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.
cal jargon by new, striking metaphors
Review by Harold Edwards of
3 Books
is a mark of good writing, isn't it?
Mac beth Murder Mystery is a hilariously
Amazingly enough,
misplaced analysis of Shakespeare's
imagine the reader du Sautoy had in
Macbeth written as though the play
mind has never heard of modular arith
were a whodunnit. Now you have pub
metic, so it seems laudable for du
James Thurber's short story The
not everyone else
uses the term "modular arithmetic": I
lished a Thurberesque review of three
Sautoy to try to come up with a fresher,
popular mathematics books about the
more insightful expression, and I think
Riemann Hypothesis (RH)-review by
his idea of "clock calculator" isn't bad
Harold M. Edwards, Mathematical In
at all. Personally, I liked du Sautoy's
telligencer, vol. 26, no. 1, 2004-written
metaphorical image of a landscape in
as though they were academic tomes.
which the zeroes of the zeta function
To
Edwards
are the points at sea level. I don't see
doesn't believe it possible to explain
any reason for complaint. As for "for
make
matters
worse,
RH to non-mathematicians: he bases
ever calling it by its new name" ... well,
this opinion on his failure to teach lib
if du Sautoy had reverted to the old
eral arts students that
V2 is irrational,
name, Edwards would have criticized
blithely ignoring the obvious alterna
him for inconsistency. Or if he hadn't,
tive hypothesis about his own teaching
I would. Edwards's struggle with du
ability. Edwards understands the dif
Sautoy's reference to "ley lines," which
ference between books aimed at pro
he eventually decides "is apparently a
fessional mathematicians and books
term used in British surveying," sug
aimed at a general readership, but de
gests that du Sautoy credits his read
cides that "it is only as a mathemati
ers with a broader general knowledge
cian that I can evaluate the books."
than is actually possessed by Edwards.
Why? Can't a mathematician be a nor
Edwards seems determined to tell
mal human being too, or at least imag
us that mathematicians are obsessed
ine what one might be like? It is as
with problems like RH entirely for their
though the Thurber character, having
own sake, without any interest at all in
tried and failed to write a tragedy, has
their history or context. He says that to
decided that tragedies are impossible
believe that the fascination of RH
to write, and is therefore reviewing one
arises from the information it would give mathematicians about prime num
as if it were a detective story. When reviewing
The Music of the
bers "is
a
profound misunderstanding
Primes by Marcus du Sautoy, which I
of our tribal culture, like believing
have read, enjoyed, and thought rather
mountaineers want to climb Mount
inspiring, Edwards grumbles, "as a
Everest in order to get somewhere."
sometime historian of mathematics,"
Well, who knows what the true motives
about the lack of citations of historical
for climbing Mount Everest are? I do
sources. But, Professor Edwards, it's
know, from the time I lived in Malaysia,
not a history of mathematics, it's a
that the first Malaysian to climb Ever
book for the general reader and posi
est was given a handsome financial re
tively shouldn't be cluttered up with
ward by the company he worked for: I rather imagine that, like the rest of us,
footnotes. Edwards
complains
about
du
he had mixed motives.
Sautoy's "habit of introducing a private
Edwards tells us that the books un
phrase to describe something and for
der review "grossly overstate the con
ever calling it by its new name rather
nection of RH to prime numbers": in
than the one used by everyone else."
support of this he points out that Rie
But why on earth shouldn't he? The re
mann himself switched his attention
placement of tired cliches and techni-
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
the Riemann hypothesis would create
more convenient to study a function in
havoc in the distribution of prime num
quite distinct from academic writing,
one form rather than another says ab
bers. This fact alone singles out the
and such books deserve to be reviewed
solutely nothing about its connection
Riemann hypothesis as the main open
on their own terms. In addition, Ed
question of prime number theory." Of
wards paints an unrealistically depress
centric if not insane to write a popular
course people who work on RH be
ing picture of mathematicians as people
(or, I should think, any other) book
come wrapped up in it-otherwise
even more inward-looking and obses
about RH without emphasizing its im
they'd have no chance of success-but
sive about their little problems than any
with prime numbers. It would be ec
books for a general readership is an art
portance in prime number theory. In
the reason that RH stands out among
group of technical experts is bound to
deed, Edwards's own book Riemann's
all the other interesting problems that
be: mathematicians aren't quite as un
Zeta Function (which, by the way, we
obsess mathematicians is precisely its
aware of the context of their work as he
should have been told about right from
history and its position in mathematics
seems to want us to think Next time you want a reviewer for
the outset of his reviews of books on
as a whole, particularly its connection
much the same subject) starts with a
with prime numbers.
reference to Riemann's paper On the Number ofPrimes Less Than a Given Magnitude and finishes with a proof of
I started to write this letter because
gest you ask a Shakespearean scholar,
I felt irritated at what seemed to me to
or a thriller writer, or perhaps even an
be a sneering attitude toward a book I
author of popular mathematics books.
