No title

«·)·"I"·' I I

Paraconsistent Thoughts About Consistency Philip J. Davis

"You will notice, Pnin said, that when on a Sunday evening in May, 1876 Anna Karenina throws herself under that freight train, she has existed more than jour years since the begin­ ning of the novel. But in the life of the Lyovins, hardly three years have elapsed. It is the best example of rel­ ativity in literature that is known to me. "-Vladimir Nabokov, Pnin, ch. 5, sec. 5 (paraphrased). Most philosophers make consistency the chief desideratum, but in mathe­ matics it's a secondary issue. Usually we can patch things up to be consis­ tent. -Reuben Hersh, What is Mathe­ matics?, Really, p. 237.

mathematicians the opportunity to write about any issue of interest to

I am an occasional writer of fiction, en­ gaging in it as an amusement and a re­ laxation. Compared to professionals, I

the international mathematical

would say that my "fictive imagination"

community. Disagreement and

is pretty weak. This doesn't bother me

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

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

should be submitted to the editor-in­

names

chief, Chandler Davis.

serted. One time it replaced President

unless

they've

bothered me after I finished was this:

Having produced a fairly long manu­ script, I was never quite sure whether it was consistent. I don't mean the consis­

tency of a pancake batter or whatever the equivalent of that would be in prose style. I mean the everyday sort of con­ sistency 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 ex­ amples 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

I

The Opinion column of f ers

I recently wrote a long short story ti­ tled "Fred and Dorothy." What really

been

pre-in­

Lincoln's War Secretary Seward with Secretary Seaweed. You know the say­ ing: 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 de­ tached sentence. For example: "Lon­ don, Cambridge, so why not Manches­ ter?" It often wants to change a passive formulation into an active one, and its

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 ju­ nior 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 novelf--or, rather, his creator­ cheats a little, recording not only what he saw and heard, or was told, but also what went on in the minds of charac­ ters 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 . . .

suggestion for doing so often ends in a terrible muck. It doesn't catch non­ sense.

As

a test I wrote, "The man re­

We learn to deal with inconsisten­ cies in books, and we do it in different

boiled the cadences through the mon­

ways. Suspension of disbelief is only

key

and my spelling and

one way. Now, whatl would find really

grammar check simply changed "re­

useful, if such a thing could exist,

boiled" to "rebelled."

would be a program that checks for

wrench,"

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

3

II

consistency as well as my copy editor

sides by 4, I get 0

=

4. Now is that an in­

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

I've now said enough about literary

She has read widely. She knows my

texts and I'm ready to get to logic and

mind. She's worth gold.

mathematics; to Boole and Frege and

that 0

Russell and Godel and Wittgenstein

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

and all those fellows. Whereas life does

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

Imagine now that I have bid Louisa goodbye and replaced her with a con­ sistency checker that I paid good money

not have a precise definition of con­

for. Call the software package CONNIE.

sistency, mathematics has a clear-cut

I run Fred and Dorothy through CONNIE. It immediately comes back

definition. A mathematical system is consistent if you can't derive a con­

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

Come to think of it, how do we know =

1 is a contradiction in 7L? Be­

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.

tradiction within it. A contradiction

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

Dorothy's eyes were blue and on page

would be something like 0

naked mathematical symbols is not

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

sistency is good and inconsistency is

interpretable (or is arbitrarily inter­

I have to spell out in my text that in the

bad. Why is it bad? Because if you can

pretable). By "naked" I mean that you

late afternoon October mist, Dorothy's

prove one contradiction, you can prove

have no indication, formal or informal,

eyes seemed brown to me?

anything. In logical symbols,

of where the writer is coming from.

with a message: "On page 8 you said

How did CONNIE handle metaphor? I wrote: "He saw the depths of the sea

(1) For all A and B,

=

1. Con­

(A & �A)� B.

Now bring in G6del's famous and notorious Second Incompleteness The­

in her eyes." Now CONNIE (a very

And so, if you allow in one measly in­

smart package) knew her (its?) Homer

consistency, it would make the whole

erary texts. To state it in a popular way,

and recalled that

program of logical deduction ridicu­

the GIT says that you cannot prove the

lous.

consistency of a mathematical system

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

Aristotle knew about equation (1)

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

by means of itself.

and had inconsistent views about it. In

If mathematics is part of the universe

the literature of logic it's called the

of natural language, and I think it is, then

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

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

get the GIT to imply that it is impossible

please make up your mind about

principle. (The Duke of Wellington

to build a universal consistency checker.

Dorothy's eyes."

