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 chur<·h. I would be rack d for tlti my
my know! dge, and must rack
lf for holding it, iliough it b
truth and all els clark. lei god
whom I
t acli d, teady me now
rumour; I t me not drift utman1ed,
against th
who nanwd your path ·; my w rd be drean1S: the v ry \\·orl The day'
duti
cannot
mo\·e on U1 m.
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
was all, to was h r
day , th
movement
t it right. Then II av n'. hand
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 a
with
crambl s and thunders over hill
voic
far b low what we can hear.
ong, bird ongs boom and twin r.
Whal
Sea, air,
v
rything'
and even tho
a chao
w 'v
of
ignals
nan1 d v
Clear Air Turbulence
r and fall
Tectonic Order
The Dakotas and th n Wy ming Wlinkl
Banff lnt mational R
tmder as th
Alb rta,
only the
air wrap
about u -
cale differs: tho
and peaks ar
th
land'
ext nd to millennia
fin
grain
flow, wh
r
lif£ bring up
ut
and boun
afloor into
b low, th Thi
onds ·aunt as the wingtips dip , br aking
'n
outhward off ridge .
cell
over tho
e
I
rra
w eping
d by th
cene
heat and cold
hundr d million y ars to turn
rang
hiUg again and pull straight.
n, the patterns stagger and br ak up; an
the air'
heated, turning cha
v n granting tlli
till know that, in flight, volum
and pre
properly described k ep
u
Perhaps tl1e view'
e n
a fit end to its local order? And
far I
annat hold
mall U1ing from it? Thi
that took fiv
what we would impose on them br a How
ki-nm. We
a tiling in nlind, but can we learn
was peaked and t
tum d and bucked in
p · and fill. hug h now
a
ight of the Wlinkled fac
n w blown
air w 'r
2004
It took five hundr d million years to tmn o larg
and up h r
tation
y ars
tr t h with a I ap
the plateau'
13,
pril
arch
anada
aliv .
too p rf
much beyond tl1e bald fa t ,
t to disc m we'v
xplain th m, if we can rely on
what's not underst od'? W
tile world
that brought
u
h r , that shape and
arn
' ay to mould
a ubtl r thing in nlind'? To h lp
us
I
am
an't. The plane drop
an instant. We're forced again to look
that liv
past the surface, th
Th'
Iillis and knotted air,
to the blank place, alway ju t ahead, where
if only for a moment, th
n told:
htmdred nlillion y ars to tum
ur living ... \ ait! Is this th
Why wi h to
b
it took fiv
h rut
tops.
m
go on but ours will n t return?
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]
8
THE MATHEMATICAL INTELUGENCER
·�·ffli•i§rr6'h£119.1,1rrlll,iihfj
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
I
Marjorie Senechal, Editor
Mathematics is an oral culture, passed
"Toward the end of the twentieth cen
down from professors to students, gen
tury, scientists in many fields, includ
eration after generation. In the math
ing math, discovered that 'disorder'
lounge, late in the evening, when the
isn't random, it's a maze of subtle pat
theorem-scribbling dwindles and e-mail
terns."
morphs into screen savers, someone
"Ammann was one of the first to dis
opens a bottle of wine, another brings
cover non-periodic tiles and tilings.
out the cake, and the stories begin.
And he showed their amazing variety,"
Kepler was mystical, Newton alchemi
I tell her. "He didn't prove much, but
cal. Hotheaded Galois died in a duel.
he had vivid insights into their nature.
Godel starved logically, to avoid being
He settled open questions, posed new
poisoned. The stories roll on without
ones, and sparked imaginations." "Artistic
end. Stories of giants, their genius and
imaginations
too,"
says. "A painter in Berlin incorporates
day. Wiener, the father of feedback,
Ammann bars in his designs. And
couldn't find his way home. The peri
patetic Erdos woke his hosts at 4 in the
they're being used in a pavilion at the Beijing Olympics.''1 "A physicist I know laid an Ammann
morning. You know, stories like that.
R
tiling, with real tiles, in the entrance obert Ammann too was a brilliant eccentric. Yes, I knew him. His
story isn't like that.
0
0
hall in his home," Richard adds. "And a vice-president at Microsoft has in corporated all of Ammann's two-di
0
0
mensional tilings in the new home he's
communities is just as unrestricted.
"Wait a minute," Jane interrupts me
building. On floors and walls
We welcome contributions from
again. "Who was Robert Ammann?"
grilles."
mathematicians of all kinds and in
Carl
foibles: yesterday's giants, giants to
"You're telling me what he did, not
The denizens of the lounge sprawl in self-organized clumps. My clump in
and
who he was," Jane reminds us.
all places, and also from scientists,
cludes Jane, a first-year graduate stu
"Robert Ammann, the person, re
historians, anthropologists, and others.
dent just learning the lore; Carl, in his
mains almost unknown," I say. "This is
third year of graduate work, who's just
his story, as I learned it."
0
passed his orals; and Richard, a col
0
0
0
league from elsewhere. Jane and Carl
I'll begin, not with his birth in Boston
sit on the rug, as befits their apprentice
on October 1, 1946, but with an an
status. Richard relaxes on the black
nouncement in Scientific American.
leather sofa, a 20-pound calculus text
The August 1975 issue, to be exact.
under his head: he gave the colloquium
"For about a decade it has been known
lecture this evening.* I slouch in an
that there are tiles that together will
armchair that has seen better days.
not tile the plane periodically but will
"You've never heard of Ammann?"
do so non-periodically. . . . Penrose
"Everyone
later found a set of four and finally a
knows about 'Ammann tiles,' and 'Am
set of just two," Martin Gardner wrote
Carl
plays
incredulous.
mann bars.' In tiling theory, anyhow." "He was a pioneer in the morphol ogy of the amorphous," says Richard. "The what of the what?" Jane asks.
in his monthly column, "Mathematical Games." That's Penrose as in Roger Penrose, the famous mathematician and gravitation theorist, son of a psy
"Non-periodic tilings, chaotic fluids,
chologist of visual paradoxes. Father
Mathematical Communities Editor,
fractal coastlines, aperiodic crystals,
and son had sent impossible figures to
Marjorie Senechal, Department
that sort of thing," Richard explains.
M. C. Escher, who used them in his lith-
Please send all submissions to the
of Mathematics, Smith College, Northampton, MA 0 1 063 USA
'Jane, Carl, and Richard are surrogates for you, the reader. Their questions-your questions, my questions
e-mail:
[email protected]
guide us through the puzzles of Ammann's work and life.
1Q
THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+Business Media, Inc.
"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. Figure 1. Left: Ammann's "octagonal" tiling in the en
"As these patterns expand, they seem
trance to Michael Baake's home; photo by Stan Sherer.
to be always striving to repeat them
courtesy of Nathan Myhrvold.
new."
Right: Ammann grille in the home of Nathan Myhrvold;
selves but instead become something Jane dumps dozens of small, thin plastic tiles onto the table, four-sided
ographs "Ascending and Descending"
Another pair of planar non-periodic
polygons with notched edges. The
and "Waterlall." Penrose's new discov
tiles? Could this be true? And the first
black ones, dart-like, are all the same
ery, to which Gardner alluded, seemed
set of non-periodic solids? Who was
size; the white ones are identical kites.
even more impossible.
this Robert Ammann? Gardner knew
She pulls up a chair and tries to put
Floor tiles-triangular tiles, paral
just about everyone who knew any
a kite and a dart together to make a par
lelogram tiles, hexagonal tiles, oc
thing about non-periodic tilings at that
allelogram. But the notches don't fit.
tagons with squares-repeat over and
time: Roger Penrose, Raphael Robin
over, like ducks in a row and rows of
son,
Graham,
I tell her. "If you could make a paral
ducks. Wall tiles do too, and tilings in
Benoit Mandelbrot, Branko Griinbaum,
lelogram with these tiles, then you
art. Even Escher's wriggling lizards,
Geoffrey Shephard. He'd never heard
could cover the plane with them, the
plump fish, haughty horsemen, and
of Ammann. Nor had they.
way square ceramic tiles cover a floor."
John
Conway,
Ron
"That's the reason for the notches,"
winsome ghosts arrange themselves
"I am excited by your discovery,"
"No, she couldn't," Carl interjects.
in regular, periodic arrays. Non-peri
Gardner replied on April 16.Ammann's
"The plane is infinite, theoretically.
odic tilings? What could they be?
tiles seemed quite different from the
She'd need infinitely many tiles. She
Gardner gave no details, drew no pic
Penrose pair Gardner planned to write
only has a hundred or so."
tures:
a
about later. "Would you object to my
"Of course. But you know what I
patent. "The subject of non-periodic
sending your tiles to Penrose for his
meant. Don't be so picky, it's after 10
tiling is one I hope to discuss in some
comments? Are you planning to write
p.m." I tum to Jane. "The notches pre
future article," Gardner concluded his
a paper about them? . . . Tell me some
vent you from making a parallelogram
thing
or a repeat unit of any kind. So every
Penrose
was
waiting
for
column. For thirty years, from 1956 to 1986, Martin Gardner intrigued young and old, amateurs and scientists, unknown
about
yourself.
How
should
you be identified. A mathematician?
tiling with kites and darts is non-peri
A student? An amateur mathemati
odic. That's why they're called non-pe
cian?"2
riodic tiles."3
and famous, geniuses and cranks, with mathematical games, puzzles, diver sions, challenges, problems. His read ers deluged him with solutions, some of them valid, some of them pseudo.
Scientific American hired assistants to help weed out the nonsense. Ammann's response to the August announcement reached Gardner's desk the following spring. "I am also inter ested in nonperiodic tiling," Ammann wrote, "and have discovered both a set of two polygons which tile the plane
F... .,
�obnr A,..,...,,,. 11•���-··· ST l-1wt>ll 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"ln<��l'\f"'
Figure 8. Another version of (a portion of) the •
C"'....., l)f
•v
I• t•r
tn
I,., rtn"
�n
t� Cl
• f'Yt' l•'""·
first tiling in Ammann's May 20, 1976, letter to
..•..,..,..,...11•, 1 "'""• ,.. ""'""'"' """"''""' "��'"'",.. -.v ""1"1 1'\lln•r� 'ln1 """• ,,.,..,� t.._n l'\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.
1
Gardner (see Fig. 7). Copies of the two small tiles can be combined into larger ones, as
·�t "''"h""'"·
;
a�·-r""V
shown by the shaded tiles. All the tiles in the
�"'"""'rt A·,.·�"
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
l
tiles are not exact enlarged copies of the
..
smaller tiles of which they are composed.
l •
J
l
i
(i �cr
r r G<·"
f"
\0
�·
'f'
��..· �
\"cl�
�
�I
..t , �er
,.,
�·�t
,A('" ,�
1 .... '"
� /e
tT.:r
t
f\kld
cr
(. "(1i ,.
11)
Figure 7. Ammann's hierarchical tilings. [Ammann to Gardner, May 20, 1976.]
Figure 9. Ammann to Gardner, April 14, 1977.
VOLUME 26, NUMBER 4, 2004
13
mathematics articles I knew of, and piece them together somehow.
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,
" Ammann bars are a grid for the tiling,'' I continue. "As Anunann ex
That hubris sustained me through
plained it, the 'pattern is based on fill
the fall and into winter. But now the
ing the plane with five sets of equidis
clock ticked toward class. I dreaded
tant parallel lines at 36- and 72-degree
facing the students: I had nothing to
angles to each other, and placing a
say. The tiling literature was incoher
small tile wherever two lines intersect
ent,
at a 36-degree angle, and a large tile
incomplete,
inconsistent,
and,
worst of all, incomprehensible. To
wherever they intersect at a 72-degree
forestall the disaster, if only for a few
angle.' If you look closely at the lines
minutes, I checked my mailbox on the
on the tiles you've laid, you'll see how
way to the classroom. I opened the
it works."
bulky package and ran to the phone.
" Some lines are closer than others,''
The authors, Branko Griinbaum and
Jane points out. " I thought you said
Geoffrey Shephard, agreed to let me
Ammann's lines were equally spaced."
and my students work through it and
"They were, in his first letter to
send comments. Tilings and Patterns
Gardner, the one I just quoted. But
became the course.
equally spaced lines can't be drawn on
when Wang posed the question, he and
The book galvanized research on
the tiles so that each tile of each kind
everyone else assumed that a decision
tilings, including my own. Griinbaum
is marked the same way. Ammann
procedure would be found. They were
and Shephard had gathered, sifted, re
modified the spacings later. The pat
wrong. Robert Berger found the first
viewed, and revised everything that
tern of intersections is the same."
non-periodic tiles in 1966. But there
had ever been written, in any language,
Jane adds tile after tile. The patch
were 20,426 different tiles, so it was
living or dead, on tilings and patterns
grows like a crazy quilt.The lines remind
in the plane. Tiles of so many kinds!
me of a children's game called pick-up
" Well, he showed that the tiling
Polygonal tiles, star-shaped tiles, tiles
sticks, but those fall any which way.
problem is formally undecidable," Jane
with straight edges, curvey tiles, tiles
" Hmm,'' Carl says. "The long and
says. "That's enough!"
symmetrically colored. These omni
short distances form a sequence, . . .L
scient authors filled in gaps, corrected
SL SLL SL . . . "
only of theoretical interest."
" If we had a jazzier name for non periodic tiles no one would ask 'who
mistakes, compared and synthesized
cares,' " Richard observes. " No one
different approaches, proposed new
asks who cares about chaos and frac
terminologies, and classified tilings
tals. Some of us tried calling tiles ape riodic if all their tilings are non-peri
with various properties.
odic. But the name never caught on.
and darts and John Conway's account of their remarkable properties had just
better."
been published. 10 In chapters sent
()
()
()
" . . .L SLL SL SLL SLL SL SL L SL SL L SL . . . ,'' he reads out. " Omigod!"
Martin Gardner's article on the kites
And no one has come up with anything
()
"Keep going," says Richad.
Jane
exclaims.
" Fi
bonacci rabbits! Where did they come from?"11 " From the hierarchical structure," I show her.
later, Griinbaum and Shephard de
"Penrose tilings are riffs on Fi
I first heard of Anunann's work through
scribed that and much more: Wang
bonacci numbers and the golden ratio,"
the grapevine, but I didn't grasp its im
tiles, Robinson's tiles, and five Am
Richard
portance until I read Tilings and Pat
mann
everywhere: it's the ratio of long to
marked with lines they called Ammann
short tile edges, the ratio of kite to dart
of an early draft. The first few chapters
bars.
areas, and the ratio of the relative num
terns.9 I was one of the lucky readers
sets, A1
through A5,
some
arrived unannounced in the mail on the
Yet except for his letters, no one
first day of the spring semester in 1978.
knew a thing about Ammann. No one,
The authors had no idea how glad I'd
not Gardner, Penrose, Griinbaum, nor
be to see it. Tilings play a key role in
Shephard, had met him.
the geometry of crystal structures, my
()
()
()
pontificates.