the prime number theorem. In his de
had enjoyed reading.But, having started
an academic mathematical tome, I sug
scription of the Riemann hypothesis
to think more carefully about Edwards's
for the Millennium prizes, Bombieri
reviews, I fmd it just plain silly that they
1 87 Sheen Lane
(whom I suppose Edwards might ad
are written from the viewpoint of some
London SW1 4 SLE
Eric Grunwald
mit as a member of the "tribe" of math
one for whom the books were not in
UK
ematicians) writes that "The failure of
tended. The writing of mathematical
e-mail:
[email protected] Harold Edwards replies:
magical ley line" to the critical line
As I believe the review makes clear, I
Du Sautoy's failure to give any indication
of the sources of his stories is a problem
Re s = 112, du Sautoy credits his read
tried to decide whether they would
ers not only with a broader general
convey inspiration, enjoyment, and a
because so many of those stories are so
knowledge than I possess but also with
reasonably accurate picture of the sub
questionable. I state my reasons for
ject to such readers. I don't deny
doubt many others. Whether through
Ameri can Heritage Dictionary of the Eng lish Language possesses.
footnotes or otherwise, he should justify
To say that "it is only as a mathe
doubting some of his statements, and I
a broader knowledge than the
don't know why he would deny mine.
his more surprising assertions. Writing
matician that I can evaluate the books"
for a naive audience does not give him a
is not to say that I
license to invent history.
in any way except as books written for
New York, NY 1 00 1 2 USA
readers who are not mathematicians.
e-mail:
[email protected] When he gives the name "Riemann's
6
THE MATHEMATICAL INTELLIGENCER
am
evaluating them
Mr.
Grunwald's right to an opinion, and
Courant Institute of Mathematical Sciences New York University
Four Poems Philip Holmes
Celestial Mechanics At dawn, when my appr nti bowl and pitch r, h
brought m
·aid the city was
tir
with talk of one Kop mik, who would hav that th
un is a ftx d tar.
My teaching, my word the
ftx d; all
un i
the
lse ·
c
thirty y
it
ru
in tracks about her which will not leav Th
bodi
, God' , m asur
the p
ru1d ·way of p riod,
in each lap
mark th ir future
in each pull again t
anoU1 r. Th
we fear d
od
and it was mine, its pivot than my gl,
ur r
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 churll 1 tf•<S Or\� J 1'301
only nonperiodically and a set of four solids which fill space only nonperiod
Figure 2. Ammann's two polygons-notched rhombs-which tile only non-periodically, and
ically."
his sketch of part of a tiling with these tiles. [Ammann to Gardner, undated, spring 1976.]
VOLUME 26, NUMBER 4 , 2004
11
ter to several experts, with Ammann's
mann's claims never found a mistake,
permission. "It seems that his discovery
though the jury's still out on a few of
was quite independent of mine!"
them. But the letters were odd. How had
Penrose explained that he'd found
his
remarkable tiles?
Why didn't he publish
his
results in
tiles in 1974; the intriguing kite and dart
mathematics journals, like
everyone
that Gardner had in mind, but also a
else? He had a droll sense of humor,
pair of rhombs, one thick and one thin.
they all could see that. But Ammann's
Penrose understood that though the
"friend" Dr. Bitwhacker must have been
tiles look very different, any tiling built
a private joke for Gardner, chronicler of
with one pair can be converted into a
Dr. Matrix's mathematical adventures.6
0
tiling by the tiles of the other.
Figure 3. A kite and a dart.
Ammann found
not one, but two pairs of non-periodic
Start, for example, with a tiling by kites and darts. Bisect the tiles into tri angles. Then recombine the triangles
in situ
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 de
into rhombs.
cidability of the tiling problem." "The tiling problem?" "It's an old, old problem. Imagine you're a tile maker, back in deep an tiquity. A rich patron hands you a fancy template and asks you to use it for thousands and thousands of tiles to
Figure 4. The deuce with two possible ex
cover her palace floor. Before you fire
tensions. For simplicity, the notches are not shown.
Figure 5. Left: a portion of a kite and dart tiling, with the tiles bisected into triangles.
Jane picks up some more tiles and fits four of them together; I recognize the
configuration
known
as
Right:
the triangles are joined to form
rhombs.
then hesitates.
together you'll be in big trouble." "What's the problem? Why not make a dozen or so and test them?" asks
the
"deuce."4 She starts to add another,
up your kiln, you'd better be sure the shape really is a tile. If copies don't fit
Jane. Ammann, who'd seen neither set,
"Even if your dozen do fit together,
Penrose's
how do you know you can add still
rhombs and rhomb tilings, but by a very
more? In fact there are cases where
the
non-periodic
can be entirely surrounded by three
zles," I remind her. "In Penrose tilings
tiles-I'll come back to those later-he
rings of copies of itself, but not four."7
you sometimes have choices."
found five new sets in the plane. He an
"So the tiling problem is: given a
"And different choices lead to dif
nounced his discoveries in a flurry of
shape or set of shapes, is there a gen
ferent tilings," Richard calls out from
letters to Gardner, with hand-drawn fig
eral procedure, one that works in every
the sofa. I'd thought he'd fallen asleep.
ures and hand-waved proofs.5
case, that determines whether you can
had
"Strange. A kite fits in this spot, but so does a dart." "Penrose tilings aren't jigsaw puz
"Penrose
tilings
aren't
indeed
rediscovered
different route. And soon, in addition to three-dimensional
you can't; Ammann found a tile that
individuals,
Gardner sent the letters on to the ex
they're species. Species with infinitely
perts, who found Ammann's construc
"You mean, of course, the infinite
many members."
tions ingenious and insightful. They
plane, not just a palace floor," Carl re
grasped his ideas immediately, from
minds us.