1769-1852, victor at the Battle of Wa­

Or, for that matter and much more im­

and it blew the whistle on me: "Hey,

I believe that with a little thought, I could

Why did "Dorothy" morph into "Dor­

terloo.) The Duke was walking down

portant these days, that it is impossible

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

the street one day when a man ap­

name the movie producers decided to

proached him.

to build a universal virus checker. If a

give her after she'd passed her screen test. CONNIE picked up a sentence in ar­ chaic English and screamed bloody mur­

consistency checker can't be produced

The Man: Mr. Smith, I believe? The Duke: If you believe that, you can believe anything. Inconsistency is (or was) the primal

der. The sentence in question was part

for mathematics with its sophisticated and conventualized textual practices and with its limited semantic field, then I have serious theoretical doubts about literary texts.

of a movie script (within my story),

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

CONNIE might catch Dorothy's eyes

whose action was placed in the 17th cen­

Harvard, I took a course in mathemati­

being simultaneously blue, brown, and

tury. It rapidly becomes clear that the no­

cal logic with Willard Van Orman Quine,

wine-dark, but there will be some in­

who in the opinion of some became the

consistencies that CONNIE misses.

tion of consistency is not context-free. And so on and on. A writer of fiction

most famous American philosopher of

Inconsistency is how things appear

can explain away post hoc what appear

his generation. Quine hadjust published

in the world. We spend part of our life

to be inconsistencies. In technical lingo,

his

Mathematical Logic and it was our

cleaning up the confusions, trying to

often employed in mathematical physics,

textbook The course startedjust before,

impose some semblance of order. To

explanations that clear up inconsisten­

or shortly after the shattering news

some extent we are successful, but

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

came that J. Barkley Rosser had found

only in a limited sense and for a lim­

an inconsistency in the axiom system

ited time. Heraclitus assured us that

has drawn up a taxonomy of textual in­

Quine had set up. Well, Quine spent the

nothing is ever the same twice, and

consistencies. It must be extremely long.

whole semester having the class patch

when things begin to get fuzzy we

Mavens who analyze language often

up the booboo in our books; crossing

think, that's not the way we had per­

split language into three systems with

out this axiom and replacing it with that;

ceived matters. So I'm afraid we all

replacing this formula with that-while

have to live with and deal with incon­

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

we logical greenhorns were anxious for

sistencies. We learn to do it. Walt Whit­

world in terms of experience and logi­

man, the poet, knew this. He said,

cal meaning), and textual (ways of com­ posing the message). I worry mostly

him to get on with it and get to the punch

Q.E.D. as regards primal sins.

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

But back to business. If 0

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

different sorts of meaning: interper­

sistency checker to work on them. 4

THE MATHEMATICAL INTELLIGENCER

line of logic, whatever that might be. =

1 is an

inconsistency, then by multiplying both

Mathematics is one way we try to

maticians are often smart enough to

"folk theorem" that bad software can

impose order, and we may do it incon­

spirit away a contradiction-just as

often be useful.

sistently. Consider the arithmetic sys­

Hersh says in the epigraph. Mathemat­

Acknowledgments

tem that is embodied in the popular

ical inconsistencies are often exor­

and useful scientific computer package

cised by the method of context-exten­

I thank Ruth A. Davis and Kay O'Hal­

known as MATLAB. Now MATLAB

sion. It is done on a case-by-case basis,

loran for providing me with some im­

yields the following statements from which a contraction may be drawn: "le - 50

=

"2 +

0 is false" (i.e., w-50

(le - 50)

=

2 is

=

0),

true."

and it is worth doing only after the con­

portant words and ideas.

tradiction has borne good fruit. So the notion

of

mathematical

consistency

may be time- (and coterie-) dependent­ just as in literature.

REFERENCES

George S. Boolos, John P. Burgess and Richard C. Jeffrey, Computability and Logic,

Well, we all recognize roundoff and

Logicians, who go for the guts of the

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

know its problems. And we know, to a

generic, and who are over-eager to for­

Chandler Davis, Criticisms of the usual ratio­

considerable extent, but not totally, how

malize everything, have come up with a

nale for validity in mathematics, in Physicalism

to deal with it; how to prevent it from

concept called

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

getting us into some sort of trouble.

has even been a World Congress to dis­

demic, Dordrecht, 1990, 343-356.

cuss the topic. Ordinary logic, as I have

Peter Gay, The Naked Heart, Norton, 1995.

noted, has the Duke of Wellington prop­

Reuben Hersh, What is Mathematics, Really ?,

cally but can't exist numerically? At

erty that if you can prove A and not A,

Oxford Univ. Press., 1 997.

one point in history it was a highly ir­

then you can prove everything. Para­

Karl Menger, Reminiscences of the Vienna Cir­

rational conclusion and one worthy of

consistent logic is a way of not having

Is it a contradiction that the diago­ nal of the unit square exits geometri­

slaughtering oxen. Was it a contradiction that there ex­ ists 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 pro­

paraconsistency.