"4> crops up
bers 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 non periodicity!" says Jane.
research field at the time, so I had an
" Are these lines the Ammann bars?"
nounced a course on them, mainly to
Jane asks, handing me a kite. She's no
" No, it's not,'' Carl deflates her. " At
teach myself. No textbook existed;
ticed the thin lines etched on the tiles,
first people thought it might be, but
most mathematicians dismissed tilings
each kind of tile etched alike.
Ammann found pairs of non-periodic tiles where all those ratios are
V2. The
as " recreational math" in those days. I
" Right," I reply. " With Ammann bars,
would pull together articles from crys
you don't need the notches. You can't
square
tallography journals, Martin Gardner's
make a parallelogram if you keep the
showed you-the one in the hallway
columns, books on design, and a few
bars straight."
is the most famous example."
•q, = (1
14
+
Vs)/2
THE MATHEMATICAL INTELLIGENCER
and
rhomb
tiling
Richard
Figure 11. A kite and a dart marked with Am mann bars. The notches are not shown. Figure 12. Left: Ammann bars; right: Penrose tiles with Ammann bars superimposed.
Jane returns to the tiling puzzle. A
Voila, the Penrose tilings! in the plural!
few minutes later she exclaims, "The
You get them all if you shift the slice
tiles don't fit any more. I've hit a dead
around. And the matching rules also
an hour and a half from my home in
end."
fall out of the sky! "12
Northampton. So I sent him a note, invit
"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."
"Is there some abracadabra so I can continue?" "Remove
ing him to dinner at my home to meet Dick de Bruijn, who was visiting from
try
the Netherlands. De Bruijn was strong
some
pieces
and
again." I pour a second glass of wine.
bait-his powerful analysis had lifted
"When you get stuck in a non-periodic
Penrose's tilings from two dimensions
tiling you can always repair it. Unlike
to five and Ammann's work from doo
life."
dle to theory. Even so, I was as sur
"How far should I backtrack?"
prised as delighted when he accepted. November 19, 1987, a cold, rainy
"No one can say."
0
"Oh."
But the mystery man's most recent
refusal was postmarked Billerica, MA
0
0
0
day. Our guest arrived after dark, three
"There's no way you could have
Over the next decade, assisted by
hours late. He was neatly dressed,
known that," I console her. "The choices
Tilings and Patterns and spurred by
short and a little stout, his very high
seemed equally valid at the time."
the startling discovery of quasicrys
forehead framed by black hair and
tals-crystals with atoms arranged in
black-rimmed glasses. I guessed his
"So very like life," she mutters. "Anyway, I'm not sure I can really tile
non-periodic patterns-tilings leaped
age about forty. He shook my hand
the infinite plane with these things. I
from the game room into the solid
limply, avoiding my gaze.
mean"-she glances at Carl-"in prin
state lab.13 Mathematicians, physicists,
Bob didn't make small talk, not
ciple, if I had an endless supply of
chemists, x-ray crystallographers, and
even hello. As I took his dripping
tiles."
materials scientists found a common
raincoat, he pulled sheets of doodles
"You can," I reply. "It's yet another
passion in non-periodic tiles. "We're all
from a brown paper bag: his latest
consequence of the hierarchical struc
amorphologists now," a physicist told
discoveries, his newest results. Dick
ture."
me. Penrose tilings and Amman tilings
and I looked at them carefully, but
were buzzwords of the day. And still no
couldn't decipher them. I asked what
one I knew had ever met Ammann.14
they
"You must have slept through my talk," says Richard. "I showed you how to get complete Penrose tilings by pro
We, the growing community of tiling
meant.
Bob's
answers
were
vague. Dick explained his pentagrid
jecting the tiles from higher dimen
specialists, attended conference after
theory. Bob showed no interest. This
sions. De Bruijn invented that method.
conference, all over the world. In those
wasn't rudeness, I sensed. He seemed
Start with five sets of equally spaced
days, before the Internet, keeping up in
far away, and ineffably sad. Fortu
parallel lines, just like Amman's origi
a hot field meant being there. Ammann
nately, dinner was waiting. Dick and
nal ones-de Bruijn calls them penta
was often invited but always declined,
I did most of the talking at dinner, but
grids. He showed that the criss-cross
if he answered at all. In the spring of
Bob seemed glad to be with us, and
pattern of lines in the plane is a slice
1987, Branko Grtinbaum again pleaded
he
of a periodic tiling in five-dimensional
with him, "Would you please recon
asked.
space."
sider? Without exaggeration, I am con
"But I'm stuck down here," Jane per sists. Richard ignores her. "Then de Bruijn
vinced that you have shown more in
answered
our
questions
when
"How did you discover your tilings?" we wanted to know.
ventiveness than the whole rest of us
"I'd been thinking about the lines of
taken together."15 Again Ammann said
red, blue, and yellow dots used to re
abracadabra-more pre
no. The mysterious Mr. Ammann," he'd
produce color photographs in newspa
cisely, he takes the pentagrid's dual
signed a letter to Gardner. Mysterious
pers," Bob replied. "I drew lines criss
and projects it down to the plane.
he remained.
crossed at appropriate angles,
does some
VOLUME 26, NUMBER 4, 2004
and
15
defy classification ...we have had our
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 nonper iodic 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 Irving as you may remember, wrote the · 1 "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.)]
eyes opened to the vast additional pos sibilities 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 0
autism a few issues back.n2
"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 stared at them for awhile.The tiles just
tion. In a world of his own, and the vi
popped out at me."
sual thinking."
In a letter to Gardner, Bob men tioned
he
had
some
"math
back
ground "; I asked what he meant. "A lit
"Not so fast," I snap."That describes most of us. Besides, thinking in images is one thing,but visual genius is another."
times." "What do you mean? The author said they had Asperger symptoms." "No one-size-fits-all diagnosis can explain such complicated people.Take
tle calculus, and some programming
"We don't need vision anymore," says
Norbert Wiener,for example. When he
languages," he said. He'd been a soft
Richard. "Ammann bars and pentagrids
wrote his autobiography, in the early
ware engineer for twelve years, but
are more important than the tiles them
1950s, the
now he worked in a post office all day,
selves.We have equations for them.We
Freud. Everyone told Wiener that his
prevailing
fashion
was
every day, sorting mail. Because, he
can feed parameters into computers,
emotional and social problems-he had
told me, civil service jobs are secure.
and the computers draw the tilings."
lots of both-were due to his father.
Had he heard about quasicrystals?
"Some of Bob's tilings don't have
Wiener rejected that explicitly, even
Yes; he'd been in touch with some
bars," I remind him."The world of non
though his father was famously diffi
physicists who were studying them.
periodic tilings exceeds every theory
He'd even gone to Philadelphia to see them once.l6 They'd told him to call
yet devised, and always will. Visual
them collect if he had any new ideas, but so far he hadn't.
imagination still has a role here." Richard rolls his eyes: "The projec tion method and hierarchical order
After dinner, as he was leaving,
cover the field."
cult. 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
Bob gave me a typescript of an arti
I get up from the armchair and take
cle he'd written, a revolutionary new
a book from the shelf. "If you don't be
prodigies back then. Another kid who
lieve me, listen to Penrose. 'The differ
entered Harvard at eleven cracked up
hoped I could help get it published. I
ent kinds of tiling arrangements that
in the spotlight."
lent him a book on fractals he hadn't
are enforced by subtly constructed
yet seen.
prototile shapes must,in a clear sense,
theory of dinosaur extinction.17 He
"What did Bob say in that dinosaur paper?" Carl asks.
"Our conversation was very touch ing,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 ask ing to interview him about how he made his discoveries; I wasn't sur prised that he didn't reply.18
0 "Bob
0 was
0
autistic, or
0 Asperger's
maybe," says Jane,looking up from the tiles. "It's obvious, from your descrip-
16
THE MATHEMATICAL INTELLIGENCER
Figure 14. Bob Ammann and N. G. de Bruijn, November 19, 1987; photograph by Stan Sherer.
"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
be a model for the quasicrystal?" Jane
penses.25 He was eager to meet John
asks. "I don't know. He intended to pub
Conway at last, and also Donald Cox eter, with whom he had corresponded.26
lish something on it but never did. It
Bob's talk, mostly on his 3-D tiles, was
might be possible."
more polished this time.Very pleased, I
"Ammann's 3-D tiles are the famous
congratulated him.He beamed.
university's new interdisciplinary re
golden rhombohedra," Carl points out.
search center.When I arrived, late in the
"You can build anything with them, even
Six years passed, with no word from
evening, the lounge was filled with the
periodic tilings. Bob must have found
Bob. Then in 1997 my fractal book
usual suspects, thirty-three assorted sci
ways to prevent that.Did he notch them,
came back in the mail, without any
entists from nine different countries.No
or what?"
note. At Christmas, I sent a card with
stories this time: the story was there.
"He marked a comer of each facet
Bob sat quietly at the edge of the crowd,
of each rhombohedron with an x or an
speaking when spoken to but not look
o, and claimed that if you match x's to a's you get a 3-D version of the rhom
ing 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
I never saw him again.
a few words of thanks. Early in Janu ary, I received a reply.
Socolar proved they force non-period
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,
bic Penrose tiles." "And do you?" Carl asks. "Yes,"
I say.
"Well,
not
exactly.
on, he mingled more easily, almost
icity, but in a weaker sense than Pen
1 994 . . .
naturally.Like the rest of us, he feigned
rose's.The whole question of matching
interest in the lectures whether he un
rules deserves a fresh look They seem
derstood them or not. He joined us for
Sincerely yours, Esther Ammann
to come in various strengths: weak,
meals. On the third day, nervous and
strong, perfect.And the connections be
She had sent me the book;
halting, he gave his first-ever talk, on
tween matching rules and hierarchical
thought I'd known. Distressed, I con
his
three-dimensional
non-periodic
she'd
structure and projections is still murkY.
tacted colleagues. None had heard
And 3-D tiles are hard to visualize. So
from Bob since the Philadelphia meet
ticular quasicrystal. I remember the
there's a thesis problem for you, if you
ing and none had known of his death.
talk as confusing and disorganized, but
need one."23
tilings and how they might model a par
others insist it wasn't too bad. We agree on Bob's wistful conclusion:
In April, when the snow had melted,
"Have any other 3-D non-periodic tiles been found?" asks Jane.
"That's all I have to say.I have no more
"Very few."24
ideas."
()
()
()
I drove the fifty miles to Brimfield to meet Bob's mother.Esther Ammann, a
()
vigorous, intelligent widow of ninety three, lived alone in a big house at the
On the fourth and last evening, at
I met Bob a third time six months later,
the conference banquet, the organizers
in the fall of 1991. No longer so painfully
the forest. The puzzle from Bielefeld
honored Bob with a special gift, a large
shy, he accepted my invitation to speak
sat proudly on her mantelpiece, sur
three-dimensional
dark
at an AMS special session on tilings in
rounded by pictures of Bobby, her only
brown wood, and photographed him
Philadelphia-if I would pay his ex-
child. Bobby in his cradle; Bobby at
puzzle
of
top of a hill with a panoramic view of
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 theo rem 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 reg ular 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 crea tures. "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
But suddenly, before he was four, t t
Bobby stopped talking. His doctors never knew why. For months, only Es
•
ther 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
absent
minded, 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. /
;_
l
/
He didn't like sports, but he loved jun
/
gle gyms, the high kind with criss crossing bars.
'
) 1�. { t J . J
5c. \ ' J _
"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
/
< 1"-,
_.. j
I
�. -----
>''
------6 !
0
'i.
...,,
'
1��� 0 � J
I
)
..-----
/
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 ca bles of the Golden Gate Bridge. He won the state math contests, and his SATs were near-perfect. MIT and Harvard in
Figure 16. Ammann's drawing of the nets for his marked rhombohedra, in his first letter to
vited him to apply, but turned him
Gardner. The two on the left are obtuse, the two on the right acute. Cut them out and fold
down after the interview. Brandeis Uni versity accepted him. He enrolled, but
them up!
he rarely left his dorm room and again got low grades. After three years, Bran deis asked him to leave.
three, with his favorite possession, a
the trucks coming in and going out. The
globe nearly his size; Bobby with his
men enjoyed his arcane, intelligent
So Bob studied computer program
Esther and August; Bobby
questions. The tot knew more geogra
ming at a two-year business college
with four or five cousins and gaily
phy than anyone. One evening a dinner
and took a low-level job with the Hon
wrapped presents, in front of a Christ
guest wondered whether the capital of
eywell Corporation near Boston. There
mas tree; Bob with Roger Penrose in
Washington was Spokane or Seattle.
he wrote and tested diagnostic rou
Germany. Over coffee she told me her
"It's Olympia," chirped Bobby.
tines for minicomputer hardware com
parents,
story.*
Bobby could read, add, and subtract
ponents. Twelve years later, the com
Bob didn't mind sorting mail: he
by the time he was three. He tied
pany let him go. He found another job
could let his mind wander. He'd always
sailors' knots, solved interlocking puz
but that company soon folded. In 1987,
liked post offices. When he was little,
zles,
Esther would hand him through the
oiled the sewing machine. He could ex
learned Indian sign language,
not long before I met him, Bob started sorting mail.
stamp window and leave him with the
plain how bulbs grew into flowers, how
"The dinner at your house was a
postal workers while she shopped.
frostbite turned into gangrene, how tis
high point of his life," Esther sighed.
Bobby loved the maps on the walls, the
sue healed in a bum, how teeth de
"No one else reached out to him."
routing books for foreign mail, the
cayed, how caterpillars changed into
stamping machines, the sorting bins,
butterflies.
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
Oregon. Esther didn't see much of her �e r \. a...:..____! nn Bir�
�1t
C1 1 t t .,rd. :
&
C.)
o,
;
H o•Hh:
aner &
lauf
e•
i�:
nnin :
Weye r :
o:
Poly
:
S t. i c
er�
bor
�. b r ew 1
:
the
�
� roa
.:;, 1 tb
•
t. 1 e a
a
.d.
:
an:
ort• : Shap l ey : B&r�a t t :
l
t • 1
o f �tho
Bob's co-workers kept their dis
� 1 o:1
!."'.,u
tics
e o s k ona
lo r l l o l llat�o l1& t l c l .
" vo.
d,-i-- ..._,._.L ..:C._., ,.... �·...,. .J7 ,o�
<-.!
L
.-i- ;. ...; "' >tJ- 1.<4, 7;
·
o • L o C l c J. & t e Q i e :d y
A. l J.
L i fe
kept
refrigerated-like
ham
burger! For up to a week! He was fond
;j
n .iorl I• Lh ll oD'onto ot llat on•t ell E i o •oa:r j�i eoc• � a� I • Tn e Nature of L n l ll& 'l'h l n
of canned spaghetti but didn't like the sauce so he'd wash the stuff in the sink or the nearest drinking fountain."
of So l o nc o
d Dr .
