The Mathematical Intelligencer volume 24 issue 4

«·)·"I"·' I I

Paraconsistent Thoughts About Consistency Philip J. Davis

The Opinion column offers

mathematicians the opportunity to

write about any issue of interest to

the international mathematical

community. Disagreement and

controversy are welcome. The views

and opinions expressed here, however,

are exclusively those of the author,

and neither the publisher nor the

editor-in-chief endorses or accepts

responsibility for them. An Opinion

should be submitted to the editor-in

chief, Chandler Davis.

"You will notice, Pnin said, that when on a Sunday evening in May, 1876

Anna Karen ina throws herself under that freight train, she has existed more than jour years since the beginning of the novel. But in the life of the Lyovins, hardly three years have elapsed. It is the best example of relativity in literature that is known to me. "-Vladimir Nabokov, Pnin, ch. 5, sec. 5 (paraphrased).

Most philosophers make consistency the chief desideratum, but in mathematics it's a secondary issue. Usually we can patch things up to be consistent. -Reuben Hersh, What is Mathematics?, Really, p. 237.

I I am an occasional writer of fiction, engaging in it as an amusement and a relaxation. Compared to professionals, I would say that my "fictive imagination" is pretty weak. This doesn't bother me much, because I'm usually able to come up with something resembling a decent plot.

I work on a word processor. The processor has a spelling check and a grammar check. The spelling check is useful but occasionally annoying. For example, it does not recognize proper names unless they've been pre-inserted. One time it replaced President Lincoln's War Secretary Seward with Secretary Seaweed. You know the saying: if something is worth doing, it's worth doing poorly. In these gray days my spelling check is the source of many laughs. Therefore it would be a pity if it were "improved."

The grammar check in my word processor is only occasionally useful and mostly annoying. It occasionally throws down a flag when I write a detached sentence. For example: "London, Cambridge, so why not Manchester?" It often wants to change a passive formulation into an active one, and its suggestion for doing so often ends in a terrible muck. It doesn't catch nonsense. As a test I wrote, "The man reboiled the cadences through the monkey wrench," and my spelling and grammar check simply changed "reboiled" to "rebelled."

I recently wrote a long short story titled "Fred and Dorothy." What really bothered me after I finished was this: Having produced a fairly long manuscript, I was never quite sure whether it was consistent. I don't mean the consistency of a pancake batter or whatever the equivalent of that would be in prose style. I mean the everyday sort of consistency of time, place, person, etc., to which I would add logical consistency.

I'm not sure how to give a general and precise definition of consistency. What do I mean by it? I'll give some examples of what I think might be seen as inconsistencies.

On page 8, of my first draft, I wrote that Dorothy's eyes were blue. On page 65, they're brown.

On page 24, Fred grad�ated from high school in 1967. On page 47, he dropped out of high school in his junior year.

On page 73 and thereafter, Dorothy, somehow, became Dorlinda.

On page 82, there is an implication that World War II occurred before World War I.

There are spelling inconsistencies: on page 34 I wrote "center" while on page 43, I wrote "centre."

There are inconsistencies in the point of view. Fred and Dorothy is a story told by a narrator; an "I." There is often an "I problem" in fiction, and here's how Peter Gay in The Naked Heart: . The Bourgeois Experience from Victoria to Freud describes it:

Often enough, the narrator [of the first person novel f--or, rather, his creatorcheats a little, recording not only what he saw and heard, or was told, but also what went on in the minds of characters who had no opportunity to reveal their workings to him. Most readers, facing these flagrant violations of the narrator's tacit contract with them, suspend their disbelief . . .

We learn to deal with inconsistencies in books, and we do it in different ways. Suspension of disbelief is only one way. Now, whatl would find really useful, if such a thing could exist, would be a program that checks for

© 2002 SPRINGER· VERLAG NEW YORK, VOLUME 24, NUMBER 4, 2002 3

consistency as well as my copy editor

Louisa does. She's smart. She's careful.

She has read widely. She knows my

mind. She's worth gold.

Imagine now that I have bid Louisa

goodbye and replaced her with a con

sistency checker that I paid good money

for. Call the software package CONNIE.

I run Fred and Dorothy through

CONNIE. It immediately comes back

with a message: "On page 8 you said

Dorothy's eyes were blue and on page

65 they're brown. What's the deal?" Did

I have to spell out in my text that in the

late afternoon October mist, Dorothy's

eyes seemed brown to me?

How did CONNIE handle metaphor?

I wrote: "He saw the depths of the sea

in her eyes." Now CONNIE (a very

smart package) knew her (its?) Homer

and recalled that

Gray-eyed Athena sent them a favorable breeze, afresh west wind, singing over the wine-dark sea,

and it blew the whistle on me: "Hey,

wine-dark isn't blue. For heaven's sake,

please make up your mind about

Dorothy's eyes."

Why did "Dorothy" morph into "Dor

linda"? That's part of my story: it's the

name the movie producers decided to

give her after she'd passed her screen

test.

CONNIE picked up a sentence in ar

chaic English and screamed bloody mur

der. The sentence in question was part

of a movie script (within my story),

whose action was placed in the 17th cen

tury. It rapidly becomes clear that the no

tion of consistency is not context-free.

And so on and on. A writer of fiction

can explain away post hoc what appear

to be inconsistencies. In technical lingo,

often employed in mathematical physics,

explanations that clear up inconsisten

cies are called interpretations. I suppose that someone, somewhere,

has drawn up a taxonomy of textual in

consistencies. It must be extremely long.

Mavens who analyze language often

split language into three systems with

different sorts of meaning: interper

sonal, ideational (i.e., ideas about the

world in terms of experience and logi

cal meaning), and textual (ways of com

posing the message). I worry mostly

about the first two, and I'd limit my con

sistency checker to work on them.

4 THE MATHEMATICAL INTELLIGENCER

II I've now said enough about literary

texts and I'm ready to get to logic and

mathematics; to Boole and Frege and

Russell and Godel and Wittgenstein

and all those fellows. Whereas life does

not have a precise definition of con

sistency, mathematics has a clear-cut

definition. A mathematical system is

consistent if you can't derive a con

tradiction within it. A contradiction

would be something like 0 = 1. Con

sistency is good and inconsistency is

bad. Why is it bad? Because if you can

prove one contradiction, you can prove

anything. In logical symbols,

(1) For all A and B, (A & �A)� B.

And so, if you allow in one measly in

consistency, it would make the whole

program of logical deduction ridicu

lous.

Aristotle knew about equation (1)

and had inconsistent views about it. In

the literature of logic it's called the

ECQ principle (ex contradictione quodlibet). But I call it the Wellington

principle. (The Duke of Wellington

1769-1852, victor at the Battle of Wa

terloo.) The Duke was walking down

the street one day when a man ap

proached him.

The Man: Mr. Smith, I believe? The Duke: If you believe that, you can believe anything.

Inconsistency is (or was) the primal

sin of logic. In 1941, in my junior year at

Harvard, I took a course in mathemati

cal logic with Willard Van Orman Quine,

who in the opinion of some became the

most famous American philosopher of

his generation. Quine had just published

his Mathematical Logic and it was our

textbook The course startedjust before,

or shortly after the shattering news

came that J. Barkley Rosser had found

an inconsistency in the axiom system

Quine had set up. Well, Quine spent the

whole semester having the class patch

up the booboo in our books; crossing

out this axiom and replacing it with that;

replacing this formula with that-while

we logical greenhorns were anxious for

him to get on with it and get to the punch

line of logic, whatever that might be.

Q.E.D. as regards primal sins.

But back to business. If 0 = 1 is an

inconsistency, then by multiplying both

sides by 4, I get 0 = 4. Now is that an in

consistency? It is in 7L, but not in 7L4!

Come to think of it, how do we know

that 0 = 1 is a contradiction in 7L? Be

cause Peano said so. Or did he? Well, if

he didn't, I would hope it can be deduced

from his axioms about the integers.

So, depending on where you're com

ing from, a set of mathematical symbols

may or may not be an inconsistency.

Just as in fiction. In fact, a set of truly

naked mathematical symbols is not

interpretable (or is arbitrarily inter

pretable). By "naked" I mean that you

have no indication, formal or informal,

of where the writer is coming from.

Now bring in G6del's famous and

notorious Second Incompleteness The

orem (the GIT). I want to apply it to lit

erary texts. To state it in a popular way,

the GIT says that you cannot prove the

consistency of a mathematical system

by means of itself.

If mathematics is part of the universe

of natural language, and I think it is, then

I believe that with a little thought, I could

get the GIT to imply that it is impossible

to build a universal consistency checker.

Or, for that matter and much more im

portant these days, that it is impossible

to build a universal virus checker. If a

consistency checker can't be produced

for mathematics with its sophisticated

and conventualized textual practices

and with its limited semantic field, then

I have serious theoretical doubts about

literary texts.

CONNIE might catch Dorothy's eyes

being simultaneously blue, brown, and

wine-dark, but there will be some in

consistencies that CONNIE misses.

Inconsistency is how things appear

in the world. We spend part of our life

cleaning up the confusions, trying to

impose some semblance of order. To

some extent we are successful, but

only in a limited sense and for a lim

ited time. Heraclitus assured us that

nothing is ever the same twice, and

when things begin to get fuzzy we

think, that's not the way we had per

ceived matters. So I'm afraid we all

have to live with and deal with incon

sistencies. We learn to do it. Walt Whit

man, the poet, knew this. He said,

Do I contradict myself? Very well then I contradict myself (I am large, I contain multitudes. )

Mathematics is one way we try to impose order, and we may do it inconsistently. Consider the arithmetic system that is embodied in the popular and useful scientific computer package known as MATLAB. Now MATLAB yields the following statements from which a contraction may be drawn:

"le - 50 = 0 is false" (i.e., w-50 = 0), "2 + (le - 50) = 2 is true."

Well, we all recognize roundoff and know its problems. And we know, to a considerable extent, but not totally, how to deal with it; how to prevent it from getting us into some sort of trouble.

Is it a contradiction that the diagonal of the unit square exits geometrically but can't exist numerically? At one point in history it was a highly irrational conclusion and one worthy of slaughtering oxen.

Was it a contradiction that there exists a function on [ -oo, + oo) that is zero everywhere except at x = 0, and whose area is 1? It wasn't among the physicists who cooked it up and used the idea productively. It was among the mathematicians until Laurent Schwartz came along in the 1940s and showed how to embed functions within generalized functions.

More recently, in connection with Hilbert's Fifth Problem, Chandler Davis has written

I cannot see why we would want a locally Euclidean group without differentiability, and yet I think that if some day we come to want it badlyin which case we will have some notions of the properties it should have -we should go ahead! After jive or ten years of working with it, if it turns out to be what we were wishing for, we will know a good deal about it; we may even know in what respect it differs from that which Gleason, Montgomery, and Zippin proved impossible. Then again, we may not . . . .

Inconsistencies can be a pain in the neck, a joy for nit-pickers, and a source of tremendous creativity.

Karl Menger, in his Reminiscences of the Vienna Circle and the Mathematical Colloquium, tells the story that in Wittgenstein's opinion, mathematicians have an irrational fear of contradiction. I've often thought as much, but I also realize that mathe-

maticians are often smart enough to spirit away a contradiction-just as Hersh says in the epigraph. Mathematical inconsistencies are often exorcised by the method of context-extension. It is done on a case-by-case basis, and it is worth doing only after the contradiction has borne good fruit. So the notion of mathematical consistency may be time- (and coterie-) dependentjust as in literature.

Logicians, who go for the guts of the generic, and who are over-eager to formalize everything, have come up with a concept called paraconsistency. There has even been a World Congress to discuss the topic. Ordinary logic, as I have noted, has the Duke of Wellington property that if you can prove A and not A, then you can prove everything. Paraconsistent logic is a way of not having an inconsistency destroy everything. Contradictions can be true. Perhaps such a system might be good for certain applications to the real world where conflicting facts are common.

Walking down the street in paraconsistent London, a man approached the Duke of Wellington.

The Man: Mr. Smith, I believe? The Duke: My dear Sir, don't let your belief bother you.

When all is said and done, and paraconsistency aside, I don't think I can defme consistency with any sort of consistency. But I'm in good company. Paralleling St. Augustine's discussion of the nature of time, though I can't define a contradiction, I know one when I see one.

In a very important paper written in the mid-1950s, the logician Y. Bar-Hillel demonstrated that language translation was impossible. This demonstration dampened translation efforts for a few years. illtimately it did not deter software factories from producing language translators that have a certain utility and that also produce absurdities. I'm sure that the software factories will soon produce a literary consistency checker called CONNIE. I will run to buy it. It might be just good enough for me. And if I've paid good money for it, then, as the saying goes, it must be worth it. The absurdities it produces will lift my spirits on gray days and serve to remind me of the

"folk theorem" that bad software can often be useful.

Acknowledgments I thank Ruth A. Davis and Kay O'Halloran for providing me with some important words and ideas.

REFERENCES

George S. Boolos, John P. Burgess and

Richard C. Jeffrey, Computability and Logic,

4th Ed. , Cambridge Univ. Press, 2002.

Chandler Davis, Criticisms of the usual ratio

nale for validity in mathematics, in Physicalism

in Mathematics (A.D. Irvine, ed.), Kluwer Aca

demic, Dordrecht, 1990, 343-356.

Peter Gay, The Naked Heart, Norton, 1995.

Reuben Hersh, What is Mathematics, Really?,

Oxford Univ. Press., 1 997.

Karl Menger, Reminiscences of the Vienna Cir

cle and the M§Jthematical Colloquium, Kluwer

AcademiC, Dordrecht, 1 994. '

Chris Mortensen, Inconsistent Mathematics,

Kluwer Academic, Dordrecht, 1 995.

A UTHO R

PHILIP J. DAVIS

Division of Applied Mathematics

Brown University

Providence, Rl 02912 USA

e-mail: [email protected]

Philip J. Davis, a native of Massachu

setts and a Harvard Ph.D., has been

in Applied Mathematics at Brown

since 1 963. He is known for applied

numerical analysis, and his tools are

typically functional analysis and classi

cal analysis: some might say, he ap

plies pure analysis to applied. But he

is known to many more as a com

mentator on mathematics. Among his

many nontechnical publications are

the widely read books with Reuben

Hersh, The Mathematical Experience

and Descartes' Dream.

VOLUME 24, NUMBER 4, 2002 5

HANSKLAUS RUMMLER

On the Distribution of Rotation Angles How great is the mean rotation angle of a random rotation?

• f you choose a random rotation in 3 dimensions, its angle is jar from being uni-�formly distributed. And the [n/2] angles of a rotation in n dimensions are strongly

correlated. I shall study these phenomena, making some concrete calculations in

volving the Haar measure of the rotation groups.

The Angle of a Random Rotation in 3 Dimensions Any rotation of the oriented euclidean 3-space IR3 has a

well-defmed rotation angle a E [0, 1r], and, in the case 0 < a < 1r, also a well-defined axis, which may be represented

by a unit vector g E S2• For the identity, only the angle a =

0 is well-defined, whereas any g E S2 can be considered as

axis; if a = 1r, there are two axis vectors ± g. By a random rotation we understand a random variable in 80(3), which

is uniformly distributed with respect to Haar measure. It is

clear that the axis of such a random rotation must be uni

formly distributed on the sphere S2 with respect to the nat

ural area measure, but what about the rotation angle?

It is certainly not uniformly distributed: The rotations by

a small angle a, let's say with 0::::; a < 1°, form a small

neighbourhood U of the identity ll E 80(3), whereas the ro

tations with 179° < a ::::; 180° constitute a neighbourhood V of the set of all rotations by 180°, which make up a surface

(a projective plane) in 80(3). It is plausible that V has a

greater volume than U, i.e., the distribution of rotation an

gles should give more weight to large angles than to small

ones. In order to calculate the distribution of the rotation

angle, I first express the Haar measure of 80(3) in appro

priate coordinates.

6 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

The Haar measure of 80(3)

Proposition 1: If one describes 80(3) by the parametrization

p: [0, 1r] X S2 � 80(3),

p(a, g):= rotation by the angle a about g,

the Haar measure of 80(3) satisfies

p*dJ1-s0(3)(a, 0 = 2�2 sin2 ( �) da dA(g)

1 = 47T2 (1 - cos a)da dA(g),

where dA is the area element of the unit sphere 82.

Proof To begin with, observe that the restriction of p to

]0, 1r[ X S2 is a diffeomorphism onto an open set U in 80(3),

and that the null set {0, 1r} X S2 is mapped by p onto

80(3) \ U, which is a null set with respect to Haar measure.

We can therefore use a and g to describe the Haar measure

of 80(3), even if they are not coordinates in the strong sense.

The mapping p is related to the adjoint representation

of the group Q of unit quatemions, and it is easy to calcu-

late the Haar measure of Q. Decomposing a quaternion into its real and imaginary parts, we may describe this group as follows:

with multiplication

(t, g) · (s, TJ) = (ts - (g, TJ), l1J + sg + g X TJ).

The natural riemannian metric on Q = S3 c IR4 is invariant, and therefore the Haar measure of Q is just a multiple of the riemannian volume element. Using the parametrization

cp : [0, 1r] x S2 � Q, c.p(y, g):= (cos y, g sin y)

and taking into account the total volume of SS, we get for the Haar measure of Q

where dA denotes the area element of the unit sphere S2. To get from this the Haar measure of S0(3), we use

the adjoint representation r = Ad : Q � S0(3), defmed by rq(O = qrii for q E Q and� E !R3. This is a twofold covering and

r*df..Lso(3) = 2df..LQ· In the parametrization ljJ: = To c.p : [0, 1r] X S2 � S0(3) we have therefore

To finish the proof, we observe that 1/J( y, g) is just the rotat'ion by 2y about the axis g, i.e., p(a, g)= 1/1(�. n 0 See also [1], pp. 327-329, and [6].

The distribution of the rotation angle

The parametrization p is well adapted to our problem, because the subset of rotations by a fixed angle a is just the image of the sphere {a} X S2. If we integrate our expression for the Haar measure of S0(3) over these spheres, we obtain the following result:

Proposition 2: The angle a E [0, 1r] of a random rotation is distributed with density f(a) = l. (1 - cos a):

7T

� :il 0.5 1 1.5 2 2.5 3

See also [7], pp. 89-93.

Generating random rotations

Integrating our expression of the Haar measure of S0(3) over the segment [0, 1r] X {g) for any g E S2 confmns that the axis g of a random rotation is uniformly distributed with respect to the natural area measure dA on S2• Using this fact and knowing the distribution of the rotation angle, we can generate random rotations by choosing axis and angle as follows:

The horizontal projection of the unit sphere S2 onto the tangent cylinder along the equator is an area-preserving map; thus we may choose a point on the cylinder and take the corresponding point on the sphere as axis. This means choosing a random point (A , h) in the rectangle [ -17, 1r] X [ -1, 1] and taking the rotation axis g = (v'f=h2 cos A , VI=h2 sin A, h).

For the rotation angle a, we choose a random number a E [0, 1] and take a : = F-1(a), where

La 1 F(a) = f(t)dt = -(a - sin a)

0 1T

is the distribution function. Linear algebra tells us how to calculate from g and a the matrix g E S0(3).

To test this generator of random rotations, I fixed x E SZ together with a tangent vector g E TxS2 and calculated with Mathematica the tangent vectors dg(x; g) for 600 random rotations g E S0(3). The mapping g � (g(x), dg(x; g)) is a diffeomorphism from S0(3) onto the unit tangent bundle of S2 and thus makes the rotations g visible by the "flags" (g(x), dg(x; g)) (Fig. 1).

For the sake of curiositr,I calculated the mean rotation angle for 5,000 random rotations: The result E5,ooo(a) = 126°13'55" matches the theory, because an easy calculation gives the answer to the question of the subtitle as a consequence of proposition 2:

Corollary: The expectation of the rotation angle of a random rotation is

1T + � = 126° 28' 32". 2 1T

Random Rotations in 4 Dimensions

The Haar measure of 50(4)

If we identify the euclidean IR4 with the skew field of quaternions IHI, the group Q = S3 of unit quaternions acts on IR4

by left and right multiplication with q E Q, Lq : 11-0 � IHI and Rq : IHI � IHI, which are linear isometries, i.e., elements of S0(4). These special rotations generate the whole group S0(4):

 : Q X Q � S0(4), <l>(p, q) := Lp o R-q

is a group epimorphism with kernel {(1, 1), ( - 1, -1)}. (See also [1], pp. 329-330.)

i#Mii;IIM

....... -:. . ... . / ·:::":f.:·;:_:::.-:_�--�<.·::\.::. . ·. . . . :. . .')� ,: .· '· .. . .. . . ·::.

,.-:. '. . . .. • :. • '•'-t . • • • . .. t.' ; ·: . ,. -· . . .·· ... . : .. . .. · . . ,. , -.. . . : . . . . . ... . : .·.: .... � .. ... · ......... ' . .. ·.··:·· :'••:

: , . : ·.

\�·:> . . ",y}

.. . .' ... · ·.�;� .... . ,. .. . �

.· .. ·�· .,..,·.:·��-· :,..

VOLUME 24, NUMBER 4. 2002 7

Using the parametrization 'P for either factor of the product Q x Q, we obtain a parametrization of S0(4):

'¥: [0, 7T1 X (0, 7T1 X S2 X S2 � S0(4), 'l"(s, t, g, TJ) : = ('P(s, g), 'P(t, TJ)).

If we admit s, t E [0, 27T1 and calculate modulo 27T, '¥ becomes a fourfold covering'¥: T2 X S2 X S2 � S0(4) with branching locus ({(0, 0)) U (1r, 1r) )) X S2 x S2:

'l"(s, t, g, TJ) = '¥(1r- s, 7T- t, -g, -TJ) = '¥( 7T + s, 7T + t, g, TJ)

= '¥(27T- s, 27T- t, -g, -TJ)

for 0 ::s: s, t ::s: 7T, and even '¥(0, 0, g, TJ) = '¥( 1r, 1r, g, TJ) = 1 for all g, TJ E 82. The Haar measure of S0(4) therefore satisfies

'l"*d�-tsoc4) = c sin2 s sin2 t ds dt dA(D dA(TJ),

with a constant c.

Pairs of rotation angles

Any rotation g E SO( 4) is cm\iugate to a standard rotation (Ro41 0 ) with Ro4 : = (c?s 1'7 - sin 1'7 )·

0 Ro42 sm 1'7 cos 1'7

Choosing the rotation angles 1'71, 1'1 2 in the interval [0, 27T1, the following pairs are equivalent, i.e., the corresponding rotations are conjugate:

(1'71> 1'7 2)- (1'7 2, 1'71)- (27T- 1'71, 27T- 1'7 2) - (27T - 1'7 2, 27T - 1'71).

The class of these equivalent pairs will be called the pair of rotation angles [1'71, 1'7 21· This is an element of T21-, where the equivalence relation - is considered on the torus y 2 = S1 x S1. Two rotations in S0(4) are cof\iugate if and only if they have the same pair of rotation angles [1'71> 1'7 21.

The following lemmas are needed to determine the pair of rotation angles for an element <l>(p, q) E S0(4).

Lemma 1: For p, q, p', q' E Q, the rotations <l>(p, q) and <l>(p', q') in S0(4) are conjugate if and only if p is conjugate to ±p' and q is conjugate to ±q' in Q, with the same sign in either case.

Proof <l>(p, q) is conjugate to <l>(p', q') if and only if there exists aTE S0(4) with <l>(p, q) = To (p', q') a y-1. As T =

<l>(u, v) for some u, v E Q, we have:

<l>(p, q) is conjugate to <l>(p', q') if and only if there exist u, v E Q with

Lp a Rfi = Lu o R:v o Lp' a Rfi· o L:u a Rv = Lu o Lp' o L:u a R;u o Rfi· a Rv

= Lup'u o Rcvq'vT = <l>(up'u, vq'v).

The kernel of contains only the two elements (1, 1) and ( -1, -1); therefore we have shown that <l>(p, q) is conjugate to <l>(p', q') if and only if there exist u, v E Q such that p = ±up'u and q = ±vq'v, with the same sign in either case. 0 Lemma 2: Let g E S2 be a purely imaginary quaternion with norm 1. Then the quaternion p = cos t + g sin t is conjugate to p' = cos t + i sin t.

8 THE MATHEMATICAL INTEWGENCER

Proof We must find a u E Q with Tu(g) := ugu = i. But as S0(3) acts transitively on 82, there exists a rotation which sends g to i, and as the adjoint representation T : Q � S0(3) is onto, there exists u E Q such that this rotation is Tu, i.e. Tu(D = i. 0

These lemmas allow us to show:

Proposition 3: Let p = cos s + g sin s, q = cos t + TJ sin t, where g and TJ are purely imaginary unit quaternions. Then the rotation <l>(p, q) E SO( 4) has the pair of rotation angles [s - t, s + t].

Proof By the two lemmas, the rotation <l>(p, q) is conjugate to ( cos s + i sin s, cos t + i sin t) and has therefore the same pair of rotation angles. Let us calculate the matrix of the latter rotation with respect to the canonical base (1, i, j, k) of !R4 = !HI:

whence

(Rots 0 ) Leos s+i sin s = O Rots ,

(ROLt 0 ) Rcos t-i sin t = O Rott ,

"'( . . . . )

(Rots-t � cos s + 1 sm s, cos t + 1 sm t =

0

which fmishes the proof. 0 �Ots+t}

Corollary (Fig. 2): The pair of rotation angles is distributed with density

fiWt, 1'7z]) = :2 sin 2 ( 1'71 ; 1'7 2 ) sin 2 ( 1'71 ; 1'1z ) 1

= 47T 2 (cos 1'11 - cos 1'1z)Z.

Herefis considered as a function on [0, 27T1 x [0, 27T1, i.e., it is normalized so that integrating it over (0, 27T1 x [0, 27T1 gives 1.

Proof Starting with the parametrization

'¥: (0, 7T1 X (0, 7T1 X S2 X S2 � S0(4)

+ildii;IIM

and using the relation [it1, it2] = [s - t, s + t], we obtain a new parametrization:

With respect to these parameters the Haar measure satisfies

I/J*dJLso(4) = C sin2 ( it1 ; it2 ) sin2 ( it1

; it2 ) dit1 dit2 dil.(g) dil.( TJ).

Integrating over { ( it1, it2)} X S2 X S2 for fixed it1, it2 gives us the density

C' = 4 (cos it1 - cos it2)2·

The constant C' = 1hr2 is obtained by integrating this function over [0, 27T] X [0, 27T]. 0

Rotations in Dimension n � 4 The results obtained in dimensions 3 and 4 can be generalized to dimension n 2::4 using Hermann Weyl's method of integration of central functions on a compact Lie group. A central function is one which is constant on cof\iugacy classes. In the case of S0(3) this is simply a function of the rotation angle, and in the case of SO( 4) of the pair of rotation angles. In dimension n > 4 we can introduce the notion of a multiangle characterizing the cof\jugacy classes.

Multiangles of rotation

Let us begin with the case of a rotation g E SO( n) for even n = 2m: as in the case n = 4, there are m rotation angles it1 ... , itm corresponding to the decomposition of gas direct sum of m plane rotations:

g = Rott't1 E9 ... E9 Ro�m· For odd n = 2m + 1, there are also m angles it1, ... , itm. Calculating modulo 27T, the list (it 1' 0 0 0 it m) is an element of the m-torus rm and is unique up to the following symmetries, which define an equivalence relation � on rm:

the iti may be permuted;

iti may be replaced by -iti, but only for an even number of indices i if n is even; for odd n there is no such restriction.

Let us call the class ma(g) := [itb ... itml E T"'/� the multiangle of the rotation g E SO(n).

Two rotations in SO( n) are col\iugate if and only if they have the same multiangle. To determine the multiangle of a rotation x E SO( n ), we fix an orthonormal base of �n and consider a cof\iugate of x in the maximal torus T c SO( n) the elements of which have, with respect to the chosen base, the form

_ (Rott't1

•• •

• 0 )

it- 0

0 ... Rott'tm

in the case n =2m; in the case n = 2m + 1, one has to add a first column and a first row with first element 1 and zeroes elsewhere. In either case we identify T with the standard torus rm. Obviously, it E rm has the multiangle ma(it) = [it], and this is the same for the whole cof\iugacy class:

ma(gitg-1) =[it] for all it E rm and g E SO(n).

The Haar measure of a compact Lie group

Let G be a compact and connected Lie group and T C G a maximal torus. There exists a natural mapping 1/J : GIT X T � G such that the diagram

GXT�G +Y

G/TX T

commutes, where cp(g, it):= rg(it) = gitg-1 and the vertical arrow is the natural projection.

The Lie algebra g is endowed with an Ad-invariant scalar product, and if t C g is the Lie algebra of the maximal torus T, its orthogonal complement ±-'- is stable under the mappings Ad g : g � g for g E.G. The restriction of �d g to±-'is denoted by Ad-'- g.

With these notations, the Haar measure of a_ can be expressed in terms of that of T together with the invariant measure of G/T:

Proposition 4: 1/J : G/T X T � G is a finite branched covering. Let dJLa and dJLr denote the Haar measures of G and T, and let dJLa;r be the G-invariant normalized measure of the homogeneous space GIT. Then

1/J*dJLa = dJLa!T X J dJLr,

where J : T � � is the function

J(it) := det(li - Ad-'-it).

For a proof of this formula, see [2], pp. 87-95.

The distribution of the multiangle

Proposition 4 may be applied in our case, with G = SO(n) and T = rm. Now 1/J((g], it)= gitg-1 has for every [g] E SO(n)/T m the same multiangle [it], i.e.,

ma(I/J([g], it)) = (it] E T/�.

Therefore the density of the multiangle [it], considered as a symmetric function on the torus rm, has the form

f(it) = C f J(it) dJLGIT = cJ(it) GIT

with a normalizing constant c. To calculate J( it) = det(ll. - Ad -'-it), we observe that in

the case n = 2m the elements of ±-'- are the symmetric matrices of the form

A=(�;� Aim

A12 A1s ... A1m) 0 A2s ... A2m 0 0 '

0 0 0 0

where the AiJ are 2 X 2-blocks.


A direct calculation shows that Ad-'-{} transforms this matrix by replacing every block Aij by the block

R111J{Ai'J) := Ro�Ai'J-Rot,&1. I' J I ) If we identify !R2x2 with IR2 0 IR2, R1Ji,1Ji becomes the tensor product of the two rotations Ro�; and Ro�I The eigenvalues are therefore e:!:iil;e:!:i-IJi = e:!:i(il;:!:ili), and we obtain

det(ll - Ril;,1l) = (2 sin ifi ; ifi t( 2 sin ifi ; ifi r

= 4(cos ifi - cos ifi?·

Now Ad-'-iJ is the direct sum of the R1l;,1li· Combining these results:

J(if) = 2m(m-l) n (cos i}i- cos i7j)2 l�i<.I�m

and

f2m(i7) = C ) fl (cos ifi- cos i7j)2. l!Si<j::=m

These formulae apply to the case of even n = 2m. In the case of odd n = 2m + 1 one has

m J(if) =2m2 n (1 - cos ifi) n (cos i}i- cos i7j)2

i=l lS.i<jsm

and m

fzm+l(if) = c n (1 - cos ifi) n (cos i}i- cos i7j?· i=l lSi<jsm

Figure 3 illustrates the functionf5(i7) for S0(5), where the normalizing factor is C = 1/(27T2):

You see a "sharper" correlation between the two angles than in the case SO( 4). The rotations with the pair of angles (arccos(l/3), 1r] = [70°31'44", 180°] are the "most frequent" ones. We shall see that the cases S0(4) and S0(5) are representative of a general phenomenon: The density of the multiangle has always a well-defined maximum with 0 :5 iJ-1 < . . . < ifm :5 7T, and for this maximum ifm = 7T, whereas iJ-1 = 0 for even nand iJ-1 > 0 for odd n.