There

an inconsistency destroy everything.

cle and the M§Jthematc i al Colloquium, Kluwer ' AcademiC, Dordrecht, 1 994.

Contradictions can be true. Perhaps

Chris Mortensen, Inconsistent Mathematics,

such a system might be good for certain

Kluwer Academic, Dordrecht, 1 995.

applications to the real world where conflicting facts are common.

ductively. It was among the mathemati­

consistent London, a man approached

cians until Laurent Schwartz came along

the Duke of Wellington.

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 lo­ cally Euclidean group without differ­ entiability, and yet I think that if some day we come to want it badly­ in 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 im­ possible. Then again, we may not. . . .

A UTHO R

Walking down the street in para­

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 para­ consistency 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 de­ fine a contradiction, I know one when I see one. In a very important paper written in the mid-1950s, the logician Y. Bar-Hil­

PHILIP

J. DAVIS

Division of Applied Mathematics Brown University Providence,

Rl 02912

USA

e-mail: [email protected]

lel demonstrated that language trans­ lation was impossible. This demon­

Philip J. Davis, a native of Massachu­

stration dampened translation efforts

setts and a Harvard Ph.D., has been

for a few years. illtimately it did not

in Applied Mathematics at Brown

deter software factories from produc­

since 1 963. He is known for applied

ing language translators that have a

numerical analysis, and his tools are

Inconsistencies can be a pain in the

certain utility and that also produce ab­

typically functional analysis and classi­

neck, a joy for nit-pickers, and a source

surdities. I'm sure that the software

of tremendous creativity.

factories will soon produce a literary

cal analysis: some might say, he ap­

Karl Menger, in his Reminiscences of the Vienna Circle and the Mathe­ matical Colloquium, tells the story

plies pure analysis to applied. But he

consistency checker called CONNIE. I

is known to many more as a com­

will run to buy it. It might be just good

mentator on mathematics. Among his

enough for me. And if I've paid good

many nontechnical publications are

that in Wittgenstein's opinion, mathe­

money for it, then, as the saying goes,

maticians have an irrational fear of

it must be worth it. The absurdities it

Hersh, The Mathematical Experience

contradiction. I've often thought as

produces will lift my spirits on gray

and Descartes' Dream.

much, but I also realize that mathe-

days and serve to remind me of the

the widely read books with Reuben

VOLUME 2 4 , 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

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

Any rotation of the oriented euclidean 3-space

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

The Haar measure of 80(3) Proposition 1: If one describes 80(3) by the parame­

trization 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

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

�

2

= 47T2

It is certainly not uniformly distributed: The rotations by

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

2

1

ural area measure, but what about the rotation angle? a small angle

=

sin2

( �) da dA(g)

(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 is easy to calcu-

of the group Q of unit quatemions, and it

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 cover­ ing and r*df..Lso(3)

In the parametrization ljJ: = have therefore

=

To

2df..LQ·

c.p

: [0, 1r]

X

S2 � S0(3) we

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

F(a)

=

1

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

0

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 ran­ dom rotations g E S0(3). The mapping g � (g(x), dg(x; g)) is a diffeomorphism from S0(3) onto the unit tangent bun­ dle 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 con­ sequence of proposition 2: Corollary: The expectation of the rotation angle of a ran­

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

+�

1T

=

1T

2

Random Rotations in

126° 28' 32".

Dimensions

4

The Haar measure of 50(4) The distribution of the rotation angle

The parametrization p is well adapted to our problem, be­ cause the subset of rotations by a fixed angle a is just the image of the sphere { a} X S2. If we integrate our expres­ sion 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):

� :il

7T

0.5

1

1.5

2

2.5

If we identify the euclidean IR4 with the skew field of quater­ nions 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

Q � S0(4),

(p, q) := Lp R-q o

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

... ... ..

-:.

..

..

":f.:·; :_::: -:_�--� 1'7 21. The following lemmas are needed to determine the pair of rotation angles for an element (p, q) E S0(4).