Hearst were the same person were, to
been
So Lo:>a 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
For the standard parameter values they are numerically
1 shows an image of the Lorenz attractor that was
not obtained by simply integrating from an arbitrary start ing condition, but by computing the one-dimensional un stable manifold tractor, plotting
wu(O). Since the origin is in the Lorenz at wu(O) gives a good picture of the attractor.
We computed both branches, one in red and one in brown,
- 22.828, - 2.667, and 1 1 .828
of the unstable manifold of the origin by integration from
in increasing order. This means that the origin is a saddle
two points on either side of
with two attracting directions and one repelling direction.
from the origin along the unstable eigendirection.
W5(0) at distance 10-7 away
1 1], there ex
It is clear from Figure 1 that each of the branches of the
ists a one-dimensional unstable manifold dimensional stable manifold
wu(O) and a two W5(0), defined as
unstable manifold visits both wings of the attractor, as is to be expected. In fact, because of the symmetry of equa
According to the Stable Manifold Theorem [2,
wu(O) = {x E �3 l lim q/(x) = O J, 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 eigen space, respectively. We call
W5(0) simply the Lorenz man
ifold. While most trajectories end up at the Lorenz attrac tor, those on W5(0) converge to the origin instead.
tions (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 man ifold
W5(0) look like, as it "wiggles" between the red and W'(O) cannot cross W5(0). brown curves of Figure 1 ? Remember that
The z-axis, the axis of symmetry, is part of the Lorenz manifold
W5(0), which is itself invariant under rotation by
Geodesic Level Sets
around this axis. Furthermore, there are two special tra
The Lorenz manifold, like any global two-dimensional in
jectories that are tangent to the eigenvector of - 22.828,
variant manifold of a vector field, cannot be found analyt
which is perpendicular to the z-axis. They form the two
ically but must be computed numerically. The knowledge
7T
branches of the one-dimensional
of global stable and unstable manifolds of equilibria and
strong stable manifold W55(0). All other trajectories on W5(0) are tangent to the z
periodic orbits is important for understanding the overall
axis.
dynamics of a dynamical system, which we take here to be
Apart from the origin, the Lorenz system (1) has two other equilibria, namely
c �Vf3CP 26
1), �
Yf3(p - 1), P -
THE MATHEMATICAL INTELLIGENCER
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 computa
1)
=
( �8.485, � 8.485, 27),
tion 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 Ameri can 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
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.
ch
chain stitch
chain stitch
de
double crochet
single crochet
Comparison with Crocheting the Hyperbolic Plane
tr
treble crochet
double crochet
dtr
double treble crochet
treble crochet
The idea to crochet a model of the Lorenz manifold was born quite suddenly in December 2002, but was indirectly
28
THE MATHEMATICAL INTELLIGENCER
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 in
which it spirals. The lower part of the manifold with z <
structions in order not to miss when to add or indeed re
has almost zero curvature. It is impossible to flatten out the
0
move an extra crochet stitch. This involves much counting
crocheted manifold on a table, as the region of strong neg
and checking of each round. In fact, Hinke crocheted the
ative curvature forms more and more folds.
Lorenz manifold in the course of two months in an esti mated
85 hours, which corresponds to about 300 stitches per hour for the total of 25, 5 1 1 stitches. (To translate this
Figure
3 shows the crocheted manifold at three differ
ent stages of progress and flattened out as much as possi ble. The images on the left show the manifold as a rippling
time estimate to your own crocheting skills, be warned that
disc; the z-axis corresponds to the vertical line through the
Hinke is an expert at crochet and counting alike!)
center of each panel. In the images on the right the cro
A Shapeless Crocheted Topological Disc
To "absorb" some of the curvature, the z-axis is then no
Initially, up to a geodesic distance of about 36. 75, the Lorenz
longer a straight line in the upper part of the images, but
cheted manifold has been folded double along the z-axis.
manifold is virtually flat as a pancake. It then starts picking
even this is not enough to avoid the increasing (with di
up a lot of negative curvature near the positive z-axis, around
ameter) 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
3. Supporting the manifold in the radial direction with a
When we first saw the crocheted but yet unmounted Lorenz
single bendable wire that runs from rim to rim and
manifold shown in Figure 3c we had some doubts whether
through the origin.
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;
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 [ 1 0]. 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 posi
two results of mounting the manifold, leading to the Lorenz
tive and negative z-axis and both branches of W88(0) go.
manifold itself with a right-handed spiral around the z-axis,
This information is collected in the weaving instructions in
or its mirror image with a left-handed spiral around the z-axis.
the Appendix. To mount the Lorenz manifold we wove an unbendable
thin kiting rod through the z-axis and bendable wires through
By giving the rim wire the right twist one can ensure that one
obtains the former solution.The final step is to bend the sup porting wires so that they are nice and smooth and the cro
the last round and the location of W88(0); details of this pro
cheted model indeed resembles the Lorenz manifold.
cedure can be found in the mounting instructions in the Ap pendix. Modulo rotations and translations in IR3, there are only
rod fixing the z-axis is vertical and in the centre of the images.
32
THE MATHEMATICAL INTELLIGENCER
The final result is shown in Figures 4-7. The carbon fibre
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
of Engineering. The mounted Lorenz manifold was carefully
enlargement of the Lorenz manifold that shows the strong sta
set up by the authors at the photo studio of the Photographic
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
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
tainly not close to the rim. In other words, it is clearly not a
Faculty of Engineering. We are very grateful to both Perry
geodesic.
and Greg for their patience and attention to detail, and for
It is our experience that the crocheted model of the Lorenz
dealing so well with our enthusiasm and perfectionism.
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
REFERENCES
[1 ] J. Gleick. Chaos, the Making of a New Science, William Heine mann, London, 1 988.
represented, so that it is possible to convey the intricate struc
[2] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynami
ture of this surface in a "hands-on" fashion. This article tries
cal Systems, and Bifurcations of Vector Fields. Springer-Verlag,
to communicate this, but for the real experience you will have
Second edition, 1 986.
to get out your own yam and crochet hook!
[3] D. W. Henderson and D. Taimina. Crocheting the hyperbolic plane.
Acknowledgments
[4] B. Krauskopf and H. M. Osinga. Two-dimensional global manifolds
The Mathematical lntelligencer, 23(2): 1 7-28, 2001 . The unmounted Lorenz manifold in Figure 3 was digitally photographed by B.K. at the Design Office of the Faculty
HINKE M. OSINGA
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
e-mad: H.M.Osinga bnstol.ac.uk
Un�tad Kingdom
Hinke Osinga learnt crocheting, and other handcraft techniques,
Bernd Krauskopf got h is Ph.D. 1n Mathematics from the UniVersity
from her mother around the age of seven. A b1t later she got a
of Gron1ngen 1n 1 995 under the d1rect1on of Floris Takens and Henk
der the d1rect10n of Henk Broer and Gert Vegter. Her research was
a two-year Postdoctoral pos1tion at Vrije Universiteit Amsterdam,
the ideas developed then have been generalized (and disguised)
versity of Bristol. Bernd works 1n the general area of dynamical sys
throughout her work. She held postdoctoral pos�tlons at the Geom
tems theory, specifically on theoretical and numerical problems 1n
Ph.D. 1n Mathematics from the University of Groningen in 1 996 un
on the computation of normally hyperbolic 1nvariant man1folds, and
etry Center, Univ9rSity of M1nnesota, and at the Cal1fomia Institute
of Technology i n Pasadena. She moved to England 1n 2000 , first to the UniverSity of Exeter. She JOined the Department of Eng1neenng
34
e-ma1l: B. Krauskopf bnstol.ac.uk
Broer. After a year as v1sit1ng professor at Cornell Un1vers1ty, and he jo1ned the Department of Eng1neering Mathematics at the Uni
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
Mathematics at the UniverSity of Bristol in 2001 The Lorenz man
neapolis- the basic idea of how to grow a global manifold emerged
ifold
over a bagel with H1nke at a bagel store on Nicolette Ma ll.
IS
her first project that comb1nes handcraft with mathematics.
THE MATHEMATICAL INTELLIGENCER
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 Equadiff2003, 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 1 971 202 , 1 999. [1 3] M. Viana. What's new on Lorenz strange attractors? The Mathe matica/ lntelligencer, 22(3):6- 1 9 , 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 a
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 5 1 56 63 68 75 80 87 92 95 96 99 104 107 108 1 2 1 (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 1 2 1 137 142 145 (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 1 70 1 72 1 75 1 76 185 189 1 9 1 (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 1 78 193 204 213 2 1 5 222 (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 1 77 1 18 2 1 6 2 1 7 (gd 50. 75) ; rnd18: 283 tr 0 2 198 202 206 207 2 1 5 2 1 6 230 235 237 238 255 268 (gd 52. 75) ; rnd19: 307 tr 16 17 21 25 32 49 56 169 1 76 193 216 225 226 229 234 237 246 258 259 269 271 276 277 280 (gd 54. 75); rnd20: 325 tr 2 7 8 80 87 104 1 1 1 128 135 152 1 59 268 269 272 274 279 285 290 (gd 56. 75) ; rnd 2 1 : 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 1 5 1 160 1 77 186 209 234 243 246 249 260 264 265 279 280 281 310 328 329 330 337 338 342 (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: 45 1 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 1 2 1 136 143 144 1 5 1 166 1 73 1 74 181 196 203 204 2 1 1 234 235 271 275 288 295 327 343 360 363 376 380 383 419 435 436 450 (gd 70. 75); rnd28: 5 1 1 tr 13 51 289 300 314 332 340 361 362 403 404 413 416 4 1 7 420 42 1 436 450 469 481 (gd 72. 75) ;
I
-2
I -1
+1
+2
+
-
-2
-
1
I r +1
0
l a
+2
+3
J I
X
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) ; rnd3 1: 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 19 148 155 174 183 190 199 208 217 224 253 260 325 360 362 363 371 377 405 414 439 470 487 488 492 501 504 507 508 509 512 513 5 14 523 535 576 577 580 583 586 590 (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 5 13 538 539 542 (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 1 78 275 292 313 3 14 345 370 380 438 441 529 540 541 552 566 589 591 592 680 (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 - 51 5 515 563 568 582 602 603 608 619 -66 1 661 -668 668 -671 671 701 714 7 19 (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 - 7 1 5 715 742 748 (gd 92. 75) ; rnd39: 804 tr 2 4 8 2 1 29 3 1 64 67 72 93 105 379 386 434 458 4 72 478 497 500 - 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 1 76 344 349 382 395 407 423 433 446 465 474 480 489 5 12 527 532 - 559 559 - 567 -572 - 579 579 - 584 584 - 59 1 639 645 653 665 666 682 686 700 705 712 714 716 - 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 3 14 381 382 389 429 526 534 550 559 564 572 - 6 1 1 - 622 -644 - 654 696 712 719 731 732 739 758 767 768 788 -853 886 (gd 1 00. 75); rnd43: 958 tr 2 37 6 1 6 6 7 6 84 206 223 254 2 6 1 262 269 300 307 308 315 346 353 354 361 392 433 509 559 588 589 613 -684 684 742 748 750 751 757 762 778 779 794 - 839 -868 903 914 915 (gd 1 02. 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 1 04. 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 8 1 1 8 14 829 830 833 863 -960 (gd 1 06. 75); rnd46: 1072 tr 0 10 15 33 1 16 126 144 279 291 302 3 14 329 341 352 364 379 391 402 414 546 579 580 590 609 636 648 661 667 674 685 - 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 1 77 535 608 653 654 658 668 689 691 699 704 705 - 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; rnd2 1 : + 1; rnd22: + 1; rnd23: + 1; rnd24: +2; rnd25: + 1 ; rnd26: + 1; rnd27: + 1 ; rnd28: +2; rnd29: + 1 ; rnd30: + 1; rnd3 1 : + 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; rnd1 7: + 1; rnd18: + 1 ; rnd19: + 1 ; rnd20: + 1; rnd2 1 : + 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; rnd l l : + 2; rnd12: + 1; rnd13: +3; rnd14: + 1; rnd15: +2; rnd16: +3; rnd17: +2; rnd18: +2; rnd19: +2; rnd20: +2; rnd2 1: +3; rnd22: +2; rnd23: +2; rnd24: +3; rnd25: +3; rnd26: +3; rnd27: +2; rnd28: +3; rnd29: +3; rnd30: +3; rnd3 1: +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; rnd2 1 : - 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
nectors are now unsuitable for electrical connections, but
Weave the unbendable thin rod of 0.9 m length through the
ideal for connecting the bendable wires. Connect the two
manifold by following the z-axis weave; we used a 5 mm
2. 70 m pieces of wire at the top and bottom with the con
carbon fibre rod used in kiting, which is lightweight and
nectors by sliding in both ends and tightening the screws.
very stiff for its diameter. Starting from the top of the z
Make a mark 0.1 m from each end of the 1.45 m length
axis, weave a 2.70 m length of the bendable wire through
of bendable wire; the middle piece of 1.25 m is the length
the outer crocheted round of the manifold until you reach
of W5"(0). Starting from the rim, weave this wire through
the bottom of the z-axis. Repeat the procedure with the
the manifold following the marking yam. Using the pliers,
second length of 2.70 m around the other half of the other
make two small loops at both ends where you made the
crocheted round of the manifold, again starting from the
mark and cut off the excess wire. Sew the ends in place
top of the z-axis.Try to spread the stitches evenly over the
with light blue yam.
wire; you will find that this introduces twist into the rim
Finally, remove the differently coloured yam.The Lorenz
wire. Make sure the twisting is clockwise near the z-axis
manifold should now be recognisable. With the help of the
in the direction of increasing z, so that you get a right
figures in this paper, tuck and bend it into its final shape,
handed helix, just like a cork screw.
making sure that the bendable wires are nice and smooth,
Cut two single electrical wire connectors from a bar and strip them of their isolating plastic cover. The stripped con-
that is, without noticable kinks. This may take some time de pending on your level of perfectionism. Good luck!
Mathematics a n d C u ltu re Mathematics and Culture m\'ei'SII') of Rome 'l:!
apienza', lral) (Ed.)
Tlu� book tresse the trong lmk bem e n mathemati
and ulrure, as mathemari
theater, literature, ar h1tecmre, art, medi 111 Th
but al
o
Mathematics, Art, Technology and Cinema
link
mema,
dance, cartoon and mu 1 .
arn I � mtrodu ed h re are meant to be
culrur
of
ien e and humani m is a thing of
the pa t. They hold that both culture are trul)• linked through idea and
thr ugh t
hn I S>· I n d mg
, the)
to non-matheman 1ans, and th of matheman lo\'er , tho e
. An m 1ghtful '' ho
feel pa
wh
reariviry, not onl)
u ceed in reachmg out ar not parri ularly f nd
k ( r math mati ian , film
ionate about 1m:1ge , and th
e
w1th
a que nonmg mmd. 2003/242 PP./HARDCOVER/$99.00/ISBN 3-54()-()()60 1-X
EASY WAYS TO ORDER: Toll-Freco 1 - 00- PRJ, . ,rR • WEB " "'"->prmgor·n).