To study the density functionsfn(if), observe that they may be written as

fn(if) = Cgn(cos iJ-1, . . . , cos ifm), m = [n/2],

10 THE MATHEMATICAL INTELUGENCER

with

gzm(XI. ... , Xm) = IT (xi - Xj)2 l�i<js.nt

and m

g2m+l(Xl,. · · , Xm) = gzm(Xl, · · ·, Xm) n (1- Xi). i=l

g2m is a well-known function, namely the discriminant of the polynomial (x - x1) · . . . · (x - Xm). Here we consider the functions gzm and gzm+ 1 on the compact simplex D : = {x E !Rm; 1;:::: x1;:::: ... ;:::: Xm :2:: -1} where they are not negative and must have a maximum.

Proposition 5: The global maximum of gn in D is also the only local maximum in D . .

For the maximum of g2m, 1 = x1 > ... > Xm = -1;for that ofgzm+b 1 > x1 > . . . > xm = -1.

Proof Let us consider the even case, i.e., the function g2m: Obviously, one has x1 = 1 and Xm = -1 for any local maximum x. Fix these two coordinates and define

On the boundary of D' := {1 :2:: Xz :2:: • • • :2:: Xm-1 :2:: - 1}, h has the value -oo, and this function is strictly concave in the interior: its Hessian is the matrix Hh(x) = (hij(x)) with

-I fori =j k=l (xi - xk)2

{ m 2

h··(x) = k*i t] 2 (xi- Xj)2 fori =F j.

The diagonal elements are strictly negative, the other elements are strictly positive but still sufficiently small to make the sum of the elements of any row negative. Therefore, the Hessian is negative definite and h is a strictly concave function and has a unique local maximum in the interior of D'. As the natural logarithm is strictly increasing, the function

g2m(1, x2, ... , Xm-1. -1) has also a unique local maximum. For g2m+ 1 the reasoning is similar. 0

As a consequence of this proposition, the density fn( i7) of the multiangle of SO(n) has always one and only one maximum in (0::; iJ-1 < . . . < iftni2J ::; 1r}; for this maximum, iJ-1 = 0 and ifm = 7T if n = 2m, whereas iJ-1 > 0 and ifm = 7T in the case n = 2m + 1.

Here is a list of the most frequent multiangles, i.e., the [if] with maximal density fn(if), for n :5 10:

80(3): [180°], 80(4): [0°, 180°], 80(5): [70°31' 44", 180°], 80(6): [0°, goo, 180°], 80(7): [46°22'41", 106°51'07", 180°], 80(8): [0°, 63°26'06", 116°33'54", 180°], 80(g): [34 °37' 55", 7g033'46", 125°07' 13", 180°], 80(10): [0°, 64°37'23", goo, 115°22'37", 180°]

A U T HO R

HANSKLAUS RUMMLER

Department of Mathematics

University, Perolles

1700 Fribourg

Switzerland


Hansklaus Rummier was born in 1942 and studied Mathe

matics at the Universities of MOnster (Germany) and Fribourg

(Switzerland), where he received his Ph.D. in 1968. In 1977/78

he spent one year at the IHES at Bures-sur-Yvette. His re

search interests are the geometric aspects of analysis, in

cluding the geometry of foliations. His hobbies are gliding and

playing viola. (Not necessary to send him viola jokes!)

It seems that there is almost no literature on the subject; however, [3], [4], and [5] treat related topics.

REFERENCES

[1 ) W. Greub, Linear Algebra (Springer-Verlag, Berlin, Heidelberg, New

York 1967)

2. W. Greub, S. Halperin and R. Vanstone, Connections, Curvature and

Cohomology, volume II (Academic Press, New York and London

1 973)

3. J. M . Hammersley, The distribution of distances in a hypersphere,

Ann. Math. Statist. 21 (1 950), 447-452

4. B. Hostinsky, Probabi/ites relatives a Ia position d'une sphere a cen

tre fixe, J. Math. Pures et Appl. 8 (1 929), 35-43

5. A. T. James, Normal multivariate analysis and the orthogonal group,

Ann. Math. Statist. 25 (1 954), 40-75

6. R. E. Miles, On random rotations in �3• Biometrika 52 (1 965),

636-639

7. D. H. Sattinger and 0. L. Weaver, Lie Groups and Algebras with Ap

plications to Physics, Geometry and Mechanics (Springer-Verlag,

Berlin, Heidelberg, New York, Tokyo 1 986)

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VOLUME 24, NUMBER 4, 2002 1 1

Mathematically Bent

The proof is in the pudding.

Opening a copy of The Mathematical Intelligencer you may ask yourself

uneasily, "What is this anyway-a

mathematical journal, or what?" Or

you may ask, "Where am !?" Or even

"Who am !?" This sense of disorienta

tion is at its most acute when you

open to Colin Adams's column.

Relax. Breathe regularly. It's

mathematica� it's a humor column,

and it may even be harmless.

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

Science Center, Williams College,

Williamstown, MA 01267 USA


Colin Adams, Editor

The Red Badge of Courage Colin Adams

I remember that day as if it were yesterday. I will never forget it. Often,

I wake up at night, in sweat-soaked sheets, screaming, "Look out, Sarge, look out!" Often my roommate is screaming, too. "Shut up, shut up!" But I can't shut up. I have to tell the story, the story of that fateful day. A day that can never be forgotten.

We were fresh out of boot camp, Leftie and me. Hardly knew an integral from a derivative. We thought the power rule was complicated. Just a pair of snot-nosed calc students. But they said we were ready for Calc II. How ridiculous that sounds now.

We arrived in country and were assigned to a unit of misfits. Sarge was the only one of us who had seen real combat before. She had fought in WWWI, a web-based trig course. And then there was Pipsqueak, Pops, Leftie, and me. They called me Kodowski. I wanted them to call me Tootsie. But they refused.

Before we had even fmished unpacking our gear, we heard a yell. "Incoming!" Grunts dove for cover. Sarge just kept eating her granola bar. "Relax," she said. "It's just a quiz." I stayed low anyway. It seemed dangerous enough to me. But it wouldn't be long before I understood the difference.

I remember that fateful morning as if it were yesterday. I woke to something dripping on my forehead. Leftie had wet the upper bunk again. He gave new meaning to the words math anxiety. I pulled him off his bunk and we had a quick shoving match .. Then we threw on our uniforms. No time to brush teeth or comb hair. Ours or any-


body else's. Destiny waits for no one. As we stumbled toward the front line, ominous clouds hung low in the sky.

We found the rest of the unit near the frontline. Pipsqueak looked like she was going to lose her breakfast, and Pops's hands were shaking. (He was a continuing student.) Sarge munched nonchalantly on a toaster pastry. Was she really that unconcerned or was that the impression she wanted us to have? I didn't know for sure, but the toaster tart sure looked good.

As we spread out over the lecture hall, hunkering down in our foxholes, I felt queasy myself. This was it. The real thing. No more training sessions, with dummy problems whizzing overhead and a solutions manual available for cover. This would be live ammunition exploding around us. Everyone else looked as frightened as I felt. Now, we fmd out what you're made of, I thought to myself, as the hour struck and the general down front signaled the beginning of the battle. I gulped once and turned over the cover page.

A couple of partial derivatives whistled overhead, and I thought to myself, I can handle this. I started firing, plugged a couple quick. Hey, no worse than an afternoon of video games, I said to myself.

Then I came up over the next page and swallowed hard as I found myself face-to-face with an armored series division. I didn't even stop to think. I just peppered them with Ratio Tests. A few went up in flames. The rest rolled forward. I switched to Root Test, spraying them indiscriminantly. A couple more went down but the rest rumbled forward. So I lobbed in a couple of Basic Comparison Tests and a Limit Comparison Test or two. Then I let loose with the Alternating Series Test and followed up with half a ton of nth Term Tests. That ought to do it, I thought, as I waited for the smoke to clear. But among the littered carcasses on the field before me, there still stood one lone se-

ries. At first I couldn't make it out. But as it lumbered forward, I suddenly realized what this monstrosity must be. It was the dreaded harmonic series. It looked right at me and then let out a howl that turned my bowels to ise.

How do you stop the harmonic series? I tried desperately to remember. We had talked about this in basic training. My instructor's voice echoed in my head, "Pray you never see a harmonic series in battle. They are the nastiest, the ugliest series you will ever see. They diverge, but just barely. There is only one thing to do if you ever find yourself looking down the barrel of the harmonic series . . . . " Yes, yes, I thought, as I waited for the voice to finish its explanation. "Use the integral test."

"See you in hell," I screamed, as I pulled the trigger. The series blew into a million pieces. I laughed maniacally. "Now that's what I call a divergent series."

Soldiers in adjacent foxholes said, "Shhh," and the general down front gave me a concerned look. I turned the page, and took a triple integral right in the gut. I rolled out of my seat and down three steps of the auditorium stairs. A medic, must have been a TA no more than 22 years old, rushed over.

"Are you all right?," she asked, a concerned look on her face.

I felt for the wound in my belly, but miraculously, my hand came out clean. "It must have hit me in the belt buckle," I said as she helped me to my feet. She handed me my helmet and gave me a strange look. She was probably wondering how anyone could survive a triple integral. But stranger things have happened. I retook my seat.

There was a noise behind me and I looked around just in time to see Leftie turning tail and heading for the exit. "Leftie, get back here," I yelled. "They'll courtmartial you for sure."

The neighboring soldiers shushed me again. Afraid I would attract ordinance. I should have known Leftie wouldn't have the guts for it. Ever" since that quiz problem on improper integrals, he had had the shakes.

I leaned over the exam and a word problem went off right in my face, something about length plus girth of a package at the post office. There was red ink everywhere. I waved the medic over and pointed at the problem, but she said, "I'm sorry. I can't help you." I guess there were grunts hurt worse than me. I pulled off my helmet and tied a bandana around my head. It was a sea of red ink out there. The noise was deafening. I started working on the problem in spite of the pain.

At one point, I happened to glance over at the Sarge. She didn't look right. I gave her the thumbs up sign, but she didn't respond. She looked like she might be sick. She was slumped down in her seat. I couldn't see it, but I had to assume there was a pool of red ink on the exam in front of her. I realized she must have taken one in the gut.

She was the one who had come up with my nickname Kodowski. Granted it was my last name, but it had meant a lot to me the first time she called me that. She had saved my ass at least a dozen times already. And now I was losing her, and there was nothing I could do about it. The frustration welled up inside me, and suddenly I roared. Something inside me snapped. I was no longer a human being. I was a calculus killing machine. I flipped the page and moved down eight partial derivatives. I turned around and nailed three limit problems before they even saw me. I took out a triple integral in cylindrical coordinates. Nothing could stop me. Three chain rule problems turned to run, but I never gave them the chance. I flipped page after page. A man with a mission, I was singlehandedly turning the tide. Suddenly I realized the battle was almost over. I triumphantly flipped the last page and found myself face-to-face with the nastiest triple integral problem I had ever seen. It was a volume inside a sphere but outside a cylinder; the famous cored apple. But it said to do it in spherical coordinates. You have to be kidding, I thought to myself. What

twisted devious mind would create such a diabolical weapon? I had no idea what to do.

But then I remembered Sarge's words. "You can't come at a problem like that directly. Come at it from below. One step at a time."

"Yeah Sarge, I remember," I said out loud. I first figured out the equations for the sphere and the cylinder in spherical coordinates. One step in front of the other, Sarge. Then I looked at the intersection. "It's described by an angle, Sarge, I know." I wrote down the triple integral, Sarge's words echoing in my ears. "Don't forget. p2sin cp dp dcp d(} in the integrand."

"Don't worry, Sarge. I won't forget that for as long as I live."

And then it came down to just pulling the .trigger. The integral could essentially do itself. I circled my answer in bright purple ink. Then I flipped the exam closed, ,stood and walked down to the front of the room. The general looked at me nervously.

"Are you proud of yourself?" I said. "All these young lives, wasted. Littered on the field of battle. Never again to raise a pencil for mathematics. Do you feel good about that?"

He looked confused. "Here is your stinking exam", I said

as I threw it down on the table. He stood open-mouthed as I turned and walked up the steps.

We lost them all that day, Sarge, Pipsqueak, Pops, and Leftie. They became Psych majors. I still see them in the halls sometimes, but they never meet my gaze. The math walking wounded.

I was awarded a silver cross to hang on my A, making it an A+. I was promoted, too. They made me a grader. They wanted me to go to officer' s training school at Princeton or maybe Berkeley. And maybe someday I will. Maybe that would make it all worthwhile. But I have to get over the nightmares first. I have to reconcile my victory with the loss of my friends. I have to see mathematics as a tool for good, not a weapon of destruction. Only then, will I be able to move on.


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1 4 THE MATHEMATICAL INTELLIGENCER

BRUCE C. BERNDT, BLAIR K. SPEARMAN, KENNETH S. WILLIAMS

Commentary on an Unpub l ished Lecture by G . N . Watson on So lvi ng the Qu int ic

he following notes are from a lecture on solving quintic equations given by the late

Professor George Neville Watson (1886-1965) at Cambridge University in 1 948. They

were discovered by the first author in 1995 in one of two boxes of papers of Pro

fessor Watson stored in the Rare Book Room of the Library at the University of

Birmingham, England. Some pages that had become separated from the notes were found by the third author in one of the boxes during a visit to Birmingham in 1999.

"Solving the quintic" is one of the few topics in mathematics which has been of enduring and widespread interest for centuries. The history of this subJ"ect is beautifully iUustrated in the poster produced by MATHEMATICA. Many attempts have been made to solve quintic equations; see, for example, [6)-[14], [17)-[21), [28]-[32), [34]-[36], [58]-[60). Galois was the first mathematician to determine which quintic polynomials have roots expressible in terms of radicals, and in 1991 Dummit [24) gave formulae for the roots of such solvable quintics. A quintic is solvable by means of radicals if and only if its Galois group is the cyclic group 71./571. of order 5, the dihedral group D5 of order 10, or the Frobenius group F2o of order

20. In view of the current interest (both theoretical and computational) in solvable quintic equations [24), [33), [43)-[46), it seemed to the authors to be of interest to publish Professor Watson's notes on his lecture, with commentary explaining some of the ideas in more current mathematical language. For those having a practical need for solving quintic equations, Watson's step-by-step procedure will be especially valuable. Watson's method applies to any solvable quintic polynomial, that is, any quintic polynomial whose Galois group is one of 71.1571., D5 or F2o·

Watson's interest in solving quintics was undoubtedly motivated by his keen interest in verifying Srinivasa Ramanujan 's determinations of class invariants, or equivalently, singular moduli. Ramanujan computed the values of over 100 class invariants, which he recorded

The first author thanks Professor Norrie Everitt of the University of Birmingham for an invitation to visit the University of Birmingham in October 1995. The third author thanks Carleton University for a travel grant which enabled him to travel to the University of Birmingham, England in December 1 999. The authors thank the staff of the University of Birmingham Library for making the papers of Watson available to them.

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 4, 2002 15

G. N. Watson

without proofs in his paper [37] and at scattered places throughout his notebooks, especially in his first notebook [39]. Although many of Ramanujan's class invariants had been also calculated by Heinrich Weber [57], most had not been verified. Class invariants are certain algebraic numbers which are normally very difficult to calculate, and their determinations often require solving a polynomial equation of degree greater than 2; and, in particular, 5.

Watson [52] used modular equations in calculating some of Ramanujan's class invariants; solving polynomial equations of degree exceeding 2 was often needed. In a series ojsixjurther papers [50]-[51], [53]-[56], he developed an empirical process for calculating class invariants, which also depended heavily on solving polynomial equations of high degree. He not only verified several of Ramanujan's class invariants but also found many new ones. For these reasons, Watson proclaimed in his lecture that he had solved more quintic equations than any other person. Despite Watson 's gargantuan efforts in calculating Ramanujan's class invariants, eighteen remained unproven until recent times. The remaining ones


were verified in two papers by Berndt, Chan, and Zhang [2], [3]; see also Berndt's book [1, Part V, Chapter 34). Chan [ 16] has used class field theory to put Watson's determinations on a firm foundation, and Zhang [61], [62] has used Kronecker's limit formula to verify Watson's calculations.

Professor Watson held the Mason Chair of Mathematics at the University of Birmingham from 1918 to 1951. He was educated at Cambridge University (1904-1908), where he was a student of Edmund Taylor Whittaker (1873-1956). He became a Fellow of Trinity College, Cambridge, in 1910. From 1 914 to 1918 he held academic positions at University College, London. Watson devoted a great deal of his research to extending and providing proofs for results contained in Ramanujan's Notebooks [39]. He wrote more than thirty papers related to Ramanujan's work, including the aforementioned papers on class invariants or singular moduli.

Most mathematicians know Watson as the co-author with E. T. Whittaker of the classic book A Course of Modem Analysis, first published in 1915, and author of the monumental treatise Theory of Bessel Functions, first published in 1922. For more details of Watson's life, the reader may wish to consult [22], [41], [48].

We now give the text of Watson's lecture with our commentary in italics. In the course of the text we give the contents of three sheets which presumably Watson handed out to his audience. The first of these gives the basic quantities associated with a quintic equation, the second gives twenty-jour pentagrams used in showing that permutations of the suffixes of

(X1X2 + X2X3 + X3X4 + X4X5 + X5X1 - X1X3 - X3X5 - X5X2 - X2X4 - X4X1)2

yield six distinct expressions, and the third gives Watson's method of solving a solvable quintic equation in radicals.

I am going to begin by frankly admitting that my subject this evening is definitely old-fashioned and is rather stodgy; you will not fmd anything exciting or thrilling about it. When the subject of quintic equations was first seriously investigated by Lagrange it really was a "live" topic; the extent of the possibility of solving equations of various degrees by means of radicals was of general interest until it was realized that numbers represented by radicals and roots of algebraic equations were about what one nowadays calls algebraic numbers.

It is difficult to know quite how much to assume that you already know about solutions of algebraic equations but I am going to take for granted . . .

Watson's notes do not state the prerequisites for the lecture!

I cannot begin without saying how much I value the compliment which you have paid me by inviting me to come from a provincial University to lecture to you in Cambridge; and now I am going to claim an old man's privilege of indulging in a few reminiscences. In order to make my lee-

SHEET 1

The denumerate form of the quintic equation is

PoY5 + P1Y4 + P2Y3 + PaY2 + P4Y + P5 = 0.

The standard form of the quintic equation is

ax5 + 5bx4 + 10cx3 + 10dx2 + 5ex + f = 0. (x = lOy)

The reduced form of the quintic equation is

z5 + 10Cz3 + 10Dz2 + 5Ez + F = 0. (z = ax + b)

The sextic resolvent is

where

K = ae - 4bd + 3c2, L = -2a2dj + 3a2e2 + 6abcf- 14abde- 2ac2e + 8acd2 - 4b3f

+ 10b2ce + 20b2d2 - 40bc2d + 15c4, M = a3cj2 - 2a3def + a3e3 - a2b2j2 - 4a2bcef + 8a2bd2f- 2a2bde2

-2a2c2dj - l la2c2e2 + 28a2cd2e - 16a2d4 + 6ab3ej -12ab2cdf + 35ab2ce2 - 40ab2d2e + 6abc3f- 70abc2de + 80abcd3 + 35ac4e - 40ac3d2 - 25b4e2 + 100b3cde -50b2c3e - 100b2c2d2 + 100bc 4d- 25c6.

ture effective, I must endeavour to picture to myself what is passing in the mind of John Brown who is sitting somewhere in the middle of this room and who came up to Trinity last October; and I suppose that, in view of the recent decision about women's membership of the University, with the name of John Brown I must couple the name of his cousin Mary Smith who came up to Newnham at the same time. To try to read their thoughts I must cast my mind back 43 years to the Lent Term of 1905 which was in my first year. If I had then attended a lecture by a mathematician 43 years my senior who was visiting Cambridge,

Watson Building, University of Birmingham. (Photo from 1995.)

an inspection of the Tripos lists would show that the most likely person to satisfy the requisite conditions would have been the late Lord Rayleigh, who was subsequently Chancellor. Probably to you he seems quite prehistoric; to me he was an elderly and venerable figure whose acquaintance I made in 1912, and with whom I subsequently had some correspondence about electric waves. You cannot help regarding me as equally elderly, but I hope that, for a number of reasons you do not �onsider me equally venerable, and that you will believe me when I say that I still have a good deal of the mentality of the undergraduate about me.

However, so far as I know, Lord Rayleigh did not visit Cambridge in the Lent term of 1905, and so my attempt at an analogy rather breaks down. On the other hand a visit was paid to Cambridge at the end of that term by a much more eminent personage, namely the Sultan of Zanzibar. For the benefit of those of you who have not heard that story, I mention briefly that on the last day of term the Mayor of Cambridge received a telegram to the effect that the Sultan and his suite would be arriving by the mid-day train from King's Cross and would be glad if the Mayor would give them lunch and arrange for them to be shown over Cambridge in the course of the afternoon. The program was duly carried out, and during the next few weeks it gradually emerged that the so-called Sultan was W. H. de Vere Cole, a third-year Trinity undergraduate. It was the most successful practical joke of an age in which practical joking was more popular than it is to-day.


If you could be transported back to the Cambridge of 1905, you would find that it was not so very different from the Cambridge of 1948. One of the differences which would strike you most would probably be the fact that there were very few University lectures (and those mostly professorial lectures which were not much attended by undergraduates); other lectures were College lectures, open only to members of the College in which they were given, or, in the case of some of the smaller Colleges, they were open to members of two or three colleges which had associated themselves for that purpose. Thus most of the teaching which I received was from the four members of the Trinity mathematical staff; the senior of them was Herman, who died prematurely twenty years ago; in addition to teaching me solid geometry, rigid dynamics and hydrodynamics, he infected me with a quality of perseverance and tenacity of purpose which I think was less uncommon in the nineteenth century than it is to-day when mathematics is tending to be less concrete and more abstract. Whitehead was still alive when I started collecting material for this lecture. Whittaker, who lectured on Electricity and Geometric Optics, whose name is sometimes associated with mine, is living in retirement in Edinburgh; and Barnes is Bishop of Birmingham. Outside the College I attended lectures by Baker on Theory of Functions, Berry of King's who taught me nearly all of what I know of elliptic functions, and Hobson on Spherical Harmonics and Integral Equations; also two of the Professors of that time that were Trinity men, Forsyth and Sir George Darwin, whom I remember lecturing on curvature of surfaces and the problem of three bodies, respectively. Two things you may have noticed, the large proportion of my teachers who are still alive, and the

Bruce Berndt with Ramanujan's Slate.


insularity, if I may so describe it, of my education. If that hypothetical lecture by Lord Rayleigh had taken place, he could have given a more striking illustration of insularity which you will probably hardly credit. In his time, each College tutor was responsible for the teaching of his own pupils and of nobody else; he was aided by one or two assistant tutors, but the pupils, no matter what subject they were reading, received no official instruction except from their own tutor and his assistants.

After spending something like ten minutes on these irrelevancies, it is time that I started getting to business.

There is one assumption which I am going to make throughout, namely that the extent of your knowledge about the elements of the theory of equations is roughly the same as might have been expected of a similar audience in 1905. For instance, I am going to take for granted that you know about symmetric functions of roots in terms of coefficients and that you are at any rate vaguely familiar with methods of obtaining algebraic solutions of quadratic, cubic and quartic equations, and that you have heard of the theorem due to Abel that there is no such solution of the general quintic equation, i.e., a solution expressible by a number of root extractions.

In modern language, if.f(x) E iQ[x] is irreducible and of degree 5, then the quintic equation.f(x) = 0 is solvable by radicals if and only if the Galois group G of .f(x) is solvable. The Galois group G is solvable if and only if it is a subgroup of the Frobenius group F20 of order 20, that is, it is Fzo, D5 (the dihedral group of order 10), or 7l./57L (the cyclic group of order 5); see for example [24, Theorem 2, p. 397], [25, Theorem 39, p. 609]. Thus a quintic equation .f(x) = 0 cannot have its roots expressed by a finite number of root extractions if the Galois group G of f is non-solvable, that is, if it is S5 (the symmetric group of order 120), or A5 (the alternating group of order 60). "Almost all" quintics have S5 as their Galois group, so the ''general" quintic is not solvable by radicals. It is easy to give examples of quintics which are not solvable by radicals; see for example [46].

You may or may not have encountered the theorem that any irreducible quintic which has got an algebraic solution has its roots expressible in the form

where w denotes exp(271i/5), r assumes the values 0, 1 , 2, 3, 4, and uY, ut u�, u� are the roots of a quartic equation whose coefficients are rational functions of the coefficients of the original quintic. If you are not familiar with such results, you will find proofs of them in the treatise by Burnside and Panton.

One can find this in Section 5 of Chapter XX of Burnside and Panton's book [5, Vol. 2] . A modern reference for this result is [24, Theorem 2, p. 397].

When I was an undergraduate, all other knowledge about quintic equations was hidden behind what modern politi-

cians would describe as an iron curtain, and it is convenient for me to assume that this state of affairs still persists, for otherwise it would be a work of a supererogation for me to deliver this lecture.

I might mention at this point that equations of the fifth or a higher degree which possess algebraic solutions (such equations are usually described as Abelian) are of some importance in the theory of elliptic functions, apart from their intrinsic interest.

Today such equations are called solvable.

There is, for instance, a theorem, also due to Abel, that the equations satisfied by the so-called singular moduli of elliptic functions are all Abelian equations.

Singular moduli are discussed in Cox's book [23, Chapter 3) as well as in Berndt's book [1, Part V, Chapter 34].

It was these singular moduli which aroused my interest some fifteen years ago in the solutions of Abelian equations, not only of the fifth degree, but also of the sixth, seventh and other degrees higher still. It consequently became necessary for me to co-ordinate the work of previous writers in such a way as to have handy a systematic procedure for solving Abelian quintic equations as rapidly as possible, and this is what I am going to describe tonight.

Methods for solving a general solvable quintic equation in radicals have been given in the 1990s by Dummit [24) a'Jild Kobayashi and Nakagawa [33]; see also [47].

To illustrate the nature of the problem to be solved, I am now going to use equations of degrees lower than the fifth as illustrations. A reason why such equations possess algebraic solutions (and it proves to be the reason) is that certain non-symmetric functions of the roots exist such that the values which certain powers of them can assume are fewer in number than the degree of the equation. Thus, in the case of the quadratic equation with roots a and {3, there are two values for the difference of the roots, namely

a- {3, f3- a.

However the squares of both of these differences have one value only, namely

(a + {3)2 - 4a{3,

and this is expressible rationally in terms of the coefficients. Hence the values of the differences of the roots are obtainable by the extraction of a square root, and, since the sum of the roots is known, the roots themselves are immediately obtainable.

The cubic equation, with roots a, {3, y, can be treated similarly. Let €3 = 1 (E i=- 1). Then we can form six expressions

a + {3E + y€2, � + yE + ail', a + {3€2 + yE, f3 + y€2 + aE,

y + aE + {3€2, y + ail' + {3E,

with the property that their cubes have, not six different values, but only two, namely

(a + {3E + y€2)3, (a + {3€2 + yE)3,

and these expressions are the roots of a quadratic equation whose coefficients are rational functions of the coefficients of the cubic. When the cubic equation is

ax3 + 3bx2 + 3cx + d = 0,

the quadratic equation is

a6X2 + 27a3(a2d - 3abc + 2b3)X + 729(b2- ac)3 = 0,

and there is no difficulty in completing the solution of the cubic.

It is easily checked using MAPLE that this quadratic is correct.

For the quartic equation, with roots a, {3, y, o, such expressions as

(a + f3 - y- o)2, (a + X - 8 - {3)2, (a + o ;- f3- y)2

have only three distinct values; similar but slightly simpler expressions are

af3 + yo - ay - ao - f3y - {3o, etc.,

or simpler still,

af3 + yo, ay + {3o, ao + f3y. When the quartic equation is taken to be

ax4 + 4bx3 + 6cx2 + 4dx + e = 0,

the cubic equation satisfied by the last three expressions is

a,X3 - 6a2cX2 + (16bd- 4ae)aX - (16b2e + 16ad2 - 24ace) = 0,

and, by the substitution

aX- 2c = -48, this becomes

4&- 8(ae - 4bd + 3c2) - (ace + 2bcd - ad2 - b2e - c3) = 0,

which is the standard reducing cubic

4& - I8 - J= 0.

This is discussed in [26, pp. 191-197; see problem 15, p. 197), where the values of I and J are given by

a b c I = ae - 4bd + 3c2, J = b c d

c d e

I have discussed the problem of solving the quartic equation at some length in order to be able to point out to you the existence of a special type of quartic equation which rarely receives the attention that it merits. In general the reducing cubic of a quartic equation has no root which is rational in the field of its coefficients, and any expression for the roots of the quartic involves cube roots; on the other


SHEET 1A

The discriminant Ll of the quintic equation in its standard form is equal to the product of the squared differences of the roots multiplied by a8/3125. The value of the discriminant Ll in terms of the coefficients is

a4j4 - 20a3bcf 3 - 120a3cdf 3 + 160a3ce2j2 + 360a3d2ef 2 -640a3de3f + 256a3e5 + 160a2b2df 3 - 10a2b2e2f 2 +360a2bc2j3- 1640a2bcdej2 + 320a2bce3f - 1440a2bd3f 2 +4080a2bd2c2f- 1920a2bde4- 1440a2c3ef 2 + 2640a2c2d2f 2 +4480a2c2de2f - 2560a2c2e4 - 10080a2cd3ef + 5760a2cd2e3 +3456a2d5f - 2160a2d4e2- 640ab3cj3 + 320ab3def 2 -180ab3c3f + 4080ab2c2ef 2 + 4480ab2cd2f 2 - 14920ab2cde2f +7200ab2ce4 + 960ab2d3ef- 600ab2d2e3 - 10080abc3df2 +960abc3e2f + 28480abc2d2ef - 16000abc2de3 - 11520abcd4j + 7200abcd3e2 + 3456ac5f 2 - 11520ac4def + 6400ac4e3 +5120ac3d3f- 3200ac3d2e2 + 256b5f 3- 1920b4cej2 -2560b4d2f 2 + 7200b4de2f - 3375b4c4 + 5760b3c2df2

-600b3c2e2f- 16000b3cd2ef + 9000b3cde3 + 6400b3d4j -4000b3d3e2 - 2160b2c4j2 + 7200b2c3def- 4000b2c3e3 -3200b2c2d3f + 2000b2c2d2e2.

hand, there is no difficulty in constructing quartic equations whose reducing cubics possess at least one rational root; the roots of such quartics are obtainable in forms which involve the extraction of square roots only. Such quartics are analogous to Abelian equations of higher degrees, and it might be worth while to describe them either as "Abelian quartic equations" or as "biquadratic equations," the latter being an alternative to the present usage of employing the terms quartic and biquadratic indifferently. (I once discussed this question with my friend Professor Berwick, who in his lifetime was the leading authority in this countcy on algebraic equations, and we both rather reluctantly came to the conclusion that the existing terminology was fixed sufficiently firmly to make any alteration in it practically impossible.)

If f(x) E Q[x] is an irreducible quartic polynomial, its cubic resolvent has at least one rational root if and only if the Galois group of f(x) is the Klein 4-group V4 of order 4, the cyclic group 7lJ47L of order 4, or the dihedral group D4 of order 8. Since D4 is not abelian, it is not appropriate to call such quartics "abelian. " For the solution of the quartic by radicals, see for example [25, p. 548].

After this very lengthy preamble, I now reach the main topic of my discourse, namely quintic equations. Some of you may be familiar with the name of William Hepworth Thompson, who was Regius Professor of Greek from 1853 to 1866, and subsequently Master of Trinity until 1886. A question was once put to him about Greek mathematics, and his reply was, "I know nothing about the subject. I have never even lectured upon it." Although there are large tracts of knowledge about quintic equations about which I am in complete ignorance, I have a fair amount of practical experience of them. For instance, if my friend Mr. P. Hall of King's College is here this evening, he will probably be horrified at the ignorance which


I shall show when I say anything derived from the theory of groups. On the other hand, while to the best of my knowledge nobody else has solved more than about twenty Abelian quintics (you will be hearing later about these solvers, and I have no certain knowledge that anybody else has ever solved any), my own score is something between 100 and 120; and I must admit that I feel a certain amount of pride at having so far outdistanced my nearest rival.

Young solved several quintic equations in [58] and [59].