"What kind of infmity?" asks Jane. "Countable, or uncountable?"
"Of course," I yawn.
his sketchy drawings.
"Un! Yet all the tilings look just
cover the plane with it or them?"
Penrose's tilings are hierarchical.
"I'd try to arrange a few tiles into
alike-as far as the eye can see. Any fi
That is, they repeat not in rows, but in
some sort of quadrilateral that I can re
nite patch of tiles in one Penrose tiling
scale: the small tiles combine into
peat in a periodic array," Jane contin
turns up in all of them. Infinitely often."
larger ones, which combine into larger
ues.
"Borges! Escher! Where are you
ones, which combine into larger ones
"That's the whole point!" I wake up.
when we need you!" Carl gasps in
. . . ad infinitum.
Ammann's tilings are
"Can you always do that? Hao Wang
mock horror.
hierarchical too. And he had devised
proved that a decision procedure ex
0
0
0
0
some intriguing variations. For exam
ists if and only if any set of shapes that
"I am most intrigued-indeed, some
ple, the large tiles in most hierarchical
tiles the plane in any manner can also
what startled-to see that someone has
tilings are larger copies of the smaller
rediscovered one of my pairs of non-pe
ones, but he found an example where
riodic tiles so quickly!" Penrose wrote
they're not.
to Gardner, who'd sent Ammann's let-
12
THE MATHEMATICAL INTELLIGENCER
The
experts
be arranged in a periodic tiling. "8
"You mean, a decision procedure
exists if and only if non-periodic tiles who
dissected
Am-
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] i
-1 �
I j
" ,.. , ... rr. r.,.r"fo'oo•r. I ,_.., •�f"lna is o
a
theory that Richard Nixon and Patty
to store food that really should have
of t t. o Fast
tne Uni�er ae
mor," David explained. "His stories about the 'penguin conspiracy' and his
der for his sanity. And he used his desk
ok
w anini of 1Yol u t. 1 .,n
TTeaoury
tance. "He had a weird sense of hu
put it mildly, enough to make one won
Abo�ou. DlnOS&\lt"S
1'n• 1>1 >o&aur
\Lt f'Z-;bo Lo k
how to conciliate.
et.
tow t.o .iot•• I L
Sl ap o o n : "
and impatient. And she hadn't known
fini. '-Y
Ctomet. r)· o f lo'J r
t:
never gotten along; August was strict
;/ . ,
.ic •H• • •
t � e ma t i cal Recr•at o n a
lol& l n
Tno
E& ct
ry A g e o r
fu n w 1 th IIA t h OII".a t l o o
A.nd.re • s :
Col
r. t.
i\ln and.
Wo4 a r .. i- U Z z. ! e a
Kra1t0r: fho
No
--
oa. a r n ' l ge t: r
t.be
Matno-s l c
n;
Xra1 t e
3u r , • y of
on .>tn•• o
oera
Fr i e nd :
r
c 1.. n• �
Re• i e " i n � t.t•
D r e a :s .l .: T ;
ca:no
son after that. Bob and his father had
Bob's apartment resembled his desk
ii o a L e i n
at the office. In 1976 the health in
Figure 11. Esther Ammann's list of Bobby's books at age 12 (first of two pages).
spector condemned it-though it was n't
that bad-and Bob was evicted. He
stored all his furniture, except his TVs, cess! Bob died when his career was
at Reed College and at MIT, but Bob is
and moved into a motel on Rte. 4, mid
only beginning.
the
way, it turned out conveniently, be
He'd have done so
much more if he'd been given the time. " Esther Ammann died i n January, 2003.
0
only person
I have personally
known who had, without question, a genius-level intellect." They lost touch when John left for
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."
tween Honeywell and the post office. "Honeywell laid him off in one of their quarterly staff reductions, regular as
college, but reconnected when he re
clockwork for almost ten years!" David
turned to MIT. "Bob began weekly vis
told me. "Bob kept coming to work any
him back on
its, always on Wednesday nights. We
way. They eventually put
had supper, talked a little bit, watched
the payroll. When he was laid off a sec
the original 'Charlie's Angels' show on
ond time a few years later, the security
an old black-and-white TV that Esther
guards were given his picture and told
had given us, and talked some more."
not to let
Bob himself owned three TVs; he watched them all at once. Those were the Honeywell days.
him back in the building. "
Bob phoned John often over the years. The invitation to Germany terri fied him. They talked about it end
"He didn't take math courses, ex
"We shared a cubicle for a few years,
lessly.
cept a few in analysis, " I reply. "Bob's
in the early 1970s," a co-worker, David
courage to go. Esther didn't know he'd
kind of math was out of fashion in the
Wallace recalled. "Bob was very shy. It
gone until he came back.