Springer www.springer-ny.com
drpanmenl. Pnu:� uhJr<.'t ro dun
""h u1 non r.
P/Nse mention 57805 when ordering to guarantM listed prices.
Pr monon
7
0
VOLUME 26. NUMBER 4, 2004
37
l:dftj.l§rr@ih$1¥fth§4£1 1 .'1 ! .1§.id
Origami Ouiz Thomas Hull
Michael Kleber and Ravi Vakil, Editors
Paper is all around us. Every day we
2.b. Does it make sense to consider
fold paper. So test your knowledge of
the point P to be chosen outside the
and your ability to explore this simple,
square? What if we instead use a rec
everyday activity.
tangle?
1 . Find a square piece of paper that is
3. Figure 3 shows how one can fold an
white on one side and colored on the
equilateral triangle in a square piece of
other. From such paper it is possible to
paper. Does it work? Is this the largest
use the contrasting colors to fold any
equilateral triangle that can be made
This column is a place for those bits of
n X n checkerboard. Trying to do this
from a square?
contagious mathematics that travel
in as few folds as possible can be a per
from person to person in the community, because they are so
plexing challenge. Figure 1 shows how to fold a 2 X 2 checkerboard in only three folds.
elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
3
{[]'
Fig. 1.
Observant
- - mountain
valley
....____._..!"\:
�
skeptics
quarrel
may
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?)
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
Fig. 4.
board? What is the fewest number of
5. We are all familiar with geometric
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?
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
P 1 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
Please send all submissions to the
Problems 3 and 4 above should be rep
Mathematical Entertainments Editor,
resented, for example.)
Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380,
6. When you did Problem 5 and con
Stanford, CA 94305-21 25 , USA
sidered the fold in Figure 3, you prob
e-mail:
[email protected]
38
Fig. 2.
THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+ Business Media, Inc.
ably included something like the fol-
lowing in your list of basic folds: Given
sidering auxiliary creases made during
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?
the folding process but not used in the
7. If we think of the paper as lying in JR2 (or if you prefer,
IC),
I
end). How many colors does it take
r- -
to color the regions in between the creases in the crease pattern, making sure that no two neighboring regions
1/
(sharing a boundary line) receive the
/
/
--;) ,
'
'
'
'
'
Fig. 6.
same color?
what type of
algebraic equation is solved by the ba
lO.b. The vertex in Figure 6 can be
sic folding operation in Problem 6?
arranged, or tessellated, with copies of itself four times to make the very
8. If we consider the piece of paper to
interesting crease pattern called the
exist in the complex plane, define
origami numbers to be those points in C that are constructible via paper
square twist (Fig. 7). Use what you de duced from lOa to compute how many
Fig. 5.
valid mountain-valley assignments ex
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?
ist for this crease pattern.
lO.a. Creases come in two types: mountains, which are convex, and val leys, which are concave (see Fig. 1). These are often distinguished in origami instructions
by
different
types
of
9. Most models in origami books are flat models. That is, when completed
just choose which crease will be moun
they can be pressed in a book without
tains and which will be valleys willy
dashed lines. But a paper-folder cannot
introducing new creases. The classic
nilly! Indeed, in Figure 6 the single
flapping bird (Fig. 5) is one example.
vertex crease pattern can fold flat, but
Take any flat origami model, unfold it,
not using the prescribed mountain
and consider the creases used in the fi
valley choices. Why is this impossible
"Solutions" follow up,
nal folded form (i.e., we are not con-
to fold flat?
61-63.
Fig. 7.
anyway-pp.
MOVING?
We need your new address so that you do not miss any issues of THE MATHEMATICAL INTELLIGENCER
Please send your old address (or label) and new address to:
P.O.
Springer Journal Fulfillment Services 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.
What Do You Need a Mathematician Fort Martinus Hortensius's
Rowe , Ed ito r
JJSpeech on the Dignity and Utility of the Mathematical Sciences" (Amsterdam 1 634) Volker R. Remmert
Send submissions to David E. Rowe,
I
I
n 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 1 1 , 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 me diae, 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 1 7th 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 certitu dine mathematicarum [Mancosu 1996; Remmert 1998, 83-90; 2004]), the math ematicae 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-
. . . d eliberate strategies to place the mathematical sciences i n the p u b l i c 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 math ematical sciences) by Martinus 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 1 found their way into print. 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-
Fachbereich 1 7 - Mathematik, Johannes Gutenberg University,
1 For a selection of these and related pieces see the bibliography I I ; cf. the discussion in [Remmert 1 998,
055099 Mainz, Germany.
1 52- 1 65; Swerdlow 1 993].
40
THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+ Business Media. Inc.
·,
•
·: .
agator of the Copernican system, Philipp
-
•
.
M A .� ·.:X- I N i
·
0···.- · .
' ·
1 ._
- · s:-
ff O .R T E N S I I
4
·: : · ·
;_ ;
·�: ·.
• ·
I
\ ::
� -�
�� l
.
y�
' ., · · f'
.
;.�<
�·:.
f�\"' �·� ll '
:· �
� ·�
.
l-�,�-6
·
-
·;�·
�4' · . ,
:.
.
�f · " 1: :-ohs ;
..
' .r ·
/
0
'
i" ,i
..
•·
V T I· ·Ld:: T A T E
ET·
· M .A �T' Jt\ :E _S
• _.
collaborated
,1.
,I
..
r:./r /.!.
' •
. • ·f _J � .., • .:: , ,{_,? .:l. . t·t. ....l
·
:.
_ Ha�ita, �ll- illufir1 Gy�afio., $�n�w� �?P.�Iique •
-
· · , X mftelociamenus · cum · ex 'autoii.t'clte rMignmco��nl DO. � o;�svt:v 1<1 .ac . !' B�AT:�a v w tjuf�·yr&is_ /-. J ,
�-
Lediones Mathe�ticas aufpicaretttt ,
..
'
V: I I £
E. l A- . � 4 1 1 ., ·•,
� ... .
; ...
C I :> • ' ..--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
Oilqum{11111tt1A D I! I tflllilf{tttimti• MIUidum CAII�t�tmht ; """ & {tctqnAI ur:tLu J,pibw, & r•Jis frt�tit[A v•'-ilw C«li l llffnt�t •fulu.JirllM fulure rlltAIII : o-ittliWftilqiiAm TmM ,;miUitilna ,.fJJIIrIl Pifiiw , & "IIICTi U"'PinnAt AtrA lllr�.i ; Fdf• H o M o tfl ; ""�""* 111 t•rr.i fllh ;,.,liM lllmdtrm, A111nnnq•e , J•i me!Urri I""' ufnra. Til,. q••qllt 111gnn• ""rt"'PWIIt ct�rnmt
FAll• D u nur-u ; {t4tffttq.U ,.D Tm,rm, Homo ad COD· N- MRI , Cr iJt{11i11s StliJ LII!Wqlll w,,, , tempWidii DUll$ i SttlUTIIIftlfat fbir•, tn...fJitfUW, U 11'1111 , Fdftrj dit- pofftt IJIIIIJIII rtiUnt lilllsllt . · N"""lf" 0 £ u s Tnr• lj...,._ M f.rttrtllto , C«kllml lltdit j/lj .,;,_, If"" urpnis •.{'i en. rinclis rtlit"' , {IIi .,, ,�, m.u.
caW.. itm1711 ,nmr . u q.. ufimdnnlllltt . Hille dt� 1% iJJ• jAm lmtport, prtritr� tmnts J:!.!ltit /UIIIT 111J1U trtlt Ntii#Ttl ctrnm l*f� ; l".f""" Tm1.trll8111 ; 14Mt t��r6Wt. ; & Altil lbatibra eLrpfnJIIIVils; llnliA{1u rirmus ;
Et hioc ortus Alhoao111ia! .
'"""'�"'IJ""Initmirm, & ragnt t•li• · Cl•r• D u rorsm,efli:u Ji"'f• pn .,w,. ; H.r tz11'1JIII P'fl rt�rits i11Ufot CIITtl t.btm :
Dm:m ;, C«,_ rldt� , Alliml/jllt nra4.. Pmtliu11 PtiMm ; vtlilt1 c.. ""'ihu Or•u Sct11tni ; 1ft rttltatf'tltiJi,. liUifi&t Pha:bi , liiiJIIt rictr MtiUit ; l'flllll lfllil ttrmiwu biN ; Cynthia m ttlni rtp•rtt{"" U.... rttwjif; bt tz•ii t-lmtil Sarurm f!tfrlttts Hila: �'"" ,;, sm,ru .,,..,. rrum {trio
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
N• l'tllit ,,._,, � : ,_,. _,;hrs iUi H.U 11mt tnJtlfis StlnJ Cl"'f''tUnt relltllt ; lptt�fqu riM LIIIU, Stt/Ji{tfllt{qtmtis
c..,.,""""' tmirfU prtl{crillnt tllf{fU Hif't-'»diligto• H l P p A .. c H I """' vifihJ ' - i.fllt• . mfnis c iA fulm q�G Cmiltr txpln"" Stlis L�t�W¥t rtetf111 , •
formam acceptt. Incnris .tlt� """Ilk' ,.,ihJ sui ,
Ylrnils ,., tAlf•ftUr . NRntilll iUc rtttptis
Figure
2. First page of Hortensius's Canto on the origin and progress of astronomy of 1632.
learning in astronomy and the mathe matical sciences is on display [Horten sius 1632] (Fig. 2). Also, Hortensius proved himself to be very well versed in classical writings and traditions-an aspect of scholarship not to be dis counted in an academic world that still felt a considerable humanistic impulse.
42
THE MATHEMATICAL INTELLIGENCER
Hortensius and his contemporaries saw metaphysics, physics, and mathe matics as parts of theoretical philoso phy, and in his Oratio he flatly asserted that "among these mathematics excels by its certainty." The notion that math ematics guaranteed the highest degree of certainty humanly attainable was a
long-standing epistemological debate, the quaestio de certitudine mathe maticarum, on which he takes a clear, self-confident position. Hortensius boldly opens by exclaiming that no "one can deny that mathematics is, in deed, 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, what ever 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 mathe matics. [ . . . ] Where other sciences, being full of uncertainty and conjec ture, can neither reach the truth by themselves, nor produce a remedy for the falsities they contain, the mathe matical sciences, lacking nothing, suf fice 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 excel lence of the mathematical sciences. But, he says, "the height of science was attained by Archimedes of Syra cuse, 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 his tory and the reading of tales offer oc casions of delight. The study of politics, ethics, logic, all have their pleasures. But the joys of the mathematical sci ences are so strong, so keen, that they attract like something seductive and ex cite the highest alacrity in the minds of their students." The mathematical sci ences, according to Hortensius, "ought to be cultivated and honoured by us and their reputation enhanced, so that through them, aspiring to the knowl edge of the stars in the sky, we may watch more carefully that book of na-
ture3 and we may read it more atten tively. [ . . . ] 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, af ter playing the humanistic parlour game of alluding to the classics for a while, he takes up the utility and practical ad vantage of the mathematical sciences. "These [the mathematical sciences] we have shown to surpass the other sci ences 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 sci ences [ . . . ] in general, to what extent it spreads itself through all orders of disciplines and faculties, and in partic ular 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 math ematical sciences. As we would ex pect, he reminds his audience that "Plato filled the books of his own phi losophy 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 medi cine, 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 sur passed. He boasts that "this is the science that has put lad ders on the world and informed as tronomers of the distance and size of the sun, moon, and planets. This has brought more light to our century than
The notion that mathematics g uaranteed the h ighest d eg ree of certai nty h u manly attai nable was a long-stan d i ng epistemolog i cal d ebate .
. on
which he takes a clear position . was given to all the schools of philos ophy 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 be cause 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 mat ters nor any trace of their investiga tion" [Hortensius 1 634, 16]. In the context of Amsterdam's repu tation as a leading centre of trade, Hort ensius pays particular attention to the advantages of practical arithmetic, ge ography, 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. "Mu sic," 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 lis teners with singular pleasure), it facili tates 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 incanta tion. Asclepiades healed with the trum pet. Theophrastus used the flute with mentally disturbed patients. Thales of Crete dispelled diseases by playing on musical instruments. There is an exam ple 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 com bining 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 eas ily 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, Ger many, Africa, and India, with the greatest variety of weights, coinage, and measures. If anyone should ask them by what art their laden goods re turn 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 compre hended in it, that they could do with out it only with clear loss of their pos sessions and harm to their families" [Hortensius 1 634, 13}. Geography, of course, is also indis pensable 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 pru dent (in other respects) and brave lead ers into ruin. The same thing has re peatedly overturned the fortunes of merchants, as, on the other hand, ex ploring 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-
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 sci ences contributed to navigation, navi gation to trade, and trade to the solid and firm prosperity of our country." On this basis, that the mathematical sci ences are essentially equal to prosper ity, he appeals to the authorities of Am sterdam to promote further the study of the mathematical sciences: "You rule a city which is very famous and powerful in the whole world. Its ex " pansion came from the study of the mathematical sciences, especially as tronomy and navigation. Use the city's energy so that the mathematical sci ences never lose their strength" [Hor tensius 1 634, 1 7f]. ' In his concluding remarks, Horten sius 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 He outlines the importance of navi mathematical contemplation; you will gation for the rise to power of Venice often find a small space of time in and Genoa, and the Spanish and Por which you may dilute the worrisome tuguese empires. However, these were troubles of business with the pleasure now superseded by the Dutch, whose of the mathematical sciences. Thales, success is also rooted in navigational one of the Seven Wise Men of Greece, skills: "we Dutch, having struck off the had time for both mathematical stud Spanish yoke, when we began to ap ies and trade. For, having foreseen proach the remotest shores of the the richness of the olive crop, he world, were inferior in eagerness and hired every press and mill in Miletus; success to none of the others. At one and afterwards when he leased them time we hardly ever entered the At out at huge prices, he showed his lantic Ocean, but sustained life on friends not only that a wise man could moderate voyages; [ . . . ] But after the be rich if he chose, but also that philo knowledge of the mathematical sci sophical and mathematical studies ences increased here, and the naviga are not at all foreign to trade" [Hor tional art began to be practiced more tensius 1 634, 19]. Hortensius's Speech on the dignity intensively, we filled all the seas with our voyages; we came to the richest and utility of the mathematical sci lands of the East and West Indies, saw ences is filled with such classical allu them and snatched them away from the sions and quotations, proclaiming not foreigners; we circumnavigated the only the dignity and practical utility of globe; we discovered lands; we found the mathematical sciences but also new straits; [ . . . ] So we have con their antiquity. It made a convincing tracted the market of all merchandise case in the prosperous city of Amster within the angle of the world, Holland, dam in the Dutch Golden Age.
gions separated by the whole sea and to frequent foreign peoples widely dis persed in all directions. Trusting to this art, mortals, among sea monsters and savage storms, among rough straits and a thousand dangers of death, com mit 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 merchan dise of Africa. Not only individual af fairs depend on navigation, but also both the continuation and the fall of the fates of kings and states."