The notation which I use is given at the top of the first of the sheets which have been distributed. The first equation, namely

PoY5 + P1y4 + P2Y3 + P3Y2 + P4Y + P5 = 0, is what Cayley calls the denumerate form, while

ax5 + 5bx4 + 10cx3 + 10&2 + 5ex + f= 0, is the standard form. The second is derived from the first by the substitution lOy = x, with the relations

a =Po, b = 2pl, c = 10p2, d = lOOp3, e = 2000p4, f = l05p5. Next we carry out the process usually described as "re

moving the second term" by the substitution ax + b = z, which yields the reduced form

z5 + 10Cz3 + 10Dz2 + 5Ez + F = 0, in which

C = ac - b2, D = a2d - 3abc + 2b3, E = a3e - 4a2bd + 6ab2c - 3b4, F = a4j- 5a3be + 10a2b2d - 10ab3c + 4b5.

The roots of the last two quintics will be denoted by Xr

and Zr respectively with r = 1, 2, 3, 4, 5.

SHEET 2

2 5 4 1 3

4 1 2 3 1 5 2 1 4 5 3 1

2 3 3 1 2 5 4 1

5 3 2 4 3 5 4 2

3

5 4 3 2 2 3 1

5 2 4 3 2 5 3 4


SHEET 3

The roots of the quintic in its reduced form are

Zr+ l = wrul + u.J.ru2 + w3ru3 + w4ru4 with w = exp(277i/5), r = 1, 2, 3, 4, 0. (1) U1U4 + U2U3 = -2C. (2) u1us + u�u1 + u�u4 + u�u2 = - 2D. (3) u1u� + u�u� - UJU2UsU4 - u1u2 - u�u4 - u3ul - U�Us = E. (4) u�[+ u� + u� + u� - 5(u1u4 - u2us)(uius - u�u1 - u�u4 + u�u2) = -F. New unknowns, (} and T, defined by

(5) U1U4 - U2U3 = 2 (}, (6) u1u3 + u�u2 - u�u1 - u�u4 = 2T.

u1u4 = -c + o, u2us = -c - o. u1us + u�u2 = -D + T, u�u1 + u�u4 = -D - T.

u1us - u�u2 = ::±:: Y(D - T? + 4(C - 0)2(C + 0) = Rb u�ul - u�u4 = ::±:: Y(D + T)2 + 4(C + 0)2(C - 0) = R2.

uru2 = (UIUs)(u�ul)/(UzUs), etc., U� = (uiusi(u�ul)/(UzUd, etc. (7) C(D2 - T2) + (C2 - 02)(C2 + 302 - E) = R1R20. (8) (D2 - T2)2 + 2C(D2 - T2)(C2 + 3()2) - 8Co2(D2 + T2)

+(Cz _ o2)2(C2 _ 502)2 + 16D03T + E2(C2 _ ()2)

- 2CE(D2 - T2) - 2E(C2 - 02)(C2 + 302) = 0. (9) (DO + CT)(D2 - T2) + T(C2 - 502)2 - 2CDEO

-ET(C2 + 02) + FO(C2 - 02) = 0.

Young's substitutions are

T = Ot, 02 = t/1.

The connexion between the (} above and the cp of Cayley's sextic resolvent is

10oV6 = a2¢.

The denumerate quintic of Ramanujan's problem is

y5 - y4 + y3 - 2y2 + 3y - 1 = 0.

For this quintic, C = 6, D = -156, E = 4592, F = -47328.

z = lOy - 2, (} = -10V5, t = -10, T = 100V5. ,------=,.-

ut u� = -13168 - 6400V5 ::±:: (2160 + 960v'5}Y79(5- 2V5),

u�, u� = -13168 + 6400V5 ::±:: (2160- 960V5}Y79(5 + 2V5).

RI, R� = 79(800 ::±:: 160V5), R1R2 = -320 x 79V5.

We remark that Young's equations for t and t/1 in this example are:

(24336 - tjJt2) + 12(24336 - tjJt2)(36 + 3t/f) - 48t/1(24336 + tjJt2) + (36 - t/1)(36 - 5tjJ)2 - 2496tjJ2t - 581898240 - 21086464t/f

+ 55104tjJt2 - 9184(36 - t/1)(36 + 3t/f) = 0


and

( -156 + 6t)(24336 - tjJt2) + t(36 - 5tjJ)2 + 6892416 - 4592t(36 + t/1) + 47328t/f = 0,

so that t = -10 and tjJ = 500 in agreement with Watson.

Our next object is the determination of non-symmetric functions of the roots which can be regarded as roots of a resolvent equation. An expression which suggests itself is

(x1 + WX"z + u?x3 + W3x4 + w4x5)5.

The result of permuting the rQots is to yield 24 values for the expression.

A permutation u E S5 acts on this element by

u((x1 + WX"z + w2x3 + w3x4 + w4x5)5) = (Xu(1) + WX"u(2) + u?xu(3) + w3Xu(4) + w4Xu(5))5.

An easy calculation shows that u preserves

a = Cx1 + WX"z + u?x3 + w3x4 + w4x5)5

if and only if u = (1 2 3 4 5)k for some k E {0, 1, 2, 3, 4}. Hence

stab85(a) = ((1 2 3 4 5)),

so that

lstabs5(a)l = 5.

Thus, by the orbit-stabilizer theorem [27, p. 139], we obtain 1" I 120 lorbs5( a )I = ? = -

5- = 24,

so that permuting the roots yields 24 different expressions.

The disadvantage of the corresponding resolvent equation is the magnitude of the degree of its coefficients when expressed as functions of the coefficients of the quintic; moreo�er it is difficult to be greatly attracted by an equation whose degree is as high as 24 when our aim is the solution of an equation of degree as low as 5.

An expression which is more amenable than the expression just considered was discovered just 90 years ago by two mathematicians of some eminence in their day, namely Cockle and Harley, and it was published in the Memoirs of the Manchester Literary and Philosophical Society. This expression is

c/>1 = X1X2 + X2X3 + X3X4 + X4X5 + X�1 - X1X3 - X3X5 - X�z - XzX4 - X4X1·

The quantities X1X2 + XzX3 + X3X4 + X4X5 + X�1 and X1X3 + X3X5 + x�z + xzx4 + X4X1 appear in the work of Harley [29] and their difference is considered by Cayley [6]. We have not located a joint paper of Cockle and Harley. When he was writing these notes, we believe Watson was reading from Cayley [6] where the names of Cockle and Harley are linked [6, p. 311 ] .

Permutations of the suffixes give rise to 24 expressions, which may be denoted by I ± X,X8, where r and s run through the values 1, 2, 3, 4, 5 with r -=F s. The choice of the signs is most simply exhibited diagrammatically, with each of the 24 expressions represented by a separate diagram. If you tum to the second page of your sheets, you will see the 24 pentagrams with vertices numbered 1, 2, 3, 4, 5 in

all possible orders (there is no loss of generality in taking the number 1 in a special place) and the rule for determination of signs is that terms associated with adjacent vertices are assigned + signs, while those associated with opposite vertices are assigned - signs.

Now the pentagrams in the third and fourth columns are the optical images in a vertical line of the corresponding pentagrams in the first and second columns, and since proximity and oppositeness are invariant for the operation of taking an optical image, the number of distinct values of 4> is reduced from 24 to 12.

Further, the pentagrams in the second column are derived from those in the first column by changing adjacent vertices into opposite vertices, and vice versa, so that the values of 4> arising from pentagrams in the second column are minus the values of 4> arising from the corresponding pentagrams in the first column. It follows that the number of distinct values of cf>2 is not 12 but 6, and so our resolvent has now been reduced to a sextic equation in cf>2, with coefficients which are rational functions of the coefficients of the quintic, and a sextic equatic,m is a decided improvement on an equation of degree t20, or even on one of degree 24.

Let a = (12345) E S5 and b = (25)(34) E 85, so that a5 =

b2 = e and bab = a4. As ac/>1 = 4>1 and bc/>1 = c/>1, we have

stabs5( c/>1) 2:: (a, b la5 = b2 = e, bab = a4) = D5,

so that

lstabs5( c/>1) l 2:: 10.

On the other hand, the first two columns of Watson's pentagram table show that

lorbs5( c/>1) ! 2:: 12.

Hence, by the orbit-stabilizer theorem, we see that

lstabs5( c/>1) l = 10, lorbs5( c/>1) l = 12

and thus

stabs5( c/>1) = (a, b la5 = b2 = e, bab = a4) = D5.

Now let c = (2 3 4 5), so that c2 = b. As acf>I = cf>I and ccf>I = 4>1. we have

stabs5( c/>I) :::2 (a, cla5 = c2 = e, c- 1ac = a3) = Fzo,

so that lstabs5(c/>I)I 2:: 20. From the first column of the pentagram table, we have

lorbs5( cf>I)I 2:: 6.

Hence, by the orbit-stabilizer theorem, we deduce that

lstabs5( c/>I)I = 20, lorbs5( c/>I)I = 6,

and thus

stabs5( c/>I) = (a, cla5 = c4 = e, c- 1ac = a3) = Fzo.

It is, however, possible to effect a further simplification; it is not, in general, possible to construct a resolvent equation of degree less than 6, but it is possible to construct a sextic resolvent equation in which two of the coefficients


are zero. We succeeded in constructing a sextic in ql- because the 12 values of 4> could be grouped in pairs with the members of each pair numerically equal but opposite in sign; but a different grouping is also possible, namely a selection of one member from each of the six pairs so as to form a sestet in which the sum of the members is zero, and it is evident that those members which have not been selected also form a sestet in which the sum of the members is zero; one of these sestets is represented by the pentagrams in the first column, the other by the pentagrams in the second column.

A sestet is a set of six objects.

Denote the values of 4> represented by the pentagrams in the first column by 4>1, c/>2, . . . , 4>6, and let

4>[ + 4>2 + · · · + 4>6 = Er.

It is then not difficult to verify that an interchange of any pair of x1, x2, . . . , x5 changes the sign of Er when r is odd, but leaves it unaltered in value when r is even.

By looking at the first column of the pentagram table we see that the even permutations (234), (243), (354), (235), (24)(35) send 4>1 to 1>2, 4>3, 4>4, 4>5, 4>6, respectively. We next show that an odd permutation cr cannot send 4>i to 4>i for any i and j. Suppose that cr( 4>i) = 4>i· By the above remarks 4>i = 04>1 for some (} E A5, and 1>1 = P4>i for some p E A5. Hence

so that per(} E stabs51>1 = D5 C A5.

Hence cr E A5, which is a contradiction. Now

{ cr( 4>!), . . . , cr( 4>6) } � orbs54>I> lorbs51>1 l = 12,

and

so that

Thus if T E s5 is a transposition,

r(Er) = r(4>[ + · · · + 4>6) = c- 1>1Y + · · · + (- 4>6Y = (- 1Y Er.

It is now evident that each of the 10 expressions Xm - Xn (m, n = 1, 2, 3, 4, 5; m < n) is a factor of Er whenever r is an odd integer.

Clearly Er E Z[xh ... , x5] and so can be regarded as a polynomial in x1 with coefficients in Z[x2, ... , x5]. Dividing Er by x1 - x2, we obtain

Er = (xl - x2)q(x2, . . . , X5) + r(x2, . . . , X5),

where


lfr is odd, the transposition (12) changes the above equation to

Adding these two equations, we obtain

0 = (XI - X2)(q(x2, X3, X4, X5) - q(X!, X3, X4, X5)) + r(x2, X3, X4, X5) + r(x1, X3, X4, X5).

Taking x1 = x2 we deduce that

r(x2, X3, X4, X5) = 0.

Hence

Thus X1 - X2 divides Er. Similarly Xm - Xn divides Erfor m, n = 1, 2, 3, 4, 5, m < n. Hence

II (Xm - Xn) l�m<n::::::;;5

divides Er when r is an odd integer.

Now the degrees of E1 and E3 in the x's are respectively 2 and 6, and so, since these numbers are less than 10, both E 1 and E3 must be identically zero, while E5 must be a constant multiple of

(x1 - x2)(x1 - x3) · · · (x3 - x5)(x4 - x5).

On the other hand, S2, S4, and S6 are symmetric functions of the x's, and are consequently expressible as rational functions of the coefficients in the standard form of the quintic.

These properties ensure that the polynomial

( 4> - 4>I)( 4> - c/>2) • . . ( 4> - 4>6)

has coefficients in ()I or Q(VI)), where D is the discriminant of the quintic, with the coefficients of 4>5 and 4>3

equal to zero.

Apart from the graphical representation by pentagrams (which, as the White Knight would say, is my own invention), all of the analysis which I have just been describing was familiar to Cayley in 1861; and he thereupon set about the construction of the sextic resolvent whose roots are 1>1> c/>2, . .. , 4>6• The result which he obtained was the equation

(0) a64>6 - 100Ka44>4 + 2000La2ql-

- 800a24>-v'M + 40000M = 0

in which the values of K, L, M in terms of the coefficients of the quintic are those given on the first sheet, while Ll is the discriminant of the quintic in its standard form, that is to say, it is the product of the squared differences of the roots of the quintic multiplied by a8/3125. Its value, in terms of the coefficients occupies the lower half of the first sheet.

The work of Cayley to which Watson refers is contained in [6], where on pages 313 and 314 Cayley introduces the

pentagrams described by Watson. Note that the usual discriminant D of the quintic is [26, p. 205]

a8(x1 - x2)2(x1 - x3)2 · · · (x4 - x5)2 = 3125Ll = 55d.

There is no obvious way of constructing any simpler resolvent and so it is only natur.§ll to ask "Where do we go from here?" It seems fruitless to attempt to obtain an algebraic solution of the general sextic equation; for, if we could solve the general sextic equation algebraically, we could solve the general quintic equation by the insertion of a factor of the first degree, so as to convert it into a sextic equation. In this connection I may mention rather a feeble joke which was once perpetrated by Ramanujan. He sent to the Journal of the Indian Mathematical Society as a problem for solution:

Prove that the roots of the equation

x6 - x3 + x2 + 2x - 1 = 0

can be expressed in terms of radicals.

This problem is the first part of Question 699 in [38]. It can be found in [40, p. 331 ] . A solution was given by Watson in [49]. It seems inappropriate to refer to this problem as a "feeble joke. "

Some years later I received rather a pathetic letter from a mathematician, who was anxious to produce something worth publication, saying that he had noticed that x + 1 was a factor of the expression on the left, and that he wanted to reduce the equation still further, but did not see how to do so. My reply to his letter was that the quintic el')uation

x5 - x4 + x3 - 2x2 + 3x - 1 = 0

was satisfied by the standard singular modulus associated with the elliptic functions for which the period iK' I K was equal to v=79, and consequently it was an Abelian quintic, and therefore it could be solved by radicals; and I told him where he would find the solution in print. I do not know why Ramanujan inserted the factor x + 1; it may have been an attempt at frivolity, or it may have been a desire to propose an equation in which the coefficients were as small as possible, or it may have been a combination of the two.

On pages 263 and 300 in his second Notebook [39], Ramanujan indicates that 2114G79 is a root of the quintic equation x5 - x4 + x3 - 2x2 + 3x - 1 = 0; see [1 , Part V, p. 193]. For a positive integer n, Ramanujan defined Gn by

where, for any z = x + iy E C with y > 0, Weber's class invariantf(z) [57, Vol. 3, p. 1 14] is defined in terms of the Dedekind eta function

00 1)(Z) = e;,.;z/12 II (1 - e211'imz)

m= l

by

A result equivalent to Ramanujan's assertion was first proved by Russell [42] and later by Watson [53]; see also [54]. The solution of this quintic in radicals is given in [49]. In [38], Ramanujan also posed the problem of finding the roots of another sextic polynomial which factors into x - 1 and a quintic satisfied by G47. For additional comments and references about this problem, see [4] and [40, pp. 400-401] . Both Weber and Ramanujan calculated over 100 class invariants, but for different reasons. Class invariants generate Hilbert class fields, one of Weber's primary interests. Ramanujan used class invariants to calculate explicitly certain continued fractions and products of theta functions.

After this digression, let us return to the sextic resolvent· it is the key to the solution of the quintic in terms of ' . radicals, provided that suCh a key exists. It is possible, by accident as it were, for the sextic resolvent to have a solution for which ¢2 is rational, and the corresponding value of ¢ is of the form p �' where p is rational. A knowledge of such a value of ¢ proves to be sufficient to enable us to express all the roots of the quintic in terms of radicals. In fact, when this happy accident occurs, the quintic is Abelian, and when it does not occur, the quintic is not Abelian.

If ¢2 E (!) it is clear from the resolvent sextic that ¢ = p� for some p E Q. We are not aware of any rigorous direct proof in the classical literature of the equivalence of ¢2 E (!) to the original quintic being solvable.

This is as far as Cayley went; he was presumably not interested in the somewhat laborious task of completing the details of the solution of the quintic after the determination of a root of his sextic resolvent.

The details of the solution of an Abelian quintic were worked out nearly a quarter of a century later by a contemporary of Cayley, namely George Paxton Young. I shall not describe Young as a mathematician whose name has been almost forgotten, because he was not in fact a professional mathematician at all. The few details of his career that are known to me are to be found in Poggendorfs biographies of authors of scientific papers. He was born in 1819, graduated M.A. at Edinburgh, and was subsequently Professor of Logic and Metaphysics at Knox College, Toronto; he was also an Inspector of Schools, and subsequently Professor of Logic, Metaphysics and Ethics in the University of Toronto. He died at Toronto on February 26, 1889. His life was thus almost coextensive with Cayley's (born August 16, 1821, died January 26, 1895). Young in the last decade of his life (and not until then) published a number of papers on the algebraic solution of equations, including three in the American Journal of Mathematics


which among them contain his method of solving Abelian

quintics.

These are papers [58], [59] and [60].

In style, his papers are the very antithesis of Cayley's. While

Cayley could not (or at any rate frequently did not) write

grammatical English, he always wrote with extreme clar

ity, and, when one reads his papers, one cannot fail to be

impressed by the terseness and lucidity of his style, by the

mastery which he exercises over his symbols, and by the

feeling which he succeeds in conveying that, although he

may have frequently suppressed details of calculation, the

reader would experience no real difficulty in filling in the

lacunae, even though such a task might require a good deal

of labour.

On the other hand, when one is reading Young's work,

it is difficult to decide what his aims are until one has

reached the end of his work, and then one has to return to

the beginning and read it again in the light of what one has

discovered; his choice of symbols is often unfortunate; in

fact when I am reading his papers, I find it necessary to

make out two lists of the symbols that he is using, one list

of knowns and the other of unknowns; finally, his results

seemed to be obtained by a sheer piece of good fortune,

and not as a consequence of deliberate and systematic

strategy. A comparison of the writings of Cayley and Young

shows a striking contrast between the competent draughts

manship of the lawyer and pure mathematician on the one

hand and the obscurity of the philosopher on the other. The rest of my lecture I propose to devote to an account

of a practical method of solving Abelian quintic equations.

The method is in substance the method given by Young,

but I hope that I have succeeded in setting it out in a more

intelligible, systematic and symmetrical manner.

Take the reduced form of the quintic equation

z5 + 10Cz3 + 10Dz2 + 5Ez + F = 0,

and suppose that its roots are

where

w = exp(2 17i/5), r = 1, 2, 3, 4, 0.

Straightforward but somewhat tedious multiplication

shows that the quintic equation with these roots is

and a comparison of these two forms of the quintic yields

four equations from which u1, u2, u3, u4 are to be deter

mined, namely

(1)


(2)

(3) u1u3 + u�u1 + u§u4 + u�u2 = -2D,

2 2 + 2 2 3 3 3 U1U4 U2U3 - U1U2U3U4 - U1U2 - U2U4 - U3U1 - U�U3 = E,

(4) uY + u� + u� + u� - 5(ulu4 - u2u3)(uiu3 - u�u1 - u§u4 + u�u2) = -F.

These coefficients were essentially given by Ramanujan in his first Notebook [39]; see Berndt [1 , Part IV, p. 38] . They also occur in [43].

We next introduce two additional unknowns, 0 and T, defined by the equations

(5) (6)

in which a kind of skew symmetry will be noticed. The nat

ural procedure is now to determine u1, u2, u3, u4 in terms of

0, T and the coefficients of the reduced quintic by using equa

tions (1), (2), (5) and (6) only. When this has been done, sub

stitute the results in (3) and (4), and we have reached the

penultimate stage of our journey by being confronted with

two simultaneous equations in the unknowns 0 and T. From (1) and (5) we have

while from (2) and (6) we have

u1u3 + u�u2 = -D + T, u�u1 + u§u4 = -D - T;

and hence it follows that

u1u3 - u�u2 = ± Y(D - T? + 4(C - 0)2(C + 0) = : R1, say;

u�u1 - u§u4 = ± Y(D + T)2 + 4(C + 0)2(C - 0) =: R2, say.

Watson makes use of the identities

(uiu3 - u�u2? = (uiu3 + u�u2? - 4(ulu4i(u2u3), (u�u1 - u§u4)2 = (u�u1 + u§u4)2 - 4(u2u3i(ulu4).

These last equations enable us to obtain simple expres

sions for the various combinations of the u's which occur

in (3) and (4). Thus, in respect of (3), we have

U2U3

with similar expressions for u�u4, u�u1, u�u3. When we sub

stitute these values in (3) and perform some quite straight

forward reductions, we obtain the equation

This shows incidentally that, when 0 and T have been de

termined, the signs of R1 and R2 cannot be assumed arbitrarily but have to be selected so that R 1R2 has a uniquely de

terminate value. The effect of changing the signs of both R1 and R2 is merely to interchange u1 with u4 and u2 with u3.

The result of rationalising (7) by squaring is the more

formidable equation

(D2 - T2)2 + 2C(D2 - T2)(C2 + 302) - 8C02(D2 + T2) (8) +(C2 - 02)(C2 - 502)2 + 16D03T + E2(C2 - 02)

-2CE(D2 - T2) - 2E(C2 - 02)(C2 + 302) = 0.

This disposes of (3) for the time being, and we turn to (4). The formulae which now serve our purpose are

5 _ (uiu3i(u�ul) t u1 - ( )2 , e c.,

U2U3

with three similar formulae. When these results are inserted in ( 4) and the equation so obtained is simplified as much as possible, we have an equati9n which I do not propose to write down, because it would be a little tedious; it has a sort of family resemblance to (7) in that it is of about the same degree of complexity and it involves the unknowns (} and T and the product R1R2 rationally.

MAPLE gives the equation as

(JJ2 - T2)(De2 + 2CTe + C2D) + 2(C2 - e2)(3CD()2 - Te3) -R1R2(Te2 + 2CD(} + C2T) + (C2 - e2)2(20T(} - F) = 0.

When we substitute for this product R1R2 the value which is supplied by (7), we obtain an equation which is worth writing out in full, namely

(De + CT)(D2 - T2) + T(C2 - 5e2)2 - 2CDEe (9) -ET(C2 + (}2) + Fe(C2 - (}2) = 0.

We now have two simultaneous equations, (8) and (9), in which the only unknowns are (} and T. When these equations have been solved, the values of u1, u2, u3, u4 are immediately obtainable from formulae of the type giving uY in the form of fifth roots, and our quest will have reached its end.

Watson means that u1 can be given as a fifth root of an e:jipression involving the coefficients of the quintic, R1 and R2.

An inspection of this pair of equations, however, suggests that we may still have a formidable task in front of us.

It has to be admitted that, to all intents and purposes, this task is shirked by Young. In place of (8) and (9), the equations to which his analysis leads him are modified forms of (8) and (9). They are obtainable from (8) and (9) by taking new unknowns in place of (} and T, the new unknowns t and 1/J being given in terms of our unknowns by the formulae

T = et, e2 = 1/J.

Young's simultaneous equations are cubic-quartic and quadratic-cubic respectively in 1/J and t. When the original quintic equation is Abelian, they possess a rational set of solutions.

Young's pair of simultaneous equations for t and 1/J are

(D2 - ljJt2)2 + 2C(D2 - 1jJt2)(C2 + 31/J) - 8CI/J(D2 + 1jJt2) + (C2 - I/J)(C2 - 51/J)2 + 16DijJ2t + E2(C2 - 1/J)

- 2CE(D2 - ljJt2) - 2E(C2 - I/J)(C2 + 31/J) = 0

and

(D + Ct)(D2 - 1jJt2) + t(C2 - 51/1)2 - 2CDE - Et(C2 + 1/J) + F(C2 - 1/J) = 0.

Young goes on to suggest that, in numerical examples, his pair of simultaneous equations should be solved by inspection. He does, in fact, solve the equations by inspection in each of the numerical examples that he considers, and, although he says it is possible to eliminate either of the unknowns in order to obtain a single equation in the other unknown, he does not work out the eliminant. You will probably realize that the solution by inspection of a pair of simultaneous equations of so high a degree is likely to be an extremely tedious task, and you will not be mistaken in your assumption. Consequently Young's investigations have not got the air of finality about them which could have been desired.

Fortunately, however, the end of the story is implicitly told in the paper by Cayley on the sextic resolvent which I have already described to you and which had been published over a quarter of a century earlier. It is, in fact, easy to establish the relations

Z1Z2 + · . . -z1Z3 - · · ·

= a2(x1x2 + · · · -x1x3- · · · ) = a2¢1,

and also to prove that the expression on the left is equal to

5(ulu4 - u2u3)V5

so that

Watson is using the relation Zi = axi + b (i E { 1, 2, 3, 4, 5}) to obtain the first equality.

With Zr = wrul + w2ru2 + w3ru3 + w4ru4 (r E { 1, 2, 3, 4, 5}) MAPLE gives

Z1Z2 + . . . -z1Z3 - . . .

= 5(ulu4 - u2u3)(w - w2 - w3 + w4)

so that

z1z2 + · · · -z1z3 - · · · = 5(ulu4 - u2u3)V5

since

Consequently, to obtaill a value of (} which satisfies Young's simultaneous equations, all that is necessary is to obtain a root of Cayley's sextic resolvent; and the determination of a rational value of ¢2 which satisfies Cayley's sextic resolvent is a perfectly straightforward process, since any such value of a2¢2 must be an integer which is a factor of 1600000000M2 when the coefficients in the standard form of the quintic are integers, and so the number of trials which have to be made to ascertain the root is definitely limited.

The quantity M is defined on Watson's sheet 1. The constant term of Cayley's sextic resolvent (0) is 40000M.

When (} has been thus determined, Young's pair of equations contain one unknown T only, and there is no difficulty at all in finding the single value of T which satisfies both of them by a series of trials exactly resembling the set of trials by which (} was determined.


Watson's metlwd of finding a real root of the solvable quintic equation:

ax5 + 5bx4 + 10c.i3 + 10dx2 + 5ex + f = 0

First transform the quintic into reduced form

x5 + 10Cx3 + 10Dx2 + 5Ex + F = 0.

Watson's step-by-step procedure gives a real root of the reduced equation in the form x = u1 + u2 + u3 + U4. The other four roots of the equation have the form wlu1 + w2iu2 + w3iu3 + w4iu4 (j = 1, 2, 3, 4), where w = exp(277i/5).

INPUT: C,D,E,F

Step 1. Find a positive integer k such that

kj16 X 108 X JJ12, eVk/a is a root of (0) for E = 1 or - 1.

Step 2. Determine 8 from

eaVk 8 =

10v5 ·

Step 3. Put the value of 8 into (7) and (9) and then add and subtract multiples of these equations as necessary to determine T.

Step 4. Determine R1 from

R1 = Y(D - Ti + 4(C - 8)2(C + 8).

Step 5. Determine R2 from R1R2 = (C(D2 - T2) + (C2 - 82)(C2 + 382 - E))/8. Step 6. Determine u1 from

_ (x2y)115 ul - z2 '

where

X = (-D + T + R1)/2, Y = (-D - T + R2)/2, Z = -C - 8.

Step 7. Determine u4 from

u1u4 = -c + 8.

Step 8. Determine u2from

u�u2 = ( -D + T - R1)/2.

Step 9. Determine usfrom

U2U3 = - C - 8.

OUTPUT· A real root of the quintic is x = u1 + u2 + Us + U4. The process which I have now described of solving an

Abelian quintic by making use of the work of both Cayley and Young is a perfectly practical one, and, as I have already implied, I have used it to solve rather more than 100 Abelian quintics. If any of you would like to attempt the solution of an Abelian quintic, you will find enough information about Ramantijan's quintic given at the foot of the third sheet to enable you to complete the solution. You may remember that I mentioned that the equation was connected with the elliptic functions for which the period-quotient was v=79, and you will see the number 79 appearing somewhat unobtrusively in the values which I have quoted for the u's.

This is the end of Watson's lecture. We have made a few corrections to the text: for example, in one place Watson wrote "cubic" when he clearly meant "quintic. " Included in this article are the three handout sheets that he refers to in his lecture. We conclude with three examples.

Three Examples Illustrating Watson's Procedure

Example 1. x5 - 5x + 12 = 0 The Galois group of x5 - 5x + 12 is D5. Here

Equation (0) is

Step 1

Step 2

Step 3


a = 1, b = 0, c = 0, d = 0, e = - 1, J = 12, C = 0, D = 0, E = - 1, F = 12, K = - 1, L = 3, M = - 1, 11 = 5 X 212, YM = 520.

4J6 + 1004J4 + 6000� - 2560004J - 40000 = 0.

k = 10.

1 (} = V5 "

2 T = V5 "

Continues on next page

Step 4

Step 5

Step 6

Step 7

Step 8

Step 9

Examples (continued)

Rz = -�Y5 - v5.

v5 + Y5 + v5 -v5 - V5 - v5 1 X = 5 , Y = 5 , Z = - v5 ,

_ -( cv5 + V5 + v5)2 cv5 + V5 - v5) ) 115 ul - 25 .

_ -( cv5 - V5 - v5)2C-v5 - Y5 + v5) )115

Uz - 25 .

_ -( cv5 + Y5 - v5)2C -v5 + V5 + v5) )115 U3 - 25 .

A solution of x5 - 5x + 12 = 0 is x = u1 + u2 + U3 + u4. This agrees with [43, Example 1] .

Example 2. x5 + 15x + 12 = 0

The Galois group of x5 + 15x + 12 is Fzo. Here

Equation (0) is

Step 1

Step 2

Step 3

Step 4

a = 1, b = 0, c = 0, d = 0, e = 3, ! = 12, C = 0, D = 0, E = 3, F = 12, K = 3 L = 27 M = 27 11 = 2�0 x 34, 'vM = 288v5.

cf>6 - 3004>4 + 54000� - 230400v54> + 1080000 = 0.

k = 180.

R = 12VIO 1 25 .

Continues on next page


Step 5

Step 6

Step 7

Step 8

Step 9

This agrees with [43, Example 2].


R - 6Vlo 2 - 25.

X = 15 + 6Vlo Y =

- 15 + 3vl0 z = -� 25 ' 25 ' 5 '

- ( -75 - 21Vlo )l/5 u1 - 125 ·

- ( -75 + 21Vlo )l/5 u4 - 125 ·

- ( 225 - 72Vlo )1/5 u2 - 125 ·

- ( 225 + 72Vlo )1/5 U3 -125 .

Example 3. x5 - 2fii3 + 50.1? - 25 = 0

The Galois group of x5 - 2fii3 + 50.1? - 25 is 7L/57L. Here

Equation (0) is

Step 1

Step 2

Step 3

Step 4

Step 5


a = 1, b = 0, c = -5/2, d = 5, e = O, f = -25, C = -5/2, D = 5, E = 0, F = -25, K = 75/4, L = 5375/16, M = -30625/64, il = 57 X 72, � = 54 X 7.

¢6 - 1875¢4 + 671875¢2 - 3500000¢ - 19140625 = 0.

k = 625.

-v5 (} = -2-.

T = 0.

R1 = Y -25 + IOV5.

R2 = Y -25 - IOV5.

Concludes on next page


Step 6

-5 + v -25 + 10V5 -5 + v -25 - 10V5 5 + V5 X = _ 2 , Y = 2 , Z = 2 ,

_ (x2y)115 _ 25 + 15V5 + 5Y - 13o - 5sV5

Step 7

ul - . z2 -4

25 + 15V5 - 5Y - 130 - 5sV5 4

Step 8

25 - 15V5 + 5Y - 13o + 5sV5 4

Step 9 Us = 25 - 15V5 - 5Y - 13o + 5sV5

4

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tion, Cambridge Philosophical Society Proceedings Ill (1 880),

1 55-1 59. [1 5, Vol. XI, Paper 7 41 , pp. 1 32-1 35.]

