Bristol. Bernd works 1n the general area of dynamical sys­

tems theory, specifically on theoretical and numerical problems 1n

b1furcation theory and their application to models arising in laser

physiCS. The collaboration w1th Hnke on global man1folds started

1n 1 997 when Bernd visited H1nke at the Geometry Center in Min­

neapolis-the basic idea of how to grow a global manifold emerged

over a bagel with H1nke at a bagel store on Nicolette Ma l l .

on global (un)stable manifolds of vector fields. SIAM Journal on

Applied Dynamical Systems, 2(4):546-569, 2003.

[6] B. Krauskopf and H. M. Osinga. The Lorenz manifold as a collec­

tion of geodesic level sets. Nonlinearity, 1 7(1 ) :C1 -C6, 2004.

[7] B. Krauskopf and H. M. Osinga. Geodesic parametrization of global

invariant manifolds or what does the Equadiff 2003 poster show?

Proceedings Equadiff 2003, to appear.

[8] E. N. Lorenz. Deterministic nonperiodic flow. Journal of the Atmo­

spheric Sciences, 20(2) : 1 30-1 48, 1 963.

[9] H. M. Osinga and B. Krauskopf. Visualizing the structure of chaos

in the Lorenz system. Computers and Graphics, 26(5) :81 5-823,

2002.

[ 1 OJ H. M. Osinga and G. R . Rokni Lamooki. Numerical approximations

of strong (un)stable manifolds. Proceedings Equadiff 2003, to appear.

[1 1 ] S. H. Strogatz. Nonlinear Dynamics and Chaos. Addison Wesley,

1 994.

[1 2] W. Tucker. The Lorenz attractor exists. Comptes Rendus de

/'Academia des Scienes. Serie I. Mathematique, 328( 1 2) : 1 1 97-

1 202 , 1 999.

[1 3] M. Viana. What 's new on Lorenz strange attractors? The Mathe­

matica/ lntelligencer, 22(3) :6-19 , 2000.

Appendix

Complete instructions

Materials: 200 g light blue and 200 g dark blue 4-ply mer­

cerized cotton yarn; 2.50 mm crochet hook; embroidery

needle; about 3 m leftover yarn of a contrasting colour;

0.9 mm unbendable 4 mm or 5 mm rod; 1 .45 m and 2 X 2. 70

m bendable 2 mm wire; 2 electrical wire connectors (come

in bars; available from DIY stores); wire cutter; pliers; small

screwdriver.

Abbreviations and Notation: see Table 1 and Figure 8.

Crochet instructions

Work 5 ch in light blue and join with a slip stitch to form

a ring. Odd rounds are worked with light blue and even

ones with dark blue yam. Work each round with the stitch

as indicated; count the stitches starting with 0 for the first

stitch and make two stitches in one for each stitch men­

tioned in the list. If the stitch appears with a minus sign,

crochet it together with the previous stitch (delete the

stitch). Geodesic distance (gd) of the Lorenz manifold af­

ter each round is given for orientation and motivation.

I I

-3 -2 - 1 + 1 +2 +

a I

X a

rnd1: foundation round of 5 ch then 10 de in loop (gd

2. 75) ; rnd2: 20 tr 0 1 2 3 4 5 6 7 8 9 (gd 4. 75) ; rnd3: 40

dtr 0 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 (gd

8. 75); rnd4: 60 dtr 0 3 4 7 8 1 1 12 15 16 19 20 23 24 27 28

31 32 35 36 39 (gd 12. 75); rnd5: 80 dtr 0 5 6 1 1 12 17 18

23 24 29 30 35 36 41 42 47 48 53 54 59 (gd 1 6. 75); rnd6:

100 dtr 3 4 1 1 12 19 20 27 28 35 36 43 44 51 52 59 60 67 68

75 76 (gd 20. 75) ; rnd7: 120 dtr 0 9 10 19 20 29 30 39 40

49 50 59 60 69 70 79 80 89 90 99 (gd 24. 75); rnd8: 122 dtr

1 1 1 1 16 (gd 28. 75) ; rnd9: 148 dtr 0 3 8 1 1 12 15 20 27 32

39 44 51 56 63 68 75 80 87 92 95 96 99 104 107 108 121 (gd

32. 75); rnd10: 171 dtr 7 8 30 3 1 44 45 58 59 72 73 86 87

100 101 108 123 124 135 138 139 140 141 144 (gd 36. 75) ;

rnd11: 189 tr 3 6 25 26 41 42 57 58 73 74 89 90 105 106 121

137 142 145 (gd 38. 75); rnd12: 192 tr 13 16 149 (gd 40. 75);

rnd13: 214 tr 0 25 28 33 36 46 51 1 18 123 133 136 141 144

167 169 170 172 175 176 185 189 191 (gd 42. 75) ; rnd14:

234 tr 3 48 61 68 71 76 79 86 89 94 97 104 107 1 12 1 1 5 122

135 199 200 206 (gd 44. 75) ; rnd15: 243 tr 23 24 1 77 178

193 204 213 215 222 (gd 46. 75); rnd16: 261 tr 47 48 69 70

135 136 157 158 199 204 2 13 2 14 221 228 229 236 237 240

(gd 48. 75) ; rnd17: 269 tr 7 8 95 96 177 1 18 216 217 (gd

50. 75) ; rnd18: 283 tr 0 2 198 202 206 207 215 216 230 235

237 238 255 268 (gd 52. 75) ; rnd19: 307 tr 16 17 21 25 32

49 56 169 176 193 216 225 226 229 234 237 246 258 259 269

271 276 277 280 (gd 54. 75); rnd20: 325 tr 2 7 8 80 87 104

1 1 1 128 135 152 159 268 269 272 274 279 285 290 (gd 56. 75) ;

rnd 21: 343 tr 7 16 44 200 209 225 235 251 261 262 267 283

284 287 297 298 300 314 (gd 58. 75) ; rnd22: 375 tr 0 6 30

56 73 82 99 108 125 134 151 160 177 186 209 234 243 246

249 260 264 265 279 280 281 310 328 329 330 337 338 342

(gd 60. 75) ; rnd23: 381 tr 54 55 265 272 314 341 (gd 62. 75) ;

rnd24: 411 tr 5 6 84 85 1 12 1 13 140 141 1 68 169 196 197

224 232 239 240 254 269 272 273 274 277 280 302 3 1 1 328

335 336 352 378 (gd 64. 75) ; rnd25: 432 tr 2 23 24 31 42

43 50 300 301 308 310 315 340 345 352 355 358 397 400 405

406 (gd 66. 75) ; rnd26: 451 tr 4 5 18 72 222 237 276 313

329 355 362 363 365 368 372 373 376 377 405 (gd 68. 75);

rnd27: 491 tr 0 16 24 35 83 84 9 1 106 1 13 1 14 121 136 143

144 151 166 173 174 181 196 203 204 2 1 1 234 235 271 275

288 295 327 343 360 363 376 380 383 419 435 436 450 (gd

70. 75); rnd28: 511 tr 13 51 289 300 314 332 340 361 362

403 404 413 416 417 420 421 436 450 469 481 (gd 72. 75) ;

I I r

- -2 - 1 0 + 1 +2 +3 l J a I

X b

Figure 8. Numbering of the holes between stitches in a round relative to a hole position (gray cross) in the previous round, as used in the

weaving instructions. Since new stitches are added in front of holes, there are two cases: one stitch in front of the hole position when no

extra stitch was added (a), and two stitches in front of the hole position when an extra stitch was added (b).

VOLUME 26, NUMBER 4, 2004 35

rnd29: 534 tr 2 17 22 25 26 48 49 60 69 78 273 282 339 348 360 375 385 427 432 433 455 475 497 (gd 74. 75); rnd30:

563 tr 5 10 13 16 17 22 46 1 13 249 258 339 342 345 348 355 356 357 358 369 372 415 439 440 446 454 458 492 529 531 (gd 76. 75) ; rnd31: 591 tr 0 4 10 13 29 47 48 104 130 155 223 232 277 284 309 332 359 367 368 381 383 414 425 466 471 472 473 488 (gd 78. 75); rnd32: 637 tr 9 10 18 45 86 1 19 148 155 17 4 183 190 199 208 217 224 253 260 325 360 362 363 371 377 405 414 439 470 487 488 492 501 504 507 508 509 512 513 514 523 535 576 577 580 583 586 590 (gd 80. 75) ; rnd33: 655 tr 2 13 89 394 420 425 431 434 506 509 520 523 533 534 537 563 573 597 (gd 82. 75); rnd34: 670 tr

5 69 1 15 322 383 388 408 423 426 436 450 513 538 539 542 (gd 84. 75); rnd35: 695 tr 4 10 27 1 19 135 324 380 398 416 423 424 428 535 540 541 545 548 554 555 558 565 575 646 658 667 (gd 86. 75) ; rnd36: 720 tr 0 8 2 1 38 138 161 178 275 292 313 314 345 370 380 438 441 529 540 541 552 566 589 591 592 680 (gd 88. 75) ; rnd37: 754 tr 12 22 50 1 1 1 168 183 206 207 222 223 244 245 260 261 284 299 351 375 376 378 435 450 451 464 -515 515 563 568 582 602 603 608 619 -661 661 -668 668 -671 671 701 714 719 (gd 90. 75) ;

rnd38: 782 tr 1 1 16 33 42 93 95 96 120 390 408 453 463 464 466 484 489 490 494 502 -524 524 -533 533 -536 -543 543 -559 559 603 605 616 617 624 627 634 642 -705 705 -715 715 742 748 (gd 92. 75) ; rnd39: 804 tr 2 4 8 2 1 29 31 64 67 72 93 105 379 386 434 458 4 72 4 78 497 500 -564 607 608 613 614 627 648 652 673 -708 - 71 1 - 72 1 -734 (gd 94. 75);

rnd40: 840 tr 4 7 22 23 26 45 47 48 128 131 176 344 349 382 395 407 423 433 446 465 474 480 489 512 527 532 -559 559 -567 -572 -579 579 - 584 584 -591 639 645 653 665 666 682 686 700 705 712 714 716 - 729 729 - 752 752 801 (gd 96. 75); rnd41: 887 tr 0 1 1 2 1 33 40 43 69 85 100 145 146 167 168 192 2 10 229 234 271 276 313 318 336 337 380 381 399 459 481 523 531 537 544 554 645 653 666 681 689 690 695 696 719 734 739 740 752 - 772 772 829 (gd 98. 75);

rnd42: 921 tr 1 1 16 51 64 69 94 102 1 14 137 161 164 202 203 226 269 270 313 314 381 382 389 429 526 534 550 559 564 572 -611 -622 -644 -654 696 712 719 731 732 739 758 767 768 788 -853 886 (gd 1 00. 75); rnd43: 958 tr 2 37 61 66 76 84 206 223 254 261 262 269 300 307 308 315 346 353 354 361 392 433 509 559 588 589 613 -684 684 742 748 750 751 757 762 778 779 794 -839 -868 903 914 915 (gd 102. 75); rnd44: 994 tr 5 9 20 23 25 44 54 59 82 92 93 135 189 204 227 427 439 528 551 561 604 605 607 627 631 632 697 740 787 790 791 792 793 803 805 844 850 -872 872 -916 916 -926 (gd 104. 75); rnd45: 1025 tr 4 7 9 19 25 33 48 1 13 230 254 419 431 5 1 1 536 543 578 582 584 589 622 645 647 659 - 730 - 741 779 780 783 791 803 81 1 814 829 830 833 863 -960 (gd 1 06. 75); rnd46: 1072 tr 0 10 15 33 1 16 126 144 279 291 302 314 329 341 352 364 379 391 402 414 546 579 580 590 609 636 648 661 667 674 685 - 703 -706 706 -712 712 -716 716 770 779 813 840 845 846 847 848 849 850 851 852 855 856 861 873 886 1022 (gd 1 08. 75);

rnd47: 1 104 tr 31 94 98 104 1 1 7 136 137 177 535 608 653 654 658 668 689 691 699 704 705 -740 740 761 777 - 787 829 842 871 894 895 906 929 934 944 1008 1017 1022 (gd 1 1 0. 75) .

36 THE MATHEMATICAL INTELLIGENCER

Weaving instructions

To mount the Lorenz manifold it is best to first indicate the

positions of the rod and the wires by weaving differently

coloured yam through the holes between stitches. Start

from the centre in the hole between the two stitches indi­

cated in rnd1 below. Then weave the yam through holes

from one round to the next, where the position of the next

hole is indicated relative to the present position as shown

in Figure 8. After weaving in the z-axis and the strong sta­

ble manifold W88(0), fold the manifold over along the z-axis

weave. You should get a result as shown on the right of

Figure 3 (c); the two branches of the W88(0) weave should

be symmetric with respect to the z-axis weave.

Positive z-axis: rnd1: 9-0; rnd2: +2; rnd3: + 1 ; rnd4: + 1 ; rnd5: + 1 ; rnd6: + 1 ; rnd7: + 1 ; rnd8: + 1 ; rnd9: + 1 ; rnd10:

+2; rndll: + 1; rnd12: + 1; rnd13: + 1; rnd14: + 1; rnd15:

+2; rnd16: + 1 ; rnd17: + 1 ; rnd18: + 1 ; rnd19: + 1 ; rnd20:

+2; rnd21: + 1; rnd22: + 1; rnd23: + 1; rnd24: +2; rnd25:

+ 1 ; rnd26: + 1; rnd27: + 1 ; rnd28: +2; rnd29: + 1 ; rnd30:

+ 1; rnd31: + 1; rnd32: + 1 ; rnd33: + 1 ; rnd34: + 1; rnd35:

+ 1; rnd36: + 1; rnd37: + 1 ; rnd38: + 1; rnd39: + 1; rnd40:

+ 1; rnd41: + 1; rnd42: + 1; rnd43: + 1; rnd44: + 1; rnd45:

+ 1; rnd46: + 1; rnd47: + 1 ;

Negative z-axis: rnd1: 4-5; rnd2: +2; rnd3: + 1 ; rnd4: + 1 ; rnd5: + 1 ; rnd6: + 1 ; rnd7: + 1 ; rnd8: + 1 ; rnd9: + 1 ; rnd10:

+ 1; rndll: +2; rnd12: + 1; rnd13: + 1; rnd14: + 1; rnd15:

+ 1; rnd16: + 1; rnd17: + 1 ; rnd18: + 1 ; rnd19: + 1 ; rnd20:

+ 1; rnd21: + 1; rnd22: + 1; rnd23: + 1; rnd24: + 1; rnd25:

+ 1 ; rnd26: + 1 ; rnd27: + 1 ; rnd28: + 1 ; rnd29: + 1 ; rnd30:

+ 1; rnd31: + 1; rnd 32: + 1; rnd33: + 1 ; rnd34: + 1 ; rnd35:

+ 1; rnd36: + 1; rnd37: + 1; rnd38: + 1; rnd39: + 1; rnd40:

+ 1; rnd41: +2; rnd42: + 1; rnd43: + 1; rnd44: + 1; rnd45:

+ 1; rnd46: + 1; rnd47: + 1 ;

Left branch of W88(0): rnd1: 1-2; rnd2: + 1 ; rnd3: + 1 ; rnd4: +2; rnd5: +2; rnd6: +2; rnd7: +2; rnd8: + 1 ; rnd9:

+2; rnd10: +3; rndll: +2; rnd12: + 1; rnd13: +3; rnd14:

+ 1; rnd15: +2; rnd16: +3; rnd17: +2; rnd18: +2; rnd19:

+2; rnd20: +2; rnd21: +3; rnd22: +2; rnd23: +2; rnd24:

+3; rnd25: +3; rnd26: +3; rnd27: +2; rnd28: +3; rnd29:

+3; rnd30: +3; rnd31: +3; rnd32: +3; rnd33: +5; rnd34:

+5; rnd35: +4; rnd36: +4; rnd37: + 5; rnd38: +6; rnd39:

+ 5; rnd40: +6; rnd41: +5; rnd42: +7; rnd43: +7; rnd44:

+5; rnd45: +5; rnd46: +5; rnd47: + 5;

Right branch of W88(0): rnd1: 7-8; rnd2: +2; rnd3: +2; rnd4: +2; rnd5: + 1; rnd6: - 1; rnd7: - 1; rnd8: - 1; rnd9:

+ 1; rnd10: - 1; rndll: - 1; rnd12: + 1; rnd13: - 1; rnd14:

- 1; rnd15: - 1; rnd16: - 1; rnd17: - 1; rnd18: - 1; rnd19:

- 1; rnd20: - 1; rnd21: - 1; rnd22: - 1; rnd23: - 1; rnd24:

-2; rnd25: - 1; rnd26: - 1; rnd27: -2; rnd28: -2; rnd29:

- 1; rnd30: -3; rnd31: -2; rnd32: -3; rnd33: -2; rnd34:

-4; rnd35: -3; rnd36: -5; rnd37: -4; rnd38: -3; rnd39:

-5; rnd40: -4; rnd41: -5; rnd42: -5; rnd43: -5; rnd44:

-5; rnd45: -4; rnd46: -4; rnd47: - 5;

Mounting instructions

Weave the unbendable thin rod of 0.9 m length through the manifold by following the z-axis weave; we used a 5 mm carbon fibre rod used in kiting, which is lightweight and very stiff for its diameter. Starting from the top of the z­axis, weave a 2. 70 m length of the bendable wire through the outer crocheted round of the manifold until you reach the bottom of the z-axis. Repeat the procedure with the second length of 2. 70 m around the other half of the other crocheted round of the manifold, again starting from the top of the z-axis. Try to spread the stitches evenly over the wire; you will find that this introduces twist into the rim wire. Make sure the twisting is clockwise near the z-axis in the direction of increasing z, so that you get a right­handed helix, just like a cork screw.

Cut two single electrical wire connectors from a bar and strip them of their isolating plastic cover. The stripped con-

nectors are now unsuitable for electrical connections, but ideal for connecting the bendable wires. Connect the two 2. 70 m pieces of wire at the top and bottom with the con­nectors by sliding in both ends and tightening the screws.

Make a mark 0.1 m from each end of the 1.45 m length of bendable wire; the middle piece of 1.25 m is the length of W5"(0). Starting from the rim, weave this wire through the manifold following the marking yam. Using the pliers, make two small loops at both ends where you made the mark and cut off the excess wire. Sew the ends in place with light blue yam.

Finally, remove the differently coloured yam. The Lorenz manifold should now be recognisable. With the help of the figures in this paper, tuck and bend it into its final shape, making sure that the bendable wires are nice and smooth, that is, without noticable kinks. This may take some time de­pending on your level of perfectionism. Good luck!