. after the
knowle d ge of the mathematical sc1ences
increase d here ,
. we fi l l e d al l
the seas with
o u r voyages .
VOLUME 26, NUMBER 4, 2004
45
BIBLIOGRAPHY
matics, in: Gergmans, Luc/Koetsier, Teun
tronomica, Nuremberg 1 537 [Reprint in: Re
Hortensius and his context
(eds): Mathematics and the Divine. A Histor
giomontanus, Johannes: Opera collectanea,
Berkel, Klaas van: Alexandrie aan de Amstel? De
ical Study, Amsterdam, to be published in
ed.
2004.
4 1 -53].
il/usies van Martinus Hortensius (1605-1639), eerste hoogleraar in de wiskunde in Amster
Swerdlow, Noel M . : Science and Humanism in
C. L./Knegtmans, P. J ./Kox, A. J.Neen, T. J .
the Dignity and Utility of the Mathematical Sci
(eds): Athenaeum //lustre. Elf studies over de
ences, in: Horwich, Paul (Hg.); World Changes:
dam, in: Haitsma Mulier, E. 0. G./Heesakkers,
the Renaissance: Regiomontanus's Oration on
Amsterdamse Doorluchtige School 1632-
Thomas Kuhn and the Nature of Science,
1877, Amsterdam 1 997, 201 -225.
Cambridge (Mass.)/London 1 993, 1 31 -1 68.
Dear, Peter: Discipline & Experience: The Math
ematical Way in the Scientific Revolutio n,
Praising the Mathematical Sciences
Chicago/London 1 995.
Barozzi, Francesco: Opusculum, in quo una
Hortensius, Martin: In viri clarissimi Philippi
Felix Schmeidler,
OsnabrOck
1 972,
A U T H O R
Oratio, & duae Quaestiones: altera de certi
Lansbergii Opus astronomicum tabulasque
tudine, & altera de medietate Mathemati
motuum caelestium dudum ab omnibus
carum continentur, Padua 1 560.
desideratas Carmen, quo ortus & progres
Brahe, Tycho: De disciplinis mathematicis Ora
sus astronomiae ad nostra usque tempora
tio (1574), in: Dreyer, John Louis Emil (ed.):
ostenditur, in: Lansbergen, Philipp: Tabulae
Tychonis Brahe Opera Omnia Tomus I, Am
motuum coelestium perpetuae; Ex omnium
FB 1 7 - Mathematik
sterdam 1 972 (Reprint of the edition Copen
Johannes Gutenberg-Unrvers1tat Ma,nz
temporum
Observationibus
temporumque
omnium
constructae,
Observationibus
consentientes. Item Novae & genuinae Mo
hagen 1 91 3), 1 43-1 73.
D - 55099 Ma1nz. Gennany
Cavalieri, Bonaventura: Trattato delle Scienze
Matematiche in generale, in: Giuntini, San
tuum coelestium theoricae & Astronomi
dra/Giusti,
carum observationum Thesaurus, Middel
Opere inedite di Bonaventura Cavalieri, in:
burg 1 632, **1 r-**4v [i.e. 1 8-24].
Bollettino di Storia delle Scienze Matem
Hortensius, Martin: Oratio de dignitate et utili
VOLKER R. REMMERT
Enrico/Uiivi,
Elisabetta
(eds):
atiche 5(1 985), 1 -350; 47-55.
e-mail:
[email protected] Volker R. Remmert was tr8Jned as a mathematician
and as a historian.
Apart from the h1story of
early
mod
em European science and culture, his main research tnterests are m t he his
tate Matheseos. Habita in il/ustri Gymnasia
Clavius, Christopher: In disciplinas mathemati
Senatus Populique Amstelodamensis, Ams
cas prolegomena, in: Clavius, Christoph:
terdam 1 634.
Opera mathematica, 5 val l . , Mainz 1 61 1 /1 2,
the first half of the twentieth century,
I, 3-9.
especially the Nazt period. Together
Mancosu, Paolo: Philosophy of Mathematics
and Mathematical Practice in the Seven teenth Century, New York/Oxford 1 996. Remmert, Volker R.: Ariadnefaden im Wis
senschaftslabyrin th. Studien zu Galilei: His toriographie- Mathematik- Wirkung,
Bern
1 998. Remmert, Volker R . : Galileo, God, and Mathe-
46
THE MATHEMATICAL INTELLIGENCER
Dee, John: The Mathematical Preface to the El
tory of mathematics
and science 1n
with Annette l mhausen
(Cambridge/
ements of Geometrie of Euclid of Megara
UK), he is currently preparing an En
(1570). With an Introduction by Allen G. De
g lish translatton and commentary of
bus, New York 1 975. Regiomontanus, Johannes: Oratio lohannis de
Monteregio, habita Patavij in praelectione Al fragani [1 464], in: Alfraganus: Rudimenta as-
Hortensius's speech. Here he ts seen with
his son Floris at the Frankfurt
Book Fair.
�e Sydney Opera House is one of The opera house on the harbour, the I the premier architectural master gardens, and the bridge create an inte
Mathematical Tour through the Sydney Opera House J oe Hammer
Does yoor hometown have any mathematical tmtrist attractions suck as statues, plaques, graves, the cafe where thefamous conjecture was made, the desk where thefamous 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 mayfollow in yoor tracks.
Please send all submissions to Mathematical Tourist Editor,
Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium
e-mail: dirk.
[email protected]
48
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 peninsulajutting 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.
THE MATHEMATICAL INTELLIGENCER © 2004 Springer Science+ Bus1ness Mecia, Inc.
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 Cly of Sydney 4 Harbour Bndge 5 Roya' Bolanc Gat
8 NO
Figure
sign a structural system that would fa cilitate serial mass production, the pre fabrication 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
SITE MAP
1. The Sydney Opera House and its site on Benelong Point.
2. Site map.
all the half shells 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 ver tex 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, experi mented with all kinds of geometric arrangements, from paraboloid to el lipsoid-type schemes, from parabolic to circular ridge. They expended hun dreds of thousands of working hours and used thousands of computer hours on the problem. They designed and ap plied computer techniques never be fore used in civil engineering, and sev eral scale models were made for laboratory tests. Despite all efforts, they obtained no satisfactory solution. The main prob lem that persisted was the lack of avail ability, in any of the models, for mass production of the elements of the sur faces. 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 cur vature is the same in all directions. This is not the case for the paraboloid or el lipsoid. Each half-shell is now a spher ical 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 cir cles of the common sphere. Simultaneously with the develop ment of the geometry of the shells, ex periments were conducted with sev eral 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 ori ental fan. Now each rib can be assembled with similar repetitive concrete segments.
50
THE MATHEMATICAL INTELLIGENCER
. . . the ri bs of a half-shel l rad iate from the pole to the rid g e . .
Figure
4. The tiled chevron lids.
.
.
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. Never theless they all have similar cross-sec tions at the same distance from the pole. Consequently, although the con crete 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 vari ous 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 di rectly on the ribs but were laid in chevron-shaped concrete panels, called "lids" (Fig. 4). The role of the lids cor responded 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 cen tre lines, the great circles of the shell ribs. The radial joints of the lids ac centuate 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 ra dial joints. This interplay of the twin parquetry of the lids and the tiles con tributes much to the visual attraction of the surfaces of the shell complex.
Figure
The Glass Walls
Utzon placed the two main theatres side by side diagonally, both lying ap proximately north to south-a brilliant geometric idea (Fig. 2). By this arrange ment the foyers and refreshment areas
The vertical bars were positione d as generato rs of the th ree su rfaces - the cyl i n d er an d the two cones . are wrapped around both theatres, so that for theatre patrons there is maxi mum 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-
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 cylin der at the top and the cone underneath, so its boundary conditions depended on the boundaries of the other two sur faces. 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,
5. The three surfaces of the large northern window.
VOLUME 26, NUMBER 4, 2004
51
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
Figure 6. Concrete beams spanning the concourse.
park, notice the double-helix-shaped ramps, allegedly the first construction caused special problems. They were
two cones also lie on the center line.
not to be visually invasive from inside
This is not only an interesting geomet
For more information about this
or obstructive from outside. Chrome
ric coincidence, it also falls in "line"
wonderful complex, consult the litera
of its kind in the world.
glazing bars were used on the outside.
with the design, which simplified com
ture below and visit the Web site
The vertical bars were positioned as
putations for prefabrication of the mul
www .sydneyoperahouse.com.
generators of the three surfaces-the
lions and glass panes.
cylinder and the two cones. The radial
The second,
smaller north-facing
joint lines of the tile lids on the shells
wall was built with the same three sur
may well have been meant to echo the
faces. None of the other walls was built
generator lines of the glass surfaces.
with three surfaces. Several other walls
Inside, the glass walls are supported
were built with two surfaces-vertical
by steel structures, the basic element
elliptical cylinders on top and an ap
of which is called a "mullion. " The
propriate cone underneath. Apparently
shapes and the appropriate positioning
a cone was used for those walls where
of the mullions presented the same vi
the greatest possible ground area was
sual problems as the chrome bars. No
required. The concave "stomach" of the
tice that the concrete fan-like shell ribs
cone provided more ground space.
and the vertically placed steel mullions reflect each other.
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 9425. [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.]
The Concourse Beams One more high point: you must see Ove
University of Sydney
sitioned and radiate from the theoreti
Amp's
NSW 2006
cal centre line of the main hall. Addi
beams over the vehicle concourse (Fig.
Australia
tionally, the theoretical apices of the
6). These have a remarkable sculptural
e-mail:
[email protected]
The mullion planes are vertically po
52
THE MATHEMATICAL INTELLIGENCER
masterpiece--the
concrete
EDWARD G. EFFROS
Matrix Revol utions : An I ntrod uction to Quantu m Variables for You ng M athematicians
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 [9f. In classical physics, quantities such momentum, and energy are regarded
as
functions. In quantum
the functions by non-commuting infinite matrices, or to be operators on Hilbert spaces. This enigmatic step remains most daunting obstacle for those who wish to under stand the subject. Although there exist many excellent mathematical intro ductions to quantum mechanics (e.g., [12], [17]), they are un derstandably focused on the development of mathematically coherent methods. As a result, mathematics students must postpone understanding why non-commuting variables ap peared 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 in formative "comic book" [1 1], 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 the
(1 )
more
theory
as
position,
one replaces
precise, self-adjoint
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 de rivation 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.
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 f(t)
(4)
=
L Cnei(n w)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
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 f(t) = g(t) =
ei(wt +a)
=
c_ 1e - i wt + c 1eiwt
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) = 54
THE MATHEMATICAL INTELLIGENCER
c_ 1e - i wt + c 1eiwt
Cnei(nw)t
L
dnei(nw)t,
nEZ
then f(t)g(t)
=
I
k,nEZ
ckdn - kei(kw +(n - k)w)t
=
I anei(nw)t,
nEZ
where an is the "convolution" (5)
cos(wt + a) = Re
L
nEZ
an = cc * d)n =
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
Cn,n 'ei(n w+n' w' )t.
I
kEZ
ckdn - k·
Furthermore sd(w) is closed under conjugation, since ](t) =
I c�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 (cwei wt) wE spoA· 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=
[force F] = [ma ] = M:£2J -2 [potential energy V] = [ - Fq] = M,;£2 2J - 2
bination principle. We may doubly index spA, i.e., we may let spoA = { Wm,n lm,n EN, in such a manner that Wm,n + Wn,p = Wm,p
(6)
[kinetic energy T] =
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)
=
=
=
where
[� Cm,keiwrn,kt dk,neiwk,ntl [� Cm,dk,neiCwrn,k+wk,rJtl [ am,neiwrn,ntJ
a = cd is the usual matrix product, and f(t)*
=
[an,me -iwn,mt]
=
[a*m,neiwm,n t]
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)
Wm'n
= 27TR
c
-
m2
-
c 27TR , n2 -
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]
=
[acceleration a] =
[ �� l 2
Idq L dt2
l
= :£ '2!- 1 = ;£2J -2
[momentum p] = [mv] = M:£'2! - 1
ri l V+
mv2 = M:£22!-2
T] = ,M;£2 2J - 2.
[total energy H] = [
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:£2 2! -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;£2 2J - 1 . In classical physics, the (restricted) action along a pa rametrized curve y is defined by the formulas J[y] =
I
'Y
pdq
=
r Tdt, 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,;£2 2J - 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
principle to predict very accurately the value of the Rydberg
J associated with an atom can take only the discrete values nh, where h is a universal constant and n E N.
constant R as well as the "radius" of a hydrogen atom.
An early task of quantum mechanics was to reconcile
Bohr's "old" quantum theory suffered from a number of defects. In particular, the increasingly technical quantum
the particle and wave properties of "quantum objects" such
conditions seemed unnatural, and it was difficult to calcu
as photons and electrons. Albert Einstein [5] related the en
late the "Fourier coefficients"
ergy
E and
momentum p of a photon to the frequency v
and the wavelength A. of the corresponding wave. Noting
am,n·
The quantity
measures the intensity of the frequencies
wm,n,
l am,nl2
or at the
level of a single atom, the probability that a jump from m
that Elv and pA. are action variables (see above), he pre
to n might occur. Just as one cannot "in principle" predict
dicted that each of these equals the "minimal action" h; i.e.,
when a radioactive atom might decay, one cannot say when
we have the Einstein relations
an electron will "jump." This is a prototypical example of
E=
hv = hw
p
h/A.
=
=
the probabilistic nature of quantum mechanics. Heisenberg concluded that the weakness of Bohr's the
hk
ory was that it was concerned with predicting the hypo
where h = h/2 7T. Subsequently L. de Broglie [4] proposed that these relations were valid for all particles exhibiting
thetical singly indexed energies En rather than the actually observed doubly indexed frequencies
wm,n· As
we have
the wave-particle dichotomy, including the electron. It was
seen above, it was this perspective that led him to consider
a short step from there to finding a wave equation for which
matrices. To carry out his program, he had to incorporate
the corresponding functions (9) are solutions. This is pre
the quantum conditions into his framework
cisely the Schrodinger equation.
In 1913 Nils Bohr used the Planck-Einstein quantum con
Phase Space and Action Angle Variables
dition to explain the spectral lines of the hydrogen atom [1].