1 3. Arthur Cayley, A solvable case of the quintic equation, Quarterly

Journal of Pure and Applietl Mathematics XVIII (1 882), 1 54-1 57.

(1 5, Vol. XI , Paper 777, pp. 402-404.]

1 4. Arthur Cayley, On a soluble quintic equation, American Journal of

Mathematics XIII (1 891), 53-58. (15, Vol. XIII , Paper 91 4, pp. 88-92.]

1 5 . Arthur Cayley, The Collected Mathematical Papers of Arthur Cay

ley, Cambridge University Press, Vol. I (1 889), Vol. I I (1 889), Vol. I l l

(1 890), Vol. IV (1 891) , Vol. V (1 892), Vol. VI (1 893), Vol. VI I (1 894),

Vol. VI I I (1 895), Vol. IX (1 896), Vol. X (1 896), Vol. XI (1 896), Vol. XII

(1 897), Vol. XI I I (1 897), Vol. XIV (1 898).

1 6. Heng Huat Chan, Ramanujan-Weber class invariant Gn and Wat

son's empirical process, Journal of the London Mathematical So

ciety 57 (1 998), 545-561 .

1 7 . James Cockle, Researches in the higher algebra, Memoirs of the Lit

erary and Philosophical Society of Manchester XV (1 858), 1 31 -1 42.

1 8 . James Cockle, Sketch of a theory of transcendental roots, The

London, Edinburgh and Dublin Philosophical Magazine and Jour

nal of Science XX (1 860) , 1 45-1 48.


A U THO R S

BRUCE C. BERNDT


University of Illinois

Urbana, Illinois

U.S.A.

BLAIR K. SPEARMAN

Department of Mathematics and Statistics

Okanagan University College

Kelowna, British Columbia V1V 1V7

Canada

KENNETH S. WILLIAMS

School of Mathematics and Statistics

Carleton University

Ottawa, Ontario K1 S 586 Canada

e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

Bruce C. Bemdt became acquainted with

Ramanujan's notebooks in February 1 974,

while on a sabbatical year at the Institute for

Advanced Study. Since then he has devoted

almost all of his research efforts toward prov

ing results from these notebooks and Ra

manujan's lost notebook. In 1 996 the Amer

ican Mathematical Society awarded him the

Steele Prize for his five volumes on Ra

manujan's Notebooks. Similar volumes on

the lost notebook, to be co-authored with

George Andrews, are in preparation. He is

most proud of his three biological children,

Kristin, Sonja, and Brooks, his seventeen

mathematical children, his five mathemati

cal children in preparation, and his current

post doc.

Blair K. Spearman completed his B.Sc. and

M.Sc. degrees at Carleton University in Ot

tawa, Canada. He received his Ph.D. de

gree in mathematics at Pennsylvania State

University underW. C. Waterhouse in 1 981 .

He currently teaches at Okanagan Univer

sity College, Kelowna, BC, Canada. His re

search interests are in algebraic number

theory.

Kenneth S. Williams did his B.Sc. degree

at the University of Birmingham, England,

attending lectures in the Watson Building.

He completed his Ph.D. degree at the Uni

versity of Toronto in 1 965 under the su

pervision of J. H. H. Chalk. After a year at

the University of Manchester he came to

Carleton University in 1 966, where he has

been ever since. He served as chair of the

Mathematics Department from 1 980 to

1 984 and again from 1 997 to 2000. He is

currently on sabbatical leave working on a

book on algebraic number theory with his

colleague Saban Alaca.

1 9. James Cockle, On the resolution of quintics, Quarterly Journal of

Pure and Applied Mathematics 4 (1 861) , 5-7.

20. James Cockle, Notes on the higher algebra, Quarterly Journal of

Pure and Applied Mathematics 4 (1 86 1 ) , 49-57.

21 . James Cockle, On transcendental and algebraic solution-supple

mentary paper, The London, Edinburgh and Dublin Philosophical

Magazine and Journal of Science XXIII (1 862), 1 35-139.

22. Winifred A Cooke, George Neville Watson, Mathematical Gazette

49 (1 965), 256-258.

23. David A Cox, Primes of the Form x2 + ny2 , Wiley, New York, 1 989.

24. David S. Dum mit, Solving solvable quintics, Mathematics of Com

putation 57 (1 99 1 ), 387-401 .

25. David S. Dummit and Richard M. Foote, Abstract Algebra, Pren

tice Hall, New Jersey, 1 99 1 .

26. W. L. Ferrar, Higher Algebra, Oxford University Press, Oxford, 1 950.

27. Joseph A Gallian, Contemporary Abstract Algebra, Fourth Edition,

Houghton Mifflin Co. , Boston MA, 1 998.

28. J. C. Glashan, Notes on the quintic, American Journal of Mathe

matics 7 (1 885), 1 78-1 79.

29. Robert Harley, On the method of symmetric products, and its appli-


cation to the finite algebraic solution of equations, Memoirs of the Ut

erary and Philosophical Society of Manchester XV (1 859), 1 72-2 1 9.

30. Robert Harley, On the theory of quintics, Quarterly Journal of Pure

and Applied Mathematics 3 (1 859), 343-359.

31 . Robert Harley, On the theory of the transcendental solution of al

gebraic equations, Quarterly Journal of Pure and Applied Mathe

matics 5 (1 862), 337-361 .

32. R. Bruce King, Beyond the Quartic Equation, Birkhauser, Boston,

1 996.

33. Sigeru Kobayashi and Hiroshi Nakagawa, Resolution of solvable

quintic equation, Mathematica Japonicae 37 (1 992), 883-886.

34. John Emory McClintock, On the resolution of equations of the fifth

degree, American Journal of Mathematics 6 (1 883-1 884), 301 -

315. 35. John Emory McClintock, Analysis of quintic equations, American

Journal of Mathematics 8 (1 885), 45-84.

36. John Emory McClintock, Further researches in the theory of quin

tic equations, American Journal of Mathematics 20 (1 898),

1 57-192.

37. Srinivasa Ramanujan, Modular equations and approximations to 7T,

Quarterly Journal of Mathematics 45 (1 9 1 4), 350--372. (40: pp.

23-39.]

38. Srinivasa Ramanujan, Question 699, Journal of the Indian Mathe

matical Society 7 (1 91 7) , 1 99. (40: p. 331 .]

39. Srinivasa Ramanujan, Notebooks, 2 vols., Tata Institute of Funda

mental Research, Bombay, 1 957.

40. Srinivasa Ramanujan, Collected Papers of Srinivasa Ramanujan

AMS Chelsea, Providence, Rl , 2000.

41 . Robert A. Rankin , George Neville Watson, Journal of the London

Mathematical Society 41 (1 966), 551 -565.

42. R. Russell, On modular equations, Proceedings of the London

Mathematical Society 21 {1 889-1 890), 351 -395.

43. Blair K. Spearman and Kenneth S. Williams, Characterization of

solvable quintics x5 + ax + b, American Mathematical Monthly 1 01

(1 994), 986-992.

44. Blair K. Spearman and Kenneth S. Williams, DeMoivre's quintic and

a theorem of Galois, Far East Journal of Mathematical Sciences 1

(1 999), 1 37-1 43.

45. Blair K. Spearman and Kenneth S. Williams, Dihedral quintic poly

nomials and a theorem of Galois, Indian Journal of Pure and Ap

plied Mathematics 30 (1 999), 839-845.

46. Blair K. Spearman and Kenneth S. Williams, Conditions for the in

solvability of the quintic equation x5 + ax + b = 0, Far East Jour

nal of Mathematical Sciences 3 (2001 ), 209-225.

47. Blair K. Spearman and Kenneth S. Williams, Note on a paper of

Kobayashi and Nakagawa, Scientiae Mathematicae Japonicae 53 (2001 ), 323-334.

48. K. L. Wardle, George Neville Watson, Mathematical Gazette 49

(1 965), 253-256.

49. George N. Watson, Solution to Question 699, Journal of the Indian

Mathematical Society 1 8 (1 929-1 930), 273-275.

�0. George N. Watson, Theorems stated by Ramanujan (XIV): a sin

gular modulus, Journal of the London Mathematical Society 6

(1 931 ), 1 26-1 32.

51 . George N . Watson, Some singular moduli (1), Quarterly Journal of

Mathematics 3 (1 932), 81-98.

52. George N. Watson, Some singular moduli (II), Quarterly Journal of

Mathematics 3 (1 932), 1 89-2 1 2 .

53. George N. Watson, Singular moduli (3), Proceedings o f the Lon

don Mathematical Society 40 (1 936), 83-1 42.

54. George N. Watson, Singular moduli (4), Acta Arithmetica 1 (1 936),

284-323.

55. George N. Watson, Singular moduli (5), Proceedings of the Lon

don Mathematical Society 42 (1 937), 377-397.

56. George N. Watson, Singular moduli (6), Proceedings of the Lon

don Mathematical Society 42 (1 937), 398-409.

57. Heinrich Weber, Lehrbuch der Algebra, 3 vols. , Chelsea, New York,

1 961 .

58. George P. Young, Resolution of solvable equations of the fifth de

gree, American Journal of Mathematics 6 (1 883-1 884), 1 03-1 14 .

59 . George P . Young, Solution of solvable irreducible quintic equations,

without the aid of a resolvent sextic, American Journal of Mathe

matics 7 (1 885), 1 70-1 77.

60. George P. Young, Solvable quintic equations with commensu

rable coefficients, American Journal of Mathematics 10 (1 888),

99-1 30.

61 . Liang-Cheng Zhang, ·Ramanujan's class invariants, Kronecker's

limit formula and modular equations (II), in Analytic Number The-

ory: Proceedings of a Conference in Honor of Heini Halberstam,

Vol. 2, B. C. Berndt, H. G. Diamond and A. J. Hildebrand, eds . ,

Birkhauser, Boston, 1996, pp. 81 7-838.

62. Liang-Cheng Zhang, Ramanujan's class invariants, Kronecker's

limit formula and modular equations (Il l), Acta Arithmetica 82 (1 997),

379-392.

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Promot1on S44 5


ip.iM$$j:J§..@hl£ili.JIIQ?-Ji Dirk H uylebrouck, Editor

Mathematics in the Hal l of Peace Norbert Schmitz

Does your lwmetown have any

mathematical toumt attractions such

as statues, plaques, graves, the cafe

where the famous conjecture was made,

the desk where the famous initials

are scratched, birthplaces, houses, or

memorials? Have you encountered

a mathematical sight on your travels?

Q so, we invite you to submit to this

column a picture, a description of its

mathematical significance, and either

a map or directions so that others

may follow in your tracks.

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium


M iinster is one of the few cities famous not for a bloody battle but

for a fruitful peace-the Peace of Westphalia. In 1648, the signing of the peace treaty in Miinster and Osnabriick marked the end to the dreadful Thirty Years' War, which had caused unimaginable suffering throughout central Europe-in particular among the German population. An additional result of the Westphalian Peace Conference was the peace treaty between Spain and the Netherlands affirming the independence of the Netherlands.

Both peace treaties were ratified in the Hall of Peace, the old council chamber of the Miinster town hall. Famous for its magnificent gable, this town hall is regarded as one of the finest existing examples of secular Gothic archi-

Figure 1 . Munster town hall.


tecture. The framework of the council chamber, which is the oldest part of the town hall, was built in the second half of the 12th century. The city itselflooks back upon an eventful history of more than 1200 years.

Around 1577, the Hall of Peace was decorated with a rich array of Renaissance woodcarvings. In the window recesses, one can see Moses the Legislator as well as the seven liberal arts (for the history of these arts and their "portraits" by sculptors and painters, see "The Liberal Arts" by B. Artmann, The Mathematical Intelligencer 20 (1998), no. 3, 40-41), in particular Ars Arithmetica with tablet and stylus and Ars Geometria with tablet and compass.

Like the other carvings, these figures are embellished with ornaments,

Figure 2. The Hall of Peace.

Figure 3. Arithmetica and Geometria.


arabesques, and pediments with angels' heads. There is an abundance of objects worth seeing, but during a short journey, one could simply follow the track of the many (crowned) heads of state who celebrated here in 1998 the 350th anniversary of the Peace of Westphalia.

There is a nice story about the visitors' book in the Town Hall. In the early 1950's J.-P. Serre is said to have signed it "Bourbaki," after a visit to the Hall of Peace. Unfortunately, this story, which was told to me by P. Ullrich (Augsburg), on the authority of M. Koecher could not be verified-either by checking hundreds of pages of the visitors'


books or by a personal inquiry to J.-P. Serre himself.

In Miinster, the "Arithmetica and Geometria" are not the only attractions for the mathematical tourist. The Mathematics Department of the University (Westfillische Wilhelms-Universitat Munster) is one of the leading departments in Germany. Here, F. Hirzebruch, H. Grauert, and R. Remmert wrote their Ph.D. and Habilitation theses, in the 1950s, as members of the school of complex analysis around H. Behnke. During the 1970s, the main field of interest switched and the department again embraced Arithmetica and Geometria.

G. Faltings wrote his Ph.D. and Habilitation theses in this department. Since 1998 a lively Sonderforschungsbereich (Special Research Field) "Geometrische Strukturen in der Mathematik" (Geometric Structures in Mathematics) is supported by the Deutsche Forschungsgemeinschaft. Yet, the mathematics building is less interesting than the town hall-architecturally, at least.

lnstitut fOr Mathematische Statistik

Universitat Munster

Einsteinstr. 62

D-48149 Munster

Germany

LEON GLASS

Looki ng at Dots

he ''Prof' at the Department of Machine Intelligence and Perception at the University

of Edinburgh, H. C. Longuet-Higgins, had just returned from a trip to the States

where he had learned of a fascinating experiment carried out by the physicist Erich

Harth. The year was 1968, and I had just completed a doctorate studying the statis-

tical mechanics of liquids, trying to apply my craft to the study of the brain. At the time, I did not realize that the experiment would have strong impact on the rest of my career.

The experiment was so simple that even a theoretician could do it. Take a blank piece of paper. Place this on a photocopy machine and make a copy of it. Now make a copy of the copy. This procedure is then iterated, always making a copy of the most recent copy.

Although the naJ:ve guess might be that all copies would be blank, this was not at all the case. Small imperfections in the paper and dust on the optics of the Xerox machine introduced "noise" that arose initially as tiny specks. As the process was iterated, these tiny specks grew up-they got bigger. They did not grow to be very big, but just achieved the size of a small dot, Figure 1. The reason for this is that the optics of the photocopy machine led to a slight blurring of each dot, so that each dot grew. On the other hand, local inhibitory fields introduced by the charge transfer underlying the Xerography process limited the growth. These local fields also inhibited the initiation of new dots near an already existing dot; so that after a while (about 15 iterations), there was a pretty stable pattern of dots.

This analogue system mimicked lateral inhibitory fields that play a role in developmental biology and visual perception, and I thought it would be a fine idea to study the spatial pattern of the dots. To do this, I decided to make a transparency of the dot patterns so that I could project the

dot patterns on a target pattern of concentric circles. By placing one dot at the center of the target pattern, I could count the number of dots lying in annuli a given distance away, this would give me an estimate of the spatial autocorrelation function of the dots.

But when I did this, I made a surprising finding. Superimposing the transparency of the dots upon the photocopy of the dots with a slight rotation, one obtained an image with an appearance of concentric circles (Figure 2). I described this effect and proposed a way that the visual system could process the images [1] .

In 1982, David Marr called these images Glass patterns in his classic text in visual perception [2]. The effect is now well-known among visual scientists, who continue to unravel the visual mechanisms underlying the perception of these images. But despite the underlying mathematical structure of these images and the potential utility of this effect to teach mathematics, the effect is not known at all by mathematicians, as witnessed by an early rediscovery of the effect [3] and also by the description of the effect in the Spring 2000 Mathematical Intelligencer [4] . Let me try here to give a glimpse into the mathematical underpinnings, and to describe some of the recent psychological studies of the perception of these images.

Perceiving Vector Fields Imagine a two-dimensional flow or vector field. We randomly sprinkle dots on the plane. Next we plot the loca-

© 2002 SPRINGER-VER LAG NEW YORK, VOLUME 24, NUMBER 4, 2002 37

_ I • • • . :. • .i . . :._._ .. . ·

·:

:

:·:

! . . . .... . • � . . ·� . . .

: . · .. · �·· ··�-. _ .:. ·r : .. . . :;\···:·· ·.;. � : : . � .

. . . ·

Figure 1 . Original images generated in the late 1 960s by making a photocopy of a blank page and then iterating the process, always taking

a photocopy of the most recent copy. Images represent the output after the 5th and 15th iterations .

. ·. · ·.· . · . ' · . . . ' . .

,· . . .. .. .. . · .. .: ,• .• . . ·. ·.:

· ··; . • , ••. � .... . , (- .�1.t:., •. :.�_.,� • • �1.,..,. < '<"'l·.,l' •.· ;t.�--:,.,.,. ·. " ,; ,. ·. I � ·.:�·.� .. �· � ;.��.:S:fl� .. :_ �:.� ,�� .. �:�cF·l·�� .. :-�.· ·.: ,�;.«t{fo��tJ-·1-;4;;1.�\ A,\:� 3-:: . ,";�: . I ·oo� I • �·!:., �� ''"' i'j,l=·t·· .• .r .. � ... , . ·: 'r.,� • ·.a.: : · .. � .. �r;- ��-'![:-,r .. .. \--: ,. .� . .. ·. � ,._··· · 1 ' :.: �';J��f'·l:j· ��;i�'f . #f;-:_:.�'5}J�!jl;;��:�\ -�{r.�!�t·i3�·�0: ! ... :. � .... :. = 1 1 � _ .,.il· .�� .. ��----:-· ·1 ..;� �9tt1'·-'•,:"J!� .. YA-i'S :'·:..:;..:."r..· -:- · ��·· ... • ·•• •• h f J �-• • l, �. J"'\�,··� "i."' ...,. ... �:.; . ... ,.� . � ... ;)f"""' i\1, · ····17 "l . · -;, . •. • , , • • •

·4�l�:i!.��f.:�J:;��: �.*��,;,�.��· �.' ·-.��4��}.{��:.;;;' !.�·-:· �l} ,/ . . " .. . . _.,.l ,l:t� ··· •<11:-.'tl' ••• 0 •-:..,. ... \' "'1!.• ' f . . r:·. ,.·�:;��,t�� ,�;.�� cv��:�·!-� �>;.:�·�'":{:��,0::� -� .. � ; : .�r 1 : ! .! .. ::':�·.::i'.b: .. ".JI'1"):� . ··· \· ... .}.��;!' -�-- . · !'(�.·� :�:-"?-�·:-: · ·�.··�,. ":__.· · ! ' • :'1': ·�. <::• ;.'J"<f..,,. -;t.?:�;- ,':i_"fi.}. •'/, -f.."'!it�"'.;:;,· •• ,;;:-.�:·:,�fi.:·,·.:.t.:-.:;.,_.'i.: . ':, � l : .... . i.:·1.! .. � ·��···:·�··'!Z·\�!-�. ::I·:!''�'jfi.';:-·::;,;;,.��:,,..il_\'/f� �:. .. �·-:;�_.'": .. I : • ��·· . .. ... " "'"., ,. . • ,. ·i'• e•.. .. . , , ·-- ..... -" '·'\: \"' ... -."7 • • f......... . .

..... .. . • }'' ·_; ·(.:. 1: .:::::.-:.··�;��-��:�;t:�'j:�p . . ·;�-�?f?;·��-��:· .. ;�.:!:; ·�:·.�:·.' ,;' ! ! ,:; :�_;._2 ;�x�<t;�1�:�J:.��-:::� �-��-��:��:7,}�\��g;f�I$ , ! 1- 1· : � . · • . • . . , · � . · • . •j • • ,.._. • '·· ' . .. -�$ .; , . .; • . . ....... . : · ' ,. . . , .. . . !

·�{�i�i5�ttr.¥.;�;�i��t�11�llr:

·

: ... -�;.· • . , ... ·�) ..... � ·:.. \f '¥t�._ �"/':.:. •• • .... _./.,,.._.,. q.,:a · = · • . . ·;�-:.. ",.,,.oc:: · • • . , · � . • :, I · � o

:-·:�:r·��}�;�:��tf�sz� �;:.(;;�Ut?:�t:;t:i.i :\)��?:if�!I?1111l � .

· ' !.-:4i;,• ··'·i£h··�¥'t' · : ·'. . � .. ..... . . . · .�··· . ==;:..::.: · � .·· · .. �:-'-:: . -��t:�J';\:T�.'; -. ... :·!�

--�-

· .:� Figure 2. The image generated by superimposing a copy of the 15th

iterate on itself in a rotated position.

38 THE MATHEMAnCAL INTELLIGENCER

tions of the original set of dots, and also the locations of

the dots a bit later, after they have moved under the action

of the flow. Provided the time interval is not too long, then

when we look at the positions of both sets of dots simul

taneously, we see the geometry of the vector field.

Figure 2 shows an example in which a set of dots is su

perimposed on itself in a rotated position to yield a circu

lar image. But other geometries can be handled [5] . First

assume that the origin is fixed, and the transformation

maps each dot to a new location by a scaling of the x

coordinate by an amount a, a scaling of the y-coordinate

by an amount b, and a rotation of the image about the ori

gin by an angle 0. Then (x,y) will be transported to the po

sition (x' ,y'), where

x' = ax cos (} - by sin (}

y' = ax sin (} + by cos (} (1)

Equation (1) is a linear map. The properties of such maps

are well understood [6] , [7].

What is amazing is that by looking at the images of the

original set of dots combined with the superimposed dots,

it is possible to perceive the underlying geometry of the

transformation defined by the map. The particular geome

try that results is defmed by the eigenvalues of the linear

transformation defined in Equation (1). The eigenvalues are

the solutions of the detenninant

I a cos (} - A

a sin (} - b sin (} I b cos (} - A

0. (2)

0.5

(a) :..· :·· .. : · . _ ,·,: :: . ·: . ... : -:· •• - ·=. t • • •• "· : •

· ·i • · · · . : . .... :� '- • '\ . . . . . . ·. ... . . . , • r. :• • • •• • .•. • •• • • • " :' . . . . .. .. : • ' . . .-. .. ·. · . .. .. . � .. �

0 · ·· ·· · ' · ·: . :# .. � .. , ·: �"'-.. . . .... . . -� " . . ... ' . . :: .. \ . .. '· . .. .

-0.5 . ( . . . · . : �.. . � ... . c.:

. . . . . .. . . . .. . .. .. . . . . : •I. •• • • • • • . . . . ��'. :· z .. · · -- · . . �.. . . . . .. . . . ' . 1 • Y • I • I • • - �------�--------�

-1 0

(b) 1 . . • •. , • • , • ., . ..: .• .. � :.: . .. : . . ... . . . . .

. . ·. ··. � . . . . '· : . . . · ' · . . . . . ·:

045 " .• . . . • • • •• ••• • ... :· �· . '· :· . . .. . ... ... . . . � :• . . . . ' ..

: . . ' : ·.-. � . .. ·. ·· -- . . :.· ·:

0 • ' • . . . . . . . :I·· , . • , : ,.. __ • • ••• • • .-i' " . • "' ' . . :: .. \ . . . '· . . . .

.�· • • : ' . • � • 4.. -0 . 5 . • •' . . •• . • .... • .. . .

. . . . . .. .. . . ... . : ·:.. • • t# •• • •• : •• •• : •

. . :-.. . . . . . . · .� . . . . � . -1 • \• • I • 1 • •

-1 0 1

0.5

(c) :..· :··.. �· . . ,.,: :.: . ·: . ... : -:.· • . .. ·=. t'

• • • • ; . : -. . ' . . . . •: " ·.• • • ·. # : . ·. .• � ... _,.. #. • '· ••• • • •• • ·!· ••• • • •

,. :• . . . " : . . . . . - ... . .. • . .. .. . · :--.• .. 0 • . • !· •• r •• • ·' • .. • , • • ,.. •

• • '·• • • ••• •

• • ..t • •• • .. •

-0.5

.. " � . : . .. � . · . '· . · . . • t#) • • • " • • � • ,. . . .... . . .. . . . .... . .. . .

: :.·. . . . .. .. . . . . ... : • • • • , .• , • . .. I • . . :--,., . . . . . . � .. . " .

• � · • ,•. • I • • -1 �------��------� - 1 0

Figure 3. (a) A random pattern of 400 dots. (b) The same pattern in which the x and y coordinates are multiplied by 1 .05. (c) The same pat

tern in which the x coordinate is multiplied by 1 .05 and the y coordinate is multiplied by 0.95.

A simple computation gives those eigenvalues as

(a + b) cos (} ± V(a - b)2 - (a + b)2 sin2 (} A ± =

2 . (3)

These effects can be beautifully illustrated using transparencies of random dot patterns, and superimposing these on an overhead projector. In Figure 3a I show a random pattern of 400 dots. The x and y coordinates of each point are multiplied by 1.05 in Figure 3(b ). In Figure 3( c), the x coordinates of each point are multiplied by 1.05 and the y coordinates of each point are multiplied by 0.95. The rotation of the photocopied patterns yielding circles in Figure 2 is one of the classic geometries (pure imaginary eigenvalues). AnQther geometry is provided by setting the center of the image as fixed, and then expanding the x and y coordinates by the same constant amount (real eigenvalues greater than 1). This yields an expanding pattern, called a "node," which is illustrated in Figure 4(a) by superimposing Figure 3(a) and 3(b ). Combining expansion with rotation gives a spiral image, called a "focus," as shown in Figure 4(b ), formed by superimposing Figures 3(a) and 3(b) in a rotated orientation (complex eigenvalues). Finally, if there is expansion in the x coordinate and contraction in the y coordinate, then there is a hyperbolic geometry called a "saddle" (the absolute value

of one eigenvalue is greater than 1 and the absolute value of the other eigenvalue is between 0 and 1). A saddle (Figure 4c), can be generated by the superposition of Figures 3(a) and 3( c). Because these geometries can be easily appreciated without using the formulae, I always use these correlated dot images to teach about the geometry of vector fields. These geometries may even be preserved when one set of dots is one color, and the second set of dots is another color (Fig. 5a). Stan Wagon has incorporated this observation to generate colorful images of vector fields in which local flows are represented by "tear drops." The visual system integrates the local tear drop flows to give a good representation of the geometry of the vector field [8] (Fig. 5b ).

(a) 1 • ,. • " • • •• • ··:... ..... ,.. ,=,�:: ··• J' "1-.,;: ... • • • .

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Neurophysiology of Perception Manipulation of visual images, combined with measurement of perception, or recording of electrical activity of nerve cells in the brain, provides powerlul techniques to probe the ftmctioning of the visual system. Because of the simple structure of the random dot images, visual scientists have often used them as a starting point for their investigations. I cannot summarize the many studies that have been carried out, but I will describe a couple and invite the reader to invent new visual effects that can be a probe of visual system ftmction.

(b) . , .,. , . . ... . .... . .

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·· � ·, '' • ' \ <· �� ..,'!I"' I t I \ .._ t"i!'•.::-,;• ,. *""' ,.., • •• • � .. .,.. .. . .. *· .. .-"' .. :'Itt "- ..

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-0 .5 . . .. .. =t ' ,,, , ' '" , • .••. •• :,h: " '" '• • , , •• .,... •. •. •. • •. .._ J'J II I •". , .... . •

• • •• �. ·: ·.· �· :- .1 II • .•,".• 1 :. � <" · · r . = 1 •• • •

- �--------�--------�

-1 0 1

Figure 4. (a) Superposition of Figure 2(a) on Figure 2(b) to generate a node geometry. (b) Superposition of Figure 2(a) and Figure 2(b) in a ro

tated position to generate a focus geometry. (c) Superposition of Figure 2(a) on Figure 2(c) to generate a saddle geometry.

(a)

(b)


- j

Figure 5 (a) A random pattern of dots generated by

tossing ink on paper is superimposed on itself in a ro

tated position, but the two sets of dots are different

colors. From a photo silkscreen print made by the au

thor in the 1970s. (b) A colorful "tear drop" represen

tation of vector fields from VisuaiDSolve (Wagon and

Schwalbe [8]). Reproduced with permission from

Wagon and Schwalbe [8].

In order to think about how the visual system might process the information in the dot patterns, it is useful to consider first the structure of the images. For each dot, there is a second dot that is correlated with the first dot. For example, for the circular images, the two dots always lie on the circumference of a circle -centered at the point of rotation. However, in addition, there are other dots that are also in the vicinity of the first dot that lie in random directions from it. In order to detect the pattern, two steps are essential: (1) to detect the locally correlated dots and (2) to integrate the local correlations to form the global percept.

Early Nobel-Prize-winning studies of the physiology of nerve cells in the visual system of the brain carried out by Rubel and Wiesel [9] provide a basis for hypothesizing a mechanism for early stages of the detection process. Rubel

this approach is making progress in linking the separations between dots in the images presented to the monkeys with the physiological properties of individual cells.

What about the interactions between the simple cells? Zucker argues that excitatory interactions between individual cells in a given "clique" of cells, all of which have similar orientation specificity and are located in a given column, might be playing an important role in contour detection [12] . In this formulation, a "clique" of cells is carrying out the averaging operations that are necessary to compute the local autocorrelations. Thus, Zucker is hypothesizing that the columnar organization may play an important role in information processing.

This work leaves open the important question of the nature of the interactions between columns that lead to global

and Wiesel showed that some nerve cells, called "simple cells," can be excited by lines of a particular orientation in a given region of the visual field. Consequently, two dots should also be able to stimulate a simple cell if they lie in the appropriate orientation. In a local region, there are many correlated dot pairs oriented along the flow, so cells spe

Because of the moi re

effect , these images

can provide a powerfu l

method to determ ine a

percept. Psychophysical studies carried out by Wilson and Wilkinson pose sharp questions about the nature of the intercolumnar information processing. By partially rell,loving some regions of the correlated dot images, they determined that the circular image, as rn Figure 2, is easier to perceive than the other types of correlated dot images.

point of rotation .

cific for that orientation in that local region would be preferentially activated compared to cells specific for other orientations. Rubel and Wiesel also found that simple cells, specific for a certain orientation but with somewhat differi,ng receptive fields all in the same general region of the visual field, were located in vertical columns. Further, there were cells they called "complex cells" that appeared to receive their input from simple cells lying in the same column [9]. Based on these observations, I hypothesized that the simple and complex cells in a column in the visual cortex could provide the anatomical loci to compute the local autocorrelation function of the dot patterns [ 1]. The integration of the outputs of the local columns to form the global percept would necessarily involve inter-columnar interactions.

Now, more than 30 years after these initial hypotheses, a large number of studies make it possible to refine and modify these ideas. Movshon and colleagues have recorded electrical activity from simple cells in the primary visual cortex (this is called area V1) of macaque monkeys while viewing dot patterns generated by superimposing a random set of dots on itself following a translation [10]. They also developed a mathematical model of the cortical cells, by assuming there were elongated excitatory and inhibitory regions of the receptive fields. A given cell would be excited (or inhibited) by dots that fell in the excitatory (or inhibitory) region of its receptive field. The good agreement between the experimentally recorded activity and the theoretical model gives support to this conceptual model of the cortical cell. Moreover, by computing the expected activity using a theoretical model, and comparing these results with the observed activity recorded experimentally,

Because the local information was the same in the various images, the differences in ability to perceive the images must be due to the integration steps. At the moment, it appears that these integration steps take place in a region of the brain called area V4 [11 ] .

Practical Implications The random dot images may be useful in a variety of other applications. Because of the moire effect, these images can provide a powerful method to determine a point of rotation, and to align images. Following the description of this effect in the Scientific American, Edward B. Seldin of Harvard Medical School developed a method to use the moire effect to help plan maxillo-facial surgery in patients who did not have ideal alignment of the upper and lower jaws. He started out with two identical dot patterns, one fixed on the upper jaw and a second fixed on the lower jaw, initially in a superimposed orientation [ 14]. By manipulating images to give a better jaw alignment, it was possible to develop a plan for the surgery. More recently, Wade Schuette of Ann Arbor, Michigan demonstrated a variety of ways these effects could be used to help in alignment tasks [ 13].