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VOLUME 26. NUMBER 4, 2004 37

l:dftj.l§rr@ih$11¥fth§4£1!.'1.1§.id Michael Kleber and Ravi Vakil, Editors

Origami Ouiz Thomas Hull

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

Paper is all around us. Every day we

fold paper. So test your knowledge of

and your ability to explore this simple,

everyday activity.

1. Find a square piece of paper that is

white on one side and colored on the

other. From such paper it is possible to

use the contrasting colors to fold any

n X n checkerboard. Trying to do this

in as few folds as possible can be a per­

plexing challenge. Figure 1 shows how

to fold a 2 X 2 checkerboard in only

three folds.

3 - - mountain

{[]' ....____._..!"\: valley

� Fig. 1.

Observant skeptics may quarrel

with the fact that we divided the paper

into thirds "free" in the above solution.

However, all we are counting are ac­

tual folds used in the end, and I adopt

the convention that any landmarks

needed (like a one-third mark) can be

made beforehand without counting in

the fold total.

How would one fold a 3 X 3 checker­

board? What is the fewest number of

folds needed?

2.a. (By Kazuo Haga, [2]) Take a

square piece of paper and let P be any

point on the square. Taking one at a

time, fold and unfold each comer of the

square to the point P (Fig. 2). When

you're finished, P should be contained

in some polygon determined by the

creases and, possibly, the sides of the

square. How many sides can this poly­

gon have? Which regions of the paper

give which polygons?

Fig. 2.

38 THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+ Business Media, Inc.

2.b. Does it make sense to consider

the point P to be chosen outside the

square? What if we instead use a rec­

tangle?

3. Figure 3 shows how one can fold an

equilateral triangle in a square piece of

paper. Does it work? Is this the largest

equilateral triangle that can be made

from a square?

Fig. 3.

4. What interesting thing is the folding

procedure in Figure 4 doing to the an­

gle (f? (Hint: what is the angle a?)

Fig. 4.

5. We are all familiar with geometric

constructions using a straightedge and

compass. Since the nineteenth century,

geometers have also been using paper­

folding as a geometric construction

medium. What are the basic folds ( op­

erations) that define paper-folding?

For example, one clear fundamental

fold is, "Given two points p1 and P2 we

can make a crease that passes through

P1 and P2-" Think of another one that

allows us to construct angle bisectors

by folding. Try to make your list as

complete as possible. (The "moves" in

Problems 3 and 4 above should be rep­

resented, for example.)

6. When you did Problem 5 and con­

sidered the fold in Figure 3, you prob­

ably included something like the fol-

lowing in your list of basic folds: Given two points P1 and P2 and a line L1, we can sometimes make a crease that passes through pz and places p1 onto L1. Why do we need to say "some­

times"? What conditions on p1, pz, and

L1 will make it always work?

7. If we think of the paper as lying in

JR2 (or if you prefer, IC), what type of

algebraic equation is solved by the ba­

sic folding operation in Problem 6?

8. If we consider the piece of paper to

exist in the complex plane, define

origami numbers to be those points

in C that are constructible via paper­

folding. How does the field of origami

numbers compare to the field of num­

bers constructible by straightedge and

compass when we consider the answer

to Problem 7? What does Problem 4 tell us?

9. Most models in origami books are

flat models. That is, when completed

they can be pressed in a book without

introducing new creases. The classic

flapping bird (Fig. 5) is one example.

Take any flat origami model, unfold it,

and consider the creases used in the fi­

nal folded form (i.e. , we are not con-

sidering auxiliary creases made during

the folding process but not used in the

end). How many colors does it take

to color the regions in between the

creases in the crease pattern, making

sure that no two neighboring regions

(sharing a boundary line) receive the

same color?

Fig. 5.

lO.a. Creases come in two types:

mountains, which are convex, and val­leys, which are concave (see Fig. 1). These are often distinguished in origami

instructions by different types of

dashed lines. But a paper-folder cannot

just choose which crease will be moun­

tains and which will be valleys willy­

nilly! Indeed, in Figure 6 the single­

vertex crease pattern can fold flat, but

not using the prescribed mountain­

valley choices. Why is this impossible

to fold flat?

MOVING? We need your new address so that you

do not miss any issues of

Fig. 6.

I

r- - --;),

/ 1/

/ ' ' '

' '

lO.b. The vertex in Figure 6 can be

arranged, or tessellated, with copies

of itself four times to make the very

interesting crease pattern called the

square twist (Fig. 7). Use what you de­

duced from lOa to compute how many

valid mountain-valley assignments ex­

ist for this crease pattern.

Fig. 7.

"Solutions" follow up, anyway-pp.

61-63.

THE MATHEMATICAL INTELLIGENCER Please send your old address (or label) and new address to:

Springer Journal Fulfillment Services

P.O. Box 2485, Secaucus, NJ 07096-2485 U.S.A.

Please give us six weeks notice.

VOLUME 26, NUMBER 4, 2004 39

lj§i(W·J·I•i David E. Rowe , Ed ito r I

What Do You Need a Mathematician Fort Marti nus Hortensius' s JJSpeech on the Dign ity and Uti l ity of the Mathematica l Sciences" (Amsterdam 1 634)

Volker R. Remmert

Send submissions to David E. Rowe,

Fachbereich 1 7 - Mathematik,

Johannes Gutenberg University,

055099 Mainz, Germany.

In early modem Europe the term

mathematical sciences was used to

describe those fields of lrnowledge that

depended on measure, number, and

weight-reflecting the much-quoted

passage from the Wisdom of Solomon 11 , 20: "but thou hast ordered all things

in measure and number and weight."

This included astrology and architec­

ture as well as arithmetic and astron­

omy. These scientiae or disciplinae mathematicae were generally subdi­

vided into mathematicae purae, deal­

ing with quantity, continuous and dis­

crete as in geometry and arithmetic,

and mathematicae mixtae or me­diae, dealing not only with quantity

but also with quality-for example as­

tronomy, geography, optics, music,

cosmography, and architecture. The

mathematical sciences, then, con­

sisted of various fields of lrnowledge,

often with a strong bent toward prac­

tical applications. These fields be­

came independent from one another

only through the formation of scientific

disciplines from the late 17th to the

early 19th century, i.e., in the aftermath

of the Scientific Revolution.

One of the important preconditions

for this transformation was the rapidly

changing status of the mathematical

sciences as a whole from the mid-16th

through the 17th century. The basis for

the social and epistemological legiti­

mation of the mathematical sciences

began to be laid by mathematicians and

other scholars in the mid-16th century.

Their strategy was essentially twofold:

in the wake of the 16th-century debates

about the certainty of mathematics and

its status in the hierarchy of the scien­

tific disciplines (quaestio de certitu­dine mathematicarum [Mancosu 1996;

Remmert 1998, 83-90; 2004]), the math­ematicae purae were taken to guaran­

tee the absolute certainty and thereby

dignity of lrnowledge produced in all

the mathematical sciences, pure and

mixed; the mathematicae mixtae, on

the other hand, confirmed the utility of

this unerring lrnowledge.

Throughout the 17th century, the

legitimation of the mathematical sci­

ences was pursued in deliberate strate­

gies to place the mathematical sciences

in the public eye. These strategies often

involved the use of print media in one

way or another-through mathemati­

cal textbooks, practical manuals, books

of mathematical entertainments, edi-

. . . del iberate

strategies to place

the mathematical

sciences in the

publ ic eye . tions of the classics, encyclopaedic

works, and orations on the mathemat­

ical sciences [Dear 1995; Mancosu

1996; Remmert 1998]. The Oratio de dignitate et utilitate Matheseos (Speech on the dignity and utility of the math­ematical sciences) by Martin us Hort­

ensius belongs to the latter genre (see

Fig. 1). To praise and promote the

mathematical sciences in inaugural

lectures was common practice, and

quite a few such orations eventually

found their way into print. 1 As Horten­

sius's speech reflects most of the stan­

dard arguments employed in the

process of legitimation-and doubly so

as he is clearly seeking not only to le­

gitimate his discipline but at the same

time to be hired by the city fathers of

Amsterdam on a permanent basis-it

is an excellent example to allow us an

overview of an elaborate array of ar­

guments from the classical Greek tra­

dition to contemporaneous develop-

1 For a selection of these and related pieces see the bibliography I I ; cf. the discussion in [Remmert 1 998,

1 52-1 65; Swerdlow 1 993].

40 THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+ Business Media. Inc.

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Figure 1 . Title-page of Hortensius's Speech on the dignity and utility of the mathematical sci­

ences, Amsterdam 1634.

ments in astronomy, including Galileo's astronomical observations.

Hortensius (1605-1639) was born as Maarten van den Hove in Delft in 1605. He was a student in the Latin school at Rotterdam, where he probably came un­der the influence of the natural philoso­pher Isaac Beeckmann. In 1625 he went to Leiden, but it was only in March 1628 that he registered as a student at the

prestigious University of Leiden, where the well-known mathematician Wille­brord Snel taught from 1613 to his early death in 1626. It was probably under Snel's guidance that Hortensius turned to the mathematical sciences and made astronomical observations in Leiden. After Snel's death Hortensius came in contact with the reformed minister, physician, astronomer, and ardent prop-

2Descartes to Mersenne, March 31 , 1 638: "il est tres ignorant" [Berkel 1 997, 2 1 9).

agator of the Copernican system, Philipp Lansbergen (1561-1632), with whom he collaborated closely in editing and translating some of Lansbergen's works from Dutch into Latin. In 1633 Horten­sius moved from Leiden to Amsterdam, hoping to get a position at the city's re­cently founded Athenaeum iUustre. Several of these "illustrious schools" had been founded throughout the Dutch Republic in the 1630s in order to pre­pare students for the universities (De­venter, Amsterdam, and Utrecht), or even to compete with them. Of these only the Amsterdam Athenaeum iUus­tre rose to a more prominent position, as the founding fathers used the im­mense wealth of the city of Amsterdam to hire away professors from Leiden. In

May 1634 Hortensius began to teach in Amsterdam, delivering his inaugural lecture on the Dignity and utility of the mathematical sciences. If we are to be­lieve his personal testimony, his daily lecture courses were a success and at­tracted quite a few listeners. At any rate, the city authorities hired him as a full professor in early 1635 [van Berkel 1997; Remmert 1998, 154-158] .

In the years that followed, Horten­sius's scientific reputation grew con­tinuously. He was known as a con­vinced Copernican and an admirer of Galileo, corresponding with such dis­tinguished scholars as Fabri de Pereisc, Galileo, Pierre Gassendi, Hugo Grotius, Constantin Huygens, Marin Mersenne, and Wilhelm Schickard. Much of his energy between 1635 and 1639 was absorbed by a futile plan to bring his hero Galileo to the Dutch Re­public. At the height of his fame, Hor­tensius received a professorship in Leiden, but he died shortly after mov­ing there in August 1639. Although he is not among the great luminaries of 17th century science-Descartes even considered him "very ignorant"2-his appointment at Leiden shows that he was highly esteemed in the Dutch re­public of letters. In his Speech on the dignity and utility of the mathemat­ical sciences as well as in his other writings, in particular the Canto on the origin and progress of astronomy, his

VOLUME 26, NUMBER 4, 2004 41

In Viri Chriffimi P H I L I P P I L A N S B E R G I I

0 p u s A S T R O N O M I C U M T A B U L A S Q_U E M O T U U M C OE L E S T I U M

dudum •b omnibus defuler•t,u C A R M E N

Qyo ortus & progrcll'us A s T 1. o N o M 1 AE ad nofua ufque tempora oftenditur,

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Figure 2. First page of Hortensius's Canto on the origin and progress of astronomy of 1632.

learning in astronomy and the mathe­matical sciences is on display [Horten­sius 1632] (Fig. 2). Also, Hortensius proved himself to be very well versed in classical writings and traditions-an aspect of scholarship not to be dis­counted in an academic world that still felt a considerable humanistic impulse.

42 THE MATHEMATICAL INTELLIGENCER

Hortensius and his contemporaries saw metaphysics, physics, and mathe­matics as parts of theoretical philoso­phy, and in his Oratio he flatly asserted that "among these mathematics excels by its certainty." The notion that math­ematics guaranteed the highest degree of certainty humanly attainable was a

long-standing epistemological debate, the quaestio de certitudine mathe­maticarum, on which he takes a clear, self-confident position. Hortensius boldly opens by exclaiming that no "one can deny that mathematics is, in­deed, of extraordinary dignity and that "mathematics guards and preserves its sublimity and dignity among the allied parts of philosophy." He continues: "That Goddess [ = mathematics], guide of the mind and actions, whom we ought to rely on and obey, what­ever we have in mind, whatever we conceive in our minds; never does she fail to shed the gleam of her noble majesty through the palace of mathe­matics. [ . . . ] Where other sciences, being full of uncertainty and conjec­ture, can neither reach the truth by themselves, nor produce a remedy for the falsities they contain, the mathe­matical sciences, lacking nothing, suf­fice for themselves; content with the guidance of nature only, they hunt and capture truth itself' [Hortensius 1634, 6] . Hortensius conjures up Apollonius, Aristotle, Euclid, Hipparchus, Pappus, Plato, Ptolemy, Proclus, Pythagoras, Thales of Miletus, and many others to prove the antiquity and early excel­lence of the mathematical sciences. But, he says, "the height of science was attained by Archimedes of Syra­cuse, everywhere admired, celebrated in so many monuments of writings."

Before he turns to showing that "the mathematical sciences do not lack practical advantage and utility," he asks, "Among pleasures, can any be greater than the mathematical sciences [for] stimulating the mind itself and flooding the inmost feelings of the spirit with fullest joy? The knowledge of his­tory and the reading of tales offer oc­casions of delight. The study of politics, ethics, logic, all have their pleasures. But the joys of the mathematical sci­ences are so strong, so keen, that they attract like something seductive and ex­cite the highest alacrity in the minds of their students." The mathematical sci­ences, according to Hortensius, "ought to be cultivated and honoured by us and their reputation enhanced, so that through them, aspiring to the knowl­edge of the stars in the sky, we may watch more carefully that book of na-

ture3 and we may read it more atten­tively. [ . . . ] Plato also said that eyes were given to men to watch the stars,

but also arithmetic and geometry were given as added wings, by which he might fly into the highest spaces of the world" [Hortensius 1634, 7f] (Fig. 3).

Still, merely praising the dignity and antiquity of the mathematical sciences was clearly not sufficient to convince the city authorities who supported the Athenaeum illustre to invest in them, i.e., to hire Hortensius. Accordingly, af­ter playing the humanistic parlour game of alluding to the classics for a while, he takes up the utility and practical ad­vantage of the mathematical sciences. "These [the mathematical sciences] we have shown to surpass the other sci­ences in the contemplation of things, by their certainty, their nobility of subject and their comfort and pleasing quality; so we will make clear that they confer the most noble benefits also upon men."

Hortensius distinguishes between "the advantage of the mathematical sci­ences [ . . . ] in general, to what extent it spreads itself through all orders of disciplines and faculties, and in partic­ular cases, according to what belongs to each part [i.e., the utility of specific branches as arithmetic or astronomy]." He discusses how the four university faculties-philosophy, theology, law, and medicine-all depend on the math­ematical sciences. As we would ex­pect, he reminds his audience that "Plato filled the books of his own phi­losophy with mathematical reasoning [ . . . and that] you will find written on the doorway of the Academy let no one ignorant of geometry enter." In the books of Aristotle too, he points out, "there are infinite matters from which no one can extricate himself without skill in the mathematical sciences" [Hortensius 1634, 10] .

Let us skip Hortensius's examples of the importance of the mathematical sciences for theology, law, and medi­cine, and tum to those which "contain particular benefits, not at all to be passed over in silence" [Hortensius 1634, 13] : practical arithmetic, geo-

desy, military architecture, mechanics and statics, music, optics, astronomy, geography and navigation. It is in the passage on optics that his Copernican fervour shines through most brilliantly, conveying the feeling that the ancients have now been most assuredly sur­passed. He boasts that

"this is the science that has put lad­ders on the world and informed as­tronomers of the distance and size of the sun, moon, and planets. This has brought more light to our century than

The notion that

mathematics

guaranteed the

h ighest degree of

certai nty humanly

attainable was a

long-stand i ng

epistemological debate . . on

which he takes a

clear posit ion . was given to all the schools of philos­ophy before us to know. I look back to that instrument, recently invented, which they call a dioptt·ic tube [i.e. , the telescope/, by which we see things far off as 'if they were close up. We have uncovered a world in the world, indeed Jupiter, accompanied by four planets orbiting a,round if at certain intervals and periods Q[ time. "

He is taking Galileo's observation of the four moons of Jupiter as clear sup­

port for the Copernican system be­cause they do not revolve around the earth. Hortensius goes on to say, "By this instrument, we perceive that

Venus, brightest of the planets, fades away into horns like the moon, that Saturn has a triple globe, that Mercury with its obscure body receives, with the rest of the planets, all its light from the sun. Among the ancients there is no mention whatsoever of all these mat­ters nor any trace of their investiga­tion" [Hortensius 1634, 16].

In the context of Amsterdam's repu­tation as a leading centre of trade, Hort­ensius pays particular attention to the advantages of practical arithmetic, ge­ography, and navigation. But before turning to these prosaic and material aspects, let's hear what he says about music as part of the quadrivium in the liberal arts. This short passage, between those on mechanics and optics, is a wonderful example of how he draws on the classics as well as on the Bible. "Mu­sic," he explains, "has various benefits, and a charm not to be despised. For (I small here pass over instruments of every kind that touch the minds of lis­teners with singular pleasure), it facili­tates the tempering of men's emotions. It excites noble minds to great actions; it softens the ferocity of behaviour and makes it smooth. Wherefore among the poets Orpheus managed to calm wild animals, lions, tigers, by the sound of his lyre; and Amphion the founder of Thebes even managed to move stones." However, music is not only one of the supreme pleasures of life but also has practical applications: it "also has great power to cure disease, which, although this is almost unknown today, was not unexplored by the ancients. For they, if we are to believe Martianus Capella, cured fevers and wounds by incanta­tion. Asclepiades healed with the trum­pet. Theophrastus used the flute with mentally disturbed patients. Thales of Crete dispelled diseases by playing on musical instruments. There is an exam­ple of this in the Bible, where David soothed the maddened Saul by singing to the lyre" [Hortensius 1634, 15] . In this perspective, music is a microcosm com­bining the dignity and utility of the mathematical sciences.

Leaving these rather fabulous flights,

3The juxtaposition of the book of Revelation and the book of nature was standard in the 17th century. and their relation stood at the core of many debates, including

the Galileo affair.

VOLUME 26, NUMBER 4, 2004 43

Figure 3. Frontispiece of Andrea Argoli's Primi mobilis tabulae (Padua 1 667). The image of arithmetic and geometry as being wings to as­

tronomy was widespread in the 1 6th and 1 7th centuries.