1960s. It wasn't taught anywhere. My
was two years in the same office be
The motel was Bob's home for the
course on tiling theory may have been
fore I found out how he pronounced
rest of his life. He ate at the fast food
the first."
his last name! Everyone in the depart
joint next door. One day the cleaning
ment pronounced it like the capital city
woman found him dead in his room. A
"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
Somehow
Bob
found
the
of Jordan: 'Ah-mahn.' He prounounced
heart attack, the coroner said. He was
it 'am-man. ' But he never corrected the
46 years old.
mispronunciation."
Steve Tague, another Honeywell co
While we specialists played with
worker and the executor of Bob's es
"In any case, Bob wasn't trainable,
Bob's tilings, studying them, applying
tate, salvaged loose sheets of doodles
was he?" Richard says. "Pass the cake."
them, extending them in new and sur
from the swirl of junk mail, old phone
prising directions, Bob's life kept hit
books and TV guides, uncashed pay
story."
0
0
0
0
"He was a kind and gentle soul," John
ting dead ends. He backtracked and
checks, and faded magazines Bob had
Thomas, a childhood and close family
tried again, over and over, but still
stuffed in the back of his car. Steve
friend, told me. "I have encountered
nothing fit.
found smaller items too, which he
many bright people during my studies
John and his wife moved back to
placed in a white cardboard box. He
VOLUME 26, NUMBER 4, 2004
1g
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
Figure 18. Request form for leave from the United States Post Office.
baum generously gave me access to their large files of Ammann correspon
stored the box and the doodles in the
I almost missed the poem tucked in
dence. I am also grateful to Michael
attic of his home in northern Massa
side a folded sheet of green construc
Baake, Ludwig Danzer, Oliver Sacks,
chusetts, near the New Hampshire bor
tion paper. "I hope you'll write more
Doris Schattschneider, Joshua Socolar,
der. Ten years later, when I drove there
like
and Einar Thorsteinn for advice and as
to talk with him, he showed them to
teacher had written on the back.
this
one!"
Bobby's
fifth-grade
I looked through the doodles. They
seemed just that. In the white box I found the shards of Bob's
shattered
childhood:
two
sistance. Michael Baake, Doug Bauer, Eleanor
me.
I'm going to Mars Among the stars The trip is, of all things, On gossamer wings.
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
cheap plastic puzzles; a little mechani cal toy; a half-dozen birthday cards, all
of this manuscript and made thought
from Mom and Dad; school report
ful suggestions, most of which I have
cards, from first grade through eighth;
adopted. I am also grateful for the en
a tiny plastic case with a baby tooth
couragement and constructive criti
and a dime; a tom towel stamped with
cisms from fellow participants in two
faded elephants and a single word,
workshops in creative writing in math
BOBBY. And some letters and clip
ematics at the Banff International Re
pings and drawings, among them a
search Station at the Banff Centre,
front-page news article, dated 1949.27
Canada.
A little boy who is probably one of the smartest three-year-olds in the coun try. . . . With a special love for geog r-aphy, he can quickly name the capi tal 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 par ents the other day by telling them that 'jour and two is six and three and three is six and five and one is six. "
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,
In the picture little Bobby, looking
king, queen, jack, deuce, and ace-for the
earnestly at the photographer, sits with
Figure 19. Undated (1949) clipping from The
seven vertex configurations allowed by
his globe.
Herald (Richland, Washington).
Penrose's rules seem permanent.
20
THE MATHEMATICAL INTELLIGENCER
5. Grunbaum and Shephard proved many of
cubes, for any positive integer n, one gets
Ammann's assertions about his tiles; he
non-periodic tilings of non-Penrose types.
March 1 8-22, 1 99 1 , ZIF (Center for Inter
joined them as co-author of Ammann, R . ,
In general, the construction gives tilings
disciplinary Research), Bielefeld University,
Grunbaum, B . , and Shephard, G. C . , "Ape
with many different tiles whose matching
riodic Tilings," Discrete and Computational
rules, if they exist, remain a mystery, but a
23. Joshua Socolar, "Weak Matching Rules for
Geometry, 1 992, vol. 8, no. 1 , 1 -25.
22. Conference, "Geometry of Quasicrystals,"
Bielefeld, Germany.
few very interesting tilings have been found
Quasicrystals," Communications in Math
6. Martin Gardner's chronicles of "Dr. Matrix"
in this way. See, e . g . , J .E.S. Socolar,
ematical Physics, vol 1 29, 1 990, 599-6 1 9 .
include The Incredible Dr. Matrix; The
"Simple octagonal and dodecagonal qua
I t should b e noted that Michael Longuet
Magic Numbers of Dr. Matrix; and Trap
sicrystals," Physical Review B, vol 39, no.