Quantization is typically applied to algebras of functions.
cular orbits by the quantum condition. To be more specific, he
is concerned with an algebra of functions on a suitable pa
orbit, and that if it drops down (respectively, jumps up) to the
particularly useful about the Hamiltonian formulation is
He proposed that the electron is constrained to particular cir assumed that the electron has a specific energy Em in the mth
nth orbit, it loses (respectively absorbs) energy
Em - En,
which is carried away or brought by a photon with frequency (J)m,n =
(10)
Because the Hamiltonian approach to classical mechanics rameter space, it is ideally suited for this process. What is that each function determines a one-parameter group of au tomorphisms, and in particular, the energy function deter mines the physical evolution of the system. Let me sum
Em - En
marize this theory as quickly as possible.
h
Let us first suppose that we are given a parameter space
When Bohr used the classical Coulomb law to calculate the angular momentum L of the electron in the mth orbit, he
!Rn. We let Slll (M) be the algebra of infinitely differentiable functions on M, and T(M) = M X [Rn be the corresponding
M=
discovered that it was given by L = mh for an integer m.
tangent space. Then we regard (x,v) E T(M) as a "tangent
In fact, by using the Hamiltonian theory from the next sec
vector at x," and it determines a corresponding directional
tion, he and Sommerfeld showed that this coincides with Planck's quantum condition J = mh, and the latter is also
derivative. Given x E M and v = I VJeJ E
true for arbitrary closed orbital motions. Within a few
D cx,v)
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 prop erty for the spectral values, which he called the corre
:
Slll (M) � IR
-n
� m orbits resulted in the
kth harmonic of a fundamental frequency
Wm,m�k
(- � ( )
= 2 7TRC
7
= 4 Tk
2
Rc
m3
+
(m
wm = 47TRc!m3:
� k)2 )
(1 - k/2m) ( 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, de
spite its nebulous character, Bohr used the correspondence 56
THE MATHEMATICAL INTELLIGENCER
VJ
:�. 1
(x).
rectional derivatives that they define, we use the notation
(x,v)
observed that for large m, the electrons behaved almost precisely, a drop of k = m
L
Because tangent vectors are only used to indicate the di
spondence principle. Returning to the Rydberg formula, he classically, in the sense that one obtained harmonics. More
: f�
!Rn, define
I·
a_ L VJ _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 t
1--0>
x : (a,b) � M : 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: DF (f)
=
lim h-->0
ah(j) - f 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 _!l F(q(t)) = m _ dt2 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
aH dq - dt ap aH dp = dt aq ' H(q,p) =
In fact, an arbitrary function a(q,p) on M2 determines a vector field sgrad a
and thus, a corresponding flow u� : Mz � Mz, where y(t) = u�(x0) = (q(t), p(t)) is a solution of the "Hamiltonian system" dp aa = dt ap dp = aa dt aq
( 1 1 ')
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 + = dt aq dt ap dt aq ap
aH a aH a gradH = - - + - aq aq ap ap ' but it is not necessary to go into details.
.!!!!_ aa ap aq
= 0. '
[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 a(q,p) aq ap
_
aQ aP ap aq _
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 (1 1):
If
=
dQ dt
dP
2
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
_
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
: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
aa a = aa a - - - - -, ap aq aq ap
dt
aH aQ aH aQ '
where by abuse of notation H(Q,P) To see this, note that aQ dQ = dt aq aQ aq aQ aq aH = aP =
dq aQ dp + dt ap dt aH aQ aH ap aq ap aH aQ + aH aP aQ ap aP ap a (Q,P) aH = a(q,p) aP '
=
H(q(Q,P),p(Q,P)).
_
(
)
_
aQ ap
(
aH aQ aH aP + aQ aq aP aq
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
(q(t),p(t)) are closed. We may assume that (q(O),p(O)) = (q(T),p(T)), where the period T depends the solution curves
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 ai
co-ordinate system" ( 8,[) with the following properties: • •
H(8,!)
= H(J),
8 increases by
i.e., H doesn't depend on 8, and
27T on
dt
and Bohr's difference formula
each closed orbit.
Given such a system, we will have di
aE ai
w=-=-
on the orbit. Our goal is to find the "simplest Hamiltonian
=
_
aH ao
To make this more explicit, let us "discretize" the action
variable I by setting I
= 0'
= mh
and
I:J.kl = kh.
Then, accord
ing to Bohr's correspondence principle, if k �
and thus I and H(J) are constant on each orbit
m,
y. It follows
that
dO dt
It thus appears that Bohr's correspondence principle is em
aH ai
w=-=-
bodied in the fact that the finite difference with respect to
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 co ordinate change of variable (x,y) � (O,p), maps closed is also constant on each orbit
curves to horizontal line segments.
(q,p) to (8,!) trans forms 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). Be The canonical transformation from
the discrete action variable I approximates the differential quotient with respect to the continuous action variable
trary quantum variables and their classical analogues. Let us use the symbolism
(13) (see
[ 18],
')'
where
(q,p).
y is
pdq = A = R = 27Tl(y),
the unique integral curve that passes through
Thus, assuming that we can find a canonical trans
formation with the desired properties, I is an action vari
pages
[13]
or
[8].
In his calculation, Heisenberg concentrated on the Fourier coefficient functions
circuit of an orbit. The action-angle variables enable us to use Fourier series
a=
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
on the
I a(f)eiCwt c
trix variable A, then for
A(m,n)
of
ei wm,nt
f=m-n
�
m the coefficient
should approximate the coefficient
(ei wt) e. The notation for such spondence will be A � a, and A(m,n) � a(f). Ifj, k � m then
of the harmonic
A(m,m-j ) � a(J) =
a( 8,!) will have period 27T in 8. It thus has a Fourier series
where a(n) is a function of I. Substituting the solution of the
a
If a is the classical function variable "reduction" of the ma
tion a on M2 and using the action-angle variables, the function
a(8,!) = I a(n) ein e,
of a function
A = [A(m,n)ei wm,nt ].
in our analysis of a periodic motion. Given an arbitrary func
(12)
a(f)
phase space M2 and the scalar matrix coefficients A(m,n) of a matrix A in the "expansions"
We define
1 / = - f pdq 2 7T ')' to be the action variable and 8 the angle variable. As one would expect, 8 is multivalued; it increases by 27T on each
The difference operator will be
(I:J.kA)(m,n) = A(m,n) - A(m- k,n - k).
able. For the proof that the transformation exists (and a formula for 8), I recommend
1 10, 56).
applied to a matrix variable by the formula
cause the purported transformation is canonical, we have
f
/.
For this reason, it seems justifiable to apply this to arbi
(I:J.kA)(m,m -j ) � kh
+ j- 1 :� (J)
a(f)
a corre
(for j =I=
0)
aa (J). aI
The equality is seen if one takes the derivative of (12) with respect to 8. The second reduction is a formal consequence of
(13).
It will tum out that if A � a and B � b, then [A,B] � h --;-{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 'L
j+k-t
(k �a (J) b(k) - a(J) j !!!!_ (k) ) ()[
()[
( I k aa w k-l !!!!_ Ck) = !!:_ ( aa ab _ aa !!!!__) (€) = � 'L
i
j+k- C,MO
()[
() ()
()[ () (}
()(}
I
j+k- tj*O
r1 aa CJ) j ab (k) () ()
()[
)
()[
h = ---:(a,b)(€). 'L
aa (0) = 0 by (12)). ab (0) = ae (see (5)-note that ae 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 corre sponding matrix elements must be negligible "since they correspond to high harmonics in the classical theory." I conclude that =
h [A ,B] � ---:(a,b). 'L
[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.
Because (p , q l = ap � _ ap � = I 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 [P,Q]
h = ---:- 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].
[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: 0-52 1 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.
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 0-48665984-4. [3] A. Connes, Noncommutative geometry. Academic Press, Inc . , San Diego, CA, 1 994. ISBN : 0-1 2- 1 85860-X.
[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 97 1 . [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
Los
Angeles,
e-mail:
IS
CA 90095 - 1 555
USA
[email protected]
After undergraduate work at MIT, Edward Effros completed h1s PhD dissertation at Harvard 1n 1961 under George Mackey. He
Mathematics
UCLA
eter
nally grateful to Mackey for steenng him to the beautiful merging
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.
of algebraic and analytic techniques with the mystery of ·mathe
W1th his wife Rita, a well-recognized immunologist, Effros en
matical quantization" 1n the work of R1chard Kadison. This area of
joys hiking and listening to classical music . The1r horizons are
mathematics, and the personal support of Kadison, set the d1rec-
11on of all Effros's work, up to his current project of quantizing Ba-
broadened by the1r daughter, a phys1cian, and their son, an archi tect.
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).
60
THE MATHEMATICAL INTELLIGENCER
Sol utions (of a sort) to the Origam i Qu iz (see
pp.
38-39)
Thomas Hull
1. The 3 X 3 checkerboard can be
"1-fold" solution to the 4 X 4 checker
folded in only 7 folds. The below solu
board puzzle.)
tion is due to Kozy Kitajima, and was presented at the Gathering for Martin Gardner conference (Atlanta, GA) in 1998. 7
6
2 I
I
5 - 1 - - - - - - -� - - - - + I 4
I --- -;I - - - - - - -;-
3 L. I I ' T - - - r -
T
I /
-
/ I /-
/
2. Imagine our piece of paper is the
valley-fold
1
plane, IR2 , and our square is drawn on the plane with vertices at ( ::':: 1 , ::':: 1). If
mountain-fold
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 con
Readers will be tempted to general ize 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 al low an origami move known as a "squash fold" to count as one fold. (A squash fold is shown in the middle fig ure below.)
taining 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 ver tices or at (O,o), then P will be con tained in a square on the paper. Other wise 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 semicir cles of radius 1 centered at the mid points of each side of the square. How these semicircles overlap determines the solution, shown below left.
In
fact,
the
question
of
what
"counts" as one fold is non-trivial. Bit ter debates
on this very question
emerge when practiced origamists face this puzzle. For example, the below crease pattern presents a " 1-fold" solu tion 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 direc tions) 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
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 re gion, 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, hep tagons can be produced. (See 3.)
3. If we let the side of the square be of
Origamist
length one, then the triangle made by
Jeremy Shafer has a similar, iso-area,
this folding procedure is equilateral be-
reversal
of the
creases.
© 2004 Springer Sc.,nce+Business Media, Inc., VOLUME 26, NUMBER 4, 2004
61
cause its sides all have length one. (Its
the perpendicular bisector to line
spect to P 1 and L1. To see what is go
left and right sides are both images of
segment P lPz. )
ing on, do the following exercise: Take
the bottom side under folding.) It is not
3. Given two lines, L1 and Lz, we
a piece of paper and let the bottom side
the biggest equilateral triangle possi
can fold line L1 onto L2. (Angle
be line L1 and take a random point p 1
bisectors.)
on the paper. Fold and unfold p1 onto L1 at many different places, making a
ble, however. The biggest is symmetric about a diagonal of the square, and a folding method for such a maximal tri angle 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.)
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
sharp crease every time you do so. The below figure
illustrates
what you
should see.
p that is perpendicular to L, in other words, folding L back onto
D
itself so that the crease passes through p. (Dropping a perpen dicular.) 6. Given two points p 1 and pz and a
L,
line L1, we can, whenever possi ble, 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
Actually proving that this is the equi lateral 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 tri secting an angle. Drawing some auxil iary 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.
and L2, we can make a crease
This exercise makes one suspect
placing p 1 onto L1 that is per
that the process of folding a point p1 to
pendicular to Lz.
whose focus is p 1 and directrix is L1.
whenever possible, make a crease
There are a number of ways to prove
that simultaneously places P1
this; for an analytic approach let p1
onto L1 and pz onto Lz.
(0, 1), let L1 be the x-axis, and find the
Operations 1-3, 5, 6, and 8 were for mulated by Humiaki Huzita. (It is not certain if he was the first to do this, but he was the first to publish these oper ations. See [5) and [6).) Move 7 is, amazingly enough, a very recent addi tion developed by Koshiro Hatori (see
~
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 some thing like the following: 1. Given two points p 1 and pz, we
62
a line L1 is actually creating a crease
8. Given two points P1 and pz and two lines L1 and L2, we can,
[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 oper ations 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 possibili ties while formalizing the degrees of
line that is tangent to the parabola
=
equation of the crease that results when p 1 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 fold ing operation 6, and clearly if the point
pz is chosen to be in the interior of the convex hull of the parabola with focus
p 1 and directrix L1. then the operation will be impossible to perform.
7. See the solution to Problem 6. Be cause we get a parabola, folding oper ation 6 is actually solving a quadratic equation for us. (Can you give an ex
2
plicit method of solving ax + bx + c = 0 where
a, b, and c are positive
freedom one has when folding one ob
integers?)
ject to another.
8. The set of numbers constructible
Also, Koshiro Hatori claims that most of these operations can be thought of as special cases of operation 8. Can you
with straightedge and compass is the smallest subfield of C that is closed un der taking square roots. So straight
can make a crease line connect
find a way to make this work?
ing them.
6. The basic origami operation cited in
equations, but certainly cannot con
2. Given two points P 1 and pz, we
Problem 6 cannot be performed if the
struct any algebraic
can fold p1 onto p2. (This creates
point p2 is poorly positioned with re-
imal polynomial is cubic. The classic
THF MATHFMATI
edge and compass can solve quadratic a
E C whose min
proof that a straightedge and compass
ferent directions when folded, insuring
Maekawa's Theorem. See [4] for more
cannot trisect an angle, for example, is
that they receive different colors. Thus
information.)
built on cos 20° being degree 3 over the
this is a proper 2-face coloring of the
rationals.
crease pattern.
In Problems 6 and 7 we saw that the
One can also prove this using only
origami operation 6 proves that paper
graph theory. First argue that all ver
folding can solve quadratic equations.
tices in the interior of the paper of a
Thus the
flat model have even degree. Thus if we
set of origami numbers
contains the set of straightedge-and
consider the crease pattern to be a
compass-constructible numbers. Fur
graph, where the boundary of the
thermore, because we know that pa
square also contributes edges to the
per-folding can trisect angles, we know
graph, the only odd-degree vertices
that the field of origami numbers
would possibly be on the paper's
strictly contains the field of straight
boundary. Create a new vertex v in the
edge-and-compass numbers. Actually,
"outside face" and draw edges from it
folding operation 8 turns out to allow
to all the odd-degree vertices on the pa
evidence, the below figure depicts the
even number of vertices of odd degree,
us to solve general cubic equations. As of possible images of P 2 = (.5, - . 5) as PI = (0, 1) is folded repeat edly onto line L 1 which is y = - 1 . This locus
graph certainly looks cubic, and deriv ing its equation can be done using
per's boundary. Graphs always have an 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
similar analytic methods to those in the
bipartite), and removing the vertex v
solution to Problem 6. For more infor
then gives a 2-face coloring of the orig
mation, see [ 1 ] .
inal crease pattern.
1
p,
,
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. Hull ed. , A. K. Peters, (2002) 307-328. [3] K. Hatori, Origami versus straight edge and compass, http://www.jade.dti.ne.jp/hatori! library/conste.html
lO.a. The problem is with the two mountain creases that surround the 45°
[4] T. Hull, The combinatorics of flat folds: a
angle at this vertex. The two angles
International Meeting of Origami Science,
neighboring the 45° angle are both goo.
Mathematics, and Education, T. Hull ed. ,
Thus, if the creases surrounding the
A . K . Peters, (2002) 29-38.