44 THE MATHEMATICAL INTELLIGENCER

Hortensius returns to the concrete when he discusses the advantages of practical aritlunetic, which are so great

"that they can hardly be described in words. Human society stands on this, and the life of men is eased by mutual exchange of goods. Without this, no state is governed, no family ordered, no war waged, nor the fruits of peace gathered. This trains men and makes them attentive to affairs, and not eas­ily liable to be defrauded by another. I ask my listeners, gaze upon your city and you will have a living example of the value of practical arithmetic. The greater part of the citizens engages in trade with Italy, France, England, Ger­many, Africa, and India, with the greatest variety of weights, coinage, and measures. If anyone should ask them by what art their laden goods re­turn safely, they will answer that it is computation, by which in exchanges and comparisons of merchandise, they overcome every problem and obscurity, and, having kept a calculation of what is received and spent, they keep their wealth in its original state, or enlarge it. If anyone should enquire about the profit of the art, they will corifess that so many conveniences are compre­hended in it, that they could do with­out it only with clear loss of their pos­sessions and harm to their families" [Hortensius 1 634, 13}.

Geography, of course, is also indis­pensable for a trading people because it "comprehends and expresses the whole world on a small table . . . . Lack of experience of places has destroyed military power and led the most pru­dent (in other respects) and brave lead­ers into ruin. The same thing has re­peatedly overturned the fortunes of merchants, as, on the other hand, ex­ploring securely the site and attribute of regions and places and knowing the condition of the merchandise there has brought them great riches" [Hortensius 1634, 15] . Hortensius reaches the apogee of his argument for the utility of the mathematical sciences in his praise of navigation. It is navigation, he reminds his audience, "that teaches and enables us to travel by ship to re-

gions separated by the whole sea and to frequent foreign peoples widely dis­persed in all directions. Trusting to this art, mortals, among sea monsters and savage storms, among rough straits and a thousand dangers of death, com­mit huge treasuries of gold and silver to the unstable ocean, and convey home in a light piece of wood the wealth of India and exotic merchan­dise of Africa. Not only individual af­fairs depend on navigation, but also both the continuation and the fall of the fates of kings and states."

" . after the

knowledge of the

mathematical '

sc1ences

increased here ,

. we fi l led al l

the seas with

our voyages . "

He outlines the importance of navi­gation for the rise to power of Venice and Genoa, and the Spanish and Por­tuguese empires. However, these were now superseded by the Dutch, whose success is also rooted in navigational skills: "we Dutch, having struck off the Spanish yoke, when we began to ap­proach the remotest shores of the world, were inferior in eagerness and success to none of the others. At one time we hardly ever entered the At­lantic Ocean, but sustained life on moderate voyages; [ . . . ] But after the knowledge of the mathematical sci­ences increased here, and the naviga­tional art began to be practiced more intensively, we filled all the seas with our voyages; we came to the richest lands of the East and West Indies, saw them and snatched them away from the foreigners; we circumnavigated the globe; we discovered lands; we found new straits; [ . . . ] So we have con­tracted the market of all merchandise within the angle of the world, Holland,

and we have stabilized it." His conclu­

sion comes in an almost mathematical guise: "What God did so that Holland

might daily expand so much, so much advantage have the mathematical sci­ences contributed to navigation, navi­gation to trade, and trade to the solid and firm prosperity of our country." On this basis, that the mathematical sci­ences are essentially equal to prosper­ity, he appeals to the authorities of Am­sterdam to promote further the study of the mathematical sciences: "You rule a city which is very famous and powerful in the whole world. Its ex­pansion came from the study of the mathematical sciences, especially as­tronomy and navigation. Use the city's energy so that the mathematical sci­ences never lose their strength" [Hor­tensius 1634, 17f].

In his concluding remarks, Horten­sius addresses his arguments for the utility and dignity of the mathematical sciences specifically to the merchants: "You [ . . . ] will have a pleasant time employing these studies, by whose benefit your wares, entrusted to the vast sea, go out and return safely. Do not object that your lives are full of cares and anxiety, and cannot admit mathematical contemplation; you will often find a small space of time in which you may dilute the worrisome troubles of business with the pleasure of the mathematical sciences. Thales, one of the Seven Wise Men of Greece, had time for both mathematical stud­ies and trade. For, having foreseen the richness of the olive crop, he hired every press and mill in Miletus; and afterwards when he leased them out at huge prices, he showed his friends not only that a wise man could be rich if he chose, but also that philo­sophical and mathematical studies are not at all foreign to trade" [Hor­tensius 1 634, 19] .

Hortensius's Speech on the dignity and utility of the mathematical sci­ences is filled with such classical allu­sions and quotations, proclaiming not only the dignity and practical utility of the mathematical sciences but also their antiquity. It made a convincing case in the prosperous city of Amster­dam in the Dutch Golden Age.

VOLUME 26, NUMBER 4, 2004 45

BIBLIOGRAPHY

Hortensius and his context

Berkel, Klaas van: Alexandrie aan de Amstel? De

il/usies van Martinus Hortensius (1605-1639),

eerste hoogleraar in de wiskunde in Amster­

dam, in: Haitsma Mulier, E. 0. G./Heesakkers,

C. L./Knegtmans, P. J ./Kox, A. J .Neen, T. J .

(eds): Athenaeum //lustre. Elf studies over de

Amsterdamse Doorluchtige School 1632-

1877, Amsterdam 1 997, 201 -225.

Dear, Peter: Discipline & Experience: The Math­

ematical Way in the Scientific Revolution,

Chicago/London 1 995.

Hortensius, Martin : In viri clarissimi Philippi

Lansbergii Opus astronomicum tabulasque

motuum caelestium dudum ab omnibus

desideratas Carmen, quo ortus & progres­

sus astronomiae ad nostra usque tempora

ostenditur, in: Lansbergen, Philipp: Tabulae

motuum coelestium perpetuae; Ex omnium

temporum Observationibus constructae,

matics, in : Gergmans, Luc/Koetsier, Teun

(eds): Mathematics and the Divine. A Histor­

ical Study, Amsterdam, to be published in

2004.

Swerdlow, Noel M. : Science and Humanism in

the Renaissance: Regiomontanus's Oration on

the Dignity and Utility of the Mathematical Sci­

ences, in: Horwich, Paul (Hg.); World Changes:

Thomas Kuhn and the Nature of Science,

Cambridge (Mass.)/London 1 993, 1 31 -1 68.

Praising the Mathematical Sciences

Barozzi , Francesco: Opusculum, in quo una

Oratio, & duae Quaestiones: altera de certi­

tudine, & altera de medietate Mathemati­

carum continentur, Padua 1 560.

Brahe, Tycho: De disciplinis mathematicis Ora­

tio (1574) , in: Dreyer, John Louis Emil (ed.):

Tychonis Brahe Opera Omnia Tomus I , Am­

sterdam 1 972 (Reprint of the edition Copen­

hagen 1 91 3), 1 43-1 73.

temporumque omnium Observationibus Cavalieri, Bonaventura: Trattato delle Scienze

consentientes. Item Novae & genuinae Mo­

tuum coelestium theoricae & Astronomi­

carum observationum Thesaurus, Middel­

burg 1 632, **1 r-**4v [i .e. 1 8-24].

Hortensius, Martin: Oratio de dignitate et utili­

tate Matheseos. Habita in il/ustri Gymnasia

Senatus Populique Amstelodamensis , Ams­

terdam 1 634.

Mancosu, Paolo: Philosophy of Mathematics

and Mathematical Practice in the Seven­

teenth Century, New York/Oxford 1 996.

Remmert, Volker R.: Ariadnefaden im Wis­

senschaftslabyrinth. Studien zu Galilei: His­

toriographie-Mathematik- Wirkung, Bern

1 998.

Remmert, Volker R . : Galileo, God, and Mathe-

46 THE MATHEMATICAL INTELLIGENCER

Matematiche in generale, in: Giuntini, San­

dra/Giusti, Enrico/Uiivi, Elisabetta (eds):

Opere inedite di Bonaventura Cavalieri, in :

Bollettino di Storia delle Scienze Matem­

atiche 5(1 985), 1 -350; 47-55.

Clavius, Christopher: In disciplinas mathemati­

cas prolegomena, in: Clavius, Christoph:

Opera mathematica, 5 val l . , Mainz 1 61 1 /1 2,

I, 3-9.

Dee, John: The Mathematical Preface to the El­

ements of Geometrie of Euclid of Megara

(1570). With an Introduction by Allen G. De­

bus, New York 1 975.

Regiomontanus, Johannes: Oratio lohannis de

Monteregio, habita Patavij in praelectione Al­

fragani [1 464], in : Alfraganus: Rudimenta as-

tronomica, Nuremberg 1 537 [Reprint in : Re­

giomontanus, Johannes: Opera collectanea,

ed. Felix Schmeidler, OsnabrOck 1 972,

41-53].

A U T H O R

VOLKER R. REMMERT

FB 1 7 - Mathematik

Johannes Gutenberg-Unrvers1tat Ma,nz

D - 55099 Ma1nz. Gennany

e-mail: vrr@mathematlk.uni-matnz.de

Volker R. Remmert was tr8Jned as a

mathematician and as a historian.

Apart from the h1story of early mod­em European science and culture, his

main research tnterests are m the his­

tory of mathematics and science 1n the first half of the twentieth century,

especially the Nazt period. Together

with Annette lmhausen (Cambridge/

UK), he is currently preparing an En­

glish translatton and commentary of

Hortensius's speech. Here he ts seen

with his son Floris at the Frankfurt Book Fair.

Mathematical Tour through the Sydney Opera House Joe Hammer

Does yoor hometown have any

mathematical tmtrist attractions suck

as statues, plaques, graves, the cafe

where the famous conjecture was made, the desk where the famous initials

are scratched, birthplaces, kmtses, or memorials? Have yoo encoontered

a mathematical sight on yoor travels?

l.f so, we invite yoo 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 yoor tracks.

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium

e-mail: dirk.huylebrouck@ping.be

�e Sydney Opera House is one of

I the premier architectural master­

pieces of the twentieth century. In 1992, two out of three respondents to a questionnaire in The Times of Lon­

don placed it first in their list of the

Seven Wonders of the Modem World.

It has become a tourist icon, and in

many ways it can be said that it repre­

sents the image of Sydney. To under­

stand why, it is helpful to know a little

about the city itself (Fig. 1).

Sydney, capital of the state of New

South Wales, is the oldest city in Aus­

tralia with a population over five mil­

lion. It lies on the southeastern Pacific

rim of the continent and enjoys a tem­

perate climate. The principal part of

the city lies between the expansive

Botany Bay and the Sydney Harbour.

The city's eastern limit on the Pacific

Ocean is dotted with several beautiful

beaches, and Sydneysiders like to say

that Sydney is the city of sun, sand, and

surf. The 300-km coastline of Sydney

Harbour is considered one of the most

beautiful natural harbours in the world.

On January 26, 1788, convicts trans­

ported from England formed a penal

colony on the southern shore of Syd­

ney Harbour. This colony was the first

settlement of Europeans on the conti­

nent. The Sydney Opera House is built

within metres of that first landing on

Benelong Point, a peninsula jutting into

the harbour. On each of the three sides

of the peninsula is a small quay. It is as

though the opera house complex, with

its white billowing sail-like roof, were

one of the sailing yachts on the har­

bour.

Overlooking the opera house is the

single-span Sydney Harbour Bridge, af­

fectionately called the "coat-hanger."

Its graceful long arch echoes the many­

faceted curvaceous roof lines of the

Opera House. On the southern side of

the opera house are the Botanical Gar­

dens. This sanctuary is home to

thousands of plant species from

throughout the world, with over one

million specimens in the herbarium.

48 THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+Bus1ness Mecia, Inc.

The opera house on the harbour, the

gardens, and the bridge create an inte­

grated environment within the city of

Sydney.

In 1957 a �year old Danish archi­

tect, Jom Utzon, won an international

competition to design a performance

centre for the government of New

South Wales on Benelong Point. His

plan comprises three basic compo­

nents (Fig. 2). The first is a terraced

platform or podium, which covers al­

most the whole site. This houses all the

technical and non-public service facil­

ities and some smaller theatres. The

second component is made up of three

groups of interlocking sail-like roof

shells and the side shells. The roof

shells cover the two principal theatres

and a restaurant; the side shells fill the

gaps between the roof shells. The third

component consists of twenty-four

glass walls enclosing the open ends of

the shells that house the foyers and re­

freshment areas.

Utzon was commissioned to be the

architect, and the world-renown struc­

tural engineer, Ove Arup, was chosen

to be consulting engineer and adminis­

trator of the project. Utzon resigned in

1966 and a new team of architects took

over, led by Peter Hall. They designed

the glass walls and the interiors of the

theatres.

The Geometry of the Roof

Vaults -A Stroke of Genius

Possibly the most difficult engineering

task in the entire complex was the de­

velopment of the unconventional

sculpture-like free shaped roof com­

plex. In Utzon's plan, as submitted for

the competition, the shapes of the roof

vaults, generally known as "shells,"

were not defined geometrically. The

first task for Arup was to discuss with

the architect the geometry governing

the shells. Obviously, only with well­

defined geometry can engineers calcu­

late forces acting on a structure and

the strains created in them. The paral­

lel problem for the engineers was to de-

3

1 Opera House 2 Syd � HarbOur 3 C ly of Sydney 4 Harbour Bndge 5 Roya' Bolanc Gat<letls 5 Sydney Cove 7 Farm Cove 8 NO<tll SOOre

SITE MAP

Figure 1. The Sydney Opera House and its site on Benelong Point.

sign a structural system that would fa­cilitate serial mass production, the pre­fabrication of the building elements needed.

As a first approach, they observed that each of the roof shells consisted of two symmetrical halves joined in a

Figure 2. Site map.

all the half­shells could be cut from the surface of a common sphere .

curve, which was called the "ridge." The vertical plane through the ridge is the longitudinal axis of the hall. The two half-shells are symmetrical with respect to this plane. In elevation, a half shell is a curvilinear triangle that descends to a theoretical point, a ver­tex of the triangle that is at the base of the podium. Structurally, the half-shell "stands" on that point. The primary problem was to define the geometry of the surface of the half shell, bounded by the curvilinear triangle. In addition to the roof shells, the geometry of the side shells had to be defined.

For more than four years, a team of architects headed by Utzon, and a team of engineers headed by Arup, experi­mented with all kinds of geometric arrangements, from paraboloid to el­lipsoid-type schemes, from parabolic to circular ridge. They expended hun­dreds of thousands of working hours and used thousands of computer hours on the problem. They designed and ap­plied computer techniques never be­fore used in civil engineering, and sev­eral scale models were made for laboratory tests.

Despite all efforts, they obtained no satisfactory solution. The main prob­lem that persisted was the lack of avail­ability, in any of the models, for mass production of the elements of the sur­faces. Then in 1961 came the break-

VOLUME 26, NUMBER 4, 2004 49

Figure 3. Model for deriving half shells from the common sphere.

through. By a stroke of genius, Utzon realised that all the half-shells could be cut from the surface of a common sphere (Fig. 3). This is feasible because the sphere has the property that its cur­vature is the same in all directions. This is not the case for the paraboloid or el­lipsoid. Each half-shell is now a spher­ical triangle. One side of the triangle is the ridge that is part of a small circle of the sphere. The other two vertices of the triangle are on the ridge. Each side shell is also a spherical triangle, the boundaries of which are small cir­cles of the common sphere.

Simultaneously with the develop­ment of the geometry of the shells, ex­periments were conducted with sev­eral structural systems. After years of deliberation, the team decided in favour of Arup's concrete rib system. Each half-shell is made up of a series of concrete ribs. The centre line of each rib is part of a great circle that passes through the pole of the sphere.

The other ends of the ribs lie on the ridge, and their centre lines are at equal distances from each other, so that the ribs of a half-shell radiate from the pole to the ridge, like an open ori­ental fan.

Now each rib can be assembled with

. . . the ri bs of a

half-shel l rad iate

from the pole

to the ridge. . . .

similar repetitive concrete segments. Figure 4. The tiled chevron lids.

50 THE MATHEMATICAL INTELLIGENCER

These segments can be thought of as the "bricks" of the spherical shells, the building blocks of the shells. The ribs of a shell are not of equal length and they widen towards the ridge. Never­theless they all have similar cross-sec­tions at the same distance from the pole. Consequently, although the con­crete bricks are not all of equal size, the spherical surface ensures that there are only a few different sizes, called "types," so that mass-production was possible, type-by-type. Obviously the symmetric half-shell has the same

rib structure, and the two halves are joined rib-by-rib at the ridge.

(Paul Erdos would have fully agreed that the spherical solution is "right from the BOOK"-of architecture.)

Needless to say, it was not a trivial engineering or computational exercise to determine that the theoretical sphere needed to have a radius of 75 metres. And there was the problem, amongst many others, of determining the appropriate small circles, the ridges of the spherical triangles for the vari­ous different shells. They had to satisfy two principal conditions. The first was to obtain the optimal visual harmony between the shells. The second was to obtain the required surface areas for the shells, which are parts of the two

main theatre halls and the restaurant. It took over a year for the engineers to produce the building plans for the shells.

Spherical geometry was applied for cladding the shell complex with square tiles. However, they were not laid di­rectly on the ribs but were laid in chevron-shaped concrete panels, called "lids" (Fig. 4). The role of the lids cor­responded to the rib segments. The concrete lids were laid over the grid segments in such a way that the joint lines between the lids followed the cen­tre lines, the great circles of the shell ribs. The radial joints of the lids ac­centuate the convexity of the spherical shells. This look was further enhanced by the way the tiles were positioned on the lids. Two types of tiles were used, glossy off-white and matte cream. The glossy tiles were placed in the center of the chevron lids in chevron patterns at 45 degrees diagonal to their vertical axes. The matte tiles were placed along the edges of the lids, following the ra­dial joints. This interplay of the twin parquetry of the lids and the tiles con­tributes much to the visual attraction of the surfaces of the shell complex.

The Glass Walls

Utzon placed the two main theatres side by side diagonally, both lying ap­proximately north to south-a brilliant geometric idea (Fig. 2). By this arrange­ment the foyers and refreshment areas

The vertical bars

were posit ioned

as generators

of the th ree

surfaces - the

cyl i nder and the

two cones . are wrapped around both theatres, so that for theatre patrons there is maxi­mum exposure to the harbour through the grand glass walls. All the other competition entrants placed the two halls back-to-hack, failing to recognise the potential of the relationship of the harbour to the buildings.