Higgins's "Nested Triacontahedral Shells,
doors, Ciphers, Penrose Tiles, and the Re
1 5, May 1 5, 1 989, 1 05 1 9-51 .
or how to grow a quasicrystal, " The Math
1 3. See M. Senechal and J. Taylor, "Qua
ematical lntelligencer, vol. 25, no. 2, Spring
7. Heesch's problem asks whether, for each
sicrystals: the view from Les Houches, "
2003, bears no relation to Ammann's con
positive integer k, there exists a tile that can
The Mathematical lntelligencer, vol. 1 2, no.
be surrounded by copies of itself in k rings,
2, 1 990, 54-64.
turn of Or. Matrix.
struction. 24. See, e.g., P. Kramer and R. Neri, "On Pe
but not k + 1 . Such a tile has Heesch num
1 4. Gardner's files show that Benoit Mandel
riodic and Non-periodic Space Fillings of Em
ber k. Robert Ammann was the first to find
brot met Ammann once in 1 980. I had not
Obtained by Projection," Acta Crystallo
met Mandelbrot then.
graphica (1 984), A40, 580-587; L. Danzer,
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 96 1 , 1 -42. 9. Branko Grunbaum and Geoffrey Shep hard, Tilings and Patterns, W. H. Freeman, New York, 1 987. tiling that enriches the theory of tiles," Mathematical Games, Scientific American, January, 1 977, 1 1 0-1 2 1 . 1 1 . See Tilings and Patterns, Chapter 1 0.6, "Ammann bars, musical sequences and 1 2. See N. G. de Bruij n, "Algebraic theory of non-periodic
Penrose tilings and quasicrystals," Discrete
are used with Grunbaum's kind permis
Mathematics, vol. 76, 1 989, 1 -7; and L.
sion.
Danzer, "Full equivalence between Soco
1 6. Ammann visited and corresponded with
lar's tilings and the (A,B,C,K)-tilings leading
Paul Steinhardt and his students, Dov
to a rather natural decoration," International
Levine and Joshua Socolar.
Journal of Modern Physics B, vol. 7, nos. 6
the Cretaceos-Tertiary Boundary Event," unpublished.
tilings
of the
plane," Proceedings of the Koninglike Ned
& 7, 1 993, 1 379-1 386.
25. Special Session on Tilings, 868th meeting of the American Mathematical Society,
1 8. For the journal Structural Topology. The
Philadelphia, Pennsylvania, October 1 2-
editor, Henry Crapo, also wrote to Am
1 3, 1 991 . The American Mathematical
mann about this but also received no
Society does not pay honoraria or travel
reply.
expenses.
1 9. Roger Penrose, "Remarks on Tiling," in R .
forced tiles," pp. 571 -580.
"Three dimensional analogues of the planar
except my letter after meeting Ammann ,
1 7. Robert Ammann, "Another Explanation of
1 0. Martin Gardner, "Extraordinary nonperiodic
Penrose's
1 5. All letters to and from Branko Grunbaum,
Moody (ed.), The Mathematics of Long
Range Aperiodic Order, Kluwer, 1 995, p.
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
468.
erlandse Akadernie van Wetenschappen
20. loan James, "Autism in Mathematics," The
ington), 1 949 (undated clipping). The Am
Series A, Vol. 84 (lndagationes Mathernat
Mathematical lntelligencer, vol. 25, no. 4,
mann family had moved from Massachu
icae, Vol. 43), 1 981 , 38-66. De Bruijn
Fall 2003, 62-65.
showed that the construction is really very general. Using n-grids and n-dimensional
2 1 . Norbert Wiener, 1 25-1 42.
setts to Washington while August Ammann,
Ex-Prodigy,
pp.
3-7,
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 B e n t
Colin Adam s , Editor
knock I sighed, lifting my feet off the
Mangum, P.l.
desk "If you won't go away, you might as well come in."
Colin Adams
T
The proof i s i n the pudding.
The door swung open, and I just about swallowed my bottle of Orang
he name's Mangum. Dirk Mangum,
ina whole. Standing in the doorway
P.l. Yeah, that's right. I am a Prin
was none other than Walter P. Parsnip,
cipal Investigator. On a National Sci
chair of the Berkeley Math Depart
ence Foundation grant. Didn't start out
ment. He was dressed suggestively, in
that way, though. You don't just decide
a white buttondown, top button un
Opening a copy of The Mathematical
to be a P.I. No, you have to earn the
done to expose his clavicle, and slacks
Intelligencer you may ask yourself
right. For me, it wasn't anything I ex
so worn you could almost seen through
pected. Just a fortuitous set of circum
them at the knee. His shirt clung to his
stances, although it didn't seem fortu
chest, the outline of his bulging stom
itous at the time. Quite the contrary.
ach obvious for all to see.
uneasily, "lthat is this anyway-a mathematical journal, or what?" Or you may ask, "�there am !?" Or even
I was working as a snotnosed post
I found it hard to believe he was
"ltho am !?" This sense of disorienta
doc out of a sleazy hole-in-the-wall of
here before me. I used to drool over
fice in LA. Actually, UCLA to be spe
this guy's articles when I was an un
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.
of a
dergraduate. He had a career built like
three-year appointment, and I didn't
a brick shipyard. And talk about legs.
cific.
It
was
my
third
year
have anything to show for the first two
He published his first article in 1932,
years except a stuffed wastebasket, a
and he was still going strong. Half the
pile of empty Orangina bottles, and a
functions in Wang Doodle theory were
whole lot of self-doubt. My story begins on one of those
named after Parsnip, and the other half were named after his dog.
days you get in LA. The sun was shin
I gave him a long look up and down
ing, a slight breeze was ruffling the
and then said as smoothly as I could,
palm trees, and it was an even 70 de
"Well come on in here and take a load
grees. Actually, I just described every
off. "
It's enough to make you
He took his time coming in, giving
want to scream. Just give me a cloud,
my eyeballs a chance to run over his
or some fog. Or god forbid, a hailstorm.
body at will. I took full advantage of
But no, there is the sun, day in, day out,
the opportunity. He slid into the over
beating a drum beat on your brain,
stuffed leather chair that sat in front of
banging out its sunny sun dance until
my desk and stretched his legs out be
day in
LA.