Similar effects also arise in color printing. Colors are often represented by dots of different colors and varying sizes. Problems in alignment of the different colors can lead to undesirable moire effects. One way to overcome these problems is for the color screens to be stochastic images. However, even when these images are stochastic, misalignment can lead to moire effects. Such problems are being addressed by Lau [ 15], who recently rediscovered these phenomena in the context of commercial printing.


Do It Yourself Some of the figures in this article are generated using the

Matlab programming language. Many readers are familiar

with Matlab or have access to a computer that has Mat

lab capabilities. The following program should be called

idots .m.

function [e]=idots(a,b,theta)

x=ones(2, 400)-2 *rand(2, 400);

R= [cos(theta) -sin(theta) ;

sin(theta) cos(theta)J ;

S=[a O;

O b] ;

xnew=R*S*x;

x=[x xnew] ;

plot(x( l , :) ,x(2, :) , ' . ');

axis([- 1 . 1 l . l - 1 . 1 1 . 1 ]) ; axis('square')

title('Glass Pattern');

e=eig(R*S);

txt= ['with eigenvalues : ' num2str(e( 1)) '

and ' num2str(e(2))J ;

xlabel(txt) ;

To run this program you need to open up Matlab. Fig

ure 3(a) shows a random pattern of dots; in Figure 3(c)

the x coordinate is multiplied by 1.05, the y coordinate is multiplied by 0.95, and there is no rotation. To superim-

0.5

0

-0. 5

-1

(a) . . . .. . . : .;' .·

..c • •• • • • • • • • • • • •

..... ... ' :·. . . .: . :· . .. . .. , : • • . · ... .. ... .. • , . .... . 4. • : • ..: • J' ,,

. . .. . . . . .. ·� "' . , ·.:· .. . . . ..... , ,, . . .

. . .. ' .. .. . : : · ··. ·. ·.· . . · .. · , : . • • • • _.,.:: •• • : ,, ,, .. • ;· • # ,

..

' . . :· .. · •''• . . ..... . ·

' :· .... : .:···:. , ... ·. . :'·: .•".

. .. :·:·. · .. '- -:., •• •• # ., · .:·· • • • • • • • • • , • I ., .• • • •• • • • ,.I' I '·· ;• .· ::,- • • •. . · , , . . . :- . . . . · : - .

N •J;. # # I I •.J • • t. • • • '• ': •.

.. •

., •

•

• :':::_:i·": �

. . � , .· . .. ,

- 1

"J • • • • •• ··� "" •• • � • •.•

• 1/# I • • .�! •• -., • • • • • • . . . . , ... . : . . . . . . . . . I

$ -i':··' .. .i; . .. . . . : , � ·· .. . · .· .· . . ·. . . . .

-0.5 0 0.5

1 .0 1 97, 0.978 1 9

pose two sets of dots with the same transformation as in

Figures 3(a) and (c), type idots(0 . 9 5 , 1 .05,0). What you

obtain should look like Figure 4(c) even though the ex

act coordinates of the random dots will be different. (Mat

lab generates a different sequence of random numbers on

each trial.)

Here's a problem. Superimpose two sets of random

dots: an original pattern and one in which the seatings in

the x and y coordinates are 1.05 and 0.95, respectively.

However, vary the rotation angle. For exan1ple, in Figure

6, I show the superposition of two patterns with a rota

tion of about 2.61 degrees (left panel), which gives a sad

dle geometry, and 5.47 degrees (right panel), which gives

a spiral geometry. As the angle of rotation is varied be

tween those two values, do you ever obtain a node geom

etry'? With Matlab, one can explore the question numeri

cally. However, it is really better to compute the

eigenvalues analytically using Eq. (3). If you do this, you

will find that there is a narrow range in which you must

pass through the node geometry. Here is the insight. In

the spiral geometry, the two eigenvalues are complex

numbers with real part less than 1, and in the hyperbolic

geometry the eigenvalues are real with one eigenvalue be

ing positive greater than 1, and the other eigenvalue pos

itive less than 1. As (} varies, the values of the eigenval

ues change continuously. Both eigenvalues first become

real and less than 1, before both eigenvalues become com

plex. This is a great way to illustrate bifurcations in dy

namical systems.

0 . 5

0

-0.5

-1

(b) . .. .. ;, .

, . . . :. . . ·.:

. •" . . :� .- ·. ·. .. ... .. . .· . . • ... 'lp .: : • • • :. ·: • • • : :. . • • ·::. :· • . .. . , . . .. .· •. . . . . . . . · . .

.. ,. � . .• .. . . : '\. ,. .. . . . . . . , . .. . .. •. ·. �

• : ··:·. •• .a • .. .. . • • • • • • • I • • • • • .., • "· •:' . . . ·.· . . . "· . . ..... . . " . . . . .... - � ' . . , .. . . . . . . ,,

. . . . ·

. • ' • • • • ,· _.,,, :1 • • •• •

.• . . , . ,. . , . .

' :. :it'• • �· •.: .. �� II # " • • : • • • f • . .. . , . ' · ' · · .. . . ...

.. . _ .. : : . � . · . : · · · · • • • • • • • • .: � • • • • l• - • • • : •• • : :.:· l ' .... • • ,: • • • ' •• ' • • . :�- ._... ...... .., :·· : ... :-.

'

..

. . . . . . � · · .. ·.• ·:. . . . · . . . . . : ·•,..:' · .

..·.

.• ..

:·

,,

. . : ' ... . .. . . . • • • • • ....._. . • • •• # • •• . . . :.·.· � · ·. · .· , ._. · , ·· . . . . . . . -��-... . . .. · . • ·= : • • ·. ·: : . • • • • -. •• :" , • • • 1

. . .

-1 -0 . 5 0 0.5 1

0 . 99544+0.081 1 86i, 0 . 99544-0.08 1 1 86i

Figure 6. (a) Saddle geometry generated from two sets of correlated random dots by using the Matlab program with a = 1 .05, b = 0.95,

6 = 2.61 °. (b) Focus geometry generated with a = 1 .05, b = 0.95, 6 = 5.47°. For a = 1 .05, b = 0.95, is there a value of 6 in the range 2.61 °

< 6 < 5.47° that gives a node geometry?

Now more than 30 years after I first observed these images composed of correlated random dots, it still seems we are just at the beginning of developing an understanding of how the visual system processes the information contained in these images. These images combine both local and global features, which can be varied independently. Observation of experimental subjects (men, monkeys, or even pigeons! [ 16]) looking at the dot patterns is providing . a window into the physiological processes of vision.

REFERENCES

[1 ] L. Glass, Moire effect from random dots, Nature 223, 578-580

(1 969).

[2] D. Marr, Vision, Freeman, San Francisco, 1 982.

[3] J. Walker, The amateur scientist, Scientific American 242 (April

1 980).

[4] Why circles?, Mathematical lntelligencer 22, no. 2, 1 8 (2000).

[5] L. Glass, R. Perez. Perception of random dot interference patterns.

Nature 246, 360-362 (1 973).

[6] R. L. Devaney, A First Course in Chaotic Dynamical Systems.

Perseus (1 992).

[7] S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to

Physics, Biology, Chemistry and Engineering, Perseus (1 994).

[8] S. Wagon and D. Schwalbe, Visua!OSolve: Visualizing Differential

Equations with Mathematica, Springer!TELOS (1 997). However, in

some cases, dots of different colors can NOT be used to capture

the geometries of vector fields. L. Glass and E. Switkes, Pattern

recognition in humans: correlations which cannot be perceived.

Perception 5, 67-72 (1 976).

[9] D. H. Hubel, T. N. Wiesel, Receptive fields, binocular interaction

and functional architecture in the eat's visual cortex. J. Physiol.

(Lond.) 1 60, 1 06-1 54 (1 962).

[1 0] M. A. Smith, W. Bair, and J. A. Movshon, Signals in macaque V1

neurons that support the perception of Glass patterns, Journal of

Neuroscience, In Press (2002).

[1 1 ] H. R. Wilson and F. Wilkinson. Detection of global structure in Glass

patterns: implications for form vision, Vision Research 38, 2933-2947 (1 998).

[1 2] S. W. Zucker, Which computation runs in visual cortical columns?

In: Problems in Systems Neuroscience, J. L. van Hemmen and T.

J. Sejnowski (eds.) Oxford University Press, in press (2002).

[1 3] W. Schuette, Glass patterns in image alignment and analysis.

United States Patent 5,61 3,013 .

[1 4] J. Walker, The amateur scientist, Scientific American 243 (No

vember 1 980).

A U THO R

LEON GLASS

Department of Physiology

McGill University

Montreal, QC H3G 1 Y6 Canada


Leon Glass, after receiving a Ph.D. in Chemistry from the Uni

versity of Chicago, was a postdoctoral fellow in Machine In

telligence and Perception at Edinburgh; in Theoretical Biology

at Chicago; and in Physics and Astronomy at Rochester. He

has been at McGill since 1 975, interspersed with visits to Har

vard Medical School and Boston University. Many readers will

have encountered his research under many disciplinary titles;

but call it physiology or theoretical biology or what you will,

it's really all mathematics. In his spare time he plays the French

hom in the I Medici Orchestra at McGill, and hikes in the

Adirondacks and other mountains.

The portrait here is a daguerrotype by Robert Shlaer; used

by permission.

[1 5] D. L. Lau , A. M. Kahn, G.· R. Arce. Minimizing stochastic moire in

FM halftones by means of green-noise masks. Journal of the Op

tical Society of America. 1 9, no. 1 1 Nov (2002).

[1 6] D. M. Kelly, W. F. Bischof, D. R. Wong-Wylie, et al. Detection of

Glass patterns by pigeons and humans: Implications for differences

in higher-level processing, Psycho/. Sci. 12, 338-342 (2001) .

VOLUME 24, NUMBER 4 , 2002 43

l$@il•i§rr6'hfil@i§4fl!:l .. i§.id Michael Kleber and Ravi Vakil , Editors !

Hat Tricks J. P. Buhler

This column is a place for those bits of

contagious mathematics that travel

from person to person in the

community, because they are so

elegant, suprising, or appealing that

one has an urge to pass them on.

Contributors are most welcome.

Please send all submissions to the

Mathematical Entertainments Editor,

Ravi Vakil, Stanford University,

Department of Mathematics, Bldg. 380,

Stanford, CA 94305-21 25, USA


I magine that you are on a team of

n > 1 people on a new reality TV game

show. After meeting your teammates,

having the rules explained, and talking

strategy with your teammates, you play

the following game, once, for a possible

shared prize of n million dollars.

The host of the game show places

black or white hats on your heads; the

hat colors are chosen uniformly at ran

dom (so that all 2n configurations are

equally likely, which could be done, for

instance, by having each hat deter

mined by a fair coin flip). All players

can see the color of every hat except

Hats are a

common device

i n mathematical

puzzles . their own. No communication is al

lowed between teammates.

Then all members of the team are

required simultaneously either to pre

dict their hat color, or to pass. The

team loses if everyone passes, or if

there are any incorrect predictions.

Otherwise-i.e., if at least one person

doesn't pass, and all non-pass state

ments are true-the team wins n mil

lion dollars.

Since every non-pass prediction is,

by the rules, a 50/50 guess, this seems

like a difficult game for the team; e.g.,

if everyone guesses, the chance of suc

cess is 112n. However, the value of the

initial strategy session comes into

sharper focus when a little thought re

veals that there is a simple plan that

gives the team a 509-6 chance of win

ning: they appoint one person to guess,

and agree that everyone else will pass.

Can you devise a strategy that gives

the team a better chance of winning?

The Hat Puzzle The hats problem circulated widely

last year, furthered by an article in The


New York Times and numerous dis

cussions on the Internet. In full gener

ality it is a fiendishly difficult puzzle;

it has many variations, most not as dif

ficult.

It is easy to misunderstand the ques

tion when you first hear it; note that af

ter the initial strategy session no com

munication is allowed between the

players, and that the players' subse

quent statements must be made simul

taneously. This could be enforced by

sending each player to a room con

taining a computer monitor listing the

names of the other players and their

corresponding hat colors, and giving

the player a choice of three statements:

"my hat is black," "my hat is white," and

"I pass."

When it is necessary to distinguish

this problem from the variants below,

I will call it the "original" hats problem

(though this is misleading: the major

ity hats problem actually predates it by

several years).

It may seem hard to believe that any

strategy could beat 50/50, since the

team can't win unless someone guesses,

and any guess has a 50% chance of be

ing wrong. Astoundingly, the optimal

strategy has a winning probability Pn that converges to 1 as n goes to infin

ity. You might want to entertain your

self by trying to solve this puzzle be

fore reading the solution below. The

case n = 3 is distinctly easier, and

makes a good puzzle to pose to your

friends (if they haven't already heard

it). The case of general n is really quite

difficult. In fact, the optimal answer is

known only for n ::::; 8, n = 2k - 1, or

n = 2k; as will be mentioned below, the

case n = 16 is especially diabolical. So

you should give the general case only

to really good friends who can tolerate

frustration, or perhaps to coding theo

rists; of course, it's probably also OK if

you are writing for The Mathematical Intelligencer and intend to describe

(most of) the solution.

Hats are a common device in math

ematical puzzles. The famous "de-

rangement" problem asks for the prob

ability that an incompetent hat-check

clerk might return n hats randomly,

none ending up with the rightful owner

(i.e., the probability that a random per

mutation of n things has QO fixed

points). The original hats puzzle first

appeared, in essence, in Todd Ebert's

1998 computer science Ph.D. thesis,

where it arose in connection with a

question in complexity theory; the

problem was phrased (see [4]) in terms

of a warden and a prisoner. Peter Wink

ler later heard it from Peter Gacs, and

immediately converted it into the hats

problem posed above. His contribution

to a recent volume in honor of Martin

Gardner [9] includes a series of puzzles

and ends with the original hats prob

lem.

An earlier "voting" puzzle due to

Steven Rudich and others [2] was also

motivated by complexity theory, and

can be phrased in terms of hats; it has

some strong connections with the orig

inal hats puzzle and will be given as the

"majority hats puzzle" in the next sec

tion.

Good puzzles often circulate faster

than gossip in mathematical circles,

and the hats puzzle is, as we will see,

remarkable in its depth and connection

with current research. Its wide dis

semination was certainly abetted by

Winkler's gregariousness, and by Sara

Robinson's charming piece in The New York Times [8].

Now we return to the puzzle itself,

so if you want to solve it, e.g., in the

case n = 3, you have to stop reading

now.

For n = 3 hats, the following strat

egy gives the team a 75% chance of win

ning: players pass if the two hats that

they see have different colors; if they

see two hats of the same color then

they assert that they have the opposite color.

There are 8 possible configurations

of the three hat colors. In 6 of them,

there are two hats of the same color

and one of the other, and a little

thought shows that in this case two

people will pass and one will make a

correct statement. In the two mono

chromatic configurations all three

players will make false statements.

Thus this strategy wins the prize with

probability 3/4, and we will see below

that this is optimal.

Somehow this surprising result is

achieved by causing all false answers

to collide and having the true answers

occur by themselves. This "unex

pected power of collaboration" is a

theme that underlies almost all of the

puzzles described in this article, and

it is really quite striking; as Ebert ob

serves, it seems that one draws infer

ences about a random variable X by

observing the values of random vari

ables that are completely independent

of X. Strategies in the hats game can be

viewed geometrically. For simplicity,

we name the colors 0 and 1, so that the

Good puzzles

often circulate

faster than gossi p

in mathematical

c ircles . configurations consist of all binary

n-tuples, i.e., the vertices of an n-di

mensional cube. All players know the

configuration, except for their own hat

color; thus a player actually knows an

edge on the cube joining the two con

figurations possible from that player's

point of view. A strategy is then a par-

iildii;JIM

tial orientation of the edges of the ncube: a direction on an edge tells a

player which configuration to guess,

and unoriented edges direct the player

to pass. The n = 3 strategy given above

is illustrated in Figure 1 .

A node is a losing configuration for

a given strategy if it has an outgoing

arrow (the corresponding player will

make a false guess) or if it has no ar

rows coming into it at all. A node is a

winning configuration if it has at least

one arrow coming into it, and no ar

rows going out.

Thus in Figure 1 there are two los

ing nodes: the antipodal points on the

3-cube, which correspond to the 2 monochromatic configurations of the

three hats. Each of the other 6 points

has one incoming arrow and no outgo

ing arrow�, and is therefore a winning

configuration. '

Let L denote the set of losing nodes

and W the set of winning -nodes. Then

L is a covering code in the sense that

every node is within Hamming distance

1 of an element of L; indeed if v is a

winning node then it has an incoming

arrow that originates in an element of

L. Here two nodes have "Hamming dis

tance 1" if one can be obtained from

the other by changing exactly one co

ordinate. (More generally, coding the

orists consider d-covering codes in

which every node is within Hamming

distance d of an element of the code;

the case d > 1 appears to be irrelevant

to hats problems.)

If L is any covering code, then it

gives a strategy: players seeing an L W edge should vote the W node, those

seeing a WW edge should pass, and

those seeing an LL edge might as well

guess.

The probability of losing is ILI/2n, and a "sphere-packing bound" gives a

lower bound on the size of L, and

hence an upper bound on the winning

probability. Namely, each losing node

v "covers" the n + 1 points that are at

Hamming distance at most one: v itself

and the n nodes obtained by reversing

a single coordinate of v. These Ham

ming spheres of radius 1 must cover

the n-cube, so


Therefore l£1 2: 2n!(n + 1), and the winning probability p of the strategy is at most

1 n < 1 - -- = -- n + l n + l .

A strategy is perfect if this bound is realized. This happens only when the covering Hamming spheres of radius 1 are also disjoint, so that elements of L have outgoing arrows on every edge and elements of W have a unique incoming arrow. In the hats game this means that there is a unique non-pass on winning configurations, and everyone makes a false statement on losing configurations. Note that the above strategy for n = 3 is perfect in this sense, so that 75% is indeed the optimal probability P3·

The problem of fmding disjoint Hamming spheres is familiar to codingtheorists: this is the problem of constructing error-correcting codes. The "dual" problem of finding Hamming spheres that cover the binary n-cube hasn't received as much attention, but it does have several applications; the defmitive reference on covering codes is [6].

Note that if L is a perfect code then

___lfL - 1 2n+ l - n + l '

so a perfect code exists only if n + 1 is a power of 2. A perfect code is both packing and covering, and codingtheorists know that perfect !-covering codes in fact exist when n + 1 = 2k; linear codes with this property are called (binary) Hamming codes.

One explicit formulation of an optimal strategy for the hats game (and construction of the Hamming codes) when n = 2k - 1 is as follows. At the strategy session, team members are assigned "names" that are nonzero k-bit 0/1 strings, perhaps thought of as the binary expansions of integers from 1 through 2k - 1. During the game, each player will then assume that the "XOR" of the players with hat color 1 is nonzero. Here XOR is "bitwise exclusive-or" or "nim addition": two 0/1


strings are combined by adding corresponding bits modulo 2, without carrying.

Thus players will XOR the names of the players they see that have hat color 1; if the result is the 0 vector, they will announce that they have hat color 1, if the result is their own name, they will announce that they have hat color 0, and in all other cases they will pass.

It is easy to verify that the team wins unless the XOR of all players with hat color 1 is zero. This set of losing positions is of course the binary Hamming code!

An equivalent, but more technical, description will be useful later in discussing the hats problem with more

I t is remarkable

that a purely

recreational

problem comes

so close to the

research frontier. than two colors. Namely, the Hamming code L is the kernel of the linear map

V* T:F 2 � V,

where V is a k-dimensional vector space over the field F 2 of two elements, VI' denotes the nonzero elements of that vector space, and F r denotes the vector space of dimension 2k - 1 of functions from VI' to F2. The vector space F r has natural basis elements [v] corresponding to (characteristic ftmctions of) vectors v E VI', and the map T takes [v] to v.

If n = 2k, then a natural extension of the above strategy turns out to be best possible. At the strategy session, one player is chosen to play dumb: that player passes, his hat color is ignored, and the remaining 2k - 1 players follow the above strategy (for proof that this is optimal, in the language of covering codes, see [6]).

For n not of the form 2k - 1 or 2k, the full story on covering codes isn't known! However, as described below,

it is possible to come close to the sphere-packing bound Pn :=::; nl(n + 1) for large n. Hamming codes L C FE ( n = 2k - 1) have the special property that they are linear, in the sense that they are vector subspaces of an F2-vector space. In all likelihood, optimal codes in other dimensions will not be linear.

It is remarkable that a purely recreational problem comes so close to the research frontier. The fundamental reference on covering codes [6] has a companion Web site [7] that contains up-to-date data on the best known covering codes. For 2k - 1 < n < 2k+ 1 -1 there is a dumb strategy, as above, based on the Hamming code for n = 2k - 1. In other words, 2k - 1 players are appointed to use the Hamming code strategy, and the remaining players play dumb. This strategy is optimal for n = 2k, but not for larger n. In addition, it is known that as n goes to infinity there are nonlinear codes with density that come very close to the sphere-packing bound for large n. More precisely, if Ln is a subset (not necessarily a subspace) of minimum cardinality of F E such that every point of FE is within Hamming distance 1 of Ln, then the density 1Lnl12n is, by definition, 1 - Pn. where Pn is the best possible probability of winning; moreover,

_1_ :=::; ILnl < _2_ n + l 2n n + l

(the lower bound comes from the sphere-packing bound above, and the upper bound can be derived from the dumb strategy). With some work (see [6]), one can show that the limit, as n goes to infinity, of (n + l)ILnl12n is 1, i.e., that for large enough n the ratio of the density of Ln to the sphere-packing bound ll(n + 1) can be made arbitrarily close to 1 (and is of course equal to 1 if and only if n = 2k - 1).

As one sees on the Web site [7], already for n = 9 the optimal strategy is unknown: the best covering code Lg has 57 :=::; l£91 :=::; 62. The sphere-packing bound says that the winning probability is at most 9/10, giving p9 < 460/512; in fact the bounds on the size of Lg imply that

Note that the value n = 16 is an especially diabolical value to give to someone as a puzzle: in addition to the possibility that your poor friend will be somehow misled by the power of 2, he will have to recover the connection with covering codes, will have to rediscover Hamming codes, and will have to rediscover a nontrivial theorem about covering codes in the case n = 2k.

I will go on in a moment to consider variants of the original hats puzzle. Meanwhile, there are a number of details and extensions that cry out for examination, but to save space I leave them as exercises for the diligent reader (with hints elsewhere in this issue; see p. 70).

1. Using some description (as above or otherwise) of the Hamming code, show that in the case n = 3 = 22 -1 it gives the solution given earlier for the hats problem, perhaps up to suitable choice of labeling.

2. Verify that for n = 2k - 1 the Hamming code solution does indeed achieve the sphere-packing bound.

:!. Show that for 2k - 1 < n < 2k+l -1 the dumb strategy above has winning probability that is at least (n -1)/(n + 1).

4. The dumb strategy for n = 5 gives a winning probability of 3/4 = 24/32 for a 5-person team in which 2 players play dumb. Show that the team can do slightly better by fmding a 7 -point covering code in the binary 5-cube, giving the team a winning probability of 25/32.

5. Verify that the Hamming code implicit in the XOR strategy is the same as the Hamming code obtained as the kernel of the linear map described above.

6. What can the team do if the game show host maliciously listens in on the strategy session and attempts to choose the hat colors nonranaomly?

7. Verify that randomization does not help in the original game, in other words, no randomized strategy can do any better than a deterministic strategy.

I I I I

........ I ........ I .......... ............ .... ....

Majority Rules

........ ... I .... ........ I 1 I I I I 1 / I I I I

Reality TV shows fade quickly, and our TV game host decides that the rules must be changed in order to boost ratings. (Perhaps he also noticed that the teams were winning too often.) The new rules do not allow players to pass, but, as compensation, a team wins if a majority of players make true statements.

In order to avoid ties this game is played with an odd number of players.

This version of the problem was originally stated in [2] as a voting problem, motivated by results on lower bounds in computational complexity arising from analyzing circuits in terms of integer polynomial "approximations" to boolean functions. In addition to the important results on approximating boolean functions by the signs of integer polynomials, the paper contains other variants of the puzzle (one of which will be described in an exercise below).

The majority hats problem is similar to the original hats problem in basic framework, and also in that there is a Hamming code solution when n = 2k - 1, as will follow from some of the arguments below. The optimal strategies for 2k - 1 < n < 2k+ 1 - 1 are not known, though they seem easier to explore than in the case of covering codes.

Elwyn Berlekamp has analyzed this majority hats problem in some detail, and finds an amusing geometric interpretation of a strategy for this game. Namely, a strategy can be described by

giving an orientation on aU of the edges of the graph of the n-cube: again, each player sees an edge and, not being allowed to pass, votes according to the direction of that player's edge. The optimal strategy for n = 3 can be obtained from the optimal strategy for the original hats game above by orienting the remaining edges in a cycle.

In Figure 2, the graph of the 3-cube decomposes into 2 tripods emanating from the losing set, and a cycle, indicated as a dotted line, joining the other 6 winning configurations (the cycle can be oriented in either direction). A little thought shows that the marked points are losses in that every player votes wrong, but that the other nodes are all wins: there are two incoming and one outgoing arrows, so that the team wins by one vote.

More generally, Berlekainp generalizes the idea of a covering code by allowing paths, possibly of -length more than 1, emanating from "sources" in the losing set L, that terminate in "sinks," which are the complementary winning set W.

Each winning node is the terminus of exactly one such path, and the paths are all edge-disjoint. Since the number of edges at every node is odd, the graph obtained by removing the chosen paths (and the vertices in L) has even valence. By Euler's theorem, this graph is a union of cycles. Therefore the original graph can be thought of as a collection of paths and cycles; the paths are directed, starting in L, and ending in W; each element of W is the endpoint of a uitique path. The cycles can be oriented arbitrarily. This ensures that at each winning node all of the votes other than the decisive vote are evenly split, and at each winning node the team wins by one vote.

Thus if we can fmd a set of edge-disjoint paths from losing nodes to each winning node, then this can be extended to an orientation of the entire graph that gives a strategy for the majority hats problem.

Note that a node in L can have at most n outgoing paths, so that a losing node "accounts for" itself and at most n winning nodes; thus the sphere-packing bound still applies, i.e., (n + 1) IL I ::::::

2n. So far, Berlekamp cannot fmd counterexamples (even for very large n) to the surprising conjecture that optimal size of L is as small as it can be consistent with this bound; i.e., that the size of the optimal L is the smallest integer bigger than or equal to 2n+ 1/ (n + 1).

8. Show that for n = 5 there is a strategy for the majority hats game with only 6 losing configurations; i.e., find an orientation of the binary 5-cube in which all but 6 vertices have an excess of incoming arrows. (Since the best that is possible for the original hats game is a 7-node covering code, this shows, as Berlekarnp notes, that democracy is preferable to consensus/perfection.)

9. Find an optimal strategy in the majority hats game for n = 9.

More Colors Again, our TV game show host wants to boost sagging ratings, and perhaps decrease winning probabilities, and decides to start using more than 2 colors of hats. What strategies should the team use when there are q colors, q > 2?

As one might guess, the team has a harder time. However, Hendrik Lenstra, Jr., and Gadiel Seroussi have shown that some of the same basic facts hold even in this case. For instance, the winning probability is arbitrarily close to 1 for large enough n. However, perfect codes do not exist, and there are several open questions.

First, let's interpret the game geometrically. The configuration space is now a q-ary n-cube Qn, where Q is a qelement set. A player sees an "edge" of the cube: the i-th player sees the configuration v E Qn except that the i-th coordinate vi is unknown. From the point of view of this player the configuration could be any of the q configurations that agree with v except possibly in the i-th coordinate. A strategy is a mapping from edges e to Q U {pass}. If a player sees the edge e he passes if the label of the edge is "pass," and announces the corresponding color if the edge is labeled with a color.

If v E Qn is a configuration of hats, let v[i] denote the set of configura-


tions, not equal to v, that agree with v in all but the i-th coordinate. Thus, each element w of v[i] has Hamming distance 1 from v, and disagrees with v in precisely the i-th coordinate.

If a strategy is given, then we get a partition Qn = W U L of configurations into winning and losing positions. The winning configurations have the following property:

(*) For all v in W there is a coordinate i such that v[i] c L.

Conversely, if a set W satisfies (*) then it produces a strategy whose losing set is precisely the complement L of W.

In order to digest this condition you might want first to convince yourself

The reader may

have d rawn the

conclusion that al l

hats problems are

i mpossib ly hard . that in the case q = 2 we recover the earlier analysis involving covering codes L. We will call a subset L C Qn a strongly covering code if its complement satisfies (*). You will also enjoy checking that in the case n = q = 3 the marked nodes in Figure 3 are a strongly covering code; to do that you have to check that for each node not in the code there is some coordinate direction in which the other two vectors are in the code.

litftll;ifi

As in the case of covering codes, the winning probability is of course

Thus we want to fmd small strongly covering codes.

Lenstra and Seroussi generalize the sphere-packing bound to show that if L c Qn is a strongly covering code then

IL l 2: qn(q - 1)

. n + q - 1

A little algebraic manipulation shows that this gives an upper bound on the winning probability p of

P = 1 _ lfd_ ::;; _ ___:_n __

qn n + q - 1

which generalizes the earlier result for q = 2.

One way to prove this is as follows. Let S be the set of ordered pairs (x, y) where x is a winning node, y is a losing node, x and y differ in exactly one coordinate, and all other nodes differing from x in that coordinate are also losing nodes; i.e.,

S = {(x, y) : x E W, there is an i with y E x[i] and x[i] c L}.

We employ the usual combinatorial device of counting this set of ordered pairs in two ways. For each x there is (at least one) coordinate direction in which all the q - 1 other elements on that edge are in L; therefore (q -1)1*1 ::;; lSI. On the other hand, if we fix y and ask how many x's could be paired with it , we note that there are n coordinates, so that y E L can be partnered with at most n x's. Therefore

(q - 1)(qn - ��) ::;; lsi ::;; n�l-A little algebraic juggling gives the upper bound claimed above.

Unfortunately, Lenstra and Seroussi also prove that for q > 2 and n > 1 , perfect strongly covering codes do not exist, so that the lower bound on L cannot actually be attained. However, they give an ingenious argument that shows that the winning probability can be made arbitrarily close to 1 by choosing n large enough. The technique somehow intertwines the binary case with the q-nary case.

Let n = 2k - 1. (For other n, we will use a "dumb" strategy as described above.) Let Q = ZlqZ be the q-element cyclic group. There are n = 2k - 1 nonzero k-vectors v in Qk with entries in {0, 1 ). Use them to label basis vectors [v] whose Q-linear combinations form a group which will be identified with Qn. Then let

T : Qn � Qk

be the group homomorphism that maps [v] to v.

So far this follows the q = 2 construction, but we now alter it in an interesting way: let L be the set of elements of Qn whose image under T is a k-tuple whose coordinates are all nonzero.

I claim that L is a strongly covering code. Indeed, if x E Qn is not in L, then T(x) has some coordinates equal to 0. If v denotes the 0/1 vector with 1's in the coordinates where T(x) is nonzero, then one checks that x[i] C L, where the coordinate i corresponds to the basis vector [ v ] . This shows that L is a strongly covering code, and it is easy to check that the winning probability is p = 1 - (q - 1)kfqk, which goes to 1 (albeit more slowly than one might like) as n goes to infinity.

Noga Alon ([1]) subsequently gave a probabilistic construction of a strongly covering code whose winning probability comes much closer to the asymptotic bound. The basic idea is to make a random choice and then alter it as necessary. This is a well-known situation in coding theory: the best known explicit constructions fall short of what can be achieved by suitably modified random codes.

By now, the reader may have drawn the conclusion that all hats problems are impossibly hard and that they aren't recreational in any sense of the word.

In an attempt to persuade you otherwise, here is a collection of (somewhat) easier hat problems. The first two can be found in Peter Winkler's.charming contribution Games People Don't

Play to the recent volume [9] of essays arising from a "Gathering for Gardner" in honor of Martin Gardner's contributions to recreational mathematics.