There are 24 glass walls surround-

Figure 5. The three surfaces of the large northern window.

ing the complex. It is remarkable that they are nearly all different in either shape or size, so is not surprising that each wall had its own design problem. The geometry of the northern wall, the largest of all, will serve for illustration. This wall demanded the most intense effort and was geometrically the most complex.

The wall is made up of three surfaces (Fig. 5). The top surface is part of an elliptical cylinder, the generators of which are vertical. The top contour of the cylinder is defmed by the boundary of the shell ribs. The bottom surface is part of a cone, the lower contour of which is defined by the geometry of the podium surface. The middle surface is also a cone, joining the elliptical cylin­der at the top and the cone underneath, so its boundary conditions depended on the boundaries of the other two sur­faces. Obviously the two cones describe different surfaces determined by the two adjacent intersection surfaces.

The basic element of the walls are the glass panes. All of them are planar; none of them is warped. Connecting the different surfaces, as well as the glass planes horizontally and vertically,

VOLUME 26, NUMBER 4, 2004 51

Figure 6. Concrete beams spanning the concourse.

caused special problems. They were

not to be visually invasive from inside

or obstructive from outside. Chrome

glazing bars were used on the outside.

The vertical bars were positioned as

generators of the three surfaces-the

cylinder and the two cones. The radial

joint lines of the tile lids on the shells

may well have been meant to echo the

generator lines of the glass surfaces.

Inside, the glass walls are supported

by steel structures, the basic element

of which is called a "mullion." The

shapes and the appropriate positioning

of the mullions presented the same vi­

sual problems as the chrome bars. No­

tice that the concrete fan-like shell ribs

and the vertically placed steel mullions

reflect each other.

The mullion planes are vertically po­

sitioned and radiate from the theoreti­

cal centre line of the main hall. Addi­

tionally, the theoretical apices of the

52 THE MATHEMATICAL INTELLIGENCER

two cones also lie on the center line.

This is not only an interesting geomet­

ric coincidence, it also falls in "line"

with the design, which simplified com­

putations for prefabrication of the mul­

lions and glass panes.

The second, smaller north-facing

wall was built with the same three sur­

faces. None of the other walls was built

with three surfaces. Several other walls

were built with two surfaces-vertical

elliptical cylinders on top and an ap­

propriate cone underneath. Apparently

a cone was used for those walls where

the greatest possible ground area was

required. The concave "stomach" of the

cone provided more ground space.

The Concourse Beams

One more high point: you must see Ove

Amp's masterpiece--the concrete

beams over the vehicle concourse (Fig.

6). These have a remarkable sculptural

quality, in addition to their structural

importance. It appears that the 52

beams strain their undulating "mus­

cles" holding the weight of the im­

mense 95-metre-wide concourse stair­

case of over 100 steps leading to the

entrances to the theatres. Expressing

the shape of these "muscles" more for­

mally, we say that the rate of change

from section to section along the axis

of the beam follows the rate of change

of a sinusoidal curve. The significance

in the design of these beams lies in the

fact that no supporting columns are

needed over a 50-metre span. The same

design was used for the concrete ribs

of the shells. This is now generally ac­

cepted engineering practice. For par­

ticulars of this important design, see

the paper of Arup and Jenkins [ 1 ] .

Finally, when you go down to col­

lect your car from the opera house car

park, notice the double-helix-shaped

ramps, allegedly the first construction

of its kind in the world.

For more information about this

wonderful complex, consult the litera­

ture below and visit the Web site

www.sydneyoperahouse.com.

REFERENCES

[1 ] Arup, 0., and Jenkins, R. S. The Evolution

and Design of the Concourse at the Syd­

ney Opera House, Proceedings of the In­

stitution of Civil Engineers No. 39, 1 968,

p . 541 -565.

[2] Arup, 0., and Zunz, J . , Sydney Opera House,

Structural Engineer 1 969, p. 99-1 32; 4 1 9-

425. [This paper summarises the develop­

ment and design of the entire complex.]

[3] Fromonot, F . , Jam Utzon- the Sydney

Opera House, Electa/Ginko 1 998. [This

book has the most extensive bibliography.]

University of Sydney

NSW 2006

Australia

e-mail: hammerg@tpg.com.au

EDWARD G. EFFROS

Matrix Revolut ions : An I ntrod uction to Quantum Variables for Young Mathematic ians

Dedicated to Richard V. Kadison and Masamichi Takesaki for TransmiUing von Neumann's Vision

he most dramatic shift in Twentieth-Century physics stemmed from Heisenberg's for­

mulation of matrix mechanics [9 f. In classical physics, quantities such as position,

momentum, and energy are regarded as functions. In quantum theory one replaces

the functions by non-commuting infinite matrices, or to be more precise, self-adjoint

operators on Hilbert spaces. This enigmatic step remains the most daunting obstacle for those who wish to under­stand the subject.

Although there exist many excellent mathematical intro­

ductions to quantum mechanics (e.g., [12], [17]), they are un­derstandably focused on the development of mathematically coherent methods. As a result, mathematics students must postpone understanding why non-commuting variables ap­peared in the first place. To remedy this, one can adopt a

more historical approach, such as that found in G. Emch's beautiful historical monograph [6], the entertaining yet in­formative "comic book" [11], or M. Born's classic text [2].

In recounting the creation of quantum mechanics, the most difficult task is to describe how Heisenberg found the canonical commutation relation

(1)

for the position and momentum operators Q and P. This equation is the final refinement of Planck's principle that a certain action variable is discrete, or, more precisely, tbat it can assmne only integer increments of a universal ron­

slant h. Heisenberg used the more sophisticated formula­

tion of Bohr and Sommerfeld tbat for periodic systems one has the "quantum condition ..

(2) fpdq=nh (see the discussion below).

In the words of Emch G6], p. 262), "one can only pro­

pose some very loose a priori juslifications., for the de­rivation of (1) from (2). Even Born, wbo was apparently the first to postuJate the genernl form of (1) (see [6], p. 264), avoided discussing it, appealing instead to the SchrOdinger model ([2), p. taO, see also [11], p. 224), and tbis is the ap-

proach that one fmds in most physics texts. I will attempt

to make Heisenberg's direct conceptual leap a little less

mysterious, by deciphering an argument that Heisenberg

presented in his 1930 survey [10] . At the heart of his com­

putation is the observation that

the analogue of the derivative for the discrete action variable is just the corresponding finite difference quotient.

(see (13) below).

Shortly after Heisenberg introduced matrix mechanics,

Schrodinger found an alternative quantum theory based on

the study of certain wave equations [ 16]. His approach en­

abled one to avoid a direct reference to Heisenberg's ma­

trices. Although it is both intuitive and computationally

powerful, "wave mechanics" is not as useful in quantum

field theory. The difficulty is that it does not fully accom­

modate the particle aspects of quanta. In quantum field the­

ory one must take into account the incessant creation and

annihilation of particles associated with the relativistic

equivalence of mass and energy. In particular, the number

of particles present must itself be regarded as an integer­

valued quantum variable. In Born's words ([2] , p. 130),

"Heisenberg's method turns out to be more fundamental."

My goal has been to maximize the accessibility of the

material. To do this I have taken liberties with the mathe­

matical, physical, and historical details. To some extent this

is justified by the fact that regardless of how much care we

might take, the discussion is necessarily tentative. Although

Heisenberg's argument is mathematically quite suggestive,

in the end we must discard these notions in favor of the

operator techniques that grew out of them.

Atomic Spectra, Fourier Series, and Matrices

The crisis that occurred in classical physics is clearly seen

in the peculiar properties of atomic spectra. If one sends

an electric discharge through an elemental gas A such as

hydrogen or sodium, the gas will emit light composed of

very precise (angular) frequencies w. The corresponding

spectrum spA of such frequencies is quite specific to the

element A. For a single frequency we have the corre­

sponding representation

cos(wt + a) = Re ei(wt+a) = c_ 1e-iwt + c1eiwt

for suitable complex constants C-h c1. Superposing these

frequencies, we may describe the radiation by the sum

(3) fA(t) = L Cweiwt, wEsp0A

where spoA = spA U -spA U {0) .

There are obvious classical analogues of this phenome­

non. If one strikes an object, the resulting sound can be de­

composed into certain specific angular frequencies. In the

case of a tuning fork, the resulting motion is harmonic, and

one obtains a corresponding Fourier series for the ampli­

tude of the sound wave in the form

f(t) = A cos(wt + a) = c_ 1e-iwt + c1eiwt

54 THE MATHEMATICAL INTELLIGENCER

for suitable complex coefficients ck. If one instead plucks a

guitar string, the resulting sound is a combination of various

frequencies, all of which are harmonics, i.e., multiples of a

fundamental frequency w. Thus one has a Fourier series

(4) f(t) = L Cnei(nw)t, nEZ

where for simplicity we assume that only finitely many of

the Cn are non-zero. We define the (full) spectrum off to

be the cyclic group 7Lw = {nw : n E 7L} . As is well-known,

one can duplicate the sound of a guitar string by super­

posing the pure frequencies as in ( 4).

More complicated systems (such as a bell) will have

more than one fundamental frequency. If there are two fun­

damental frequencies w,w', there will be an "almost peri­

odic" expansion

f(t) = L Cn,n'ei(nw+n'w')t. n,n'ElL

Let us restrict our attention to the periodic expansions ( 4).

The linear space sd(w) of all functions of the form (4)

with finitely many non-zero terms is closed under multi­

plication, for if we are given

then

f(t) = L Cnei(nw)t nEZ

g(t) = L dnei(nw)t, nEZ

f(t)g(t) = I ckdn-kei(kw+(n-k)w)t = I anei(nw)t, k,nEZ nEZ

where an is the "convolution"

(5) an = cc * d)n = I ckdn-k· kEZ Furthermore sd( w) is closed under conjugation, since

](t) = Ic�ei(nw)t,

where c';. = C-n· In more technical terms, the *-algebra

sd(w) is a representation of the group *-algebra C[7L] . This

result, of course, stems from the fact that spf = 7Lw is a

group under addition.

Returning to atomic spectra, it is tempting to regard (3)

as some kind of Fourier series. There are several problems

with this interpretation.

First of all, we are actually interested in analyzing the

property of a single atom. In this case it is inappropriate

to "add up" the series (3). For example (getting a little ahead

of ourselves), a hydrogen atom will radiate only one fre­

quency at a time corresponding to the electron taking a par­

ticular orbital jump. Thus superpositions do not occur when

one "watches" a single atom. For this reason it is more ac­

curate to letf(t) stand for the array (cweiwt)wEspoA· Second, in striking contrast to the classical models, it is

not useful to consider the additive group generated by spoA.

Given w E spoA, one need not find any of the harmonics

nw in spoA. Nevertheless the set spoA does display an ex­

quisitely precise algebraic structure, called the Ritz com-

bination principle. We may doubly index spA, i.e., we may

let spoA = { Wm,nlm,nEN, in such a manner that

(6) Wm,n + Wn,p = Wm,p

for all m, n, p E 1\l. In particular, Wm,m + Wm,m = Wm,m and

thus Wm,m = 0. Furthermore, Wm,n + Wn,m = Wm,m = 0, and

therefore wn,m = - wm,n· Using this double indexing of the

spectrum, our array becomes a matrix function of t:

(7)

The set M( w) of matrices (7) is already a linear space.

Owing to (6), M( w) is closed under matrix multiplication

and the adjoint operation; indeed,

f(t)g(t) = [ � Cm,keiwrn,kt dk,neiwk,ntl = [ � Cm,dk,neiCwrn,k+wk,rJtl = [ am,neiwrn,ntJ

where a = cd is the usual matrix product, and

f(t)* = [an,me-iwn,mt] = [a*m,neiwm,nt]

with a* the adjoint matrix. In fact one can regard M(w) as

a representation of the *-algebra C[N X 1\l ] of the full

groupoid 1\l X 1\l. This point of view has been explored by

Connes [3], but will not be pursued further in this paper.

It is easy to prove that any doubly indexed family wm,n satisfying (6) must have the form

for suitable constants Cm. The values for the hydrogen atom

are given by Balmer's equation

(8) c c Wm n = 27TR -2 - 27TR -2 , ' m n

where c is the speed of light, and R is known as Rydberg's

constant.

Long before matrices were introduced, Bohr justified

Rydberg's equation by combining Rutherford's model of the

atom with a quantum condition on the action variable. This

"old" quantum theory was to play a crucial role in the evo­

lution of matrix mechanics.

Action and Quantization Conditions

Action is perhaps the least intuitive of the standard notions

of classical mechanics. As usual, the easiest way to under­

stand a physical quantity is to consider its units or "di­

mensions." Let M, :£, and '2J denote units of mass m, length

(or position) q, and time t (e.g., one can use grams, meters,

and seconds). Given a physical quantity P, let [P] denote

its units. We have, for example,

[velocity v] = [ �� l = :£'2!- 1

[acceleration a] = I d2q l = ;£2J-2 L dt2 [momentum p] = [mv] = M:£'2!- 1

[force F] = [ma] = M:£2J-2 [potential energy V] = [ -Fq] = M,;£22J-2

[kinetic energy T] = r imv2l = M:£22!-2

[total energy H] = [V + T] = ,M;£22J-2.

Noting that they have the same dimensions, we simply re­

gard V, T, and E = V + T as "different forms" of energy. We

will often consider derivative and integral versions of these

quantities, such as v and a above and the potential energy

V = -JF(q)dq.

The dimensions frequently mirror physical laws. For ex­

ample, the equation for force corresponds to Newton's sec­

ond law. On the other hand the relativistic equation E =

mc2 corresponds to M:£22!-2 = M x (:£'2!- 1)2. The usual form of a travelling wave (in one spatial di­

mension) is given by

(9) f(t,q) = A cos(wt + kq), where w is the angular frequency (radians per second) and

k is the angular wavenumber (radians per meter). The cor­

responding dimensions are

[angular frequency w] = [radians]/[ time] = '2J- 1 . [angular wavenumber k] = [radians]/[ distance] = ;£- 1.

We recall that these are related to the frequency v (cycles

per second) and wavelength (of a cycle) A. by w = 2 7TV and

k = 2 7T/A..

Given an angular co-ordinate (} measured in radians, we

have the dimension

[angular velocity w] = r �� l = '2!- 1.

By analogy with the momentum formula p = mv, the an­

gular momentum is defined by L = �w, where � is the "mo­

ment of inertia"; equivalently, L is the signed length of the

vector L = r X p, where r is the position vector and p is

the momentum vector. Thus we have

[angular momentum L] = ,M;£22J- 1.

In classical physics, the (restricted) action along a pa­

rametrized curve y is defined by the formulas

J[y] = I pdq = r Tdt, 'Y a

and the actual motion taken by the particle is determined

by finding the stationary values of suitable variations of J

with fixed energy (alternatively one can use a different vari­

ational principle involving the Lagrangian, see [7], [8]). The

corresponding dimensions are given by

[action J] = [energy] X [time]

= [momentum] X [distance] = M,;£22J- 1.

We see from above that action has the same dimensions as

angular momentum. Following [ 13], I will also use the ac­

tion I = (1/27T)J.

Quantum mechanics began in 1900 with Max Planck's pa­

per [14]. He discovered that he could predict the radiation

properties of black bodies provided he assumed a "quantum

VOLUME 26, NUMBER 4, 2004 55

condition." He essentially postulated that the action variable J associated with an atom can take only the discrete values nh, where h is a universal constant and n E N.

An early task of quantum mechanics was to reconcile the particle and wave properties of "quantum objects" such as photons and electrons. Albert Einstein [5] related the en­ergy E and momentum p of a photon to the frequency v and the wavelength A. of the corresponding wave. Noting that Elv and pA. are action variables (see above), he pre­dicted that each of these equals the "minimal action" h; i.e., we have the Einstein relations

E = hv = hw p = h/A. = hk

where h = h/2 7T. Subsequently L. de Broglie [4] proposed that these relations were valid for all particles exhibiting the wave-particle dichotomy, including the electron. It was a short step from there to finding a wave equation for which the corresponding functions (9) are solutions. This is pre­cisely the Schrodinger equation.

In 1913 Nils Bohr used the Planck-Einstein quantum con­dition to explain the spectral lines of the hydrogen atom [1]. He proposed that the electron is constrained to particular cir­cular orbits by the quantum condition. To be more specific, he assumed that the electron has a specific energy Em in the mth orbit, and that if it drops down (respectively, jumps up) to the nth orbit, it loses (respectively absorbs) energy Em - En, which is carried away or brought by a photon with frequency

(10) Em - En (J) = m,n h

When Bohr used the classical Coulomb law to calculate the angular momentum L of the electron in the mth orbit, he discovered that it was given by L = mh for an integer m. In fact, by using the Hamiltonian theory from the next sec­tion, he and Sommerfeld showed that this coincides with Planck's quantum condition J = mh, and the latter is also true for arbitrary closed orbital motions. Within a few years, Bohr's theory was used to predict the frequencies of the spectral lines for a variety of systems.

Bohr also formulated a fundamental asymptotic prop­erty for the spectral values, which he called the corre­spondence principle. Returning to the Rydberg formula, he observed that for large m, the electrons behaved almost classically, in the sense that one obtained harmonics. More precisely, a drop of k = m - n � m orbits resulted in the kth harmonic of a fundamental frequency wm = 47TRc!m3:

Wm,m�k = 2 7TRC (- �2 + (m � k)2 )

= 47Tk ( Rc ) (1 - k/2m) m3 (1 - 2klm + k2!m2)

� kwm.

A similar principle applies if k is negative. Here the notation k � n indicates relatively small positive or negative jumps.

In principle it would seem that we might have to consider infinitely many fundamental frequencies Wm· However, de­spite its nebulous character, Bohr used the correspondence

56 THE MATHEMATICAL INTELLIGENCER

principle to predict very accurately the value of the Rydberg constant R as well as the "radius" of a hydrogen atom.

Bohr's "old" quantum theory suffered from a number of defects. In particular, the increasingly technical quantum conditions seemed unnatural, and it was difficult to calcu­late the "Fourier coefficients" am,n· The quantity lam,nl2 measures the intensity of the frequencies wm,n, or at the level of a single atom, the probability that a jump from m to n might occur. Just as one cannot "in principle" predict when a radioactive atom might decay, one cannot say when an electron will "jump." This is a prototypical example of the probabilistic nature of quantum mechanics.

Heisenberg concluded that the weakness of Bohr's the­ory was that it was concerned with predicting the hypo­thetical singly indexed energies En rather than the actually observed doubly indexed frequencies wm,n· As we have seen above, it was this perspective that led him to consider matrices. To carry out his program, he had to incorporate the quantum conditions into his framework

Phase Space and Action Angle Variables

Quantization is typically applied to algebras of functions. Because the Hamiltonian approach to classical mechanics is concerned with an algebra of functions on a suitable pa­rameter space, it is ideally suited for this process. What is particularly useful about the Hamiltonian formulation is that each function determines a one-parameter group of au­tomorphisms, and in particular, the energy function deter­mines the physical evolution of the system. Let me sum­marize this theory as quickly as possible.