45° angle have the same mountain valley (MV) parity, then the two goo an L
lO.b. The answer is 16. There are sev
eral ways to enumerate the valid MV
survey, Origarn1'J: Proceedings of the Third
[5] H.
Huzita,
"Understanding
Geometry
Through Origami Axioms: is it the most ad
gles will both be forced to cover up the
equate method for blind children?" in the
45° on the same side of the paper. If
Proceedings of the First International Con
these creases are then pressed flat, the
ference on Origami in Education and Ther
two large angles will be forced to in
apy, J. Smith ed. , British Origami Society,
tersect one another, and self-intersec
1 992, pp. 37-70.
tions of the paper are not allowed (un
[6] H. Huzita and B. Scimemi, ''The Algebra of
9. All flat origami crease patterns are
less one is folding in the fourth dimension, which we assume we are
ings of the First International Meeting of
2-face colorable. The proof is simple:
not!).
take your flat-folded origami model
It turns out that mountains and val
and lay it on a table. Color all regions
leys can be assigned to these crease
of the paper yellow if they face up (i.e.,
lines and be flat-foldable if (1) the
Paper-folding (Origami)." In the Proceed
Origami Science and Technology, H . Huzita ed. , 1 989, pp. 205-222. [7] R. Lang, personal communication.
away from the table) and all regions
creases surrounding the 45° angle are
Department of Mathematics
pink if they face down. Any two neigh
not the same and (2) the number of
Merrimack College
boring regions of the crease pattern
mountains and the number of valleys
North Andover, MA 01 845
will have a crease line in between
differ by 2. (This last result holds for
USA
them, and thus they will point in dif-
general flat vertices and is known as
e-mail:
[email protected]
VOLUME 26, NUMBER 4, 2004
63
lil§'h§l,'iJ
Osmo Pekonen, Editor
I
how pragmatic and incoherent science
Gentzens Problem. Mathematische Logik im nationalsozia I istischen Deutschland.
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 in tellectuals: "Unfortunately one needs
b y Eckart Menzler- Trott
them. Otherwise, one might-I don't
BASEL, BOSTON, BERLIN: BIRKHAUSER VERLAG, 2001 , xviii + 41 1 pp. €43. ISBN 3-7643-6574-9.
REVIEWED
BY PETR
know-wipe them out or something. But unfortunately one needs them" [ 1 ] . Need one say more?
HAJEK AND
Menzler-Trott shows how compli
DIRK VAN DALEN
cated, incoherent, and often inconsis tent the Nazi philosophy of science is.
Feel like writing a review for The
G
For Gentzen and logic the oft-men erhard Gentzen (1909-1945) was
tioned German or "Aryan" mathemat
undoubtedly one of the most im
ics, as opposed to Jewish mathematics,
portant mathematical logicians of the
is the central issue. The author takes
twentieth century, the founder of mod
the reader through the confusing un
em mathematical proof theory. His
dergrowth of political arguments and
you would welcome being assigned
work is of great importance, not only
machinations ( ch. 4).
a book to review, please write us,
for pure mathematical logic but also
For mathematics and logic the key
for computer science, in particular for
figure in politics is Ludwig Bieberbach,
Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if
telling us your expertise and your predilections.
The
who formulated a form of racial clas
book under review is a detailed biog
theorem-proving by computer.
sification of mathematics and mathe
raphy of Gerhard Gentzen and at the
maticians. But even Bierberbach had
same time a penetrating analysis of the
difficulty
situation of mathematical logic (and of
consequences of his views when not
mathematics) in Nazi Germany. The topic of science and the Third
just academic issues but real people were concerned. The formalistic side
Reich has been discussed repeatedly
of mathematics, "symbol pushing," was classified as being Jewish, whereas the
and extensively, but the position of
accepting
the
unpleasant
logic in Nazi Germany is somewhat
intuitive approach belonged to the
apart from the familiar subjects, as it
Aryan domain. This would obviously
represents a clash, or at worst a com
speak against David Hilbert's formal
promise, between the realm of ultimate
ism (which in its extreme and slightly
transparency and that of a dark and
caricatural form claimed that mathe
opaque political philosophy.
matics was a meaningless play with
The author takes great pains to an
symbols); but when from the more
alyze the ambiguous relationship be
philosophical
tween logic (and thus logicians) and
Hilbert was attacked for this, Bieber
side
(Steck,
Dingler)
the Party. One might wonder what a
bach chose the side of the great Got
neutral,
tinger. It was the logician Scholz who
"value-free"
discipline
like
logic had to fear from any regime what
(encouraged by Bieberbach) under
soever. The answer is not all that sim
took the defence of logic and Hilbert's
ple, mainly because a German science
program.
Column Editor: Osmo Pekonen, Agora
and education, as promoted by the
This is the social-political stage
Center, University of Jyvaskyla, 400 1 4 Finland
Party, was a fairly intangible notion.
where Gerhard Gentzen had to make
e-mail:
[email protected]
The present book once more illustrates
his
64
THE MATHEMATICAL INTELLIGENCER © 2004 Spnnger Scrence+ Business Media, Inc.
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 "teachers join 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 Weyl 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 wehrun tauglich (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, un worldly 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 de sertion? 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 de scription of the last days of Gentzen as reported (mostly) by Krammer (p. 273-278). The German prisoners were subjected to cruelties, lack of food, un hygienic circumstances, no medical as sistance, 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 com parably mild military regime of the sig nal corps, had no chance. Only the strong and (mentally) fit could with luck survive. On August 4 Gentzen died in prison of total exhaustion. Appar ently it was not the rampant typhoid that killed him, but it was just his phys ical and mental constitution that could not endure. Before Menzler-Trott's in vestigations cleared up most of the cir cumstances 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 incom plete information, and at the time, the correspondence quoted by Menzler Trott 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 great est logicians. It always is-and the more so in the case of a scholar who lived and died in uncertain times and circum stances-surprising how much evi dence a clever researcher can fmd. Menzler-Trott has created a fitting mon ument to an introverted and naive sci entist, 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 gen erous sprinkling of quotations, re minding 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 de serves. Indeed Gentzen's systems of Natural Deduction and his Sequent Cal culus are outstanding specimens of an almost architectural beauty. Matters of this sort are, however, difficult to con vey; 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 Peen emiinde, 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 de mand considerations from their vic tims; 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 sci ence called Nazi mathematics. Menzler Trott has provided a wealth of facts and details, and he has gone a long way to wards their interpretation. One does not have to agree with every single conclu sion in order to appreciate his contri bution to our awareness of the dangers of the role of politics in science. In par ticular, we may be certain that Gentzen belongs forever to the giants of mathe matical logic. The book teaches us, and coming generations, a lesson that we should keep in mind: the giants of sci ence 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 1 866- 1 945. 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:
[email protected]
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 7 1 1 9-8
REVIEWED BY TERRY J. LYONS
his book aims to be an introduc
Ttion to the probabilistic methods
used in finance. It targets undergradu ate and graduate mathematicians in terested in the area of mathematical finance rather than mathematical prac titioners, 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 mea sures (risk-neutral measures) and the absence of arbitrage. The remaining chapters of this part treat the value of a single risky transaction, dealing re spectively with utility, portfolio, opti misation, 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 eco nomic problems of utility optimisation and equilibrium. The authors limit their attention to the case of one time inter val, and, for example, give a version of Chris Rogers's result that the absence of arbitrage is equivalent to the exis tence of an optimal consumption pat tern; they make the connections be tween exponential utility and relative entropy. The text is well-paced, clear, and methodical, and it will be easy for a rel atively advanced student with a rea sonable amount of time to learn the material well. There is a kind of stu dent who will find this material very at tractive: well-trained pure mathemati cians, happy with the basics of modem analysis (they should understand con vexity, 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 in vestor behaviour in such an environ ment. 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 re stricting themselves to the discrete set ting 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 introduc tions to many of the more sophisti cated ideas currently under considera tion in the mathematical analysis of incomplete markets. It is a theoretical tour de force and will equip the reader well to under stand much of the contemporary liter ature, if he or she is willing to add a lit tle bit of continuous time. I have no hesitation in recommending the book to students who have already done rig orous courses in probability and analy sis and would like to understand some of the mathematical modelling that de velops out of considering incomplete financial models. However, it is striking that the word "volatility" appears only once in the in dex 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 trad ing 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 in sightful-it definitely does aid one's understanding of the general picture. But, in my experience this book should
be complemented if a student is seri ously interested in mathematical fi nance. In particular, students should not be afraid of complete models! Or of models chosen to be low-dimen sional in order to keep them computa tionally 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 approxi mate well enough can be far more use ful than perfect models of excessive complexity. The situation is more so phisticated and more robust than an outsider might appreciate. In many parts of their business, banks and traders are in effect retail ers: 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 ap proach is to use a well-tested model that is broadly correct, calibrate it to market prices, and then use it to inter polate 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 per fect 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 fre quently 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 hedg ing). 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
people in the same room, see what
not have to be! This is the residual po
before.
sition-it needs to be contained so as
recognized (again) as an integral part
to represent a small liability to the busi
of human culture, something to be re
To start things happening, members
ness and should be protected from se
garded with slightly more curiosity and
of both groups deliver a fair number of
rious downside behavior-but it does
slightly less suspicion than before, and
talks (on average, twenty-five per year)
not need to be perfectly hedged to zero.
not just as an (un)necessary evil to for
on topics ranging from architecture
This robustness in the way Finan
get as soon as possible after finishing
and math to zoology and math. There
Mathematics is increasingly
happens.
cial Mathematics is used by the major
high school. Furthermore, this shift of
are also mathematically related art
financial trading institutions is, I be
perception is driven by the idea that
shows, plays, movies, and concerts.
lieve, a major factor in its overall suc
mathematics might be interesting, not
Usually people make connections:
cess and reliability, and explains its
only because of its applications, but
both intellectual connections, among
long-lasting effectiveness.
per se.
apparently unrelated subjects, and per
To summarize, the book is great for
Changes like this do not happen by
sonal connections, among intellectu
a mathematically competent beginner to
themselves; they need a lot of effort
ally curious people who may work in
acquire knowledge of finance, equilib
and preparation, usually going unno
apparently unrelated subjects, but who
rium, incomplete markets, optimal port
ticed for a long time. Mathematicians
find that they have more in common
It has perhaps
started trying to explain what they
than they had thought. And this kindles
folio management, etc.
the best account of utility, and uses the
were doing; people from other sciences
conversations, and confrontations, and
mathematical tools at the correct level
and from humanities started to listen;
the diffusion of ideas, mathematical
to get results quickly and effectively.
and somebody in between started or
and non-mathematical. Afterwards, as
ganizing venues where mathematicians
any good virus should,
appreciate
and non-mathematicians could meet
spread elsewhere, back home or at
from this monograph; a student seri
and exchange ideas about mathemat
work, infecting unsuspecting friends
ously interested in finance will need
ics, culture, and everything else.
and co-workers with unexpected con
Complete models are far more im portant than
one
might
supplementary studies.
In Italy, the divide between sciences and humanities is traditionally deep;
such ideas
nections between mathematics and, well, just about anything else.
Mathematical Institute
actually, culture has often been con
The published volumes of proceed
University of Oxford
sidered to be synonymous with hu
ings of the "Mathematics and Culture"
24-29 St Giles'
manities. Even in the "scientific" high
conferences are an integral part of Em
Oxford OX1 3LB
schools, a sizable number of lectures
mer's project. Seven volumes have ap
England
are devoted to humanities (including
peared, and they give a good idea of the
e-mail:
[email protected]
the compulsory study of Latin). So
range of topics at the Venice confer
when in 1997 Michele Emmer (origi
ences. I decided to concentrate this
Matematica e Cultura 2000
nally with P. Odifreddi and E. Casteln
review on the volumes that are (or
uovo, but later by himself) organized a
soon will be) available in English; the
series of conferences on "Mathematics
others, published by Springer-Verlag
and Culture," held annually in Venice,
Italia, are for the moment available in
MILAN SPRINGER-VERLAG ITALIA, 2000, pp. vii1 + 342,
for Italy it was a complete novelty. To
Italian only.
ISBN 88·470·01 02·1 .
some critics, it was an oxymoron and
The first volume contains the pro
doomed to failure. Luckily, the critics
ceedings of the 1999 conference, held
were wrong, and Emmer's creature,
while the Kosovo bombings were be
edited by Michele Emmer
Matematica e Cultura 2003 edited by Michele Emmer
The structure of these congresses is
MILAN SPRINGER-VERLAG ITALIA, 2003, pp. viii + 279, ISBN 88·470-02 1 0·9.
REVIEWED BY MARCO ABATE
I
eight years later, is alive and kicking.
ginning (the
2003 conference was held
just after the Iraq bombing started-and
easy to explain. The idea is to put in
"Mathematics and war" is one of the re
the same room for two days a number
curring themes in the conferences). The
of mathematicians (university profes
twenty-eight papers are subdivided into
sors, high school professors, and, last
eleven sections, each containing two or
but not least, students) ef\ioying a broad
three essays: Mathematicians; Mathe
n the last ten years, the perception of
notion of what can be mathematically
matics and History; Mathematics and
mathematics by the general public
(or
non-mathematically)
interesting,
Economics; Mathematics, Arts, Aesthet
(or, at least, by the general cultured
and a number of non-mathematicians
ics; Mathematics and Movies; Math Cen
public) has been changing.
(artists, journalists,
ters; Mathematics and Literature; Math
Movies
scientists,
what
about mathematicians have won Acad
ever) who somehow have found that
ematics and Technology, an homage to
emy Awards; articles about mathemat
mathematics can be relevant to what
Venice; Mathematics and Music; Mathe
ical results have appeared on the front
they do-or who are willing to be sur
matics and Medicine. The authors range
page of major newspapers; and books
prised by the fact the mathematics can
from Claudio Procesi and Enrico Giusti
concerning mathematics, both novels
be relevant to what they do. The goal
to Harold W. Kuhn and Peter Green
and essays, have sold as possibly never
is also simple: after having put these
away; nineteen are Italians, and fifteen
68
THE MATHEMATICAL INTELLIGENCER
are from the rest of the world (yes, a few
wants to destroy the unrealistic repre
structural ideas that guided him in
papers had more than one author, and
sentation of reality provided by the
composing these pieces. Particularly
Emmer wrote two of them, which ex
classical rules of perspective;
Mos
interesting are the parallels he finds be
plains why nineteen plus fifteen yields
quera R. is fascinated by the metaphor
tween his work and M. C. Escher's
ical uses of the M<:Ebius band and the
paintings, parallels of a structural
Let me describe some of the more in
topological terminology; and Green
and hence mathematical-nature. The
teresting (to me) papers. The section on
away is so attracted by the intrinsic
other papers in this section deal with
"Mathematics
contains
beauty of the rigid and yet rich struc
mathematical models of musical sounds
three essays. The first one, by Giorgio Is
tures that can be derived by numerical
(Giovanni De Poli and Monica Dorfler),
rael, describes Italian mathematics dur
sequences that he builds most of his
philosophical
ing the Fascist years, and in particular
movies around them, using sequences
notion of "listening" (Laura Tedes
the reactions of Italian mathematicians
both as structural devices and for their
chini Lalli), and fractal music (Stefano
to racial laws. The second, by Jochen
metaphorical power. And I am sure
Busiello).
