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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: [email protected]
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: [email protected]
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: [email protected]
·�·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: [email protected]
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: [email protected]
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
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Ylrnils ,., tAlf• ftUr . N Rntilll iUc rtttptis
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: [email protected]
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: [email protected]
�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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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 [email protected]
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: [email protected]
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