Let us first suppose that we are given a parameter space M = !Rn. We let Slll(M) be the algebra of infinitely differentiable functions on M, and T(M) = M X [Rn be the corresponding tangent space. Then we regard (x,v) E T(M) as a "tangent vector at x," and it determines a corresponding directional derivative. Given x E M and v = I VJeJ E !Rn, define

Dcx,v) : Slll(M) � IR : f � L VJ :�. (x). 1

Because tangent vectors are only used to indicate the di­rectional derivatives that they define, we use the notation

(x,v) = L VJ _a_ I · axJ x

A vector field is a mapping

F : M � T(M) : x � F(x) E Tx(M) = {x) X !Rn,

and we may write

Given f E Slll(M), the function DF : x � F(x)f is again a smooth function on M, and the mapping

is a derivation of the algebra Slll(M), meaning that

D(fg) = D(J)g + JD(g).

As is well known, all derivations of q]J(M) arise in this man­

ner (see [ 18]).

A curve

x : (a,b) � M : t 1--0> x(t) = (x1(t), . . . , Xn(t))

is an integral curve for a vector field F if for each t, x' (t) = F(x(t)). Thus x(t) = (x1(t), . . . , Xn(t)) is just the so­

lution to the system of first-order differential equations

dx ·(t) -it = Fj(X(t)).

Under appropriate conditions, we may find an integral flow

for the vector field; i.e., a family of mappings u1 : M � M such

that for each x E M, t 1--0> u1x is an integral curve for j, and

furthermore Ut+t ' = u1 o u1· , u0 = I. This in tum determines

a one-parameter group of algebraic automorphisms a1 of

the algebra q]J, where atf(x) = f(u-1x). Using power series,

one finds a simple relationship between the derivation DF

and the automorphism group a1:

D (f) = lim ah(j) - f

F h-->0 h

Turning to physics, let us consider a single oscillating

particle with one degree of freedom. The Newtonian equa­

tion of motion is given by F = ma. Let us assume that the

force F only depends on the position q. Thus we are con­

sidering the second-order equation

d2 F(q(t)) = m _!l_2 dt

Because we have restricted to one spatial dimension, F is

automatically conservative; i.e., F(q) = - V'(q) for some

function V, namely V(q) = -f F(q)dq. We begin by replacing Newton's equation with two first­

order equations. Although there are many ways this can be

done (e.g., one can let dq/dt = v, and dvldt = F/m), Hamil­

ton found a particularly elegant way. Specifically we use the

variables q and p = mv. The corresponding equations are

( 1 1)

where

dq -dt

dp =

dt

aH -ap

aH aq '

2 H(q,p) = :

m + V(q).

We may regard the solution curves y(t) = (q(t),p(t)) as the

integral curves of the symplectic gradient vector field

aH a aH a sgradH = - - - - -

ap aq aq ap

in the phase space M2 = IR2 of variables (q,p). This quan­

tity is the "symplectic" analogue of the usual gradient

aH a aH a gradH = - - + - -

aq aq ap ap '

but it is not necessary to go into details.

In fact, an arbitrary function a(q,p) on M2 determines a

vector field

aa a aa a sgrad a = - - - - -,

ap aq aq ap

and thus, a corresponding flow

u� : Mz � Mz,

where y(t) = u�(x0) = (q(t), p(t)) is a solution of the

"Hamiltonian system"

( 1 1 ')

dp =

aa dt ap dp = aa dt aq

The Poisson bracket of two functions a and b is defined by

aa ab aa ab {a,bj = (sgrad a)(b) = - - - - -.

ap aq aq ap

In particular, we note that if { a,b} = 0, then letting

(q(t),p(t)) be an integral curve of ( 1 1 ') ,

db =

.!!!!_ dq + .!!!!_ dp = .!!!!_ aa _ .!!!!_ aa = 0.

dt aq dt ap dt aq ap ap aq '

i.e., the function b is constant on the orbits of a. Since { a,a} = 0, we see that a is constant on its own integral curves.

Perhaps the most striking attribute of the phase space

parametrization is that the area pq of a rectangle has the

dimensions

[momentum] X [distance] = .M1.:2:J- l:

area is an action variable. This link between the notion

of area (or more precisely the area two-form !1 = dp 1\ dq)

and a physical parameter is one of the most powerful fea­

tures of the Hamiltonian theory. We say that a change of

variable Q(q,p), P(q,p) is canonical if it preserves the area

form, i.e., if the Jacobian is 1 :

1 = a(Q,P) = aQ aP _ aQ aP

_ a(q,p) aq ap ap aq

If that is the case, then the dynamical system Q(t) =

Q(q(t),p(t)), P(t) = P(q(t),p(t)) is also Hamiltonian; i.e., it

has the same form as (11):

dQ aH dt aQ dP aH dt aQ '

where by abuse of notation H(Q,P) = H(q(Q,P),p(Q,P)). To see this, note that

dQ =

aQ dq + aQ dp dt aq dt ap dt

aQ aH _ aQ aH aq ap ap aq

= aQ ( aH aQ + aH aP ) _ aQ ( aH aQ + aH aP) aq aQ ap aP ap ap aQ aq aP aq

= aH a(Q,P)

= aH

aP a(q,p) aP '

VOLUME 26. NUMBER 4. 2004 57

and a similar calculation for the second equation. It is also easy to see that a canonical change of variables will leave the Poisson brackets of functions invariant. If the system (Q,P) is Hamiltonian, we say that Q and P are cof\iugate variables.

Let us assume that our system is oscillatory; i.e., all of the solution curves (q(t),p(t)) are closed. We may assume that (q(O),p(O)) = (q(T),p(T)), where the period T depends on the orbit. Our goal is to find the "simplest Hamiltonian co-ordinate system" ( 8,[) with the following properties:

• H( 8,!) = H(J), i.e., H doesn't depend on 8, and • 8 increases by 27T on each closed orbit.

Given such a system, we will have

di = _ aH = 0 dt ao '

and thus I and H(J) are constant on each orbit y. It follows that

dO aH w = - = -dt ai is also constant on each orbit y; i.e. , w = w(I) = w( y), and O(t) = wt + C for some constant C. We may assume C = 0, and from the second property, w = 27TIT.

The mapping (q,p) � (8,!), analogously to the polar co­ordinate change of variable (x,y) � (O,p), maps closed curves to horizontal line segments.

The canonical transformation from (q,p) to (8,!) trans­forms the area A enclosed by an orbit y(t) = (q(t),p(t)) to the area R of the rectangle 0 :::; 8 :::; 27T, 0 :::; I :::; I( y). Be­cause the purported transformation is canonical, we have

f pdq = A = R = 27Tl(y), ')'

where y is the unique integral curve that passes through (q,p). Thus, assuming that we can find a canonical trans­formation with the desired properties, I is an action vari­able. For the proof that the transformation exists (and a formula for 8), I recommend [ 13] or [8] . We define

/ = -21 f pdq 7T ')'

to be the action variable and 8 the angle variable. As one would expect, 8 is multivalued; it increases by 27T on each circuit of an orbit.

The action-angle variables enable us to use Fourier series in our analysis of a periodic motion. Given an arbitrary func­tion a on M2 and using the action-angle variables, the function a( 8,!) will have period 27T in 8. It thus has a Fourier series

(12) a( 8,!) = I a( n )eine,

where a( n) is a function of I. Substituting the solution of the Hamiltonian equations, we obtain a as a function of time:

a(t) = I a(n)einwt,

where a(n) is constant on the orbit. (Here and below, I use

58 THE MATHEMATICAL INTELLIGENCER

for the Fourier coefficients of the function the same letter as for the function.)

The Commutation Relation

We will identify the energy variables H and E. There is a close parallel between the classical formula

aH aE w = - = -ai ai and Bohr's difference formula

To make this more explicit, let us "discretize" the action variable I by setting I = mh and I:J.kl = kh. Then, accord­ing to Bohr's correspondence principle, if k � m,

It thus appears that Bohr's correspondence principle is em­bodied in the fact that the finite difference with respect to the discrete action variable I approximates the differential quotient with respect to the continuous action variable /. For this reason, it seems justifiable to apply this to arbi­trary quantum variables and their classical analogues. Let us use the symbolism

(13)

(see [ 18], pages 1 10, 56). The difference operator will be applied to a matrix variable by the formula

(I:J.kA)(m,n) = A(m,n) - A(m-k,n-k).

In his calculation, Heisenberg concentrated on the Fourier coefficient functions a( f) of a function a on the phase space M2 and the scalar matrix coefficients A(m,n) of a matrix A in the "expansions"

a = Ia(f)eiCwt c

A = [A(m,n)eiwm,nt ] . If a is the classical function variable "reduction" of the ma­trix variable A, then for f = m - n � m the coefficient A(m,n) of eiwm,nt should approximate the coefficient a(f) of the harmonic ( eiwt)e. The notation for such a corre­spondence will be A � a, and A(m,n) � a( f).

If j,k � m then

A(m,m-j) � a(J) = + j-1 :� (J) (for j =I= 0)

(I:J.kA)(m,m-j) � kh aa (J). a I

The equality is seen if one takes the derivative of (12) with respect to 8. The second reduction is a formal consequence of (13).

h It will tum out that if A � a and B � b, then [A,B] �

--;-- {a,b}: non-commutativity of operators flows from Poisson � brackets!

Let us suppose that we are given matrices A and B and functions a and b with A � a and B � b. If f = m - n � m,

(AB - BA)(m,n) = I A(m,m-J)B(m-j,m-j-k) - I B(m,m-k)A(m-k,m-k-J) j+k-t j+k-t

= I [A(m,m-J) - A(m-k,m-j-k)]B(m-j, m-j-k) -j+k-t

I A(m-k,m-j-k)[B(m,m-k) - B(m-j,m-j-k)] j+k-t I (t:..kA)(m,m-J)B(m-j,m-j-k) - A(m-k,m-k-J)(t:..jB)(m,m-k)

j+k-t

� � I (k �a (J) b(k) - a(J) j !!!!_ (k)) 'L j+k-t ()[ ()[

= � ( I k aa w k-l !!!!_ Ck) - I r1 aa CJ) j ab (k)) 'L j+k-C,MO ()[ ()(} j+k-tj*O () () ()[

= !!:_ ( aa ab _ aa !!!!__) ( €) i ()[ () (} () () ()[

h = ---:- (a,b)(€). 'L ab aa (see (5)-note that ae (0) = ae (0) = 0 by (12)).

As Heisenberg points out in a footnote, this calculation is problematical even as a heuristic guide. Although n -m = e = j + k is assumed "relatively small" with respect to m and n, we are summing over arbitrary j,k with j + k =

€. Heisenberg explains this away by pointing out that ifj is large it will follow that k is large (usually with opposite sign) and vice versa, and thus all the matrix positions (m,m-J), (m-j,m-j-k), (m,m-J), and (m-k,m-k-J) will be distant from the diagonal. He states that the corre­sponding matrix elements must be negligible "since they correspond to high harmonics in the classical theory."

I conclude that

Because

h [A,B] � ---:- (a,b). 'L

( l = ap � _ ap � = I p,q iJp iJq iJq iJp ' if we let P and Q be the quantized momentum and position

matrices, i.e., P� p and Q � q, we are led to postulate the commutation rule

h [P,Q] = ---:- I. 'L

This relation is the most essential algebraic ingredient of quantum mechanical computations. The reader may find early instances of these calculations in [2].

REFERENCES

[1 ] N. Bohr, On the constitution of atoms and molecules: Introduction

and Part I- binding of electrons by positive nuclei, Phil. Mag. 26

(1 9 1 3), 1 -25 .

[2] M. Born, Atomic physics, Dover, New York, 1 969 ISBN 0-486-

65984-4.

[3] A. Connes, Noncommutative geometry. Academic Press, Inc . , San

Diego, CA, 1 994. ISBN : 0-1 2- 1 85860-X.

[4] L. de Broglie, Sur Ia definition generale de Ia correspondance en­

tre onde et mouvement, CR Acad. Sci. Paris 1 79, 1 924.

[5] A. Einstein, On a heuristic point of view about the creation and con­

veion of light (English translation of title), Ann. Phys. 1 7 (1 905),

1 32-1 48.

[6] G. Emch, Mathematical and conceptual foundations of 20th­

century physics. North-Holland Mathematics Studies, 1 00. Notas

de Matematica, 1 00. North-Holland Publishing Co. , Amsterdam,

1 984. ISBN: 0-444-87585-9

[7] I . Gelfand and S. Fomin, Calculus of variations. Revised English

edition translated and edited by Richard A. Silverman, Prentice­

Hall, Inc . , Englewood Cliffs, N.J . 1 963. ISBN 0-486-41 448-5 (pbk).

[8] H. Goldstein, Classical mechanics, Addison Wesley, 1 950. ISBN

0-201 -02510-8.

[9] W. Heisenberg, Quantum-theoretical reinterpretation of kinematic and

mechanical relations (translation of title), Z. Phys. 33, 879-893, 1 925.

[1 0] W. Heisenberg, Physical principles of the quantum theory, Dover,

New York, 1 949. ISBN : 486-601 1 3-7.

[1 1 ] Transnational College of Lex, What is quantum mechanics, a

physics adventure, translated by J. Nambu, Language Research

Foundation, Boston, 1 996. ISBN 0-9643504- 1 -6.

[1 2] G. Mackey, The mathematical foundations of quantum mechan­

ics, W.J. Benjamin, New York, 1 963.

[1 3] I . Percival and D. Richards, Introduction to dynamics. Cambridge

University Press, Cambridge-New York, 1 982. ISBN: 0-521 -

23680-0; 0-521 -281 49-0.

[1 4] M. Planck, On an improvement of Wien's equation for the spec­

trum (translation of title), Verhandlungen der Deutschen Physik.

Gese//s. 2 , 202-204.

[1 5] A. Sommerfeld , Munchener Berichte, 1 91 5 , 425-458.

[1 6] E. Schrbdinger, E. Quantization as an eigenvalue problem (trans­

lation of title), Ann. Physik 1 926 79, 361 -376.

[1 7] V. Varadarajan, Geometry of quantum theory, Springer-Verlag,

New York, 1 968.

[1 8] F. Warner, Foundations of differentiable manifolds and Lie groups,

Scott Foresman, 1 971 .

[1 9] T. Wu, Quantum mechanics, World Scientific, Singapore-Philadel­

phia, 1 985, ISBN 9971 -978-47-4.

VOLUME 26, NUMBER 4, 2004 59

A U T H O R

EDWARD G. EFFROS

Department of Mathematics UCLA

Los Angeles, CA 90095- 1 555

USA e-mail: ege@math.ucla.edu

After undergraduate work at MIT, Edward Effros completed h1s PhD dissertation at Harvard 1n 1961 under George Mackey. He IS eter­

nally grateful to Mackey for steenng him to the beautiful merging

of algebraic and analytic techniques with the mystery of ·mathe­

matical quantization" 1n the work of R1chard Kadison. This area of

mathematics, and the personal support of Kadison, set the d1rec-

11on of all Effros's work, up to his current project of quantizing Ba-

nach space theory in collaboration with Zhong·Jin Ruan. The pres­

ent artiCle IS a d1st1llation of the many rapid treatments of quanti­

zation Effros has attempted over the years.

W1th his wife Rita, a well-recognized immunologist, Effros en­

joys hiking and listening to classical music. The1r horizons are

broadened by the1r daughter, a phys1cian, and their son, an archi­

tect.

60 THE MATHEMATICAL INTELLIGENCER

Note added in proof: Paul Chernoff has reminded us of another important historical source: Van der Waerden's Sources of Quantum Mechanics, North Holland Publishing Company, Amsterdam, 1967. In particular, it includes a 1924 paper of H. A. Kramers, in which the author explicitly states the derivative/ finite difference correspondence (13).

Solut ions (of a sort) to the Origam i Qu iz (see pp. 38-39)

Thomas Hull

1. The 3 X 3 checkerboard can be folded in only 7 folds. The below solu­tion is due to Kozy Kitajima, and was presented at the Gathering for Martin Gardner conference (Atlanta, GA) in 1998.

6 7 2 I I

5 - 1 - - - - - - -� - - - - + ­I

I I 4 - -; - - - - - - -;- - - - T -I / 1

3 L. I I I /-/ ' T - - - r -/ ;-;

valley-fold

mountain-fold

Readers will be tempted to general­ize this puzzle to n X n checkerboards but it quickly becomes extraordinaril; difficult. For the 4 X 4 case, the best solution known to the author requires 14 folds, and this assumes that we al­low an origami move known as a "squash fold" to count as one fold. (A squash fold is shown in the middle fig­ure below.)

In fact, the question of what "counts" as one fold is non-trivial. Bit­ter debates on this very question emerge when practiced origamists face this puzzle. For example, the below crease pattern presents a " 1-fold" solu­tion to the 2 X 2 checkerboard puzzle. That is, if each crease is carefully made beforehand, then all of the creases (in their proper mountain/valley direc­tions) must be folded simultaneously to obtain the checkerboard pattern. Because only one motion is required, does this count as one fold? (This is tricky to do; readers are encouraged to try it persistently. And the reward is great because it actually makes a 2 X 2 checkerboard on both sides of the paper. This is an example of what origamists call an iso-area model where both sides of the paper are do� ing the same thing, up to rotation and reversal of the creases. Origamist Jeremy Shafer has a similar, iso-area,

"1-fold" solution to the 4 X 4 checker­board puzzle.)

2. Imagine our piece of paper is the plane, IR2, and our square is drawn on the plane with vertices at (::':: 1 , ::':: 1). If we ignore the square, it is clear that if we fold and unfold the vertices to our random point P, the crease lines will form four sides of a quadrilateral con­taining P. Then answering 2a reduces to determining how the sides of the square intersect this quadrilateral. If P

is located at one of the square's ver­tices or at (O,o), then P will be con­tained in a square on the paper. Other­wise if P is close enough to one of the sides of the square, then that side will cut across the quadrilateral made by the folding. We can determine when this will happen by drawing semicir­cles of radius 1 centered at the mid­points of each side of the square. How these semicircles overlap determines the solution, shown below left.

As for 2b, thinking of the paper as being the infinite plane allows us to consider P to be chosen outside the square. However, after we make our folds, P will be located in an infinite re­gion, and it is an interesting game to consider how we can redefine what we choose to "count" as our polygon in such cases. In any case, the differences will be determined by extending our semicircles in the solution to 2a to full circles.

Rectangular paper is handled in the same way as in 2a. Interestingly, hep­tagons can be produced. (See 3.)