10. The TV game show host introduces the following more extreme variation of the game. The hats game is played as described originally except that passes are not allowed, and players making false statements are executed. What "worstcase" strategy can the team adopt that gives them the largest number of guaranteed survivors?

1 1 . What is the team's best worst-case strategy in the following variation? The team members are lined up in a manner that allows players to see only the hats in front of them in the line, e.g., the front player sees no hat colors, and the player at the back of the line sees all colors but one-the one she is wearing. The players are required to state their hat colors, one at a time starting at the back of the line. Players making false statements are executed. All players hear all of the statements, but not their consequences.

12. Same as the previous problem, except that the game show host uses q > 2 colors of hats.

13. [2] What is the team's best strategy if the host uses the majority hat game, except that "Chicago-style" voting is allowed in which players can cast as many votes as they like?

14. (Gadiel Seroussi) What strategy would you follow if the game show host, in a fit of desperation, did not allow a strategy session, and did not turn the lights on after the hats were placed? Thus team members cannot see any hat colors; they are complete strangers. (Individual team members are allowed to assume that their teammates are highly rational, and the rules permit flipping coins in the dark to generate random numbers, by feel-

ing the top of a coin to see whether it is heads or tails).

Acknowledgments

I thank Peter Winkler for telling me the original hats problem, Michael Kleber for encouraging me to write this article, and Elwyn Berlekamp, Hendrik Lenstra, Jr., and Gadiel Seroussi for some delightful conversations about the puzzle. Elwyn Berlekamp, Danalee Buhler, Michael Kleber, Hendrik Lenstra, Gadiel Seroussi, Ravi Vakil, and Peter Winkler all made helpful comments on various drafts of this piece.

REFERENCES

[1 ] Noga Alon, "A comment on generalized

covers," note to Gadiel Seroussi , June 2001 .

[2] James Aspnes, Richard Beigel, Merrick

Furst, and Steven Rudich, The expressive

powec.of vo'ting polynomials, Combinatorica 14

(1 994), 1 35-1 48.

[3] Mira Bernstein, The Hat Pro_,blem and Ham

ming Codes, in the Focus newsletter of the

MAA, November, 2001 , 4-6.

[4] Todd Ebert and Heribert Vollmer, On the

Autoreducibility of Random Sequences, in

Proc. 25th International Symposium on Math

ematical Foundations of Computer · Science,

Springer Lectures Notes in Computer Science,

v. 1 893, 333-342, 2000.

[5] Hendrik Lenstra and Gadiel Seroussi,

On Hats and other Covers, preprint, 2002,

www.hpl.hp.com/infotheory/hats_extsum.pdf

[6] G. Cohen, I. Honkala, S. Litsyn, and A. Lob

stein, Covering Codes, North-Holland, 1 997.

[7] Simon Litsyn's online table of covering

codes: www.eng.tau.ac.il/�litsyn/tablecr/

[8] Sara Robinson, Why Mathematicians Now

Care About Their Hat Color, New York Times,

Science Tuesday, p. D5, April 1 0, 2001 . On

line at http://www.msri.org/activities/jir/sarar/

01 041 ONYTArticle.html

[9] Peter Winkler, Games People Don't Play,

301-313 in Puzzlers' Tribute, edited by David

Wolfe and Tom Rodgers, A. K. Peters, Ltd. , 2002.


Reed College

Portland, OR 97202

USA



l@ffli•i§rr6hlf119.1rr1rr11!.1h14J Marjorie Senechal , Editor I

MASS Program at Penn State Anatole Katok, Svetlana Katok,

and Serge T abachnikov

This column is a forum for discussion

of mathematical communities

throughout the world, and through all

time. Our definition of "mathematical

community" is the broadest. We include

"schools" of mathematics, circles of

correspondence, mathematical societies,

student organizations, and informal

communities of cardinality greater

than one. What we say about the

communities is just as unrestricted.

We welcome contributions from

mathematicians of all kinds and in

all places, and also from scientists,

historians, anthropologists, and others.

Please send all submissions to the

Mathematical Communities Editor,

Marjorie Senechal, Department

of Mathematics, Smith College,

Northampton, MA 01 063 USA


The MASS program-Mathematics Advanced Study Semesters-is an

intensive program for undergraduate students recruited every year from around the USA and brought to the Penn State campus for one semester. MASS belongs to a rare breed; we know of two somewhat similar mathematics programs for American undergraduates, both based abroad: Budapest Semesters in Mathematics, and Mathematics in Moscow; the former is in its "teens" (started in 1985) while the latter is just 1 year old. MASS at Penn State has turned 6, and this seems to be a good time to reflect on the MASS community.

How It Started All three founders of the MASS program (the first two authors of this article and the first MASS director, A Kouchnirenko) are steeped in the Russian tradition where interested students are exposed to a variety of mathematical endeavors, often of nonstandard kind, at an early age. By their senior undergraduate years such students are already budding professionals. We briefly describe this tradition in the Appendix. The US educational system is built on completely different principles, and interested young students are routinely encouraged to progress quickly through the required curriculum. Here a typical mathematically gifted high school student takes courses in a local university and often is considered a nerd by his peers. The founders felt that there was a way to combine some of the best features of both traditions within the US academic environment, namely, to gather a group of mathematics majors and to expose them to a substantial amount of interesting and challenging mathematics from the core fields of algebra, geometry, and analysis, going way beyond the usual curriculum.

The second author's first exposure to an intensive program for US undergraduates was at the Mills College


Summer Mathematics Institute for mathematically gifted undergraduate women. But why not a co-educational program along the same lines, whose participants would contribute a variety of experiences and backgrounds? The number of undergraduate students in the USA, interested in mathematics and advanced enough for such a program, is rather limited, and we decided not to restrict the pool of potential participants. The result was the SURI (Summer Undergraduate Research Initiative) program at Penn State in the summer of 1993, where all three future founders of the MASS program came together. During this program it became clear that a semester-long format would be even more productive for an intensive program organized mostly around advanced learning with elements of research initiation.

And so we envisioned a semesterlong program for undergraduate students from across the country. We thought it crucial for the success of the program that the cost for the participants should not exceed that at their home universities. It took 3 years to get the original financial commitment from the Penn State administration at various levels and to solve numerous logistic problems before the MASS program could begin.

Program Description The main idea of the MASS Program, and its principal difference from various honors programs, math clubs, and summer educational or research programs, is its comprehensive character. MASS participants are immersed in mathematical studies: since the program is intensive, its full-time participants are not supposed to take other classes. All academic activities for a semester are specially designed and coordinated to enhance learning and introduce the students to research in mathematics. This produces a quantum leap effect: the achievement and enthusiasm of MASS students increases

much more sharply than if they had been exposed to a similar amount of material over a longer time in a more conventional environment.

A key feature of the MASS experience is an intense and productive interaction among the students. The environment is designed to encourage such interaction: a classroom is in full possession of MASS (quite non-trivial to arrange in a large school such as Penn State!) and furnished to serve as a lounge and a computer lab outside of class times. Each student has a key and can enter the room 24 hours a day. The students live together in a contiguous block of dormitory rooms and they pursue various social activities together. The effect is dramatic: the students find themselves members of a cohesive group of like-minded people sharing a special formative experience. They quickly bond, and often remain friends after the program is over. They study together, attack problems together, debug programs together, collaborate on research projects, and,

most importantly, talk mathematics most of the time. Of course, this is exactly how "mature" mathematicians operate in their professional life! A necessary condition for this environment is the gathering of a critical mass of dedicated and talented students, which is one of the chief accomplish ments of MASS.

Let us describe the main components of MASS:

• Three core courses on topics chosen from the areas of Analysis, Algebra/Number Theory, and Geometry!ropology. Each course features three 1-hour lectures per week, a weekly meeting conducted by a MASS Teaching Assistant, weekly homework assignments, a written midterm exam, and an oral final examination/presentation.

• Individual student research projects ranging from theoretical mathematics to computer implementation. Most of the projects are related to the core courses; some are devel-

oped independently according to the interests and abilities of the student.

• A weekly 2-hour interdisciplinary seminar run by the director of the MASS program (the third author of this article), which helps to unify all other activities.

• The MASS colloquium, a weekly lecture series by distinguished mathematicians, visitors, or Penn State research facuity.

All elements of MASS (3 courses, the seminar, and the colloquium) total 16 credit hours, all listed as Honors classes that are transferable to MASS participants' home universities. Additional recognition is provided through prizes for best projects and merit fellowships. Each student is issued a Supplement to the MASS Certificate, which includes the list of MASS courses with credits, grades, final presentations, and special achievements. It also includes the descriptions of MASS courses, the list of MASS colloquia, and the description of MASS program exams.

photo © S. Katok


These supplements are useful for the

student's home institution equiva

lences and enhance the student's ap

plications to graduate schools.

The core courses are custom de

signed for the program and are avail

able only to its participants. Each

course addresses a fundamental topic

which is not likely to be covered in the

usual undergraduate (and, in many

cases, even graduate) curriculum. For

example, the core courses offered at

MASS 2001 were

Mathematical Analysis of Fluid Flow by A. Belmonte, Theory of Partitions by G. Andrews, and Geometry and Relativity: An Introduction by N. Higson.

Designing and teaching such a

course, an instructor is challenged to

reach a delicate balance between cov

ering the basics, with which the stu

dents might be unfamiliar, and intro

ducing advanced material typically

taught in topics courses.

Consider, for example, a MASS 2000 course Finite Groups, Symmetry, and Elements of Group Representations by

A. Ocneanu. This class started with


fundamental facts about finite groups

and their representations and pro

ceeded to what is often referred to as

"quantum topology": invariants of

knots and 3-dimensional manifolds as

sociated with statistical physics and

"The MASS

prog ram has

been the best

semester of

my l ife . " the Yang-Baxter equation. The course

was received by the students with great

enthusiasm and is likely to direct some

of them toward this active area of re

search.

The final exams (three, in total) have

a unique format. It is quite unusual for

a US university and represents a cre

ative development of a European tradi

tion where examinations are often oral.

A student draws a random "ticket"

which typically contains a theoretical

question from the course and a prob

lem. Then the student has an hour to

prepare the answers, with no access to

literature or lecture notes during this

hour. The answers to the ticket ques

tions constitute only about a third of the

oral examination. Another third is a pre

sentation of the research project asso

ciated with the course; this presentation

is prepared in advance and may involve

slides, computer, etc. The last third of

the exam is a discussion with the com

mittee of three (the course instructor,

the teaching assistant, and another

Penn State faculty).

A MASS colloquium is similar to a

usual colloquium at a department of

mathematics, with an important differ

ence: a speaker cannot assume much

background material. Although this

makes the speaker's task harder, we

find that the quality of the talks usually

benefits from this restriction. To quote

the opening sentences of an inspiring

article by J. McCarthy "How to give a

good colloquium" (see at www.math.

psu. edu/colloquium/goodcoll.pdf) :

"Most colloquia are bad. They are too

technical and aimed at too specialized

photo © S. Katok

an audience." This is precisely a sin that MASS colloquium is free of. As a result, along with MASS students, it is well attended by graduate students and faculty at the Department.

To preserve the intellectual effort that goes into MASS colloquium talks, a group of 2 or 3 MASS students is assigned to take notes and prepare a readable exposition of the talk. We also experiment with videotaping the talks.

Choosing the speakers, we always invite mathematicians known for their expository skills. We also try to represent as broad a spectrum of mathematical research as possible. We find it beneficial to combine very well-known mathematicians with those in the early stage of their careers. A complete list of MASS colloquium talks can be found on the web site www.math.psu.edu/mass.

The MASS seminar plays many roles in the program. One of them is to introduce the students to the topics that, otherwise, are likely to "fall between cracks in the floor." For example, one of the seminar topics in 2001 was the classical configuration theorems of projective geometry: Pappus, Desargues, Pascal, Brianchon, and Poncelet. Once wojective geometry was a core subject in the university curriculum, but nowadays it is perfectly possible to obtain a doctoral degree in mathematics without a single encounter with these facts. Another example: the theory of evolutes and involutes was a crowning achievement of Calculus to be included into textbooks. Alas, a contemporary student is not likely to see these things any more. The MASS seminar is a natural place to learn such a topic.

Another purpose of the seminar is to prepare the students for the upcoming MASS colloquium talks. A colloquium speaker is asked whether certain material should be covered in advance so that the students get the most from the talk. For example, as preparation for A. Kirillov's talk on Family Algebras in 2001, a 2-houJ seminar was devoted to the basics of Lie groups and Lie algebras. Still another function of the seminar is to rehearse the students' presentations of the research projects on the ,final exam. This usually occupies the last quarter of the semester. Probably an even more im-

portant function of the seminar is to bring out elements of unity of modem mathematics. Often identical or similar notions appear in different courses in various guises, and the seminar is the place to explore, develop, and clarify these connections.

The Summer Program: REU and MASS Fest The Penn State Summer RED (Research Experiences for Undergraduates) program started in 1999 as an extension of MASS. Unlike MASS, this program is not unique: currently, there are about 50 RED programs in mathematics available to undergraduate students in the USA. The Penn State RED is closely related to MASS: about half of its participants stay for the MASS semester in the fall. This makes it possible to offer research projects that require more than 7 weeks (the length of RED program) for completion.

Mathematical research usually includes three components: study of the subject, solving of a problem, and presentation of the result. These three components are present in the RED program: in addition to the traditional individual/small group research projects supervised by faculty members, the program includes two short courses, a weekly seminar, and the MASS Fest.

MASS Fest is a 3-day conference at the end of the REU period at which the participants present their research. This is also a MASS alumni reunion. Along with the RED students, a num-

photo © S. Katok

ber of guest speakers, mostly Penn State ..faculty, give expository talks at the conference.

Here are two examples of REU students' research projects. ·

"Simplices with only one integer point" (2 students; faculty mentor A. Borisov). The students found an effective procedure that allows them to describe all classes of simplices with vertices that have only integer coordinates and only one point with integer coordinates inside. Using computers they found all classes in dimensions 3 and 4.

"New congruences for the partition function" (1 student; faculty mentor K. Ono ). This project started before the RED program began. Using the theory of Heeke operators for modular forms of half� integral weight, the student found an algorithm for primes 13 :::;; m :::;; 31 which reveals 70,266 new congruences of the form p(An + B) == 0 (mod m), where p( n) denotes the number of unrestricted partitions of a non-negative integer n. As an example, she proved that p(3828498973n + 1217716) == 0 (mod 13) for every integer n. The first three congruences were found in 1919 by Ramanujan, and after that finding new ones was considered a very difficult problem. The paper written by this student has been accepted for publication.

We would like to emphasize a unique role played by the RED coordinator, M. Guysinsky, who has been coming to Penn State for the summer since 1999


as a visiting Assistant Professor supported by VIGRE funds.1 He organizes all the REU activities, including MASS Fest, runs the seminar, and supervises research projects, some suggested by other faculty not present during the REU period, and some by him. This requires an unusual combination of mathematical and pedagogical talents, and we are very fortunate to have found this combination in Guysinsky.

Participants MASS participants are selected from applicants currently enrolled in US colleges or universities who are juniors, seniors, or sometimes sophomores. They are expected to have demonstrated a sustained interest in mathematics and a high level of mathematical ability. The required background includes a full calculus sequence, basic linear algebra, and advanced calculus or basic real analysis. The search for participants is nationwide. Participants are selected based on academic record, recommendation letters from faculty, and an essay.

The number of MASS participants varies from year to year, with an average of 15 per semester. Some are Penn State students, but most are outsiders. It is interesting to analyze where they come from. For this purpose we divide American universities into four categories: (1) small, mostly liberal arts, schools; (2) state universities (mostly large); (3) elite private universities; ( 4) Penn State. The breakdown over the last 6 years is as follows: about 20% of the participants belong to the first category, about 400Al to the second, only 3% to the third, and 37% to the fourth. One should take into account that some Penn State students are part-time participants (they take one or two courses), but a few of them participate in MASS more than once.

These numbers are probably not very surprising (although we strongly feel even students from elite schools benefit significantly from the program). Another statistic: women represented about 300;6 of the enrollment (with considerable deviations: in 2000, the ratio was 50/50).

About 700Al of MASS graduates have gone on to graduate programs in mathematics (one should keep in mind that some recent participants are still continuing their undergraduate studies). The distribution of the graduate schools is very wide. Without providing a comprehensive listing, we mention some: Harvard, Cornell, Stanford, Princeton, Yale, University of Chicago, University of Michigan, University of California at Berkeley, University of Wisconsin, Indiana University, University of Utah, University of Georgia. About 15% of MASS graduates chose Penn State for graduate school.

Here is what Suzanne Lynch, a MASS 96 participant who is about to receive her Ph.D. from Cornell, wrote in an unsolicited letter:

The MASS program has been the best semester of my life. I was immersed in an environment of bright motivated students and professors and challenged as never before. I was pushed by instructors, fellow-students, and something deep inside myself to work and learn about mathematics, and my place in the mathematical world. I loved my time there, and never wanted to leave. I believe the MASS program helped to prepare me for the rigors of graduate school, academically and emotionally. . . . The MASS program has been very instrumental in opening grad school doors to me, and in giving me the courage to walk through them.

Talking of MASS participants, one must mention the teaching assistants involved. T As are chosen from among the most accomplished Ph.D. students of the Penn State Department of Mathematics. Their work is demanding but also rewarding. TAs are required to sit in the respective class and take notes; once a week they have a 1-hour meeting with the students that is devoted to problem-solving, project discussion and, sometimes, individual tutoring. In some cases the material of a MASS course may be new for the TA as well as the students. This gives the assistant a welcome opportunity to learn a new

topic but makes the work even more challenging. Some MASS TAs are themselves MASS graduates.

Student Research During the semester, each MASS participant works on three individual projects. Usually a project consists in learning a certain topic in depth, working on problems (ranging from routine exercises to research problems, usually related to the subject of the respective course), and making a presentation during the final examination. For many MASS participants who also attend the REU program, a project is a continuation of one started in summer.

In some cases, a research project produced a significant piece of mathematical research. Here are two examples:

An Nguyen, a MASS 96 student and now a graduate student in Computer Science at Stanford, rediscovered the famous value of A = 1 + Vs for the appearance of period-three orbits in the logistic family f(x,A) = Ax(1 - x), and then went on to discover a previously unknown bifurcation point where the second period-four orbit appears:

A = 1 + Y 4 + 3vTo8.

James Kelley, a MASS 98 participant, now a graduate student at UC Berkeley, studied the representation of integers by quadratic forms, a classical problem in number theory. In particular, he studied a well-known problem posed by Irving Kaplansky: What integers are of the form x2 + y2 + 7z2 where x, y, and z are integers? Obviously, if N is of this form, then so is Nk2• However, the converse is not necessarily true. James proved, us

ing the theory of elliptic curves and modular forms, that every "eligible" integer N which is not a multiple of 7 and not of this form, is square-free! This result has appeared in print: J. Kelley, "Kaplansky's ternary quadratic form," Int. J. Math. Sci. 25 (2001), 289-292.

The research project topics may be related to the student's major, different from mathematics. For example, a biology major in the 2001 course "Mathe-

1VIGRE: Grants for Vertical Integration of Research and Education, a program designed to promote educational experiences of undergraduate and graduate students in the context of ongoing mathematical research within the university.


matical analysis of fluid flow" has a research project "A mathematical analysis of fluid flow through the urinary system."

MASS students present their research projects at the Undergraduate Student Poster Sessions at thi!.January AMSIMAA joint meetings. For example, N. Salvaterra and B. Wiclanan (REU and MASS 1999) were among the winners in Washington, DC, January 2000, with the poster "The Growth of Generalized Diagonals in a Polygonal Billiard" (advisors: A. Katok and M. Guysinsky). Another example: B. Chan (REU and MASS 2000) was a winner in New Orleans, January 2001 , with the poster "Estimation of the Period of a Simple Continued Fraction" (advisors: R. Vaughan and M. Guysinsky).

Funding MASS is jointly funded by Penn State and the National Science Foundation. Penn State provides fellowships for outof-state students that reduce their tuition to the in-state level. Further support comes through the NSF VlGRE grant. In particular, MASS participants whose tuition in their home institution is lower than Penn State in-state tuition receive grants for the difference. The balance of the VlGRE funds are used to further decrease out-of-pocket expenses of the participants, and is distributed individually based on merit and need. In particular, several merit fellowships are awarded at the end of the MASS semester. The VlGRE grant also supports the MASS colloquium series by covering the speakers' travel expenses.

Perspectives We are confident that MASS will continue to grow. Here are some ideas for the program's future.

• One of the key issues is funding. We hope to attract private money to complement the current NSF support of the program. There is a considerable interest in mathematics among private and corporate aonors, and the contribution of the MASS program to undergraduate mathematics education is substantial. Ideally, we would like to see the whole program endowed. '

• We envision a larger, 2-level MASS

program that runs two consecutive semesters: one oriented toward freshmen and sophomores, the other, more advanced, for juniors and seniors.

• With a broader financial base, MASS could include a certain number of foreign students. The available NSF funds can support only US citizens and permanent residents. However, there is an interest in the program among foreign students attending American universities, and a few such students have attended MASS paying from their own funds. As a first step, we would like to extend the program to undergraduates in Canada

• An important issue is preservation of MASS materials. Each MASS core course developed for the program can be used elsewhere. We envision an ongoing series of small books containing course material in a lecture notes style, detailed enough to serve as guidelines for a qualified instructor to design a similar course. As a first step, we are preparing a MASS presentation volume that will be published by the American Mathematical Society. This book will present all components of the program (core courses, REU courses, MASS colloquia, students' research), and it will appear late in 2002 or early in 2003. We also hope to record MASS colloquium talks and make them available to the public, possibly online, in the MSRI style.

Our optimism about the future of MASS is based on the enthusiasm of the students, instructors, and TAs, and on the general public interest in improving the mathematical education in the USA.

Appendix: On the Russian Tradition of Mathematical Education Russian mathematics constitutes one of the most vital and brilliant mathematical traditions of the 20th century. Mathematicians trained in Russia are very well represented in the top echelon of the world mathematical community. Behind this flourishing stands a powerful tradition of spotting and training mathematical talent, which is

not without its downside. The subject is certainly too complex for a detailed discussion, but we will try to present a brief outline.

A typical path of a mathematically talented student would start rather early. It would include participation in mathematical olympiads of various levels, from school district to the allUnion one (the first Mathematical Olympiad in the Soviet Union was held in Leningrad in 1934, and Moscow followed suit the next year; the first allUnion Olympiad took place in 1961). Another activity for an interested school student was a kruzhok (literally, "circle"; a closer English equivalent is probably "workshop"); kruzhki also appeared in the mid-1930s. They usually met at the university once a week in the evening and were run by dedieaten undergraduate or graduate students with a tremendous enthusiasm for mathematics, very ·often themselves alumni of a kruzhok-a good example of "vertical integration"! The material discussed usually went well beyond the secondary school curriculum and included challenging problems and nonstandard topics from elementary to higher mathematics.

Beginning in the early 1960s, special high schools for mathematics and physics were organized in major cities. Many benefited from the help of the local university faculty; for example, E. B. Dynkin and I. M. Gelfand played a prominent role in running the legendary Moscow School No. 2, whose many alumni are now professors of mathematics in universities across the globe. Another well-known high school, the Boarding School for Mathematics No. 18 at Moscow State University, was established by A N. Kolmogorov. Unlike other mathematical schools in Moscow which essentially sprang from private initiative and had no special funding, this school was a special institution affiliated with the university and specially funded by the state. Still other celebrated Moscow schools for mathematics were No. 7, No. 57 and No. 444 (the second and third authors are alumni of these schools, No. 7 and 2, respectively, and the first and the third authors taught in School No. 2). The mathematics curriculum of a special school was more in-


tensive and systematic than that of the

kruzhki, and this influenced our think

ing about the structure of the MASS pro

gram. An essential part of the tradition was

the participation of prominent mathe

maticians of various ages in teaching

and popularizing mathematics. A typical

example is the magazine Kvant (mean

ing "Quantum") on physics and mathe

matics for school students published

since 1970. Kvant had 12 issues a year

and, at the peak of its popularity in the

mid-1970s, boasted more than 300,000

subscribers. Among the authors were

well-known mathematicians A. D.

Alexandrov, V. I. Arnold, D. B. Fuchs, I. M. Gelfand, S. G. Gindikin, A. A. Kirillov,

A. N. Kolmogorov, M. G. Krem, Yu. V.

Matiyasevich, S. P. Novikov, and L. S.

Pontryagin, among many others. For

many generations of students, Kvant opened new horizons and determined

their choice of mathematics as a pro-

A UTH O R S

ANATOLE KATOK


Pennsylvania State University

University Park, PA 1 6802

USA

e·mail: [email protected]

Anatole Katok was educated in the

"Moscow mathematical school," as were

his co-authors; A.N. Kolmogorov was ref

eree of his doctoral thesis. After immigrat

ing in 1 978 to the United States (the coun

try of his birth), he taught at Maryland and

Caltech before coming in 1 990 to Penn

State. Among his numerous publications

are two books with his former student Boris

Hasselblatt: Introduction to the Modern

Theory of Dynamical Systems and the

forthcoming The Rrst Course in Dynamics

with a Panorama of Recent Developments.


fession. Along with Kvant, there was a

rich popular literature; nwnerous col

lections of problems for all ages, and

books on various topics in "serious"

mathematics. We would like to mention

some people who made a very substan

tial contribution to popularization of

mathematics: N. B. Vasiliev, N. Ya

Vilenkin, I. M. Yaglom. The third author

of this article was for a nwnber of years

the Head of Kvant's Mathematics De

partment.

At the university level, the emphasis

on creative thinking continued, some

times to the detriment of systematic

leaining. For example, the standard

mandatory courses often did not fully

reflect the most current thinking in their

subjects, and were looked down on by

the top students. A very important role

was played by topics courses, offered in

a wide variety of subjects and attended

by a mixture of undergraduate and grad

uate students. Similarly, specialized

SVETLANA KATOK




USA


Svetlana Katok (daughter of B.A. Rosen

feld, a "grand old man" of Moscow geom

etry) immigrated to the United States in

1 978 with her husband and children, and

got her Ph.D. in 1 983 at the University of

Maryland. Her research is on automorphic

forms, dynamical systems, and hyperbolic

geometry. She is author of Fuchsian

Groups and Oointly with A. Katok) of the ar

ticle "Women in Soviet Mathematics," No

tices of the American Mathematical Soci

ety 40 (1 993), 1 08-1 1 6.

seminars were usually attended by a

mix of undergraduates, graduates, and

established mathematicians. Starting

from the third year of the university,

every student had an advisor and was

considered a member of a research

community in his or her field. It was not

unusual for the best undergraduate stu

dents at major universities to have pa

pers published in first-rate research

journals by the end of their 5 years of

undergraduate studies.

This system had multiple effects. On

the one hand, it stimulated early de

velopment of research interests and

mathematical precocity. On the other

hand, it often led to inflated standards

and expectations, and eventually to a

great waste of talent. A student with

considerable talent but not very high

self-esteem might be crushed by the

system. Still, it succeeded spectacu

larly in producing creative and techni

cally powerful mathematicians.

SERGE TABACHNIKOV




USA


Serge Tabachnikov wrote his thesis (1 987)

at Moscow State University on differential

topology and homological algebra. Later his

interests shifted to symplectic geometry and

Hamiltonian dynamics, as reflected in his

book Billiards. Before coming to Penn State

he taught at the University of Arkansas.

l]¥1f9·i.(.j David E . Rowe , Editor j

Einstein's Gravitational Field Equations and the Bianchi Identities David E. Rowe

Send submissions to David E. Rowe,

Fachbereich 1 7 - Mathematik,

Johannes Gutenberg University,

055099 Mainz, Germany.

In his highly acclaimed biography of Einstein, Abraham Pais gave a fairly

detailed analysis of the many difficulties his hero had to overcome in November 1915 before he finally arrived at generally covariant equations for gravitation ([Pais], pp. 250-261 ). This story includes the famous competition between Hilbert and Einstein, an episode that has recently been revisited by several historians in the wake of newly discovered documentary evidence, first presented in [Corry, Renn, Stachel 1997].

In his earlier account, Pais emphasized that "Einstein did not know the [contracted] Bianchi identities

(RJJ-V - .!gJJ-V R) = 0 (1) 2 ;v when he wrote his work with Grossmann." (The symbol ';' denotes covariant differentiation, which here is used as the generalized divergence operator.)

In 1913 Einstein and the mathematician Marcel Grossmann presented their Entwurjfor a new general theory of relativity. Guided by hopes for a generally covariant theory, they nevertheless resolved to use a set of differential equations for the gravitational field that were covariant only with respect to a more restricted group of transformations. However, when Einstein abandoned this Entwurf theory in the fall of 1915, he once again took up the quest for generally covariant field equations. By late November he found, though in slightly different form, the famous equations:

GJJ-V = - KTJJ-V, p,, IJ = 1, . . . ' 4 (2)

where 1 GJJ-V == RJJ-V - - gJJ-V R (3) 2

is the Einstein tensor. (Here TJJ-v is the energy-momentum tensor and gJJ-v the metric tensor that determines the properties of the space-time geometry. The contravariant Ricci tensor RJJ-v is obtained by contracting the RiemannChristoffel tensor; contracting again yields the curvature scalar RJJ-vg JJ-V = R.

The symmetry of gJJ-v, RJJ-v, and TJJ-v means that (2) yields only 10 equations rather than 16.)

Applying the covariant divergence operator to both sides of the Einstein equations (2) yields, according to (1),

Gtvv = Ttvv = 0. (4)

This tells us that actually only 10 - 4 = 6 of the field equations (2) are independent, as should be the case for generally covariant equations. Ten equations for the 10 components of the metric tensor gJJ-vwould clearly over-determine the latter, since general covariance requires that a.Ry smgle solution gJJ-"(xi) of (2) corresponds to a 4-parameter family of solutions obtained simply as the gJJ-v(xi) induced by arbitrary coordinate transformations. Choosing a specific coordinate system thus singles out a unique solution among this family.

Einstein for a long time resisted drawing this seemingly obvious conclusion. Instead he concocted a thought experiment-his infamous hole argument-that purported to show how generally covariant field equations will lead to multiple solutions within one and the same coordinate system (see [NorJ989] and [Sta 1989]). His initial efforts therefore aimed to circumvent this paradox of his own making, for, on the one hand, physics demanded that generally covariant gravitational equations must exist, whereas logic (mixed with a little physics) told him that no such equations can be found (see his remarks in [Einstein 1914], p. 574). Luckily, Einstein had the ability to suppress unpleasant conceptual problems with relative ease. And so in November 1915 he plunged ahead in search of generally covariant equations, unfazed by his own arguments against their existence! Once he had them, he quickly found a way to climb out of the hole he had created (as explained in [Nor 1989] and [Sta 1989]).

By 1916 Einstein was also quite aware that his field equations led directly to the conservation laws for matter T:Vv = 0. Nevertheless, he was rather vague about the nature of this


connection in [Einstein 1916a] , his first summary account of the new theory. There he wrote, "the field equations of gravitation contain four conditions which govern the course of material phenomena. They give the equations of material phenomena completely, if the latter are capable of being characterized by four differential equations independent of one another" (ibid., p. 325). He then cited Hilbert's note [Hilbert 1915] for further details, suggesting that he

was not yet ready to make a fmal pronouncement on these issues.

Still, by early 1916 Einstein had come to realize that energy conservation can be deduced from the field equations and not the other way around. Pais remarks about this in connection with the tumultuous events of November 1915:

Einstein stiU did not know [the contracted Bianchi identities] on November 25 and therefore did not re-

alize that the energy-momentum conservation laws

Tf;," = 0 (5)

follow automatically from (1) and (2). Instead, he used these conservation laws [ (5)] as a constraint on the theory! ([Pais 1982], p. 256).

Pais's 20-20 hindsight no doubt identifies this particular source of Einstein's difficulties, but it hardly helps to explain

Mathematische Gesel lschaft 1 902.

'ula. Hansen. BIWDeuthal.

c. Miller. Dane7. SdUilinr. JWIIert.

H1111el. &bMidt.