Biiining, describes what happened to
that Greenaway loves the description
Fractals also appear in the works of
the Berlin mathematical school after the
of the relationship between numbers,
Escher, Paul Klee, and Marcel Duchamp,
advent of Nazism. The third, by Silvana
colors, and music in ancient Asia given
according to Roberto Giunti (but I must
Tagliagambe,
in the essay by Tran Quang Hai.
twenty-eight).
and History"
describes the
develop
ment of philosophical and mathematical studies in Russia from Peter the Great
behind
the
admit that in the case of Duchamp I
Other papers describe less metaphor ical
problems
applications of mathematics.
I
found Giunti's arguments not that con vincing); and are somewhat implied by
to Stalin. In all three cases, the descrip
would like to mention at least the papers
the labyrinthine structure of Venice it
tion of the use of "aseptic" mathemati
by Laura Tedeschini Lalli on the math
self, as described by Michele Emmer.
cal arguments to support extremist po
ematics of Indonesian musical instru
On the opposite side of geometrical
litical
and
ments; by Enrico Casadio Tarabusi on
complexity, the excursus of Manuel
horrifying-and instructive. Keeping in
the Radon transform and computer
Corrada on the possible definitions of
positions
is
fascinating
mind the present-day rhetorical uses of
ized tomography; and by Camillo De
straight lines sheds an unusual light on
mathematical terminology and "theo
jak and Roberto Pastres on a mathe
Fred Sandback's sculptures (unfortu
rems" to support economic politics, as
matical study of high tides in Venice.
nately not shown in the book).
hinted at in the papers by Marco Li Calzi
The second volume contains the
Another large section of the book
and Achille Basile, much can be learned
proceedings of the
2002 conference.
deals with mathematics and China.
from their accounts of the (mostly fruit
The twenty-four papers are subdivided
Two very interesting essays, by Jean
ful) relationship between mathematics
in the following eight sections, con
Claude Martzloff and Anjing Qu, deal
and economics.
taining from one to five essays each:
with the history of mathematics and as
The short essay by Lucio Russo on
Mathematicians; Mathematics and Mu
tronomy in ancient China; a third one,
Mathematics and Literature provides
sic; Mathematics and Arts; Mathemat
by Francesco D 'Arelli, discusses the
still other connections between math
ics
and
false perceptions of Chinese astron
ematics and rhetoric; and another in
Venice; toward Beijing
2002; Mathe
omy in sixteenth-century Europe; and
and
Movies;
Mathematics
teresting walk in the rhetorical and
matics and Theatre; Mathematics and
the last one describes Michele Em
metaphorical use of mathematics is de
Comics. The authors range from Gio
mer's trip to a mathematical congress
scribed in the paper by Piergiorgio
vanni Gallavotti and Aljosa Volcic to
held in Lhasa, Tibet.
Odifreddi, discussing numerology, the
Harold W. Kuhn (again: it is not un
The section on mathematics and
ology, and mathematics.
common for some speakers to come
comics contains a description (by Stew art Dickson) of the computer graphics
manifestations
back after a few years to talk about
and uses of mathematics in the arts is
something else) and Sergio Escobar;
techniques used in the Disney movie
a theme common to the papers by
thirteen are Italians and nine are from
Dinosaurs; and a list (by Luca Boschi)
Achille Perilli, an Italian painter with a
the rest of the world.
The
metaphorical
of numerological and arithmetical cu
body of work which he describes as
The largest section is devoted to
riosities in Disney comics. Further
"the theory of the geometric irrational,"
Mathematics and Music. The volume is
more, the participants to the confer
where he plays with and undermines
sold with a CD containing three short
ence are now the happy owners of a
the classical use of perspective in
musical pieces for guitar by the com
copy of a comic book created by Luca
paintings; Gustavo Mosquera R., direc
poser Claudio Ambrosini, collected un
Boschi expressly for this occasion;
Mcebius, where Argen
der the unifying title "Three studies on
knowing the world of comics collec
tinian society just after the end of the
perspective" (which reminds me not
tors, this comic will soon become valu
dictatorship is mixed with topology;
only of the essay by Achille Perilli de
able (alas, it is described but not en closed in the proceedings).
tor of the film
famous
scribed above, but of the joke on art
artist and movie director. Each of these
critics which says that writing about
Of course, not all the presentations
artists describes his own work, and it
paintings is like dancing about archi
at these conferences are of the same
is interesting to compare what attracts
tecture). Ambrosini himself describes
quality. This year
them to mathematical themes. Perilli
his work in an essay, illustrating the
talk on topology and architecture in
and
Peter Greenaway,
the
(2004), I attended a
VOLUME 26. NUMBER 4 , 2004
69
which the speaker managed to convey
imization and Some Modem Twists; (4)
the impression that she (and the ar
The Forgotten War of Descartes and
example, determining where to locate
chitects whose work she was describ
Fermat; (5) Calculus Steps Forward,
a fire station within a community to
ing) had no idea of the actual meaning
Center Stage; (6) Beyond Calculus; and
minimize the maximum distance from
of the word "topology." In another talk,
(7) The Modem Age Begins. These head
the fire station to any of the surround
tical form of this problem would be, for
on fractals in Pollock's paintings, I had
ings cover a total of fifty sections. For
ing homes." The author cites the work
the distinct feeling that the speaker just
example, the main calculus chapter con
of Franco P. Preparata and Michael Ian
found a clever way to sell a word (frac
sists of sections (5. 1) The Derivative:
Shamos on the minimum spanning cir
tal) to unsuspecting art critics, and that
Controversy and Triumph; (5.2) Paint
cle, and he points out that they also dis
he was well aware that he was faking
ings Again, and Kepler's Wine Barrel;
cuss the dual problem: "what is the
it. But, again, there have also been very
(5.3) The Mailable Package Paradox;
exciting talks (I remember in particular
(5.4) Projectile Motion in a Gravitational
largest circle inside the convex hull of the given n points (think of the points
one describing techniques to teach
Field; (5.5) The Perfect Basketball Shot;
as vertical posts, and a rubber band
arithmetic and geometry to primary
(5.6) Halley's Gunnery Problem; (5. 7) De
snapped all around them [as shown])
and
L'Hospital and His Pulley Problem, and
singing-a new and unexpected twist on
a New Minimum Principle; and (5.8) De
would tell us, for example, where to
the joke about architecture above); and
rivatives and the Rainbow. These sec
place an objectionable service facility
the overall mixture worked very well. So
tion headings represent notable features
for the town, e.g., a centrally located
I am looking forward to next year's con
of the book: timely and interesting
waste-treatment plant that nobody
ference; and meanwhile, I cannot but
choices of topics, conversational tone,
wants to live near!"
recommend reading the available pro
practical perspectives, and the develop
Chapter 6, Beyond Calculus, is prob
ceedings volumes.
ment of concepts historically as well as
ably as compelling an introduction to
mathematically.
the calculus of variations as you can
school
children
by
dancing
Dipartimento di Matematica
Several standard calculus problems
that contains none of the points? That
fmd
anywhere.
In
particular,
the
isoperimetric problem, already woven
Universita di Pisa
are presented and then usefully ex
Via Buonarroti 2
tended beyond what you will find in
into the first two chapters, resurfaces
561 27 Pisa
a calculus text. For example, "Projec
in section 6.8, titled "The Isoperimetric
Italy
tile Motion in a Gravitational Field"
Problem, Solved (at last!)."
e-mail:
[email protected]
starts with the usual differential equa tions dxldt
When Least Is Best by Paul J. Nahin
=
v0 cos((}) and
dyldt =
Many calculus books discuss the catenary as the curve of an ideal hang
gt and establishes that the
ing chain. Many calculus books also
path of motion is a parabola. This and
fail to link that chain to the "other" out
the familiar questions regarding opti
standing property of a catenary, the
Vo sin((})
-
mal height and range are posed in
one that pertains to the St. Louis Gate
terms of athletic events, first shot put
way Arch. The author takes this up el
$29.95, ISBN 0-691 -07078-4
and javelin throw, then golf. Finally,
egantly on page 250:
REVIEWED BY CLARK KIMBERLING
seven pages are devoted to The Perfect Basketball Shot, leading into Halley's
[The hanging chain] is, at every
Gunnery Problem. (This is Edmund
point, in tension only, i.e., there
PRINCETON UNIVERSI1Y PRESS, 2004, 370 pp. US
T
his attractive book is, of course,
Halley, as in Halley's
the
clearly is no point where a hanging
about much more than minimiza
surname rhymes with "Sally," not
chain is in compression. This was
tion. One might describe it as a book in popular-mathematics tone about op
Comet;
"Cayley.") In Derivatives and the Rainbow, the
apparently first pointed out in 1675 by Newton's contemporary (and
timization, written by an engineering
author analyzes primary, secondary,
sometimes
professor whose work is well known
and tertiary rainbows. This section,
(1635-1703) . . . . Further, Hooke went on to observe, if the hanging
rival)
Robert
Hooke
in his field (and also in science fiction).
like all others, includes computer-gen
As such, the work is of great value to
erated plots created by the author us
catenary was "frozen in place" (e.g.,
many, but most especially to the thou
ing MATLAB, and other figures by
glue the links of the flexible chain
sands of people who teach and learn
Christopher L. Brest. The book's over
together) and then inverted, the re
calculus. (The same can be said for an
all up-to-dateness is typified by a cor
sulting arch would be in compres
other of the author's books that may
rection of Marilyn Savant's account of
sion only, and at no point would
have crossed your desk: An Imaginary
the tertiary rainbow in Parade Maga
there be tension. Thus, an inverted
zine, August 4, 2002.
catenary is the best (strongest)
Tale: the Story oj v=l, Princeton Uni
versity Press, 1998.)
The "precalculus chapters" consider
curve for a stone arch.
When Least Is Best has seven chap ters: (1) Minimums, Maximums, Deriv
many enticing optimization problems. One of these, in Chapter 2, is to deter
atives, and Computers; (2) The First
Chapter 7, The Modem Age Begins,
mine the smallest circle that spans a
Extremal Problems; (3) Medieval Max-
opens with a favorite problem of tri
set of n given points in a plane. "A prac-
angle geometry, originating in Fermat's
70
THE MATHEMATICAL INTELLIGENCER
1629 Methodfor 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
An Invitation to Algebraic Geometry Kahanpaa, Pekka Kekalainen, and William Treves UNIVERSITEXT 1 st ed. 2000. Carr. 2nd printing, 2004, XVI,
programming.
power of their elaborate formalism is the false idea that nothing is accessible ties by Heisuke Hironaka, one of the major
achievements
of
algebraic
geometry in the last century, is a good
1 61 pp., ISBN: 0-387-98980-3 US $49.95
example of a result that was proved
MARC CHARDIN
with very little formalism.
lem precedes final sections of the book on linear programming and dynamic
Zariski, Andre Weil, Jean-Pierre Serre, and Alexandre Grothendieck.
without it. The resolution of singulari
BERLIN, HEIDELBERG, NEW YORK, SPRINGER-VERLAG.
BY
time, among them David Hilbert, Oscar
The unfortunate side of the evident
by Karen E. Smith, Lauri
REVIEWED
most influential mathematicians of the
Let us also recall that many impor
A
lgebraic geometry is a very active branch
tant results on the classification of
that is
curves and surfaces were obtained by
linked to many other fields-in partic
the Italian school in the nineteenth
frequent appearance of "an extrema"
ular to arithmetic, one of the most fas
(cf. "an apples"), and, on page
cinating areas in mathematics, but also
century, at a time when Hilbert's Null stellensatz was not yet established: can
Some readers will wonder about the
1 12, "ex tremas" (cf. "geeses"). Perhaps ex tremum is following datum ("piece of data") out of English. In contrast, min imum remains intact-and yet minima is minimized, as evidenced by Mini mums in the heading of Chapter 1. There i s one type o f 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 fail
ure,
is an application of the well-or
for instance to complex analysis or to
one imagine doing algebraic geometry
theoretical physics.
without Hilbert's theorems today?
At its origins, algebraic geometry is
This book, based on notes of lec
the study of the zero set defined by a
tures by Karen Smith at the University
The u nfortu nate
that it is indeed possible to present im
side of the evi
element-which is equivalent to the principle of mathematical induction. That is to say, "first failure," as used in number
theory,
combinatorics,
and
matics.
known fied by
axioms
achievements
demonstrate in
geometry without much
algebraic formalism.
Doing so necessitates modesty and
their elaborate
often need to be made. Also some con
necessary restrictions and hypotheses cepts cannot be defined with complete
formal ism is
rigor. It is frustrating, especially for an
the false i d ea
way of providing a comprehensive in
that not h i ng is accessible
of mathe
notions are least greatest known, exempli greatest known prime. Some
portant
Finland,
some hard choices; in particular, un
probability theory, is closely associ ated with one of the
of Jyviiskylii,
d ent power of
dering principle-that every nonempty set of positive integers contains a least
of mathematics
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
without it .
Related
and
algebraist, but otherwise there is no
vide the minimal background knowl edge in algebra, a little of the formalism,
collection of polynomials. The reason
and a good collection of examples, so
readers may wish that the author had
for its power is probably the interplay
that the reader understands what are
applied his witty insights to a selection
between geometrical intuition and the
the challenges, what is the meaning of
algebraic formalism.
the theorems and conjectures,
of lesser known and well-known
knowns
and
greatest knowns.
least
On the
Geometry is a
and
guideline for defining the proper con
where the motivations come from. The first chapters of the book are dedicated
other hand, the book is well focused on
cepts
extrema of the sort encountered in cal
means for proof; the algebraic formal
culus and engineering. To summarize:
ism makes these ideas applicable to
this book is highly recommended.
cases where the geometric picture is
and
often
suggests
possible
to this delicate task Chapter
1
presents a short account
of affine algebraic varieties, their mor
not obvious, and in many cases it clar
phisms, the Zariski topology, and the
Department of Mathematics
ifies the initial ideas by extracting the
notion of dimension. It is illustrated by
University of Evansville
essence of the argument.
several examples and counter-exam
1 800 Lincoln Avenue
The search for a good algebraic
ples. Chapter
2 is devoted to a more
Evansville, IN 47722
framework was a major factor in alge
substantial presentation of the alge
USA
braic geometry in the last century. This
braic notions attached to affine vari
e-mail:
[email protected]
(r)evolution is due to several of the
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 NuUstellen satz), 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 d o wel l to accept this n ice i nvitation 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:
[email protected]
Kjfi i.MQ.iQ.I§i ..
Rob i n Wilson
The Philamath's Alphabet-F F
I
narrow to contain' of the statement that
ods into the solution of geometrical
n (> 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