3. If we let the side of the square be of length one, then the triangle made by this folding procedure is equilateral be-

© 2004 Springer Sc.,nce+Business Media, Inc., VOLUME 26, NUMBER 4, 2004 61

cause its sides all have length one. (Its left and right sides are both images of the bottom side under folding.) It is not the biggest equilateral triangle possi­ble, however. The biggest is symmetric about a diagonal of the square, and a folding method for such a maximal tri­angle is shown below. (Note that the angle fL equals 15°. This "proof without words" construction was devised by Emily Gingrass, Merrimack College class of 2002.)

Actually proving that this is the equi­lateral triangle of maximal area that can be inscribed in a square is a fun trig/elementary calculus problem.

4. This is an origami method of tri­secting an angle. Drawing some auxil­iary lines and unfolding the paper can prove that the trisection works. In the next diagram, argue that the segments AB, BC and CD are all of the same length.

~ p, D

5. Lists of basic origami operations may vary. A lot depends on how one sets things up, and we do not want our list to be redundant. But an initial list of folding operations might look some­thing like the following:

1. Given two points p1 and pz, we can make a crease line connect­ing them.

2. Given two points P1 and pz, we can fold p1 onto p2. (This creates

62 THF MATHFMATI<CAI INTFI I I�FN<CFR

the perpendicular bisector to line segment PlPz.)

3. Given two lines, L1 and Lz, we can fold line L1 onto L2. (Angle bisectors.)

4. We can locate points where two non-parallel lines intersect.

5. Given a line L and a point p not on L, we can make a fold through p that is perpendicular to L, in other words, folding L back onto itself so that the crease passes through p. (Dropping a perpen­dicular.)

6. Given two points p1 and pz and a line L1, we can, whenever possi­ble, fold P1 onto line L1 so that the resulting crease passes through point p2. (This was part of the construction in Problem 3, where p2 was one corner of the paper.)

7. Given a point p1 and two lines L1 and L2, we can make a crease placing p1 onto L1 that is per­pendicular to Lz.

8. Given two points P1 and pz and two lines L1 and L2, we can, whenever possible, make a crease that simultaneously places P1 onto L1 and pz onto Lz.

Operations 1-3, 5, 6, and 8 were for­mulated by Humiaki Huzita. (It is not certain if he was the first to do this, but he was the first to publish these oper­ations. See [5) and [6).) Move 7 is, amazingly enough, a very recent addi­tion developed by Koshiro Hatori (see [3]). Most readers will not have thought of operation 8, although it does appear in Problem 4. Also recently, Robert Lang ([7]) has proven that these oper­ations exhaust all that origami can do. He does this by beginning with the premise that all we can do in origami is fold points and lines to each other, and he runs through all the possibili­ties while formalizing the degrees of freedom one has when folding one ob­ject to another.

Also, Koshiro Hatori claims that most of these operations can be thought of as special cases of operation 8. Can you find a way to make this work?

6. The basic origami operation cited in Problem 6 cannot be performed if the point p2 is poorly positioned with re-

spect to P1 and L1. To see what is go­ing on, do the following exercise: Take a piece of paper and let the bottom side be line L1 and take a random point p1 on the paper. Fold and unfold p1 onto L1 at many different places, making a sharp crease every time you do so. The below figure illustrates what you should see.

D L,

This exercise makes one suspect that the process of folding a point p1 to a line L1 is actually creating a crease line that is tangent to the parabola whose focus is p1 and directrix is L1. There are a number of ways to prove this; for an analytic approach let p1 =

(0,1), let L1 be the x-axis, and find the equation of the crease that results when p1 is folded to an arbitrary point (t,O) on L1. Then take the envelope of this family of lines; a parabola should result.

Now, this situation is at play in fold­ing operation 6, and clearly if the point pz is chosen to be in the interior of the convex hull of the parabola with focus p1 and directrix L1. then the operation will be impossible to perform.

7. See the solution to Problem 6. Be­cause we get a parabola, folding oper­ation 6 is actually solving a quadratic equation for us. (Can you give an ex­plicit method of solving ax2 + bx + c = 0 where a, b, and c are positive integers?)

8. The set of numbers constructible with straightedge and compass is the smallest subfield of C that is closed un­der taking square roots. So straight­edge and compass can solve quadratic equations, but certainly cannot con­struct any algebraic a E C whose min­imal polynomial is cubic. The classic

proof that a straightedge and compass cannot trisect an angle, for example, is built on cos 20° being degree 3 over the rationals.

In Problems 6 and 7 we saw that the origami operation 6 proves that paper­folding can solve quadratic equations. Thus the set of origami numbers contains the set of straightedge-and­compass-constructible numbers. Fur­thermore, because we know that pa­per-folding can trisect angles, we know that the field of origami numbers strictly contains the field of straight­edge-and-compass numbers. Actually, folding operation 8 turns out to allow us to solve general cubic equations. As evidence, the below figure depicts the locus of possible images of P2 =

(.5, - .5) as PI = (0, 1) is folded repeat­edly onto line L1 which is y = - 1 . This graph certainly looks cubic, and deriv­ing its equation can be done using similar analytic methods to those in the solution to Problem 6. For more infor­mation, see [ 1 ] .

1 p,

L,

9. All flat origami crease patterns are 2-face colorable. The proof is simple: take your flat-folded origami model and lay it on a table. Color all regions of the paper yellow if they face up (i.e., away from the table) and all regions pink if they face down. Any two neigh­boring regions of the crease pattern will have a crease line in between them, and thus they will point in dif-

ferent directions when folded, insuring that they receive different colors. Thus this is a proper 2-face coloring of the crease pattern.

One can also prove this using only graph theory. First argue that all ver­tices in the interior of the paper of a flat model have even degree. Thus if we consider the crease pattern to be a graph, where the boundary of the square also contributes edges to the graph, the only odd-degree vertices would possibly be on the paper's boundary. Create a new vertex v in the "outside face" and draw edges from it to all the odd-degree vertices on the pa­per's boundary. Graphs always have an even number of vertices of odd degree, so the degree of v is even, and the new graph we've created has all vertices of even degree. It is an elementary graph theory fact that all such graphs are 2-face colorable (prove that its dual is bipartite), and removing the vertex v then gives a 2-face coloring of the orig­inal crease pattern.

lO.a. The problem is with the two mountain creases that surround the 45° angle at this vertex. The two angles neighboring the 45° angle are both goo. Thus, if the creases surrounding the 45° angle have the same mountain­valley (MV) parity, then the two goo an­gles will both be forced to cover up the 45° on the same side of the paper. If these creases are then pressed flat, the two large angles will be forced to in­tersect one another, and self-intersec­tions of the paper are not allowed (un­less one is folding in the fourth dimension, which we assume we are not!).

It turns out that mountains and val­leys can be assigned to these crease lines and be flat-foldable if (1) the creases surrounding the 45° angle are not the same and (2) the number of mountains and the number of valleys differ by 2. (This last result holds for general flat vertices and is known as

Maekawa's Theorem. See [4] for more information.)

lO.b. The answer is 16. There are sev­eral ways to enumerate the valid MV assignments. One way is to look at the inner "diamond" whose MV assignment will force the MV parity of the rest of the creases. (Why?) The inner diamond creases can have any combination of mountains or valleys, giving 24 = 16 possibilities.

REFERENCES

( 1 ] R.C. Alperin, A mathematical theory of

origami constructions and numbers, New

York Journal of Mathematics, Vol. 6 (2000),

1 1 9-1 33 (available online at http://nyjm.

albany.edu).

[2] K. Haga, Fold paper and enjoy math:

origamics, Origarni3: Proceedings of the

Third International Meeting of Origami Sci­

ence, Mathematics, and Education, T. Hul l

ed. , A. K. Peters, (2002) 307-328.

[3] K. Hatori , Origami versus straight edge and

compass, http://www.jade.dti.ne.jp/hatori!

l ibrary/conste.html

[4] T. Hull, The combinatorics of flat folds: a

survey, Origarn1'J: Proceedings of the Third

International Meeting of Origami Science,

Mathematics, and Education, T. Hull ed. ,

A . K . Peters, (2002) 29-38.

[5] H. Huzita, "Understanding Geometry

Through Origami Axioms: is it the most ad­

equate method for blind children?" in the

Proceedings of the First International Con­

ference on Origami in Education and Ther­

apy, J. Smith ed. , British Origami Society,

1 992, pp. 37-70.

[6] H. Huzita and B. Scimemi, ''The Algebra of

Paper-folding (Origami)." In the Proceed­

ings of the First International Meeting of

Origami Science and Technology, H . Huzita

ed. , 1 989, pp. 205-222.

[7] R. Lang, personal communication.

Department of Mathematics

Merrimack College

North Andover, MA 01 845

USA

e-mail: Thomas.Hull@merrimack.edu

VOLUME 26, NUMBER 4, 2004 63

lil§'h§l,'iJ Osmo Pekonen, Editor I

Feel like writing a review for The

Mathematical Intelligencer? You are

welcome to submit an unsolicited

review of a book of your choice; or, if

you would welcome being assigned

a book to review, please write us,

telling us your expertise and your

predilections.

Column Editor: Osmo Pekonen, Agora

Center, University of Jyvaskyla, 400 1 4 Finland

e-mail: pekonen@mit.jyu.fi

Gentzens Problem. Mathematische Logik im nationalsozia I istischen Deutschland. by Eckart Menzler-Trott

BASEL, BOSTON, BERLIN: BIRKHAUSER VERLAG, 2001 ,

xviii + 41 1 pp. €43. ISBN 3-7643-6574-9.

REVIEWED BY PETR HAJEK AND

DIRK VAN DALEN

Gerhard Gentzen (1909-1945) was undoubtedly one of the most im­

portant mathematical logicians of the twentieth century, the founder of mod­em mathematical proof theory. His work is of great importance, not only for pure mathematical logic but also for computer science, in particular for theorem-proving by computer. The book under review is a detailed biog­raphy of Gerhard Gentzen and at the same time a penetrating analysis of the situation of mathematical logic (and of mathematics) in Nazi Germany.

The topic of science and the Third Reich has been discussed repeatedly and extensively, but the position of logic in Nazi Germany is somewhat apart from the familiar subjects, as it represents a clash, or at worst a com­promise, between the realm of ultimate transparency and that of a dark and opaque political philosophy.

The author takes great pains to an­alyze the ambiguous relationship be­tween logic (and thus logicians) and the Party. One might wonder what a neutral, "value-free" discipline like logic had to fear from any regime what­soever. The answer is not all that sim­ple, mainly because a German science and education, as promoted by the Party, was a fairly intangible notion. The present book once more illustrates

64 THE MATHEMATICAL INTELLIGENCER © 2004 Spnnger Scrence+ Business Media, Inc.

how pragmatic and incoherent science policy was in the Third Reich (p. 123ff.). In a way this should not surprise us; the leading personalities of the Party had no high opinion of the world of scholars. In a press conference in 1938

Hitler frankly gave his opinion on in­tellectuals: "Unfortunately one needs them. Otherwise, one might-I don't know-wipe them out or something. But unfortunately one needs them" [ 1 ] .

Need one say more? Menzler-Trott shows how compli­

cated, incoherent, and often inconsis­tent the Nazi philosophy of science is. For Gentzen and logic the oft-men­tioned German or "Aryan" mathemat­ics, as opposed to Jewish mathematics, is the central issue. The author takes the reader through the confusing un­dergrowth of political arguments and machinations ( ch. 4).

For mathematics and logic the key figure in politics is Ludwig Bieberbach, who formulated a form of racial clas­sification of mathematics and mathe­maticians. But even Bierberbach had difficulty accepting the unpleasant consequences of his views when not just academic issues but real people were concerned. The formalistic side of mathematics, "symbol pushing," was classified as being Jewish, whereas the intuitive approach belonged to the Aryan domain. This would obviously speak against David Hilbert's formal­ism (which in its extreme and slightly caricatural form claimed that mathe­matics was a meaningless play with symbols); but when from the more philosophical side (Steck, Dingler) Hilbert was attacked for this, Bieber­bach chose the side of the great Got­tinger. It was the logician Scholz who (encouraged by Bieberbach) under­took the defence of logic and Hilbert's program.

This is the social-political stage where Gerhard Gentzen had to make his career. Helmut Hasse, Hilbert,

Scholz, Paul Bernays, Kurt Godel, and

many more experts play larger or

smaller roles in the story.

The really remarkable fact about

Gentzen's life and career remains his

lack of sensibility and judgment where

politics was concerned. Gentzen ac­

cepted the new regime and its conse­

quences as one of those facts of life you

cannot do much about. It is not some­

thing he wished for or approved of, but

something that simply happened, much

like the weather or fashion. He joined

Nazi organizations, because his friends

told him to, or because he considered it

one of those standard obligations­

"teachersjoin a teacher's union" (p. 47).

Political color did not seem to be a

point. The author offers us an impres­

sive amount of historical information on

Gentzen's life and work, and none of it

seems to hold any sinister details that

hint at political motives or social re­

sentment. The conclusion seems to be

that Gentzen's finer instincts were to­

tally of a scientific nature.

We will comment briefly on the

chapters of the biography, but in view

of the extraordinary richness of the

material, the reader should explore the

treasure chest that Menzler-Trott has

filled for us.

1 909-1 932. Youth and study up to

the program of his thesis

Gentzen's childhood was by no means

remarkable. His interest in mathematics

was awakened at the age of 13. He en­

tered the university at 19. His main

teachers were Helmuth Kneser (Greif­

swald), Hilbert (Gottingen), Constantin

Caratheodory, Oskar Perron, and Hein­

rich Tietze (Munich). Returning to Got­

tingen (1931), Gentzen fell in with Saun­

ders Mac Lane. He studied Hilbert's ideas

under the guidance of Paul Bernays.

1 933-1 938. Six years of National

socialism in peace time

This chapter treats an eventful and

confusing period in which Gentzen's

incredibly ingenious logical researches

took place. The introduction of the so­

called Gentzen-systems, which brought

system and elegance into the hacker's

toolshed of logic. In a subject that was

so roughly shaken up by Godel's mirac-

ulous incompleteness theorem, Gentzen

managed to restore order so that re­

fined proof-theoretic analysis became

possible. The cut-elimination, the con­

sistency proof, and a natural approach

to intuitionistic logic belong to this

period. The supervision of Gentzen

was somewhat vaguely determined;

Bernays looked after the young man,

and Hermann W eyl became the official

Ph.D. advisor. Weyl and Bernays left

Gottingen for obvious reasons, and

Gentzen was rather left to his own de­

vices. In spite of his allegiance to the

Nazi Party, he remained in correspon­

dence with Bernays, who had moved

to Zurich. It is a relief to read that he

did not close his letters to Bernays with

the obligatory "Heil Hitler."

Gentzen played no role in dogmatic

discussions, partly because he was until

1942 in the army, partly because he

was-strange as it may seem-apolitical.

1 939-1 942. From the beginning

of the war until his release

from the army

In 1939 Gentzen became an ordinary

assistant in Gottingen-Hasse had no

problem recognizing mathematical

quality when he saw it, even if it was

in logic-and there his career met a

temporary halt. Genius or not, Gentzen

was drafted at the outbreak of war

like everybody else. Menzler-Trott has

managed to trace the (scarce) material

about Gentzen's military service. He

was assigned to the signal corps of the

air defence. No front duty-but never­

theless the service heavily taxed his

nerves. The chapter provides most in­

structive reading, the author manages

to illustrate how chaotic and unpre­

dictable the political world was for

mathematics. The unworldly mathe­

matician Gentzen could perhaps try to

accommodate an enigmatic political

environment, but keeping sane in the

army was more than he could manage.

In 1942 he was discharged as wehrun­tauglich (not fit for the army), with a

nervous breakdown. After his return to

Gottingen, Gentzen remained for some

time in uncertainty, until he was called

to Prague to teach at the Charles Uni­

versity, where he started his lectures in

the spring of 1943.

1 940-1 945. The fight for a "German

mathematics"

Looking back at the war years, one is

tempted to ask, "Were there no urgent

matters to discuss, in view of the pos­

sible annihilation of the German na­

tion?" The first years were, however, a

golden time for the regime; military

successes convinced even the most

pessimistic critics, and it seemed quite

in order, or even a "historical neces­

sity" to complete the total Nazification

of all sections of civil order. Menzler­

Trott's narrative takes us along a sad

and partly incomprehensible route to

show us how at the margins of world

history private dogmatic wars of a po­

litical-philosophical nature were being

waged. In the shadow of the all-power­

ful Party, half-baked philosophies were

marketed in pursuit of a truly German Mathematics. The personnel side had

been taken care of by the Party; the ex­

odus, and worse, of German talent was

a fact, albeit one fervently denied by the

Party. The remaining mathematicians­

logicians were trying to salvage what

was left, leaving the battle of words to

the faithful party followers. The chapter

makes gruesome reading. Enough has

been said about the Hitler period, but it

will never cease to shock and warn us.

The author does not spare the sensitive

soul ( cf. p. 176).

1 942-1 944. Recovery and a teaching

position.

Being discharged from the army was

not as definitive as it looked; there was

always a chance that Gentzen could be

recalled to arms. Fortunately for him,

Gentzen was offered a teaching posi­

tion at the Prague German university

by H. Rohrbach. He was indeed occu­

pied with teaching, but it had nothing

to do with logic. The position also

brought a research contract for the SS,

statistical computations for the ballis­

tic station at Peenemiinde. In this role

Gentzen had the supervision of a group

of female students (p. 59). In Septem­

ber 1944 he refused to leave Prague

(from an exaggerated sense of duty),

thinking wrongly that nothing could

happen to him. As late as April 28, 1945,

his colleague F. Krammer tried to con­

vince him to leave. Gentzen refused,

VOLUME 26, NUMBER 4, 2004 65

"Dr. G. was always an idealist, un­worldly like most mathematicians." Was Gentzen so naive, or was he loyal to his oath to the Fuhrer, or perhaps afraid to be shot on the spot for de­sertion?

Arrest, imprisonment, and death;

what was left; the estate

On May 7, 1945, Gentzen was arrested at the Charles square in Prague and taken into protective custody by Czech militia. The biography ends with a de­scription of the last days of Gentzen as reported (mostly) by Krammer (p. 273-278). The German prisoners were subjected to cruelties, lack of food, un­hygienic circumstances, no medical as­sistance, treatment that, intended or not, could be taken as repayment for the horrid crimes on the German side. Under the circumstances, Gentzen, who could not even live under the com­parably mild military regime of the sig­nal corps, had no chance. Only the strong and (mentally) fit could with luck survive. On August 4 Gentzen died in prison of total exhaustion. Appar­ently it was not the rampant typhoid that killed him, but it was just his phys­ical and mental constitution that could not endure. Before Menzler-Trott's in­vestigations cleared up most of the cir­cumstances of Gentzen's death, there were accounts of his last days that claimed involvement of the Russian army in Gentzen's arrest and eventual death. Cf. Gentzens Problem p. 270. These accounts were based on incom­plete information, and at the time, the correspondence quoted by Menzler­Trott was not available. The Russian army in fact entered Prague on May 9,

after the Czech uprising (May 5).