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 sec
fore 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.
ond bottle of Orangina for the day. I had
"I'm . . . , " he started to say.
been wrestling with the proof of a lemma
"Oh," I said, cutting him off, "I know
all afternoon, but it had me in a double
who you are. What I don't know is what
overhook headlock and the chances I
someone as hot as you wants with
would end up anywhere but on the mat
someone as cold as me."
were slim indeed. The constant drone of
"I'm in trouble," he said.
Column editor's address: Colin Adams,
the air-conditioner sounded like a UPS
"Who isn't?" I retorted.
Department of Mathematics, Bronfman
truck tackling the Continental Divide.
Science Center, Williams College,
There was a knock at my door.
''I'm in deep trouble," he said. He fixed me with a look that would have
Williamstown, MA 01 267 USA
"I'm not in," I yelled.
made
e-mail:
[email protected] There was a pause; then a second
hadn't been chewing on it at the time.
22
THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+Business Media, Inc.
me swallow my tongue if I
He leaned forward conspiratorially,
"It is exactly what is needed to solve
"No, I can't wait," he said. "Please
giving me a nice view down the inside
my dilemma. What will it take to get
fax it to me now. I'll come down Mon
of his well-used pocket protector. "I've
you to help me, Dirk?"
day."
got a theorem. It's a big one." "I bet it is," I said, trying to sound
He placed his hand on mine. I felt the warmth of his gnarled knuckles. I smiled my most captivating smile.
casual. But I knew that if Parsnip thought it was big, it would make Riemann Roch
"Who in his right mind would tum down a chance to publish with you?"
ing 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
He smiled back.
look like Zorn's Lemma.
I should have smelled a double-deal
splashed across the page. "PARSNIP
"It implies Canooby." Over the next eight months I de
AND KAZDAN SOLVE CANOOBY." This
the biggest open problem in all of
voted myself to the problem. I should
time I did swallow my tongue, but luck
Pinched Rumanian Monofield Theory.
have been writing papers based on my
ily I quickly coughed it up. There was a
You solve Canooby, and they deliver
thesis, getting published to ensure a
huge picture of the two of them shaking
the presidency of the American Math
follow-up job. Instead, I thought of
hands with the governor. I had been
Society to your doorstep.
nothing but the lemma. I worked on it
played for a fool.
The Canooby Conjecture, perhaps
"Doesn't sound like a problem to me," I said. "It's joint work with Kazdan." I lifted an eyebrow. Kazdan was the
in the shower.I worked on it in the tub.
Figuring out what had happened
I even worked on it at the office. It be
took me less time than it takes a bam
came an obsession.
fly to find sustenance. Parsnip and Kaz
I started to dream about it. There
dan were working on Canooby the en
current darling of the math community.
was one dream in which Parsnip and I
tire time, but they got stuck. They
Twenty-six years old, Belgian, and bril
were dancing the rhumba.
needed help, but they weren't about to
liant. So hot that if he were a waffle
Vichy danced over, laughed in that
let a pissant postdoc like me get my
iron, you could pour batter and get
falsetto laugh of his, and said, "Oh, no,
name on a theorem as big as this. So
fully cooked waffles in an instant. Bel
you are not doing math here." I woke
they devised their ruse: Parsnip comes
gian waffles.
up in a cold sweat.
Shwase
to see me, acting the jilted collaborator, wouldn't
desperate for my aid. Sucker that I am,
legs, his pant cuff riding up enough to
budge. Parsnip notwithstanding, I was
I fall head over heels.They figure I can't
expose some hairy leg just above his
ready to give up. It seemed hopeless.
resist his charms, and they're right.
sheer black socks. He caught me tak
But then, one day, as I was stepping off
ing a gander.
the bus, it hit me. I had an epiphany.
I watched as Parsnip crossed his
"So, what's the problem with work ing with Kazdan?" I asked.
And
still,
the
lemma
Once they have the fax, I'm history. Nobody will believe a postdoc without
Suddenly realizing what I had been
a single publication to his name, and
missing, I couldn't believe my stupid
with a job disappearing faster than the
ity. All this time I had been working on
woolly mammoth.In a year, I would be
for Vichy."
semiupperpseudohypermultitudinal
pumping Slurpees at the local Seven
Shwase Vichy was the youngest faculty
fluxions. When I should have been
Eleven.
member ever to get tenure a Chicago;
thinking about multihyperpseudoup
he was still packing a lunch box. This
persemitudinal fluxions. I had been
office and cried into my Orangina. Al
must be hard on Parsnip.
looking at it exactly backward. With
though diluted, the salt in the tears
"Kazdan isn't working with me any more. He
dumped
me
For the first three days, I sat in my
"How can I help?" I asked, looking
this realization, I knew that I had not
added zest. For the following three
deep into his milky brown eyes. They
only solved the problem, but I had cre
days I tried to figure out how to fran
were eyes you could spend a lot of time
ated a whole new field of mathematics.