H. lflllln. r�hiYf.

St:hwantelalhl. R�t�n. 1-,l'i:�<•l:t•r. llml"'tl'i•

Dlrl<trL 1"'"�11>. Felix Klein presiding over the Gottingen Mathematical Society in 1902. The seating arrangement reflects more than just the need to have the

tall men stand in back. Here, David Hilbert could affirm his undisputed position as Klein's "right-hand man," counterbalanced on the left by

Klein's star applied mathematician, Karl Schwarzschild. Taking up the wings in the front row were two of Gottingen's most ambitious younger

men, Max Abraham and Ernst Zermelo. Klein's attention seems to have riveted on Grace Chisholm Young, the charming Englishwoman who

took her doctorate under him in 1 895. She and her husband, the mathematician W. H. Young, in fact resided in Gottingen for several years.

Schwarzschild Nach/ass; courtesy of the Niedersiichsische Staats- und Universitiitsbibliothek Gottingen.

what happened in November 1915. Nor does it shed much light on subsequent developments. Paging ahead, we see that Pais returns to the Bianchi identities in his discussion of energy-momentum conservation, which in 1918 was one of the most hotly debated topics in general relativity theory (GRT) (see [Cattani, De Maria 1993]). There he points out that, "from a modem point of view, the identities (1) and (5) are special cases of a celebrated theorem of Emmy Noether, who herself participated in the Gottingen debates on the energy-momentum law" ([Pais 1982], p. 276). But back in November 1915 "neither Hilbert nor Einstein was aware of this royal road to the conservation laws" ([Pais 1982], p. 274). Pais might have added that even in 1918 no one in Gottingen seems to have realized the connection between Noether's results on identities derived from variational principles and the classical Bianchi identities.

A little bit of contextualization can go a long way here. During the period 1916-1918 only a few individuals were in a position to see the connection between Einstein's equations and the Bianchi identities, even though the latt�r were quite familiar to those immersed in Italian differential geometry. Among these experts, only Levi-Civita seems to have seen the relevance of the Bianchi identities immediately. But, as we shall see below, by 1918 a handful of others began to rediscover what the Italians had already largely forgotten. Emmy Noether, however, was not among them. Her work on GRT was mainly rooted in Sophus Lie's theory of differential equations, as applied to variational problems, an area Lie left untouched. Most importantly, Noether's efforts came as a response to a set of problems first raised by Hilbert, who tried to synthesize Einstein's theory of gravitation with Gustav Mie's theory of matter (see [Rowe 1999]).

Hilbert's approach to energy conservation in 1915--16 used a gener�y invariant variational principle, which (he claimed without proof) led to four differential identities linking the Lagrangian derivatives [Hilbert 1915]. No one understood this argument at the time, and only recently have historians managed to disentangle its many threads

(see [Sauer 1999] and [Renn, Stachel 1999]). Several other related issues remained murky, as well. The relationship between Einstein's theory and Hilbert's adaptation of Mie's matter theory, for example, was by no means clear. Nor was it easy to discern whether one could formulate conservation laws in general relativity that were fully analogous to those of classical physics.

Aided by Emmy Noether, in 1918 Felix Klein eventually managed to fmd a simpler way to construct Hilbert's invariant energy vector in [Klein 1918a] and [Klein 1918b]. He also urged Noether to explore Hilbert's assertion regarding the four identities that he saw as the key to energy conservation in GRT. In July 1918 she generalized and proved this result as one of two fundamental theorems on invariant variational problems [Noether 1918]. Although famous today, Noether's theorems evoked very little interest at the time they were published. Moreover, unlike Pais, no contemporary writer linked Noether's results with the Bianchi identities so far as I have been able to find.

If RJrvK is the Riemann-Christoffel tensor, then lowering the index u yields the purely covariant curvature tensor

The latter satisfies the following three properties:

RAJ.LVK = RvKAJ.L (6) RAJ.LVK = -RJ.LAVK = -RAJ.LKV

= +RJ.LAKv (7) RAJ.LVK + RAKJ.LV + RAVKJ.L = 0 (8)

These algebraic conditions imply that RAJ.LvK has only Cn = 1� n2(n2 - 1) independent components. For the space-time formalism of GRT, where n = 4, the Riemann-Christoffel tensor thus depends on c4 = 20 parameters. Using its covariant form, the classical Bianchi identities read:

RAJ.LVK;TJ + RAJ.LTJV,K + RAJ.LKTJ;V = 0, (9)

where the last three indices v, K, TJ are permuted cyclically. The connection between these identities and Einstein's equations follows immediately from two basic results of the tensor calculus: (1) raising and lowering indices commutes with covariant differentiation, and (2) Ricci's lemma, which as-

serts that the covariant derivative of the fundamental tensor gJ.Lv vanishes, g�v

= 0. Thus, by multiplying (9) by gAv and contracting, we obtain in view of (6) and (7):

RJ.LK;TJ - RJ.LTJ;K + Rj;_KTJ;V = 0. (10)

Multiplying by gJ.LK and contracting again yields

R;TJ - R�;J.L - R�;v = 0,

(R� - �B� R);v = 0,

which are the contracted Bianchi identities (1):

(RJ.LV - lgJ.LV R).v = G�';,V = 0. 2 , , These conditions simply assert that

the divergence of the Einstein tensor vanishes, a relation already derived by Weyl in 1917 using variational methods. �et neither he nor hi:; Gottingen mentors, Hilbert and Klein, recognized that these identities could be obtained directly as above using elementary tensor calculus. As I will indicate below, disentangling these differential-theoretic threads from variational principles took considerable time and effort. Even as late as 1922 the Bianchi identities and their significance for GRT were still being "rediscovered" anew.

Clearly, Pais's retrospective account skirts all the real historical difficulties, telling us more about what didn't happen between November 1915 and July 1918 than about what actually did. In the meantime, however, a number of new studies have cast fresh light on the early history of G RT (see especially the articles in Einstein Studies, vols. 1, 3, 5, 7). Moreover, the crucial period 1914-1918 has become more readily accessible through the publication of volumes 6 and 8 of the CoUected Papers of Albert Einstein ([Einstein 1996] and [Einstein 1998]; volume 7, covering his work from 1918 to 1921, is now in press). Michel Janssen's commentaries and annotations in [Einstein 1998] are particularly helpful when it comes to contextualizing the topics I address below. This new source material has helped sustain a flurry of recent research on the early history of general relativity, some of which has filled important gaps left open by earlier researchers. Still, no one since Abraham Pais has addressed the issues


surrounding the interplay between the

Bianchi identities and GRT, especially

Einstein's equations, energy-momen

tum conservation, and Noether's theo

rems. So without any pretense of do

ingjustice to this rich topic, let me take

it up once again here.

Einstein's Field Equations Roughly a half year after Einstein deliv

ered six lectures on general relativity in

Gottingen, Hilbert entered the field with

his famous note [Hilbert 1915] on the

foundations of physics, dated 20 No

vember 1915. Until quite recently, histo

rians had paid little attention to the sub

stance of this paper, which makes

horribly difficult reading. Its fame stems

from one brief passage in which Hilbert

asserted that his gravitational equations,

derived from an invariant Lagrangian,

were identical to Einstein's equations

(2), submitted to the Berlin Academy 5

days later. It was long believed that

Hilbert's "derivation" was more elegant,

and to many it appeared that he and Ein

stein had found the same equations vir

tually simultaneously. Since they had

also corresponded with one another

during November 1915, it was natural to

speculate about who might have influ

enced whom. Curiously, this interest

centered exclusively on a post hoc adju

dication of priority claims, as historians

pondered who should get credit for find

ing and/or deriving the Einstein equa

tions: Einstein, Hilbert, or both? Proba

bly Jagdish Mehra went furthest in

pushing the case for Hilbert in [Mehra

1973], but until recently the balance of

opinion was aptly sunrmarized by Pais,

who wrote, "Einstein was the sole cre

ator of the physical theory of general rel

ativity and . . . both he and Hilbert should

be credited for the discovery of the fun

damental equation" ([Pais 1982], p. 260).

Today we know better: when Hilbert

submitted his text on 20 November 1915,

it did not contain the equations (2) (see

[Rowe 2001]). In fact, Hilbert's original

text contained no explicit form for his

10 gravitational field equations, which

he derived from a variational principle

by varying the components of the met-

ric tensor g�-tv. Only later, some time af

ter 6 December, did he add the key pas

sage containing a form of (2) into the

page proofs. Presumably he did so with

out any wish to stake a priority claim,

for he cited Einstein's paper of 25 No

vember. Moreover, the explicit field

equations play no role whatsoever in the

rest of Hilbert's paper. It therefore

seems likely that he added this para

graph merely in order to make a con

nection with Einstein's results, which

were by no means clear or easily acces

sible at that time (only in his fourth and

final November note did Einstein pre

sent generally covariant field equations

with the trace term). So the equations

(2) are rightly called "Einstein's equa

tions" and not the "Einstein-Hilbert

When and why

d id physicists and

mathematicians

become

interested i n

issues l ike

proving Einstein ' s

equations?

equations." This "belated priority issue"

was definitely put to rest in [Corry, Renn,

Stachel 1997]. Unfortunately, some of

the authors' other more speculative

claims have now been spun into a highly

romanticized account of these events in

God's Equation [Acz 1999].

As to who first derived the Einstein

equations, the answer is less clear. If

by a derivation we mean an argument

showing that the equations (2) uniquely

satisfy a certain number of natural prop

erties, then for Einstein and Hilbert one

can only reach the conclusion "none of

the above" (see [Rowe 2001], pp.

416-418). Hilbert, in particular, failed to

show how the Einstein equations could

be obtained from those of his own the

ory, citing a bit of folklore about second

rank tensors that he probably got from

Einstein. One finds scattered hints in

Einstein's published and unpublished

papers indicating that the only possible

second-rank tensors obtainable from the

metric tensor and its first and second de

rivatives and linear in the latter must be

of the form:

a R�-tv + b g�-tv R + c g�-tv = 0. (11)

Einstein knew very well that this math

ematical result was crucial when it came

to narrowing down the candidates for

generally covariant field equations. And

without these mathematical underpin

nings, his claim ([Einstein 1916a], pp.

318-319) that the equations (2) repre

sent the most natural generalization of

Newton's theory would have been se

riously weakened. Still, he simply took

this for granted, probably because he

relied heavily on Grossmann's (lim

ited) expertise in the theory of differ

ential invariants.

By 1917 Felix Klein asked his assis

tant Hermann Vermeil to give a direct

proof of this fundamental result to

which both Einstein and Hilbert had ap

pealed. By employing so-called normal

coordinates, as first introduced by Rie

mann, Vermeil was able to prove that

the Riemannian curvature scalar R was

the only absolute invariant that satisfied

the above conditions (see [Vermeil

1917]). In 1921 Max von Laue completed

Vermeil's argument in [Laue 1921] , pp.

99-104, and Hermann Weyl gave an

even more direct proof of Vermeil's re

sult in [Weyl 1922], Appendix II, pp.

315--317. Wolfgang Pauli also referred to

Vermeil's work in his definitive report

[Pauli 1921] ; but one otherwise fmds

very few references to such formal is

sues in the vast literature on GRT.1

So who first proved Einstein's equa

tions? If this were a game show ques

tion, one might be tempted to answer:

Hermann Vermeil. But a more serious

response would begin by reformulating

the question: when and why did physi

cists and mathematicians become in

terested in foundational issues like

1An exception is the work of David Lovelock, who proved that the only divergence-free, contravariant second-rank tensor densities in dimension four are of the form aVgG�<" + bVgg"" in [Lovelock 1 972].

.

Making Music in Zurich. During his early struggles with general relativity, Einstein often liked

to relax in the home of his colleague Adolf Hurwitz, shown here pretending to conduct his

daughter Lisi and their physicist friend as they play a violin duet. Hurwitz was a pure math

ematician of nearly universal breadth. Though only four years Einstein's senior, he had served

as the principal mentor to both David Hilbert and Herman Minkowski during their formative

years in Konigsberg. Source: George Polya, The Polya Picture Album: Encounters of a Math

ematician, ed. G. L. Alexanderson (Boston: Birkhauser, 1987), p. 24. Reprinted with permis

sion of Birkhauser Publishers.

proving Einstein's equations? To answer this, it is again helpful to look carefully at local contexts. Among the more important centers for research on GRT were Leiden, Rome, Cambridge, and Vienna. In the case of Einstein's equations, this was largely a moppingup operation, part of a communal effort orchestrated by Felix Klein in Gottingen. Klein's initial interest in general relativity focused on the geometrical underpinnings of the theory, including the various "degrees of curvature" in space-times (Eddington's terminology in [Eddington 1920], p. 91). By early 1918, however, Klein became even more puzzled by the various results on energy conservation in GRT that had been obtained by Einstein, Hilbert, Lorentz, Weyl, and Emmy Noether. He was not alone in this regard.

General Relativity in Gottingen As Einstein himself conceded, energymomentum conservation was the one facet of his theory that caused virtually all the experts to shake their heads. Back in May 1916, he had struggled to

understand Hilbert's approach to this problem, the topic of a lecture he was preparing for the Berlin physics colloquium. Twice he wrote Hilbert asking him to explain various steps in his complicated chain of reasoning (24 and 30 May, 1916, [Einstein 1998], pp. 289-290, pp. 293-294). Einstein expressed gratitude for Hilbert's illuminating replies, but to his friend Paul Ehrenfest he remarked: "Hilbert's description doesn't appeal to me. It is unnecessarily specialized regarding 'matter,' is unnecessarily complicated, and not straightforward (=Gauss-like) in set-up (feigning the super-human through concealment of the methods)" (24 May, 1916 [Einstein 1998], p. 288). But Hilbert couldn't feign that he understood the connection between his approach to energy conservation and Einstein's. About this, he intimated to Einstein that "[m]y energy law is probably related to yours; I have already given this question to Frl. Noether" (27 May 1916, [Einstein 1998], p. 291). She apparently made some progress on this problem at the time, as Hilbert later acknowledged: "Emmy

Noether, whose help I called upon more than a year ago to clarify these types of analytical questions pertaining to my energy theorem, found at that time that the energy components I had set forthas well as those of Einstein-could be formally transposed by means of the Lagrangian differential equations . . . into expressions whose divergence vanished identicaUy . . . . " ([Klein 1918a], pp. 560-561).

By late 1917 Klein reengaged Noether in a new round of efforts to crack the problem of energy conservation (see [Rowe 1999], pp. 213-228). Klein's discomfort with energy conservation in GRT had to do with his knowledge of classical mechanics in the tradition of Jacobi and Hamilton. There, conservation laws help to describe the equations of motion. of physical systems which would otherwise be too hopelessly complicated to handle as an n-body problem. In GRT, by contrast, the conservation laws for matter (5) could be derived directly from the field equations (2) without any recourse to other physical principles. Hilbert's work pointed in this direction, but his ''purely axiomatic" presentation only obscured what was already a difficult problem. Klein later described [Hilbert 1915] as "completely disordered (evidently a product of great exertion and excitement)" (Lecture notes, 10 December 1920, Klein Nachlass XXII C, p. 18).

In early 1918 Klein succeeded in giving a simplified derivation of Hilbert's invariant energy equation, which involves. a very complicated entity ev known as Hilbert's energy vector satisfying Div (e") = 0. Klein emphasized that this relation should be understood as an identity rather than as an analogue to energy conservation in classical mechanics. He noted that in mechanics the differential equation

d(T + U) = 0 (12)

dt

cannot be derived without invoking specific physical properties, whereas in GRT the equation Div (ev) = 0 follows from variational methods, the principle of general covariance, and Hilbert's 14 field equations for gravity and matter.


Klein's article was written in the form of a letter to Hilbert. After discussing the main mathematical points, Klein remarked that "Fri. Noether continually advises me in my work and that actually it is only through her that I have delved into these matters" [Klein 1918a], p. 559. Hilbert expressed total agreement not only with Klein's derivation but with his interpretation of it as well. He even claimed one could prove a theorem that ruled out conservation laws in GRT analogous to those that hold for physical theories based on an orthogonal group of coordinate transformations. Klein replied that he would be very interested "to see the mathematical proof carried out that you alluded to in your answer." He then turned to Emmy Noether, who resolved the issue six months later in her fundamental paper [Noether 1918].

In the meantime, Einstein had taken notice of this little published exchange, and in March 1918 he wrote Klein, "With great pleasure I read your extraordinarily penetrating and elegant discussion on Hilbert's first note. Nevertheless, I regard what you remark about my formulation of the conservation laws as incorrect" (13 March, 1918, [Einstein 1998], p. 673). Einstein objected to Klein's claim that his approach to energy-momentum conservation could be derived from the same formal relationships that Klein had applied to Hilbert's theory. Instead, Einstein insisted that "exactly analogous relationships hold [in GRT] as in the non-relativistic theories." After explaining the physical import of his own formalisms, he added, "I hope that this anything but complete explanation enables you to grasp what I mean. Most of all, I hope you will alter your opinion that I had obtained for the energy theorem an identity, that is an equation that places no conditions on the quantities that appear in it" (ibid., p. 674).

Eight days later, Klein replied with a ten-step argument aimed at demolishing Einstein's objections. His main point was that Einstein's approach to energy-momentum conservation expressed nothing beyond the information deducible from the variational apparatus and the field equations that can


Tullio Levi-Civita was the leading expert on

the absolute differential calculus in Italy. To

gether with his teacher Gregorio Ricci, he co

authored an oft-cited paper on the Ricci cal

culus published in Mathematische Annalen in

1901 . In 191 5, Einstein confided to a friend

that Levi-Civita was probably the only one

who grasped his gravitational theory com

pletely: "because he is familiar with the math

ematics used. But he is seeking to tamper

with one of the most important proofs in an

incessant exchange of correspondence. Cor

responding with him is unusually interesting;

it is currently my favorite pastime" (Einstein

to H. Zangger, 10 April, 1915, Collected Pa

pers of Albert Einstein, vol. 8, pp. 1 1 7-1 18).

Their correspondence broke off, however,

about one month later when Italy entered the

war against the Axis powers.

be derived from it. Einstein countered by asserting that his version of energymomentum conservation was not a trivial consequence of the field equations. Furthermore, if one has a physical system where the energy tensors for matter and the gravitational field, T:;and t:;, vanish on the boundary, then from the differential form of Einstein's conservations laws

I acr:; + t:;) = o (13) ., ax., one could derive an integral form that was physically meaningful:

:!x4 {J (T! + t!)dV} = 0,

for u = 1, 2, 3, 4. (14)

Einstein stressed to Klein that the constancy of these four integrals with respect to time could be regarded as analogous to the conservation of energy and momentum in classical mechanics.

Klein eventually came to appreciate Einstein's views, though only after giving up on an alternative approach suggested by his colleague Carl Runge. Several experts, including Lorentz and Levi-Civita, objected to Einstein's use of the pseudo-tensor t:; to represent gravitational energy. Klein and Runge briefly explored the possibility of dispensing with this t:;, but Noether threw cold water on Runge's proposal for doing so ([Rowe 1999], pp. 217-218). By July 1918, Klein wrote Einstein that he and Runge had withdrawn their publication plans, and that he was now investigating Einstein's formulation of energy conservation based on T:; + t:;. To this, Einstein replied: "It is very good that you want to clarify the formal significance of the t:;.. For I must admit that the derivation of the energy theorem for field and matter together appears unsatisfying from the mathematical standpoint, so that one cannot characterize the t:; formally" ([Einstein 1998], p. 834).

Einstein and Klein quickly got over their initial differences regarding the status of Einstein's (13), and afterward Klein dealt with this topic and the various approaches to energy conservation adopted by Einstein, Hilbert, and Lorentz in [Klein 1918b]. Einstein responded with enthusiasm: "I have already studied your paper most thoroughly and with true amazement. You have clarified this difficult matter fully. Everything is wonderfully transparent" (A. Einstein to F. Klein, 22 October, 1918 ([Einstein 1998], p. 917). The contrast between this response and Einstein's reaction to Hilbert's work on GRT (noted above) could hardly have been starker. Perhaps the supreme irony in this whole story lies here. For [Klein 1918b] is nothing less than a carefully crafted axiomatic argument, set forth by a strong critic of modem axiomatics largely in order to rectify

the flaws in Hilbert's attempt to wed GRT to Mie's theory of matter via the axiomatic method.

In this paper Klein developed ideas that were closely linked with [Noether 1918] , though he mentions this.parallel work only in the concluding paragraph. Emmy Noether, on the other hand, gave several explicit references to [Klein 1918b] that make these interconnections very clear. In his private lecture notes, Klein later wrote that it was only "through the collaboration of Fri. Noether and me" that [Hilbert 1915] "was completely decoded" (Lecture notes, 10 December 1920, Klein Nachlass XXII C, p. 18). Today, this jointly undertaken work would normally appear under the names of both authors, but back in 1918 Emmy Noether wasn't even allowed to habilitate in Gottingen, despite the backing of both Hilbert and Klein.

On Rediscovering the Bianchi Identities All of these events, it must not be forgotten, took place against the backdrop of the Great War that nearly brought European civilization to its knees. Einstein's revolutionary theory of gravitation interested almost no one prior to 1916, and before November 1919 only a handful of experts had written about it. But afterward, Einstein and relativity emerged as two watchwords for modernity. On 6 November, just before the Versailles Treaty was to take effect, the British scientific world announced that Einstein's prediction regarding the bending of light in the sun's gravitational field had been confirmed. Thereafter, the creator of general relativity was no longer merely a famous physicist: he emerged as one of the era's leading cultural icons. But let's now wind back the reel and look again at GRT during the Great War.

Once Einstein's mature theory came out in 1916-alongside Hilbert's paper and the pioneering work of Karl Schwarzschild containing the firSt exact solutions of the Einstein equationsmany mathematicians and physicists began to take up GRT and the Ricci calculus. Doing so in wartime, however, presented real difficulties. Communi-

cation between leading protagonists in Italy and Germany proved next to impossible, as the lapse in Einstein's correspondence with Tullio Levi-Civita demonstrated. Through his friend Adolf Hurwitz, whom he visited in Zurich in August 1917, Einstein managed to get his hands on Levi-Civita's paper [LeviCivita 1917b], which briefly reignited their earlier correspondence. Like many others, Levi-Civita found Einstein's for-

mulation of energy conservation unacceptable due to his use of the pseudotensor t� for gravitational energy [Cattani, De Maria 1993]. What Einstein (and presumably everyone else in Germany) overlooked was that in this paper LeviCivita employed the classical Bianchi identities. In [Levi-Civita 1917a] he introduced an even more fundamental concept: parallel displacement of vectors in Riemannian spaces, a notion

Dirk Jan Struik, ca. 1920, when he was working as an assistant to Jan Amoldus Schouten in

Delft. Earlier Struik had studied in Leiden with Paul Ehrenfest, a close personal friend of Ein

stein's who therefore realized the importance of Ricci's calculus for general relativity at an

early stage. At a crucial stage, Ehrenfest arranged a meeting for Struik with Schouten, Hol

land's leading differential geometer. Thus began a collaboration that led to several books and

articles during the 1920s and 30s.


quickly taken up by Hermann Weyl and Gerhard Hessenberg.

These fast-breaking mathematical developments raised staggering difficulties, and not just for physicists like Einstein. None of the mathematicians in Gottingen was a bona fide expert in

differential geometry, which helps explain why no one in the Gottingen crowd recognized the central importance of the Bianchi identities. Had Klein suspected that the Einstein tensor satisfied four simple differential identities (corresponding to the four

:From the PHII.OSOPHICAT. MAGAZrNF., vol. ::dvii. 1lllarcl� 1924.

Note on J.l[r. Harwa1·d's Paper on tlte Identical Rel(�tions in Einstein's Theory.

Tv the Edito·rs of tfte Philosophical Maga::ine .

GE�TLE¥E�,-

IN a paper 011 " The Identical Relations in Einstein's Theory '' iu the August 1922 number (pp. aS0-382) o f

the Philosophical Magazine, Mr . Harward proves a general theorem which he discovered for himself, although be did riot believe it to be undiscovered before. Tht: theorem is :

(B�ve1)r + (B�ar)v + (B�,v)e1= 0. It may be of interest to mention that this theorem is kno wn,

especially in Germany and Italy, as " Bianchi's Iden tity/' having been p ublished by L. Bianchi in the Rendic:ont-i Ace. Lincei, xi. (5) pp. 3-17 (1902) . In this paper, Bianchi already deduced from this theorem the identity Mr. Harward refers to :

Gv - to G }J.V - o·�>l-' ·

The identity of Bianchi seems, ho·wever, to be :published for the first time by E. Padova (Rendiconti .Ace. Lincei, v. ( 4) pp. 174-178, 1889) , who obtained it from G. Ricci. Compare for this the book of Struik ( Grundzuge der mehrdimensionalen Dijfe1·entialgeometrie : Berlin, ,J. Springer, 1!j22, p. 141:l) , where the theorem is proved in a similar way as Mr. Harward does. A similar proof was already given by Schouten (Mathern. Zeitschr. xi. PP: 58-88, 1921).

The theorem can be generalized by taking geometries with a more general parallel displacement than in ordinary Riemann geometry. We then obtain the generalized geometries of Schouten (.1.1-latltem. Zeitschr. xiii. pp. 56-81 , 1922), of which the geometries of Weyl aud Eddington (if. Eddington's ' Mathematic�\! Theory of Relativity,' Oh. vii.) are special cases. Schouten, in a recent pape-r (Mathem. Zeitscltr. xvii. pp. 111-115, 1923), proved the generalization of Bianchi's Identity for a geometry of which a geometry with a symmetrical displacement (that is, a geometry in which r� .. = r�., cf. Eddington, Zoe. cit. p. 214) is a special case. This �pecial case is treated by A. Veblen (Proc. Nat. Ac. of Sciences, July 1 922). R. .Bach already gave the generalization for the geometry o£ vVeyl (:iYiathem. Zeitsch1·. ix. pp. 110-135, 1921 ).

A simple proof of Bianchi's Identity, which holds for a.

Struik's copy of the open letter he and Schouten wrote to the Philosophical Magazine on 28 April, 1923. Their account clarified several historical issues involving the Bianchi identities. It

also contained a simple proof of these identities for spaces with a symmetrical connection

suggested by the Prague mathematician, Ludwig Berwald. Struik presumably knew Berwald

through his wife, Ruth, who studied mathematics in Prague during happier days. In 1941,

Berwald and his wife were transported to the ghetto in Lodz, where they died from mal

nourishment.


parameters in a generally covariant system of equations), he might have turned to his old friend Aurel Voss for advice. Had he asked him, Voss likely would have remembered that he had published a version of the contracted Bianchi identities back in [Voss 1880] ! Thus, the Gottingen mathematicians clearly could have found references to the Bianchi identities, in either their general or contracted form, in the mathematical literature. They just didn't know where to look. As Pais pointed out, the name Bianchi does not appear in any of the five editions of Weyl's Raum, Zeit, Materie, nor did Wolfgang Pauli refer to it in his Encyclopii.die article [Pauli 1921] .

With regard to the Bianchi identities in their full form (9), we have it on the authority of Levi-Civita that these were known to his teacher, Gregorio Ricci ([Levi-Civita 1926], p. 182). Ricci passed this information on to Emesto Padova, who published the identities without proof in [Padova 1889]. They were thereafter forgotten, even by Ricci, and then rediscovered by Luigi Bianchi, who published them in [Bianchi 1902]. Both Ricci and Bianchi had earlier studied under Felix Klein, who solicited the · now-famous paper [Ricci, Levi-Civita 1901] for Mathematische Annalen; but this classic apparently made little immediate impact. Indeed, before the work of Einstein and Grossmann, Ricci's absolute differential calculus was barely known outside Italy [Reich 1992].

By 1918 a number of investigators outside Italy had begun to stumble upon various forms of the full or contracted Bianchi identities. Two of them, Rudolf Forster and Friedrich Kottler, even passed their findings on to Einstein in letters (see [Einstein 1998], pp. 646, 704). Forster, who published under the pseudonym Rudolf Bach, took his doctorate under Hilbert in Gottingen in 1908 and later worked as a technical assistant for the Krupp works in Essen. In explaining to Einstein that the identities (1) follow directly from (9), he noted that the latter "relations appear to be still completely unknown" (ibid.). Forster contemplated publishing these results, but did so only in [Bach 1921], which dealt with Weyl's generalization of Riemann-

ian geometry. Even at this late date he presented these identities as "new" (ibid., p. 114).

During the war years, the Dutch astronomer Willem De Sitter introduced the British scientific community to Einstein's mature theory. This helped spark Arthur Stanley Eddington's interest in GRT and the publication of [Eddington 1918], which contains the contracted Bianchi identities. Two years later, in [Eddington 1920], he expressed doubt that anyone had ever verified these identities by straightforward calculation, and so he went ahead and carried this out himself for the theoretical supplement in the French edition of [Eddington 1920].

In 1922 Eddington's calculations were simplified by G. B. Jeffery, and almost immediately afterward the English physicist A. E. Harward reproved Bianchi's identities (9) and used them to derive the conservation of energymomentum in Einstein's theory in [Harward 1922]. He also cof\iectured that he was probably not the first to have discovered (9). In [Schouten and Struik 1924], an open letter to the Philosophical Magazine, dated 28 April 1923, J. A. Schouten and Dirk siruik confirmed Harward's co{\jecture, noting that (9) "is known, especially in Germany and Italy, as 'Bianchi's Identity.' " More importantly, they emphasized that similar identities hold in affme spaces (those that do not admit a Riemannian line element ds2 = gJJ-vdxJJ-dxv). Regarding these, they referred explicitly to [Bach 1921 ] for Weyl's gauge spaces as well as a 1923 paper by Schouten for non-Riemannian spaces with a symmetric connection r�v = r�w These results and many more appeared soon afterward in Schouten's 1924 textbook Der RicciKalkiil. By this time, of course, the dust had largely cleared, as a number of leading experts-including Schouten and Struik, Veblen, Weitzenbt:ick, and Berwald-had by now shown the importance of Bianchi-like identities in non-Riemannian geometries.

What should be made of all this groping in the dark? No doubt a certain degree of confusion arose due to the importance Hilbert and' others attached to variational principles in mathemati-

cal physics. Not surprisingly, within Gt:ittingen circles there was considerable expertise in the use of sophisticated variational methods. Emmy Noether coupled these with invariant theory to obtain her impressive results. But she and her mentors had relatively little familiarity with Italian differential geometry. Ironically, this widespread lack of fundamental knowledge of tensor analysis had at least one important payoff. It gave the aged Felix Klein an inducement to explore the mathematical foundations of general relativity theory. He did so by drawing on ideas familiar from his youth, most importantly Sophus Lie's work on the con-

Mathematicians

often prefer to

figure out some

th ing on thei r

own rather than

read someone

else 's work.

nection between continuous groups and systems of differential equations. Moreover, his efforts helped clarify one of the most baffling and controversial aspects of Einstein's theory: energymomentum conservation. Even Hilbert's muddled derivation of an invariant energy vector found its proper place in the scheme set forth in [Klein 1918b]. Through Klein, Emmy N oether became deeply immersed in these complicated problems, and she succeeded in extracting from them two fundamental theorems in the calculus of variations that would later provide field physicists with an important tool for the derivation of conservation laws.

Mathematicians often prefer to figure out something on their own rather than read someone else's work, so we need not be surprised that the classical Bianchi identities escaped the notice of such eminent mathematicians as Hilbert, Klein, Weyl, Noether, and of course Einstein himself. Had they known

them, the early history of the general theory of relativity would not have have looked quite the same.

Acknowledgments The author is grateful to Michel Janssen and Tilman Sauer for their perceptive remarks on an earlier version of this column. The editor deserves a note of thanks, too, for posing questions that helped clarify some obscure points. Remaining errors and misjudgments are, of course, my own, and may even reflect a failure to heed wise counsel.

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[Levi-Civita 1 926] Tullio Levi-Civita, The Ab

solute Differential Calculus, trans. Marjorie

Long, London: Blackie & Son, 1 926.

[Lovelock 1 972] David Lovelock, "The Four-Di

mensionality of Space and the Einstein Ten

sor," Journal of Mathematical Physics 1 3

(1 972), 8 7 4-876.