Gentzen was arrested on May 7.

The author deserves credit for the enormous task of doing justice to the life and personality of one of our great­est logicians. It always is-and the more so in the case of a scholar who lived and died in uncertain times and circum­stances-surprising how much evi­dence a clever researcher can fmd. Menzler-Trott has created a fitting mon­ument to an introverted and naive sci­entist, who had so much to offer to sci-

66 THE MATHEMATICAL INTELLIGENCER

ence, but who utterly failed to grasp the problems and obligations of humanity.

The book contains a selection of photographs, a list of publications, and the texts of three lectures given by Gentzen. The book also contains a gen­erous sprinkling of quotations, re­minding us of the historical reality in which Gentzen tried to find his way. Furthermore Jan von Plato has added a brief exposition of Gentzen's logical contributions.

If we can find any shortcoming, it is that the aesthetic aspect of Gentzen's work does not get the attention it de­serves. Indeed Gentzen's systems of Natural Deduction and his Sequent Cal­culus are outstanding specimens of an almost architectural beauty. Matters of this sort are, however, difficult to con­vey; one does not start to enjoy Beethoven's music by reading reviews, one has to hear and to play it oneself. This applies to logic as well.

Finally a few remarks on Gentzen's personal choices in life, and on the wartime atrocities.

We believe that modem readers will agree that the treatment of Gentzen in the post-Nazi Prague of 1945 cannot be justified, but at these violent turning points in history law and rationality are usually victims of emotional reactions fed by memories of suffered injustices, and worse. "An eye for an eye" is the rallying cry in war and revolution. The author does not evade the atrocities in the name of Germany; he mentions the Osenberg-action and its role in slave labor for the rocket industry at Peen­emiinde, and the murderous reaction to student demonstrations in Prague in October 1939. He is fully justified in his disgust with the postwar "blind eye" practice-"The culprits not only de­mand considerations from their vic­tims; they blame them for the fact that they could do this to them" (p. 243). It

is questionable how much the present generation knows about the severe conditions of the German occupation of Czechoslovakia during World War II; some additional information on this point would have been helpful.

We can do no better than quote G. Kreisel who writes, in his review of Gentzen's collected works [2) about the

letter of F. Krammer from November 1946: " . . . the writer simply splutters with indignation at the atrocities in the camp, so much that he probably really had no thought left for the wartime atrocities by Germans in nearby Lidice and Theresienstadt (or, for that matter, their antecedents), which made some violent reaction inevitable."

Summing up: Gentzens Problem is a valuable contribution to the history of an enigmatic logician and his work, and to the singularity in the history of sci­ence called Nazi mathematics. Menzler­Trott has provided a wealth of facts and details, and he has gone a long way to­wards their interpretation. One does not have to agree with every single conclu­sion in order to appreciate his contri­bution to our awareness of the dangers of the role of politics in science. In par­ticular, we may be certain that Gentzen belongs forever to the giants of mathe­matical logic. The book teaches us, and coming generations, a lesson that we should keep in mind: the giants of sci­ence are also, as the Scripture says, "of like passions with you" -sometimes naive or confused.

Kreisel ends his above-mentioned review with the following words (quoted also by Menzler-Trott): "From all I heard I get the impression that Gentzen lived within his moral and emotional means and never harmed a fly." Kreisel's epitaph is fully borne out by this biography.

REFERENCES

[1 ] Gordon A. Craig. Germany 1866-1945. Ox­

ford Paperbacks. Oxford University Press,

1 981 . Oxford. p. 638.

[2] G. Kreisel. Review of M. E. Szabo: Collected

papers of Gerhardt Gentzen. Journal of Phi­

losophy 68 (1 971 ) 238-265, note 22.

Petr Hajek

Institute of Computer Science

Academy of Sciences of the Czech Republic

1 82 07 Prague

Czech Republic

e-mail: hajek@cs.cas.cz

Dirk van Dalen

Department of Philosophy

Utrecht University

Utrecht 3508, The Netherlands

e-mail: dirk.vandalen@phil .uu.nl

Stochastic Finance. An Introduction in Discrete Time by Hans Follmer and

Alexander Schied

BERLIN: WALTER DE GRUYTER, 2002 €54 00, 422 pp., ISBN 3-1 1 -01 71 1 9-8

REVIEWED BY TERRY J. LYONS

This book aims to be an introduc­tion to the probabilistic methods

used in finance. It targets undergradu­ate and graduate mathematicians in­terested in the area of mathematical finance rather than mathematical prac­titioners, although the authors hope that experts will find value in the book as well.

The book is substantial, with 415 pages, and has two parts. Within the first part, the first chapter focuses on the duality between martingale mea­sures (risk-neutral measures) and the absence of arbitrage. The remaining chapters of this part treat the value of a single risky transaction, dealing re­spectively with utility, portfolio, opti­misation, and risk measures.

The material of the first part of the book is genuinely fresh. Its novelty and attraction come from the mature and stimulating way that it tackles the eco­nomic problems of utility optimisation and equilibrium. The authors limit their attention to the case of one time inter­val, and, for example, give a version of Chris Rogers's result that the absence of arbitrage is equivalent to the exis­tence of an optimal consumption pat­tern; they make the connections be­tween exponential utility and relative entropy.

The text is well-paced, clear, and methodical, and it will be easy for a rel­atively advanced student with a rea­sonable amount of time to learn the material well. There is a kind of stu­dent who will find this material very at­tractive: well-trained pure mathemati­cians, happy with the basics of modem analysis (they should understand con­vexity, Fatou's theorem, and various inequalities), and interested in applica-

tions. These readers will appreciate the crisp and precise conversion of basic economic principles into mathematical statements with clear assumptions and very little pedantry.

Because the first part of the book is substantially confined to what happens over one time step, one is naturally led to consider incomplete models, and to try to find rational approaches to in­vestor behaviour in such an environ­ment.

The remaining 209 pages take up the discrete dynamic setting where there is an opportunity to hedge and to average risks over successive times. By re­stricting themselves to the discrete set­ting throughout the book, the authors are able to discuss, in far more detail than is usual for an introductory text, the issues involved when one tries to hedge in an incomplete market. One finds thoughtful and careful introduc­tions to many of the more sophisti­cated ideas currently under considera­tion in the mathematical analysis of incomplete markets.

It is a theoretical tour de force and will equip the reader well to under­stand much of the contemporary liter­ature, if he or she is willing to add a lit­tle bit of continuous time. I have no hesitation in recommending the book to students who have already done rig­orous courses in probability and analy­sis and would like to understand some of the mathematical modelling that de­velops out of considering incomplete financial models.

However, it is striking that the word "volatility" appears only once in the in­dex and plays almost no role in the book. The Black and Scholes model quite correctly appears as a limit of discrete models as the number of trad­ing intervals increases to infinity, but it receives only a short discussion. It is a general theme of the book that the real world is full of incomplete markets. There is no mention of computational aspects.

These omissions do not detract from the book, which is self-contained, interesting mathematically, and in­sightful-it definitely does aid one's understanding of the general picture. But, in my experience this book should

be complemented if a student is seri­ously interested in mathematical fi­nance. In particular, students should not be afraid of complete models! Or of models chosen to be low-dimen­sional in order to keep them computa­tionally tractable. There is a touch of idealism about the disdain in this book for models that are not quite correct.

One might think that, in finance, prices have to be determined with great precision, and yes they do. But still, incorrect models that approxi­mate well enough can be far more use­ful than perfect models of excessive complexity. The situation is more so­phisticated and more robust than an outsider might appreciate.

In many parts of their business, banks and traders are in effect retail­ers: they buy and sell many contracts, making small margins which, with big volumes, allow them to make profits; prices are determined by supply and demand. However, the contracts they buy and sell are parametrised (e.g., by strike). So in practise a trader faced with a contract at a new strike will need to price it using the available market information about contracts with different strikes. One robust ap­proach is to use a well-tested model that is broadly correct, calibrate it to market prices, and then use it to inter­polate to the new parameter value. The model is used to interpolate rather than to give absolute prices. In this way high precision can be obtained without per­fect models.

Hedging is also of vital importance on most trading desks, and it is built into the computer systems so that traders are continually aware of their exposure. However, as mentioned above, in many cases traders are playing a very similar role to a retailer. They buy and sell fre­quently with small margins; profits come from volume and are relatively riskless. But, like any retail trader, they could get into a lot of trouble if they were left with a lot of stock. The trader's stock is his residual position; he trades to keep it small (this is hedg­ing). The modem theory of replicating derivatives is absolutely vital to this part of the business. Certainly the mathematics is not perfect. But it does

VOLUME 26, NUMBER 4, 2004 67

not have to be! This is the residual po­

sition-it needs to be contained so as

to represent a small liability to the busi­

ness and should be protected from se­

rious downside behavior-but it does

not need to be perfectly hedged to zero.

This robustness in the way Finan­

cial Mathematics is used by the major

financial trading institutions is, I be­

lieve, a major factor in its overall suc­

cess and reliability, and explains its

long-lasting effectiveness.

To summarize, the book is great for

a mathematically competent beginner to

acquire knowledge of finance, equilib­

rium, incomplete markets, optimal port­

folio management, etc. It has perhaps

the best account of utility, and uses the

mathematical tools at the correct level

to get results quickly and effectively.

Complete models are far more im­

portant than one might appreciate

from this monograph; a student seri­

ously interested in finance will need

supplementary studies.

Mathematical Institute

University of Oxford

24-29 St Giles'

Oxford OX1 3LB

England

e-mail: tlyons@maths.ox.ac.u

Matematica e Cultura 2000 edited by Michele Emmer

MILAN SPRINGER-VERLAG IT ALIA, 2000, pp. vii1 + 342,

ISBN 88·4 70·01 02·1 .

Matematica e Cultura 2003 edited by Michele Emmer

MILAN SPRINGER-VERLAG IT ALIA, 2003, pp. viii + 279,

ISBN 88·470-021 0·9.

REVIEWED BY MARCO ABATE

In the last ten years, the perception of

mathematics by the general public

(or, at least, by the general cultured

public) has been changing. Movies

about mathematicians have won Acad­

emy Awards; articles about mathemat­

ical results have appeared on the front

page of major newspapers; and books

concerning mathematics, both novels

and essays, have sold as possibly never

68 THE MATHEMATICAL INTELLIGENCER

before. Mathematics is increasingly

recognized (again) as an integral part

of human culture, something to be re­

garded with slightly more curiosity and

slightly less suspicion than before, and

not just as an (un)necessary evil to for­

get as soon as possible after finishing

high school. Furthermore, this shift of

perception is driven by the idea that

mathematics might be interesting, not

only because of its applications, but

per se.

Changes like this do not happen by

themselves; they need a lot of effort

and preparation, usually going unno­

ticed for a long time. Mathematicians

started trying to explain what they

were doing; people from other sciences

and from humanities started to listen;

and somebody in between started or­

ganizing venues where mathematicians

and non-mathematicians could meet

and exchange ideas about mathemat­

ics, culture, and everything else.

In Italy, the divide between sciences

and humanities is traditionally deep;

actually, culture has often been con­

sidered to be synonymous with hu­

manities. Even in the "scientific" high

schools, a sizable number of lectures

are devoted to humanities (including

the compulsory study of Latin). So

when in 1997 Michele Emmer ( origi­

nally with P. Odifreddi and E. Casteln­

uovo, but later by himself) organized a

series of conferences on "Mathematics

and Culture," held annually in Venice,

for Italy it was a complete novelty. To

some critics, it was an oxymoron and

doomed to failure. Luckily, the critics

were wrong, and Emmer's creature,

eight years later, is alive and kicking.

The structure of these congresses is

easy to explain. The idea is to put in

the same room for two days a number

of mathematicians (university profes­

sors, high school professors, and, last

but not least, students) ef\ioying a broad

notion of what can be mathematically

(or non-mathematically) interesting,

and a number of non-mathematicians

(artists, journalists, scientists, what­

ever) who somehow have found that

mathematics can be relevant to what

they do-or who are willing to be sur­

prised by the fact the mathematics can

be relevant to what they do. The goal

is also simple: after having put these

people in the same room, see what

happens.

To start things happening, members

of both groups deliver a fair number of

talks (on average, twenty-five per year)

on topics ranging from architecture

and math to zoology and math. There

are also mathematically related art

shows, plays, movies, and concerts.

Usually people make connections:

both intellectual connections, among

apparently unrelated subjects, and per­

sonal connections, among intellectu­

ally curious people who may work in

apparently unrelated subjects, but who

find that they have more in common

than they had thought. And this kindles

conversations, and confrontations, and

the diffusion of ideas, mathematical

and non-mathematical. Afterwards, as

any good virus should, such ideas

spread elsewhere, back home or at

work, infecting unsuspecting friends

and co-workers with unexpected con­

nections between mathematics and,

well, just about anything else.

The published volumes of proceed­

ings of the "Mathematics and Culture"

conferences are an integral part of Em­

mer's project. Seven volumes have ap­

peared, and they give a good idea of the

range of topics at the Venice confer­

ences. I decided to concentrate this

review on the volumes that are (or

soon will be) available in English; the

others, published by Springer-Verlag

Italia, are for the moment available in

Italian only.

The first volume contains the pro­

ceedings of the 1999 conference, held

while the Kosovo bombings were be­

ginning (the 2003 conference was held

just after the Iraq bombing started-and

"Mathematics and war" is one of the re­

curring themes in the conferences). The

twenty-eight papers are subdivided into

eleven sections, each containing two or

three essays: Mathematicians; Mathe­

matics and History; Mathematics and

Economics; Mathematics, Arts, Aesthet­

ics; Mathematics and Movies; Math Cen­

ters; Mathematics and Literature; Math­

ematics and Technology, an homage to

Venice; Mathematics and Music; Mathe­

matics and Medicine. The authors range

from Claudio Procesi and Enrico Giusti

to Harold W. Kuhn and Peter Green­

away; nineteen are Italians, and fifteen

are from the rest of the world (yes, a few

papers had more than one author, and

Emmer wrote two of them, which ex­

plains why nineteen plus fifteen yields

twenty-eight).

Let me describe some of the more in­

teresting (to me) papers. The section on

"Mathematics and History" contains

three essays. The first one, by Giorgio Is­

rael, describes Italian mathematics dur­

ing the Fascist years, and in particular

the reactions of Italian mathematicians

to racial laws. The second, by Jochen

Biiining, describes what happened to

the Berlin mathematical school after the

advent of Nazism. The third, by Silvana

Tagliagambe, describes the develop­

ment of philosophical and mathematical

studies in Russia from Peter the Great

to Stalin. In all three cases, the descrip­

tion of the use of "aseptic" mathemati­

cal arguments to support extremist po­

litical positions is fascinating and

horrifying-and instructive. Keeping in

mind the present-day rhetorical uses of

mathematical terminology and "theo­

rems" to support economic politics, as

hinted at in the papers by Marco Li Calzi

and Achille Basile, much can be learned

from their accounts of the (mostly fruit­

ful) relationship between mathematics

and economics.

The short essay by Lucio Russo on

Mathematics and Literature provides

still other connections between math­

ematics and rhetoric; and another in­

teresting walk in the rhetorical and

metaphorical use of mathematics is de­

scribed in the paper by Piergiorgio

Odifreddi, discussing numerology, the­ology, and mathematics.

The metaphorical manifestations

and uses of mathematics in the arts is

a theme common to the papers by

Achille Perilli, an Italian painter with a

body of work which he describes as

"the theory of the geometric irrational,"

where he plays with and undermines

the classical use of perspective in

paintings; Gustavo Mosquera R., direc­

tor of the film Mcebius, where Argen­

tinian society just after the end of the

dictatorship is mixed with topology;

and Peter Greenaway, the famous

artist and movie director. Each of these

artists describes his own work, and it

is interesting to compare what attracts

them to mathematical themes. Perilli

wants to destroy the unrealistic repre­

sentation of reality provided by the

classical rules of perspective; Mos­

quera R. is fascinated by the metaphor­

ical uses of the M<:Ebius band and the

topological terminology; and Green­

away is so attracted by the intrinsic

beauty of the rigid and yet rich struc­

tures that can be derived by numerical

sequences that he builds most of his

movies around them, using sequences

both as structural devices and for their

metaphorical power. And I am sure

that Greenaway loves the description

of the relationship between numbers,

colors, and music in ancient Asia given

in the essay by Tran Quang Hai.

Other papers describe less metaphor­

ical applications of mathematics. I

would like to mention at least the papers

by Laura Tedeschini Lalli on the math­

ematics of Indonesian musical instru­

ments; by Enrico Casadio Tarabusi on

the Radon transform and computer­

ized tomography; and by Camillo De­

jak and Roberto Pastres on a mathe­

matical study of high tides in Venice.

The second volume contains the

proceedings of the 2002 conference.

The twenty-four papers are subdivided

in the following eight sections, con­

taining from one to five essays each:

Mathematicians; Mathematics and Mu­

sic; Mathematics and Arts; Mathemat­

ics and Movies; Mathematics and

Venice; toward Beijing 2002; Mathe­

matics and Theatre; Mathematics and

Comics. The authors range from Gio­

vanni Gallavotti and Aljosa Volcic to

Harold W. Kuhn (again: it is not un­common for some speakers to come

back after a few years to talk about

something else) and Sergio Escobar;

thirteen are Italians and nine are from

the rest of the world.

The largest section is devoted to

Mathematics and Music. The volume is

sold with a CD containing three short

musical pieces for guitar by the com­

poser Claudio Ambrosini, collected un­

der the unifying title "Three studies on

perspective" (which reminds me not

only of the essay by Achille Perilli de­

scribed above, but of the joke on art

critics which says that writing about

paintings is like dancing about archi­

tecture). Ambrosini himself describes

his work in an essay, illustrating the

structural ideas that guided him in

composing these pieces. Particularly

interesting are the parallels he finds be­

tween his work and M. C. Escher's

paintings, parallels of a structural­

and hence mathematical-nature. The

other papers in this section deal with

mathematical models of musical sounds

(Giovanni De Poli and Monica Dorfler),

philosophical problems behind the

notion of "listening" (Laura Tedes­

chini Lalli), and fractal music (Stefano

Busiello).