chise salted Orangina.
looking into. Why you would want to
The other passengers waiting to get
On the seventh day, I received a
do that, I don't know, but people pick
off the bus began to push, but I didn't
grant proposal for review from the Na
strange hobbies.
care. I knew I was right.
tional Science Foundation. And won ders of wonders, it was from Kazdan
"It is a lemma," he said. "Just one
I rushed to my office, overwhelmed
lemma I need. With the lemma, I will
with excitement.I would have Parsnip's
and Parsnip. They wanted five million
have my proof."
undying gratitude. A tenured position
dollars to study multihyperpseudoup
at Berkeley might be in my future.
persemitudinal fluxions. Now, why the
"What makes you think I can help you with your lemma?" I asked, lean
Parsnip picked up his phone on the
National Science Foundation sent the
ing back in my chair, trying to appear
first ring."Hello, Parsnip?I solved your
proposal to me for review, I'll never
disinterested.
problem."
"They tell me you are the best when it comes to the theory of semiupper pseudohypermultitudinal fluxions." "Well, that was the title of my Ph.D.
"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
know. They certainly didn't know I in vented the field. And it's unlikely they realized there was a connection be tween
multihyperpseudouppersemitu
dinal fluxions and semiupperpseudohy
thesis. But you're the first person who
show it to you. Then you can tell me
permultitudinal
ever pronounced it correctly."
how great I am."
whatever reason, the osprey of oppor-
fluxions.
But
VOLUME 26, NUMBER 4, 2004
for
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 as sumes 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 ad
ditional 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.
H I N KE M. OSINGA AND BERND KRAUSKOPF
Croch eti ng the Lorenz M an ifo d ou have probably seen a picture of the famous butterfly-shaped Lorenz attractoron 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 man ifold for short. This surface organizes the dynamics in the
Hasselt, Belgium, in July 2003 [ 7 ] . The model is quite large,
three-dimensional phase space of the Lorenz system. It is
for transportation.
about 0.9 m in diameter, and has to be flattened and folded
invariant under the flow (meaning that trajectories cannot
In this article we explain the mathematics behind the
cross it) and essentially determines how trajectories visit
crocheted Lorenz manifold and provide complete instruc
the two wings of the Lorenz attractor.
tions that allow you to crochet your own. The images
We have been working for quite a while on the devel
shown here are of a second model that was crocheted in
opment of algorithms to compute global manifolds in vec
the Summer of 2003. We took photos at different stages,
tor fields, and we have computed the Lorenz manifold up
and it was finally mounted with great care and then pho
to considerable size. Its geometry is intriguing, and we ex
tographed
plored different ways of visualizing it on the computer [6,9].
mounted permanently, while we use the first model for
However, a real model of this surface was still lacking.
touring.
During the Christmas break 2002/2003 Rinke was relax
professionally.
This
second
model
stays
We would be thrilled to hear from anybody who pro
ing by crocheting hexagonal lace motifs when Bernd sug
duces another crocheted model of the Lorenz manifold.
gested, "Why don't you crochet something useful?"
an incentive we offer a bottle of champagne to the person
The algorithm we developed "grows" a manifold in steps. We start from a small disc in the stable eigenspace of the
As
who produces model number three. So do get in touch when you are done with the needle work!
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
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
y
z
= u(y - x), =
px - y -
xz,
= xy - {3z.
© 2004 Spnnger Sc1ence+ Bus1ness Media, Inc., VOLUME 2 6 , NUMBER 4, 2004
(1)
25
We consider here only the classic choice of parameters, namely
u=
10,
p
= 28, and
the symmetry
(x,y,z) that is, rotation by
7T
f3 =
�
l The Lorenz system has
2
( - x, - y,z),
(2)
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 condi tion, 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, result ing 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 accu racy, 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 actu
ally a chaotic attractor. For an account of the mathemat ics involved see the
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.
lntelligencer article by Viana [ 13]. which are also saddles. They sit in the centres of the "wings" of the Lorenz attractor and are each other's image
Stable and Unstable Manifolds
The origin is always an equilibrium of
(1). The eigenvalues
Figure
of the linearization at the origin are
- f3 and
-
u_ +_1 _ 2
+ -
under the symmetry (2).
.!. v--:?'+ -4 • ' ..--1 •.
�
I :> C I
.:cf:''
· ·
·
''
editing
and
'- .'
.. i �·-
from Dutch into Latin.
In 1633 Horten
sius moved from Leiden to Amsterdam, hoping to get a position at the city's re cently founded
Athenaeum iUustre.
Several of these "illustrious schools" had been founded throughout the Dutch Republic in the 1 630s in order to pre pare students for the universities (De venter, Amsterdam, and Utrecht), or even to compete with them. Of these only the Amsterdam Athenaeum
iUus tre rose to a more prominent position, as the founding fathers used the im
to hire away professors from Leiden.
In
May 1634 Hortensius began to teach in
�-� •
.J. -�
in
mense wealth of the city of Amsterdam
X X X I V,
� .
' -
closely
translating some of Lansbergen's works
I
T .
,•
Lansbergen (1561-1632), with whom he
•
:.:.,.