[Mehra 1 973] Jagdish Mehra, "Einstein, Hilbert,

and the Theory of Gravitation," in The Physi

cist's Conception of Nature, Jagdish Mehra,

ed. Dordrecht: Reidel, 1 973, pp. 92-1 78.

[Noether 1 91 8] Emmy Noether, "Invariants Vari

ationsprobleme," Nachrichten der K6niglichen

Gesellschaft der Wissenschaften zu G6ttingen,

Mathematisch-Physikalische Klasse, 1 91 8,

235-257.

[Nor 1 989] John Norton, "How Einstein Found his

Field Equations," in [Ho-St 1989] , pp. 1 01 -1 59.

[Padova 1 889] Ernesto Padova, "Sulle defor

mazioni infinitesime," Rendiconti della Reale

Accademia dei Uncei (IV), vol. 5(1), 1 889, pp.

1 74-178.

[Pais 1 982] Abraham Pais, 'Subtle is the Lord

. . . ' The Science and the Life of Albert Ein

stein, Oxford: Clarendon Press, 1 982.

[Pauli, 1 92 1 ] Wolfgang Pauli, "Relativitatstheo

rie," in Encyklopadie der mathematischen

Wissenschaften, vol. 5 , part 2 (1 921 ), pp.

539-775; Theory of Relativity, G. Field, trans.

London: Pergamon , 1 958.

[Reich 1 992] Karin Reich, Die Entwicklung des

Tensorkalkuls. Vom absoluten Differentialkal-

kul zur Relativitatstheorie (Science Networks,

vol. 1 1 ), Basel: Birkhauser, 1 992.

[Renn, Stachel 1 999] JOrgen Renn and John

Stachel, "Hilbert's Foundation of Physics:

From a Theory of Everything to a Constituent

of General Relativity, " Max-Pianck-lnstitut fOr

Wissenschaftsgeschichte, Preprint 1 1 8, 1 999.

[Ricci, Levi-Civita 1 901 ] Gregorio Ricci and Tul

lio Levi-Civita, "Methodes de calcul differen

tiel absolu et leurs applications," Mathema

tische Annalen 54 (1 901) , 1 25-201 .

[Rowe 1 999] David Rowe, "The Gottingen Re

sponse to General Relativity and Emmy Noe

ther's Theorems," The Symbolic Universe.

Geometry and Physics, 1890-1930, Jeremy

Gray, ed. (Oxford: Oxford University Press),

1 999, pp. 1 89-233.

[Rowe 2001 ] David Rowe, "Einstein meets

Hilbert: At the Crossroads of Physics and

Mathematics," Physics in Perspective 3 (2001) ,

379-424.

[Sauer 1 999] Tilman Sauer, "The Relativity of

Discovery: Hilbert's First Note on the Foun

dations of Physics," Archives for History of

Exact Sciences 53 (1 999), 529-575.

[Schouten and Struik 1 924] J. A Schouten and

Dirk Struik, "Note on Mr. Harward's Paper on

the Identical Relations in Einstein's Theory,"

Philosophical Magazine 47 (1 924), 584-585.

[Sta 1 989] John Stachel, "Einstein's Search for

General Covariance, 1 9 1 2-1 915 , " in [Ho-St

1 989], pp. 63-1 00.

[Vermeil 1 9 1 7] Hermann Vermeil, "Notiz ber das

mittlere Krummungsmass einer n-fach aus

gedehnten Riemann'schen Mannigfaltigkeit, "

K6nigliche Gesellschaft der Wissenschaften

zu G6ttingen. Mathematisch-physikalische

Klasse. Nachrichten 1 9 1 7 , 334-344.

[Voss 1 880] Aurel Voss, "Zur Theorie der Trans

forrnation quadratischer DifferentialausdrOcke

und der Krurnmung hoherer Mannigfaltigke

tien," Mathematische Annalen 16 (1 880),

1 29-1 78.

[Weyl 1 922] Hermann Weyl , Space-Time-Mat

ter, 4th ed. , trans. Henry L. Brose, London:

Methuen, 1 922.

GIORGIO GOLDONI

A Visual Proof for the Sum of the Fi rst n Squares and for the Sum of the Fi rst n Factorials of Order Two

It is well lrnown that the sum of the first n numbers may be

seen as the area of a stairs-shaped polygon, and that two of

these polygons may be arranged in a rectangle n X (n + 1):

This gives a visual proof for the identity

_ n(n + 1) 1 + 2 + 3 + . . . + n -2

.

This identity can be generalized in at least two ways:

• the sum of the first n squares:

• the sum of the first n factorials of order two:

(2) 1 . 2 + 2 . 3 + 3 . 4 + . . .

+ n(n + 1) = n(n + 1)(n + 2) 3

Here is a visual proof of these identities, representing the

sums as volumes of certain solids.

The sum of the first n squares 12 + 22 + 32 + . . . + n2 may be seen as pyramid-shaped stairs:

We can arrange six of these pyramids into a parallelepiped:

Step 1 :

© 2002 SPRINGER· VERLAG NEW YORK, VOLUME 24, NUMBER 4 , 2002 67

Step 2:

Step 3:

Step 4:

Step 5:


We have obtained a parallelepiped n X ( n + 1) x (2n + 1) and this immediately yields (1 ).

For the sum of the first n factorials we need three pyramids a little different from the ones used in the previous case, but not all three the same. One of them must be the mirror image of the other two-.

Now we can arrange the pyramids in a parallelepiped: Step 1

Step 2

Step 3

We have obtained a parallelepiped n x (n + 1) X (n + 2), and formula (2) has been proven too.

A U T H O R

GIORGIO GOLDONI

Civico Planetaria "F. Martino"

Viale J. Barozzi, 31

411 00 Modena

Italy


Giorgio Goldoni, after training in engineering and mathemat

ics, has been for twenty years a high-school mathematics

teacher and a member of the Centro Sperimentale per Ia Di

dattica deii'Astronomia in Modena. He lives in the nearby town

of Rolo with his wife and two children. He thanks his high

school students for stimulating him to find new simple ways

to see things he thought he already knew well.


Some Hints on Problems 1-14 (presented in Mathematical Entertainments, "Hat Tricks," p. 47)

1. The only solutions with winning probability 3/4 correspond to antipodal points on the 3-cube, no matter what the labeling of the cube. 2. If the Hamming code is given as the

kernel of the map T:Fr � V described in "Hat Tricks," then T is surjective and its kernel has dimension n - k = 2k -

1 - k which has density 2n-k;2n =

1f2k = ll(n + 1). 3. The density of the code L coming

from the dumb strategy is 1/2k, and the desired inequality is equivalent to 1/(n + 1) ::::; 1f2k < 2/(n + 1). l'lii!id!lllil!;l+l--lr--------------------------4. Check that the 7 marked nodes in

Figure 4 are a covering code. (The 5-cube is given as two 4-cubes; corresponding vertices in the two halves must be joined by edges, which are left out of the figure for the sake of clarity.)

5. If l[vi] is in the kernel of T then the sum of the vi is 0 in V = F�.

6. Nothing, if the host listens in on all communication in the strategy session. However, if the team can surreptitiously generate random numbers during the strategy session, outside of earshot of the host, then they can establish a mapping between colors and 0/1 (for each player), which can be used during the game to defeat any nonrandomness that the host introduces.

1§1311;11+

70 THE MATHEMATICAL INTELLIGENCEA

7. From any player's point of view the probability of winning using a random strategy is a linear function of the availability parameters, which are subject to linear constraints. The optimum is attained at a vertex of the corresponding convex polyhedron, i.e., at a deterministic strategy.

8. Let L denote the set of six marked black points in Figure 5. Gray paths from two of the points in L end in six gray points, and the other points are all at distance 1 from L (taking into account the understood edges connecting the two halves). A strategy is then

given by orienting all edges "away from L" on the gray paths or paths of length 1 , and decomposing the rest of the graph into cycles which can be oriented arbitrarily. Thus L is the set of losing points for this strategy for the majority hats game for 5 people.

9. Berlekamp finds a "code" that is as small as possible, i.e. , with 1 + [512110] = 52 points. 10. The team can guarantee that [n/2) players would survive. One could think of couples pairing off and guessing so that one (and only one) of them would survive, or of two halves (sub-teams) wagering on opposite parities of, say, the number of black hats. 1 1. The player in back announces the parity of the hats that he sees. 12. The player in back announces the sum modulo q. 13. Players order themselves, and vote only if all prior hats are white, in which case they vote black with a large enough number of votes to swamp all earlier votes for black. 14. The only thing that can be done is to guess with probability p. The optimal probability is asymptotic to log(4)/n, and the team wins with probability approaching 114.

I il§i) t§i.llJ Jet Wi m p , Editor I

Feel like writing a review for The

Mathematical Intelligencer? You are

welcome to submit an unsolicited

review of a book of your choice; or, if

you would welcome being assigned

a book to review, please write us, telling us your expertise and your

predilections.

Column Editor's address: Department

of Mathematics, Drexel University,

Philadelphia, PA 1 91 04 USA.

Geometry Civi l ised: H istory, Culture and Technique by J. L. Heilbron

OXFORD, CLARENDON PRESS 1 998, 309 pp. U.S. $35, ISBN 01 9-850-6902

REVIEWED BY MICHAEL LONGUET-HIGGINS

The Elements, Euclid's famous treatise on geometry, has been hailed

as a cornerstone of Western culture, indeed as a fme proof of idealistic philosophy. Nevertheless, when used as a textbook it has been the bane of gen

erations of high-school students, who mostly fail to see the point of Euclid's pedantic proofs of the seemingly obvious. Numerous attempts have been made to improve upon the Elements, either by introducing more natural, though less rigorous, demonstrations or by interspersing the text with important applications to physics and engineering and to familiar problems in everyday life. Professor Heilbron's richly illustrated volume is perhaps the most attractive attempt at improvement yet. But it is more than that. In a long introductory chapter (headed "An Old Story"), the author provides a wide-ranging and readable survey of

the place of geometry in Western culture, with fascinating excursions into the traditional geometry of Indian, Chinese, Egyptian, and Babylonian civilisations.

In both Egypt and China, accurate land measurement became a necessity for the equitable assessment of land taxes, especially following the yearly flooding of river valleys. Practical prescriptions for calculating areas were not always accurate. For example, an Egyptian rule for finding the area of a given triangle was to multiply the lengths of two adjacent sides of a triangle and divide by 2-not mentioning that the included angle must be a right angle. It was the Greeks, surely motivated by a desire to eliminate such errors, who intro-

duced the now familiar method of systematically deducing successive theo

rems from initial, irrefutable axioms. How much pleasure and surprise have since been given by the beautiful and unexpected results contained in Greek texts such as Euclid's Elements or the Conics of Apollonius! Nevertheless, today's high-school students "should not become impatient if they do not immediately understand the point of geometrical argument. Whole civilisations have done the same."

During the Dark Ages the West forgot Greek and soon lost all but a few scraps-of the Elements. Roman textbooks of agrimensura (land measurement) transmitted a certain amount of serviceable information from classical times to mediaeval Europe. But "fortunately the Arabs took a strong interest

During the

fifteenth century

the Greek texts ,

preserved in

Byzant ium ,

came West . in geometry and preserved Euclid. When, during the twelfth century European scholars began to make useful contacts with their better-educated Islamic counterparts, the Elements stood ready for study, in Arabic. A few Westerners, ambitious for learning, mastered the tongue of Islam and translated Euclid into Latin." Later during the fifteenth century the original Greek texts, preserved in Byzantium, came West at the beginning of the Renaissance. After the invention of printing the Elements was one of the fust books to be published, initially in Latin and then, in 1570, in English.

Euclid came to be a part of school and college education in the West, par-


ticularly in England, where it was pro

moted by the Cambridge Platonists, and

subsequently also in America Ironically,

with the advent of universal education a

reaction set in. Some reasons why geom

etry puts off the average student were

neatly summarised three centuries ago

(in 1701) by an anonymous writer:

"The aversion of the greater part of

Mankind to serious attention and close

arguing; Their not comprehending the

necessity or great usefulness of these in

other parts of Learning; an Opinion that

this study requires a particular Genius

and turn of head, which few are so

happy as to be Born with: and the want

of . . . able Masters." Sound familiar?

The long series of attempts to im

prove on the Elements as a textbook

began as early as 1794 with the French

mathematician A. M. Legendre, who

mixed geometry with algebra and

trigonometry, introduced practical ex

amples, and omitted proofs of the

obvious. But British and American

schools still clung to Euclid as an in

tellectual discipline. At Oxford Univer

sity, a distinguished mathematics pro

fessor L. C. Dodgson (who as Lewis

Carroll wrote Alice in Wonderland) made a study of a dozen modem rivals

to the Elements and declared, in 1883,

"Euclid's treatise is, at present, not

only unequaled, but unapproached."

After World War II, despite the avail

ability of some excellent texts (such as

N. Altschiller-Court's CoUege Geometry, Barnes and Noble 1925, 1952) the

proportion of American students tak

ing geometry declined dramatically.

Chapter 1 of Geometry Civilised in

cludes the story of how deductive

geometry has been virtually banished

from the American curriculum. In Eu

rope and Canada the situation is simi

lar, though the subject is still alive.

Professor Heilbron gives as his rea

son for writing this book " . . . to con

tinue to reap and in a measure to

repay, the pleasure that studying geom

etry has given me, . . . delight in finding

a clear, tidy proof, in seeing a powerful

application of simple principles, in per

ceiving unexpected spacial relation

ships in buildings, in patterns on

ceramics and textiles, in highway inter

changes . . . in advancing the Elements


to higher geometries, spherical astron

omy and geodesy. A further satisfaction

is gaining confidence in systematic rea

soning . . . . Another source of pleasure

is the integration of the pictorial and the

verbal . . . . Finally, pursuing geometry

opens the mind to relationships among

learning, its applications, and the soci

eties that support them."

How well has the author suc

ceeded? About as well as possible,

given the difficulty of the task As Eu

clid is said to have answered to

Ptolemy I of Egypt when asked if there

was a shorter way to mastering geom

etry than working through the Elements, "There is no royal road to geom

etry." Nevertheless the path is here

eased by numerous illustrations, cho

sen for artistic merit or historical in

terest; by applications of geometrical

theorems to everyday life; by compar

ison of Euclidean proofs with different

methods from other cultures; by alter

native derivations using algebra or tri

onometry; and by posing further prob

lems to intrigue and test the reader. A

number of these are drawn from a lit

tle-lmown publication, the Ladies ' Diary, which prospered in England be

tween 1704 and 1840. Women took part

in both setting and solving the prob

lems. One editor wrote in 1718 that his

women correspondents had "as clear

Judgements, as sprightly quick Wit, as

penetrating Genius, and as discerning

and sagacious Faculties as men's."

An interesting section of Chapter 1

discusses the role of women in geome

try. In Chapter 5 it is recounted how in

1643 Rene Descartes corresponded

with Princess Elizabeth of Bohemia on

the problem of drawing a circle tangent

to three others, and how she astonished

him with a complete solution. Another

feminine tour de force might have been

mentioned: Alicia Boole Stott, third of

the five daughters of the mathematician

George Boole, from 1878 onwards de

veloped her own method of construct

ing the 3-dimensional cross-sections of

the regular polytopes in 4 dimensions

(see H. S.M. Coxeter, Regular Polytopes, pp. 258-259).

The book, after Chapter 1, follows

roughly the order of Books I to IV of

Euclid's textbook, except that circles

are introduced after areas, and then on

to Book VI (Book V contains Euclid's

theory of incommensurables ). A sepa

rate chapter is devoted to Pythagoras's

theorem, which was lmown to both In

dian and Chinese mathematicians. Lit

tle in the book is unrelated in some way

to the material in the Elements. Since

a classic work is often justified by its

later developments, some mention of

the elegant theorems of projective

geometry such as Desargues's theorem

on triangles in perspective would have

been welcome. However, a theorem of

Pascal's, essentially one of projective

geometry, is offered as an exercise in

Chapter 3.

In view of the author's stated ob

jectives, it is surprising to find no men

tion of Japanese temple geometry,

which flourished especially during the

18th and 19th centuries (see Japanese Temple Geometry Problems, by H.

Fukagawa and D. Pedoe, Winnipeg,

1989). Some very remarkable theorems

were discovered and illustrated (usu

ally without proof) on wooden or stone

tablets suspended in Japanese temples.

This at a time when Japan was quite

isolated from the rest of the world.

The book ends with an account of

the Tantalus Problem, a geometrical

teaser involving the isosceles triangle

having angles of 20°, 80° and 80°. When

it was published by the Washington Post in 1995, the man who set the puz

zle, on being challenged for a solution,

said he had forgotten how to do it and

could not repeat his lost performance.

"I contacted 40 geniuses around the na

tion and they all gave me insights into

the problem without being able to

solve it," he said. With these insights

and a weekend's labour he managed a

solution, which involves a clever but

non-intuitive geometrical construction.

The problem indeed deserves its name.

In such a broadly conceived and

splendidly produced volume, it might

seem ungrateful, though necessary, to

point out a few errors. Figure 1.1. 7 is clearly meant to illustrate a general sca

lene triangle, but it is drawn as equilat

eral. In Figure 1. 1.8 the lines AB and CD

are not parallel diameters as stated in

the caption, though the theorem is true

even if they are only parallel chords. On

p. 258 a Freudian slip, perhaps, on the part of the Oxford University Press, ends a proof with "O.E.D." On p. 260, should not para 5.5.21 start, "seven men bought equal shares"? Near the end of the proof of the Tantalus Problem on p. 294, b2/2a should be b2/a. 1n the problems APS 5.2.12 on p. 215, the factor 2 should be on the left of the equation, not on the right. And so on.

None of this detracts significantly from the author's main achievement. Not only will the book be enjoyed by mathematical specialists interested in broadening their knowledge of other cultures, but it may serve to draw mathematically untrained readers into the pleasures of a subject, deductive geometry, so sadly expelled from the curriculum in some countries.

Institute for Nonlinear Science

University of California, San Diego

La Jolla, CA 92093-0402 USA

e-rnail: [email protected]

My Numbers, My Friends: Popular Lectures on Number Theory by Paulo Ribenboim

NEW YORK: SPRINGER-VERLAG, 2000

US $39.95, 375 pp, paperback ISBN: 03-8798-91 1 0

REVIEWED BY DAVID BRESSOUD

Number Theory is endlessly fascinating. No other field of mathe

matics can match it for its range of problems and the variety of its techniques. It has problems that can be explained to a child still struggling with the rudiments of arithmetic and problems that can be comprehended only after years of directed post-doctoral study. There is much to learn and explore at every level between these extremes. Clever amateurs can still-make significant contributions, but the answers to simple-sounding questions can require results from the very forefront of mathematical research. Ribenboim's book reflects that spread.

Paulo Ribenboim is enamored of Number Theory, a fact that shines through this flawed but exuberant book The title may suggest otherwise, but what really excites him is the theory that enables us to explore and say interesting things about numbers. Certain areas hold a particular fascination, and he returns to them repeatedly: primality, Fibonacci sequences and the more general Lucas sequences, Diophantine analysis, class numbers, irrationality, and transcendence.

I particularly enjoyed his second essay, "Representation of Real Numbers by Means of Fibonacci Numbers." I learned Kakeya's result [ 1 ] that if (si) is a monotonically decreasing sequence that approaches 0 and if I�= 1 si diverges, then every positive real number can be written as a sum of some subsequence of the si. Ribenboim uses this as a leadin to Landau's theorem [2] that explicitly evaluates I:=t 1/F2n CFm is the mth Fibonacci number) in terms of the Lambert series L(x) = I;'=1 xn/(1 -xn), and I;'=1 1/F2n-l in terms of thetafunctions. The proofs are almost selfcontained. The only result that he needs to quote is Jacobi's sum of two squares formula: Given any positive integer m, the number of pairs of integers (s,t) for which s2 + t2 = m is equal to four times the difference between the sum of the divisors of m that are congruent to 1 modulo 4 and the sum of the divisors of m that are congruent to 3 modulo 4. The essay concludes, as most of them do, with references to some of the many related questions.

The omission of a proof of Jacobi's sum of two squares formula-a result that is not hard to prove given the audience that this book will draw-is symptomatic of a serious problem with this book: It is a collection of random essays with no attempt to fmd a consistent voice or level of detail, no concern to fill in significant omissions or avoid significant repetitions. Wieferich's proof that the first case of Fermat's Last Theorem is true for prime p when 2P-1 �1 (modp2) is mentioned on page 192, again on page 220, and yet again on page 237, occurring in three consecutive essays.

His essays range from the trivial-

light entertainment for an after-dinner talk-to some fairly sophisticated mathematics. "Selling primes" is so lightweight it is almost embarrassing. The author assumes the reader knows nothing about the distribution, or even the infinitude of the primes. 1n "What kind of a number is v2v2?", he considers it necessary to define complex and algebraic numbers with considerable care. "Gauss and the class number problem" is an extended and thorough essay on the subject. It assumes the reader is familiar with characters and quadratic reciprocity. "Powerless facing powers" looks at powerful numbers-an integer n is kpowerful if prime p divides n implies that pk divides n-and perfect powers. It is a romp through a wide assortment of theorems and conjectures, including a detailed section on the ABC co�ecture (also stated in two other ess�ys ). One of the few proofs in this essay assumes familiarity with p-adic arithmetic.

More than anything else, this is a profuse collection of interesting results in Number Theory, which is why repetitions and omissions are so frustrating. One keeps wishing that instead of just collecting his essays, he had mined them to put together thematic exhibits. As an example, eight of the eleven essays use or make reference to quadratic extensions. Quadratic extensions are defined in two of the essays, but not in the first essay in which they are encountered. 1n the first essay of the book, "The Fibonacci numbers and the Arctic Ocean," Ribenboim describes Lucas sequences, a discussion that could benefit from the language of quadratic extensions. Here there is no mention of them.

To cap off one's frustration with this book, most page numbers listed in the index are off by one.

Many of the essays, on their own, are excellent. "What kind of a number is V2v'2?" is one of Ribenboim's best, ranging through continued fractions, measures of irrationality, and proofs of transcendence, pointing out the wellknown as well as many obscure but interesting results. But as a collection, this book is disappointing.

REFERENCES

[1 ] S. Kakeya. 1 941 . On the partial surn of an


infinite series. Science Reports Tohoku Imp.

Univ. ( 1 ) 3: 1 59-1 63.

[2] E. Landau. 1 899. Sur Ia serie des inverses

de nombres de Fibonacci. Bull. Soc. Math.

France 27: 298-300.

Mathematics and Computer Science

Department

Macalester College

1 600 Grand Avenue

St. Paul, MN 551 05-1 899 USA


Cinderella: The ·Interactive Geometry Software by J. Richter-Gebert and

U. H. Kortenkamp

1 999 NEW YORK: SPRINGER-VERLAG

US $59.95. ISBN 35-401 4-7195

REVIEWED BY GILL BAREQUET

I happily took on myself the task of reviewing this software package be

cause I like to play with such geometric toys, and also because from time to time I have to draw geometric figures for my papers. This is what Cinderella is about. It enables you to draw dynamic geometric constructions, view them through several types of lenses, create animations, capture a scene in a Postscript file, and export your creation into HTML.

Some readers may be familiar with The Geometer's Sketchpad (manual available from Key Kurriculum Press). Cinderella is similar in some aspects, superior in some, and inferior in others. The first thing that attracted me in Cinderella was its human interface. It is very intuitive and easy to learn. Once you have learned how to defme a line by two existing points and how to define a point as the intersection of two given lines, you can easily guess how to perform many other operations, e.g., how to create a line passing through a given point and parallel to an existing line.

So one can play with the software and learn it without any instructions. Other people like to read the entire manual before performing the first


mouse click. The manual is targeted at both types of users. It takes you stepby-step through a getting-started section, showing you how to draw and move points and lines, parallels and perpendiculars, etc. You immediately see the difference between free and dependent objects. For example, define a point as the intersection of two previously created lines. The point depends on the lines, so moving either line will change the location of the point. However, moving the point is not possible, as it is the dependent object. Then you get a quick overview of how to control the appearance of objects (sizes, colors, etc.) and of the possible views of a scene.

At this point you face one of the strongest features of Cinderella: its sup-

The fi rst th ing that

attracted me in

Cinderella was its

h uman interface . port of different geometric views. In fact, the software supports two distinct, not to be confused with each other, features. The first feature is the support of two non-Euclidean geometries (hyperbolic and elliptic) in addition to the regular Euclidean geometry. The second feature is the support of different views of the geometry: Euclidean, spherical, and hyperbolic. A good example is the spherical view in Euclidean geometry. In this view the entire plane is mapped to a hemisphere, shown on the screen as a circle. Points at infinity lie on the boundary of that circle, so that this boundary is "the line at infinity." Here it is very easy to draw, manipulate, and view objects at or close to infmity. The "price" is, naturally, the fast-decreasing resolution of details as you approach infmity.

Now you experiment with Cinderella's nice theorem-proving mechanism. That is, its ability to detect geometric tautologies. For example, theorems that say that three seemingly independent lines in some geometric constructions always meet at one

point. Denote the three lines as a, b, and c, and let P be the intersection point of a and b. Cinderella is able to come up with the statement that c and P are incident to each other. The manual describes how the software does this.

In a dynamic geometric construction, it is fun to move around free objects and visualize the effect on the entire scene. Moreover, it is educational to understand the nature of the changes. This is provided by the locus feature. You choose a "mover" and a "road" (e.g., a point and line). While the mover slides along the road, the construction keeps changing. To visualize the dynamics of the scene, you choose a "tracer." In case the tracer is a point, the result is the path traversed by the tracer while the mover advances along the road. You can either see the final trace or switch to an animation mode, in which the mover slides slowly along the road, and you gradually see the creation of the trace. This is, more or less, the end of your getting-started session.

A very interesting section of the manual reveals information "behind the scenes." In particular, it tells you how the software uses projective geometry, homogeneous coordinates, and complex numbers to maintain the various geometric entities and to solve continuity problems. For example, assume you defme a point as one of the two common points of two existing intersecting circles, and use that point for further constructions. Now assume you move the circles around such that they first do not intersect any more, and then intersect again. A few interesting questions arise. For example, when the two circles become intersecting again, which of the two intersection points would you expect to resume the role of the intersection point you originally defmed? In addition, what should happen to (not to mention how can you internally represent) those objects that depend on the temporarily disappearing intersection point?

In a nutshell, the authors' solution is to represent everything with complex numbers. Naturally, you define and see objects with real data, but that's not how the software sees it. For

example, say you define a point with coordinates (x,y), where x and y are real numbers. The software stores this as (x + Oi, y + Oi). In the previous example, the intersection point is nothing but one of the solutions Qf a system of two quadratic equations. When the circles cease to intersect, the solutions still exist, but now they are complex. And so is everything that depends on them. We simply see on the screen only the real objects, or if you like, the intersection of a 4-dimensional space (where each of the x and y axes has 2 degrees of freedom) with the real plane (no imaginary components).

The continuity problem is solved by cursor tracking. Putting it simply, Cinderella interprets how you move the mouse as a guide for what the desired continuous move of objects is. Again, the manual provides the details.

Let me now refer to some disadvantages of Cinderella. The problem is not with what it contains (at least I was unable to find any bug or strange be-

havior), but with what is missing. To me it is obvious that the version I played with (version 1.0) is immature and not ready (yet) for distribution. For example, the only way to defme a conic in this version is by specifying 5

points. This is almost useless; normally you (and I) would like to define, say, an ellipse by its 2 focal points and the sum of distances from them to every point on the ellipse. (Or, say, by specifying the lengths and positions of the 2 axes of the ellipse.) The authors have assured me that this is just a matter of providing a human interface, and that a more natural mechanism for defining conics is planned for a near-future release of the software.

I would also like to see other types of entities, e.g., higher-degree polynomials and even trigonometric functions. The authors claim that some limited functionality (e.g., no theorem-proving) for any externally defined function (e.g., by Mathematica) will be provided in version 2.0 of CindereUa. Moreover, the authors promise more flexibility with the locus and animation features. (With Ver-

Erratum

sion 1.0 only points and lines can serve as movers, roads, and tracers.) Another promised feature for the next version of the software is a scripting language for textual input, batch files, logging, etc.

Finally, I have mixed emotions about the manual. On the one hand, as mentioned above, its contents are concise and well organized, and they make the software-learning process smooth and easy. On the other hand, it has (to my taste) two stylistic drawbacks. First, it is a bit too self-congratulatory about the use of the sophisticated mathematical tools in the implementation of the software. Second, it contains spelling errors and seems to require one more proofreading. I believe that these two drawbacks can be readily fixed. Let me conclude with a good word about the software installation: It's a piece of cake. After '3'0 seconds or so, you

'can play.

Faculty of Computer Science

Technion - Israel Institute of Technology

Haifa 32000

Israel


In his "Rediscovering a family of means" (Mathematical Intelligencer 24 (2002), no. 2, 58-65),

Stephen R. Wassell begins his account with the use of the arithmetic mean by the ancient Babylo·

nians. Readers may have been surprised, as I was, by the very early date given (and by its preci

sion).

One reader, Robert Davis of Southern Methodist University, wrote to query this dating. Profes

sor Wassell thanks him, as do I.

Wassell was relying on an article by Maryvonne Spiesser, which he cited. Prompted by Profes

sor Davis's query, he went back to Spiesser and found he had misread her, taking an identification

number for a date! He gives this corrected text for the sentence following equation (5) in his

article:

The arithmetic mean, the simplest of the three, was known and used by the Babylonians, per

haps as early as 1900 BCE.

For this he refers to Spiesser and to

0. Neugebauer and A. Sachs, Mathematical Cuneiform Texts, American Oriental Society, New

Haven, CT, 1945.

The Editor


41fi,I.MQ·h·i§i Robin Wilson

Two Serbian Mathematicians by Slobodanka Jankovic

and Tatjana Ostrogorski

Please send all submissions to

the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes,

MK7 6AA, England

e-mail: r.j [email protected]

In this column we celebrate the work

of the Serbian mathematician Jovan Karamata and of his teacher Mihailo Petrovic, both of whom have recently

been commemorated on stamps.

Mihailo PetroviC (1868--1943), math

ematician and philosopher, was pro

fessor at the University of Belgrade and

founder of the Belgrade mathematical

school. He studied in Paris at the Ecole

Normale Superieure and obtained his

doctoral degree in 1894. The examin

ers for his thesis, Sur les zeros et les irifinis des integrales des equations differentieUes algebriques, were Her

mite, Picard, and Painleve. He worked

in differential equations, · real and com

plex analysis, and algebra, and also in

physics, chemistry, and astronomy, and

wrote many papers in all these areas.

Petrovic constructed several ma

chines and measuring instruments. He

also constructed an integrator, a kind

of analogue computer based on hydro

dynamic principles for solving first-or

der differential equations; for this, he

obtained a special award at the Paris

Exhibition in 1900, and in London in

1907. He wrote several essays on math

ematical phenomenology, as well as

books on travel. He had many students,

the most famous of whom was Jovan

Karamata. This stamp commemorates

the fiftieth anniversary of Petrovic's

death.

Jovan Karamata


Jovan Karamata (1902-1967) is

best known for his theory of "regularly

varying functions," from the early

1930s. This is a class of real functions

that behave nicely in asymptotic rela

tions. Much later, these functions

found various applications in other ar

eas of mathematics-especially proba

bility theory, but also in number the

ory, the theory of analytic functions,

and the theory of generalized func

tions.

Karamata was a professor at the

University of Belgrade, where he

founded an important school in math

ematical analysis. From 1951, he was a

professor at the University of Geneva.

He wrote many papers, mainly in clas

sical analysis, but also in number the

ory, Fourier analysis, inequalities, and

geometry. The most important part of

his work, which included his best re

sults, was related to the summability

theory of divergent series and to

Tauberian-type theorems. He became

famous for his short and elegant proof

of Hardy and Littlewood's Tauberian

theorem, which he published in 1930.

This stamp commemorates the cente

nary of his birth.

Mathematical Institute SANU

Kneza Mihaila 35

1 1 000 Belgrade

Yugoslavia

Mihailo Petrovic

The Mathematical Intelligencer volume 24 issue 4

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Transcript of The Mathematical Intelligencer volume 24 issue 4