Fractals also appear in the works of

Escher, Paul Klee, and Marcel Duchamp,

according to Roberto Giunti (but I must

admit that in the case of Duchamp I

found Giunti's arguments not that con­

vincing); and are somewhat implied by

the labyrinthine structure of Venice it­

self, as described by Michele Emmer.

On the opposite side of geometrical

complexity, the excursus of Manuel

Corrada on the possible definitions of

straight lines sheds an unusual light on

Fred Sandback's sculptures (unfortu­

nately not shown in the book).

Another large section of the book

deals with mathematics and China.

Two very interesting essays, by Jean­

Claude Martzloff and Anjing Qu, deal

with the history of mathematics and as­

tronomy in ancient China; a third one,

by Francesco D'Arelli, discusses the

false perceptions of Chinese astron­

omy in sixteenth-century Europe; and

the last one describes Michele Em­

mer's trip to a mathematical congress

held in Lhasa, Tibet.

The section on mathematics and

comics contains a description (by Stew­

art Dickson) of the computer graphics

techniques used in the Disney movie

Dinosaurs; and a list (by Luca Boschi)

of numerological and arithmetical cu­

riosities in Disney comics. Further­

more, the participants to the confer­

ence are now the happy owners of a

copy of a comic book created by Luca

Boschi expressly for this occasion;

knowing the world of comics collec­

tors, this comic will soon become valu­

able (alas, it is described but not en­

closed in the proceedings).

Of course, not all the presentations

at these conferences are of the same

quality. This year (2004), I attended a

talk on topology and architecture in

VOLUME 26. NUMBER 4, 2004 69

which the speaker managed to convey the impression that she (and the ar­chitects whose work she was describ­ing) had no idea of the actual meaning of the word "topology." In another talk, on fractals in Pollock's paintings, I had the distinct feeling that the speaker just found a clever way to sell a word (frac­tal) to unsuspecting art critics, and that he was well aware that he was faking it. But, again, there have also been very exciting talks (I remember in particular one describing techniques to teach arithmetic and geometry to primary school children by dancing and singing-a new and unexpected twist on the joke about architecture above); and the overall mixture worked very well. So I am looking forward to next year's con­ference; and meanwhile, I cannot but recommend reading the available pro­ceedings volumes.

Dipartimento di Matematica

Universita di Pisa

Via Buonarroti 2

561 27 Pisa

Italy

e-mail: abate@dm.unipi.it

When Least Is Best by Paul J. Nahin

PRINCETON UNIVERSI1Y PRESS, 2004, 370 pp. US

$29.95, ISBN 0-691 -07078-4

REVIEWED BY CLARK KIMBERLING

This attractive book is, of course, about much more than minimiza­

tion. One might describe it as a book in popular-mathematics tone about op­timization, written by an engineering professor whose work is well known in his field (and also in science fiction). As such, the work is of great value to many, but most especially to the thou­sands of people who teach and learn calculus. (The same can be said for an­other of the author's books that may have crossed your desk: An Imaginary Tale: the Story ojv=l, Princeton Uni­versity Press, 1998.)

When Least Is Best has seven chap­ters: (1) Minimums, Maximums, Deriv­atives, and Computers; (2) The First Extremal Problems; (3) Medieval Max-

70 THE MATHEMATICAL INTELLIGENCER

imization and Some Modem Twists; ( 4) The Forgotten War of Descartes and Fermat; (5) Calculus Steps Forward, Center Stage; (6) Beyond Calculus; and (7) The Modem Age Begins. These head­ings cover a total of fifty sections. For example, the main calculus chapter con­sists of sections (5. 1) The Derivative: Controversy and Triumph; (5.2) Paint­ings Again, and Kepler's Wine Barrel; (5.3) The Mailable Package Paradox; (5.4) Projectile Motion in a Gravitational Field; (5.5) The Perfect Basketball Shot; (5.6) Halley's Gunnery Problem; (5. 7) De L'Hospital and His Pulley Problem, and a New Minimum Principle; and (5.8) De­rivatives and the Rainbow. These sec­tion headings represent notable features of the book: timely and interesting choices of topics, conversational tone, practical perspectives, and the develop­ment of concepts historically as well as mathematically.

Several standard calculus problems are presented and then usefully ex­tended beyond what you will find in a calculus text. For example, "Projec­tile Motion in a Gravitational Field" starts with the usual differential equa­tions dxldt = v0 cos((}) and dyldt =

Vo sin((}) - gt and establishes that the path of motion is a parabola. This and the familiar questions regarding opti­mal height and range are posed in terms of athletic events, first shot put and javelin throw, then golf. Finally, seven pages are devoted to The Perfect Basketball Shot, leading into Halley's Gunnery Problem. (This is Edmund Halley, as in Halley's Comet; the surname rhymes with "Sally," not "Cayley.")

In Derivatives and the Rainbow, the author analyzes primary, secondary, and tertiary rainbows. This section, like all others, includes computer-gen­erated plots created by the author us­ing MATLAB, and other figures by Christopher L. Brest. The book's over­all up-to-dateness is typified by a cor­rection of Marilyn Savant's account of the tertiary rainbow in Parade Maga­zine, August 4, 2002.

The "precalculus chapters" consider many enticing optimization problems. One of these, in Chapter 2, is to deter­mine the smallest circle that spans a set of n given points in a plane. "A prac-

tical form of this problem would be, for example, determining where to locate a fire station within a community to minimize the maximum distance from the fire station to any of the surround­ing homes." The author cites the work of Franco P. Preparata and Michael Ian Shamos on the minimum spanning cir­cle, and he points out that they also dis­cuss the dual problem: "what is the largest circle inside the convex hull of the given n points (think of the points as vertical posts, and a rubber band snapped all around them [as shown]) that contains none of the points? That would tell us, for example, where to place an objectionable service facility for the town, e.g., a centrally located waste-treatment plant that nobody wants to live near!"

Chapter 6, Beyond Calculus, is prob­ably as compelling an introduction to the calculus of variations as you can fmd anywhere. In particular, the isoperimetric problem, already woven into the first two chapters, resurfaces in section 6.8, titled "The Isoperimetric Problem, Solved (at last!)."

Many calculus books discuss the catenary as the curve of an ideal hang­ing chain. Many calculus books also fail to link that chain to the "other" out­standing property of a catenary, the one that pertains to the St. Louis Gate­way Arch. The author takes this up el­egantly on page 250:

[The hanging chain] is, at every point, in tension only, i.e., there clearly is no point where a hanging chain is in compression. This was apparently first pointed out in 1675 by Newton's contemporary (and sometimes rival) Robert Hooke (1635-1703) . . . . Further, Hooke went on to observe, if the hanging catenary was "frozen in place" (e.g., glue the links of the flexible chain together) and then inverted, the re­sulting arch would be in compres­sion only, and at no point would there be tension. Thus, an inverted catenary is the best (strongest) curve for a stone arch.

Chapter 7, The Modem Age Begins, opens with a favorite problem of tri­angle geometry, originating in Fermat's

1629 Method for Determining Maxima and Minima and Tangents to Curved Lines, namely, how to locate, relative

to an arbitrary triangle ABC, the point

P that minimizes the sum PA + PB + PC. This problem obviously lends itself

to a wide variety of generalizations

known as facility location problems (e.g., where to locate the town fire de­

partment). Other types of problems are

where to dig the optimal trench and

least-cost paths through directed

graphs. The Traveling Salesman Prob­

lem precedes final sections of the book

on linear programming and dynamic

programming.

Some readers will wonder about the

frequent appearance of "an extrema"

( cf. "an apples"), and, on page 112, "ex­

tremas" ( cf. "geeses"). Perhaps ex­tremum is following datum ("piece of

data") out of English. In contrast, min­imum remains intact-and yet minima is minimized, as evidenced by Mini­mums in the heading of Chapter 1.

There is one type of least problem

that is barely represented, as when the

author presents Euclid's wonderful

demonstration that there is no largest

prime. The method of demonstration,

sometimes nowadays called first fail­ure, is an application of the well-or­

dering principle-that every nonempty

set of positive integers contains a least

element-which is equivalent to the

principle of mathematical induction.

That is to say, "first failure," as used in

number theory, combinatorics, and

probability theory, is closely associ­

ated with one of the axioms of mathe­

matics. Related notions are least known and greatest known, exempli­

fied by greatest known prime. Some

readers may wish that the author had

applied his witty insights to a selection

of lesser known and well-known least knowns and greatest knowns. On the

other hand, the book is well focused on

extrema of the sort encountered in cal­

culus and engineering. To summarize:

this book is highly recommended.

Department of Mathematics

University of Evansville

1 800 Lincoln Avenue

Evansville, IN 47722

USA

e-mail: ck6@evansville.edu

An Invitation to Algebraic Geometry by Karen E. Smith, Lauri

Kahanpaa, Pekka Kekalainen, and

William Treves

BERLIN, HEIDELBERG, NEW YORK, SPRINGER-VERLAG.

UNIVERSITEXT 1 st ed. 2000. Carr. 2nd printing, 2004, XVI,

1 61 pp., ISBN: 0-387-98980-3 US $49.95

REVIEWED BY MARC CHARDIN

Algebraic geometry is a very active

branch of mathematics that is

linked to many other fields-in partic­

ular to arithmetic, one of the most fas­

cinating areas in mathematics, but also

for instance to complex analysis or to

theoretical physics.

At its origins, algebraic geometry is

the study of the zero set defined by a

The u nfortu nate

side of the evi ­

dent power of

their elaborate

formal ism is

the false idea

that noth ing is

accessib le

without it . collection of polynomials. The reason

for its power is probably the interplay

between geometrical intuition and the

algebraic formalism. Geometry is a

guideline for defining the proper con­

cepts and often suggests possible

means for proof; the algebraic formal­

ism makes these ideas applicable to

cases where the geometric picture is

not obvious, and in many cases it clar­

ifies the initial ideas by extracting the

essence of the argument.

The search for a good algebraic

framework was a major factor in alge­

braic geometry in the last century. This

(r)evolution is due to several of the

most influential mathematicians of the

time, among them David Hilbert, Oscar

Zariski, Andre Weil, Jean-Pierre Serre,

and Alexandre Grothendieck.

The unfortunate side of the evident

power of their elaborate formalism is

the false idea that nothing is accessible

without it. The resolution of singulari­

ties by Heisuke Hironaka, one of the

major achievements of algebraic

geometry in the last century, is a good

example of a result that was proved

with very little formalism.

Let us also recall that many impor­

tant results on the classification of

curves and surfaces were obtained by

the Italian school in the nineteenth

century, at a time when Hilbert's Null­stellensatz was not yet established: can

one imagine doing algebraic geometry

without Hilbert's theorems today?

This book, based on notes of lec­

tures by Karen Smith at the University

of Jyviiskylii, Finland, demonstrate

that it is indeed possible to present im­

portant achievements in algebraic

geometry without much formalism.

Doing so necessitates modesty and

some hard choices; in particular, un­

necessary restrictions and hypotheses

often need to be made. Also some con­

cepts cannot be defined with complete

rigor. It is frustrating, especially for an

algebraist, but otherwise there is no

way of providing a comprehensive in­

troduction to algebraic geometry, to­

gether with examples and open prob­

lems, in only a few lectures.

Before getting into up-to-date re­

search advances, it is necessary to pro­

vide the minimal background knowl­

edge in algebra, a little of the formalism,

and a good collection of examples, so

that the reader understands what are

the challenges, what is the meaning of

the theorems and conjectures, and

where the motivations come from. The

first chapters of the book are dedicated

to this delicate task

Chapter 1 presents a short account

of affine algebraic varieties, their mor­

phisms, the Zariski topology, and the

notion of dimension. It is illustrated by

several examples and counter-exam­

ples. Chapter 2 is devoted to a more

substantial presentation of the alge­

braic notions attached to affine vari­

eties, and the dictionary between alge-

VOLUME 26, NUMBER 4, 2004 71

bra and geometry. It contains two fun­

damental theorems of Hilbert (the fi­

nite basis theorem and the NuUstellen­satz), and it presents the notions of

spectrum of a ring and pullback of a

morphism of affine varieties. Many ex­

amples are given, and also hints of the

history and an example of a recent re­

sult: the effective Nullstellensatz by

Dale Brownawell and Janos Kolhir.

The next two chapters are dedi­

cated to projective and quasi-projec­

tive varieties and morphisms. These

are key concepts in algebraic geome­

try; the first corresponds to the notion

of compact varieties and the second to

open subspaces of these. It turns out

that a natural setting for many results

is projective schemes over the com­

plex numbers. These chapters also

contain the definition of regular mor­

phisms and some additional material

on Zariski topology.

Chapter 5, on "classical construc­

tions," gives a collection of classical

examples that supplement, or detail,

the ones given in the previous chap­

ters. These might be thought too stan­

dard, but on the other hand they are in­

deed fundamental. Perhaps other

examples, some toric varieties for in­

stance, would have been a good com­

plement. The section on Hilbert func­

tions has an interesting discussion on

Hilbert schemes. A proof of a few easy

facts on the Hilbert function of finite

sets of points would have been a good

illustration of the connection between

algebra and geometry (but choices

about what to include are hard to

make). Smoothness and the tangent

space are the subject of Chapter 6. The

notions are very clearly illustrated; the

second part is on families, the Bertini

theorem, and the Gauss map.

The last two chapters present im­

portant advances and challenges in al­

gebraic geometry. The first subject is

72 THE MATHEMATICAL INTELLIGENCER

birational geometry. Two varieties are

birationally equivalent if there exists

an isomorphism between a non-empty

open subset of the first and another of

the second, in other words if they are

essentially the same almost every­

where. For example, a parametrized

curve is birationally equivalent to a

line. Birational geometry tries to un­

derstand the families of varieties bira­

tional to a given one: what invariants

do they have in common? is it possible

to distinguish a nice representative of

the family? how to parametrize the el­

ements in the family?

A fundamental result due to Heisuke

Hironaka shows that any of these fam­

ilies contains a smooth variety (a man-

You wi l l do wel l

to accept th is

n ice invitat ion

to algebraic

geometry. ifold). In fact the result is much more

precise and shows a sequence of geo­

metric operations that leads to a

smooth variety from a singular one; in

particular, these operations never alter

the locus where the initial variety is

smooth. This general result was pre­

ceded by the work of his advisor Oscar

Zariski, who proved the result in di­

mensions two and three.

Karen Smith presents this theorem

and interesting remarks on its proof.

She then shows by examples what a

blow-up is and why it is of interest for

desingularization. The geometric sig­

nificance of the blow-up is clearly ex­

plained. The different sides of the clas­

sification problem are described at the

end of the chapter. It gives an elemen-

tacy introduction to the theory of mod­

uli spaces of curves (continued in

Chapter 8) and the minimal-model pro­

gram.

The last chapter is dedicated to vec­

tor bundles, line bundles, and embed­

dings of projective varieties, especially

curves. The motivation here is to un­

derstand how a variety can be embed­

ded in a projective space, in particular

the existence of an embedding that de­

pends only on the isomorphism class

of the variety. This is of importance for

many reasons, one of which is that it

gives representatives of the isomor­

phism classes and opens the way to

construct a space that parametrizes all

curves sharing some common invari­

ants, up to isomorphism. The funda­

mental vector bundles associated to a

smooth variety, the connection be­

tween vector bundles, and the study of

embeddings are presented in the first

sections. The last section is dedicated

to (pluri-)canonical embeddings of

smooth projective curves and the mod­

uli space of curves of a given genus.

You will do well to accept this nice

invitation to algebraic geometry. You

will need very little baggage in algebra,

but some notions of complex analysis,

geometry, and topology are useful.

From this book, most graduate students

in mathematics will be able to get a fla­

vor of what algebraic geometry is all

about. Also, working mathematicians

who are not familiar with the field can

certainly benefit from this series of lec­

tures. It may leave them with a desire

to go on to discover many other facets

of algebraic geometry and the funda­

mental concepts of its formalism.

lnstitut Mathematique de Jussieu

Universite Pierre et Marie Curie

75252 Paris Cedex 05

France

e-mail: chard in@math.jussieu.fr

Kjfi .. i.MQ.iQ.I§i Robin Wilson I

The Philamath' s Alphabet-F

Fermat: Pierre de Fermat (1601?-

1665) spent most of his life in

Toulouse following a legal career. He

considered mathematics a hobby, pub­

lished little, and communicated with

other mathematicians by letter. His

two main areas of interest were ana­

lytic geometry, analysing lines, planes

and conics algebraically, and number

theory, proving the 'little Fermat theo­

rem' that for each positive integer a and prime p, aP - a is divisible by p. Fermat's 'last theorem': In his copy of

Diophantus's Arithmetica, Fermat

claimed to have 'a truly marvellous

demonstration which this margin is too

:c•• y• .. z• " .. pu iU 3oluti01l p::_ur tiu rntitrs n �-•

Fermat

Fermat's "last theorem"

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

narrow to contain' of the statement that

for any integer n (> 2) there do not ex­

ist non-zero numbers x, y and z for

which xn + yn = z11• Fermat proved this

for n = 4, using his 'method of infinite

descent,' but it is highly unlikely that he

had a general argument. Fermat's last

theorem was eventually proved in 1995,

after a long struggle, by Andrew Wiles.

Fibonacci: Leonardo of Pisa (c.

1 170-1240), known as Fibonacci, is re­

membered mainly for his Liber abaci [book of calculation] which he used to

popularise the Hindu-Arabic numerals,

largely unknown in Europe, and pre­

sent a wide range of mathematical puz­

zles. The best known of these is on the

breeding of rabbits and leads to the Fi­

bonacci sequence 1, 1, 2, 3, 5, 8, 13, . . .

in which each successive term is the

sum of the preceding two.

Folium of Descartes: With his solu­

tion of a problem of Pappus, Rene

Descartes introduced algebraic meth-

Fibonacci

Folium of Descartes

76 THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+Business Media, Inc.

ods into the solution of geometrical

problems. He also discussed various

curves, such as the 'folium of Descartes'

with equation .x3 + y3 = 3axy.

Foucault's pendulum: In 1851 the

French physicist Jean Foucault pre­

sented his famous pendulum experi­

ment, designed to demonstrate the ro­

tation of the earth. A 28-kg ball was

suspended from the roof of the Pan­

theon in Paris and allowed to swing.

After a short time the swinging pendu­

lum's path shifted, showing that the

earth must be rotating.

Fractal pattern: When a recurrence

of the form Zn+ 1 = Zn 2 + c is applied to

each point z0 in the complex plane, the

boundary curve between those points

that remain fmite and those that 'go to

infinity' is a fractal pattern, called a 'Ju­

lia set' after the French mathematician

Gaston Julia. This stamp shows a de­

tail of the fractal pattern that arises

when c = 0.2860 + 0.01 15i.

Foucault's pendulum

Fractal pattern