...
Amsterdam, delivering his inaugural lecture on the Dignity and utility of the mathematical sciences. If we are to be lieve his personal testimony, his daily lecture courses were a success and at
,.: . "'
tracted 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] . .
..··:
�- .:-� .. . -
. '
· .·
:
-
.
'
'
: . � ...
_: ' >�. .
-- 1 .
tj�(·
In the years that followed, Horten sius's scientific reputation grew con
•'
'
;.:;:- ;.:_;; .
·"�...
:· ·
_: , -
�
tinuously. He was known as a con
· · ··:
vinced Copernican and an admirer of Galileo, corresponding with such dis tinguished
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 Re Figure
1 . Title-page of Hortensius's Speech on the dignity and utility of the mathematical sci
ences, Amsterdam
1634.
public. At the height of his fame, Hor tensius received a professorship in Leiden, but he died shortly after mov ing there in August 1639. Although he
ments in astronomy, including Galileo's
prestigious University of Leiden, where
is not among the great luminaries of
the well-known mathematician Wille
1 7th century science-Descartes even
Hortensius ( 1605-1639) was born as
brord Snel taught from 1613 to his early
considered him "very ignorant"2-his
Maarten van den Hove in Delft in 1605.
death in 1626. It was probably under
appointment at Leiden shows that he
He was a student in the Latin school at
Snel's guidance that Hortensius turned
was highly esteemed in the Dutch re
Rotterdam, where he probably came un
to the mathematical sciences and made
der the influence of the natural philoso
astronomical observations in Leiden.
pher Isaac Beeckmann. In 1625 he went
After Snel's death Hortensius came in
to Leiden, but it was only in March 1628
contact with the reformed minister,
that he registered as a student at the
physician, astronomer, and ardent prop-
public of letters. In his Speech on the dignity and utility of the mathemat ical sciences as well as in his other writings, in particular the Canto on the origin and progress of astronomy, his
astronomical observations.
2Descartes to Mersenne, March 31 , 1 638: "il est tres ignorant" [Berkel 1 997, 2 1 9).
VOLUME 26, NUMBER 4, 2004
41
In Viri
P H I L I P P I 0
Chriffimi L A N S B E R G I I
A S T R O N O M I C U M T A B U L A S Q_U E M O T U U M C OE L E S T I U M
p
u
s
dudum •b omnibus defuler•t,u
C
A
R M E progrcll'us A s ortus T 1. o N o M 1 AE & Qyo tempora oftendi tur,
N ad nofua ufque
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Sttllmull ptflttu ·mils , Ktdi{9111 •tt�tlll �prilllii he· Etllpfos ; t•l'ft ,..,kl ,.,.,•l'/n tcflbll$ iochoata , s.un. A1itJu i,p«i ,,...,, {tinli 2) there do not ex
problems. He also discussed various
for any integer
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
curves, such as the 'folium of Descartes'
.x3 + y3 = 3axy. Foucault's pendulum: In 1851 the
with equation
French physicist Jean Foucault pre
had a general argument. Fermat's last
sented his famous pendulum experi
theorem was eventually proved in 1995,
ment, designed to demonstrate the ro
ermat: Pierre de Fermat ( 1 60 1 ?-
after a long struggle, by Andrew Wiles.
tation of the earth. A 28-kg ball was
1665) spent most of his life in
Fibonacci:
(c.
suspended from the roof of the Pan
Toulouse following a legal career. He
1 1 70-1240), known as Fibonacci, is re
theon in Paris and allowed to swing.
considered mathematics a hobby, pub
membered mainly for his
Liber abaci
After a short time the swinging pendu
lished little, and communicated with
[book of calculation] which he used to
lum's path shifted, showing that the
other mathematicians by letter. His
popularise the Hindu-Arabic numerals,
earth must be rotating.
two main areas of interest were ana
largely unknown in Europe, and pre
lytic geometry, analysing lines, planes
sent a wide range of mathematical puz
Fractal pattern: When a recurrence of the form Zn + 1 = Zn2 + c is applied to
Leonardo
of
Pisa
and conics algebraically, and number
zles. The best known of these is on the
each point z0 in the complex plane, the
theory, proving the 'little Fermat theo
breeding of rabbits and leads to the Fi
boundary curve between those points
a aP - a is divisible by p. Fermat's 'last theorem': In his copy of Diophantus's Arithmetica, Fermat
Folium of Descartes: With his solu
Gaston Julia. This stamp shows a de
claimed to have 'a truly marvellous
tion of a problem of Pappus, Rene
tail of the fractal pattern that arises
demonstration which this margin is too
Descartes introduced algebraic meth-
when
rem' that for each positive integer
and prime p,
bonacci sequence 1, 1, 2, 3, 5, 8, 13, . . .
that remain fmite and those that 'go to
in which each successive term is the
infinity' is a fractal pattern, called a 'Ju
sum of the preceding two.
lia set' after the French mathematician
c = 0.2860 + 0.01 15i.
:c•• y• .. z• " .. pu iU 3oluti01l p::_ur tiu rntitrs n�-•
Fermat
Foucault's pendulum
Fibonacci
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] 76
Folium of Descartes
THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+Business Media, Inc.
Fractal pattern