Corollary (Fig. 2): The pair of rotation angles is dis­

Lemma 1: For p, q, p', q' E Q, the rotations (p, q) and (p', q') in S0(4) are conjugate if and only if p is con­

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.

jugate to ±p' and q is conjugate to ±q' in Q, with the same sign in either case. Proof (p, q) is conjugate to (p', q') if and only if there exists aTE S0(4) with (p, q) = To (p', q') y-1 . As T = (u, v) for some u, v E Q, we have: a

(p, q) is conjugate to (p', q') u, v E Q with Lp

a

Rfi

o

o

a

o

R

a

= Lu R:v Lp' = Lu Lp' o L:u = Lup'u cvq'v o

if

tributed with density fiWt, 1'7z])

=

T

=

L:u

a

a

(up' , vq' v ).

The kernel of contains only the two elements (1, 1) and (- 1, -1); therefore we have shown that (p, q) is conjugate to (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 S 2 be a purely imaginary quaternion with norm 1. the quaternion p= cos t + g sin t is conjugate to p' = cos t i sin t.

Then

8

+

THE MATHEMATICAL INTEWGENCER

s

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

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

Rfi· Rv R;u Rfi· Rvu o

:2 sin 2 ( 1'71 ; 1'7 2 ) in 2 ( 1'71 ; 1'1z )

Proof Starting with the parametrization

and only if there exist

o

=

+ildii;IIM

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

+ t], we obtain a

With respect to these parameters the Haar measure satisfies I/J*dJLso(4) =

( ; it2 ) sin2 ( it1 ; it2 ) dit1 dit2 dil.(g) dil.(

it1 C sin2

TJ).

Integrating over {(it1, it2)} XS2XS2 for fixed it1, it2 gives us the density

=

ma(gitg-1)=[it] for

The constant C' = 1hr 2 is obtained by integrating this func­ tion over [0, 27T] X [0, 27T]. 0 n �4

The results obtained in dimensions 3 and 4 can be gener­ alized 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 ro­ tation angles. In dimension n > 4 we can introduce the no­ tion of a multiangle characterizing the cof\jugacy classes.

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

commutes, where cp(g, it):= rg(it)=gitg-1 and the verti­ cal 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 map­ pings 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 ex­ pressed in terms of that of T together with the invariant measure of G/T: Proposition 4: 1/J : G/TXT � G is a finite branched cov­

ering. Let dJLa and dJLr denote the Haar measures of G and T, and let dJLa;r be the G-invariant normalized mea­ sure of the homogeneous space GIT. Then

Multiangles of rotation

Let us begin with the case of a rotation g E SO(n) for even n = 2 m: as in the case n=4, there are m rotation angles it1 ..., itm corresponding to the decomposition of gas di­ rect sum of m plane rotations: g=Rott't1 E9 ...E9 Ro�m· For odd n = 2 m + 1, there are also m angles it1, ..., itm.Calculating modulo 27T, the list (it1' itm) is an element of the m-torus rm and is unique up to the following symmetries, which define an equiva­ lence relation on rm: 0

0

�

the iti may be permuted; iti may be replaced by -iti, but only for an even num­ ber 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 mul­ tiangle 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 _

it-

(

Rott't1 • •

•

•

0

0

0 ... Rott'tm

it E rm and g E SO(n).

+Y

=4 (cos it1 - cos it )2· 2

0

all

G/TXT

C'

Rotations in Dimension

in the case n=2 m; in the case n 2 m + 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:

)

1/J*dJLa = dJLa!TXJ 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) =

f J(it) dJLGIT = cJ(it)

C

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 ma­ trices of the form A=

(

�

�

;

A im

A12 A1s ...A1 m 0 A2s ...A2m 0

0

0

0

0

)

'

0

where the AiJ are 2 X2-blocks.

VOLUME 24, NUMBER 4, 2002

9

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

)

with gzm(XI. ... ,Xm)

R111J{Ai'J ) := Ro�Ai'J-Rot,&1. J I

I'

If we identify !R2x2 with IR2 0 IR2, R1Ji,1J becomes the ten­ i sor product of the two rotations Ro�; and Ro�I The eigen­ values 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

= 4(cos ifi - cos ifi?·

ifi

; ifi r

Now Ad-'-iJ is the direct sum of the R1l;,1l·Combining these i results: J(if)

=

2m(m-l)

n

l�i ofgz m+b 1 > x1 > . . . > xm -1.

... > Xm= -1;for

that

=

Proof Let us consider the even case, i.e., the functiong2m: Obviously, one has x1=1 and Xm =-1 for any local max­ imum x. Fix these two coordinates and define

{

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

(cos i}i- cos i7j?·

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 an­ gles (arccos(l/3), 1r]= [70°31'44", 180°] are the "most fre­ quent" 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

IT

l�i<js.nt

and

h

and

=

··(x)= ht]

•

m

2

-I

2 k=l (xi - xk) k*i

2

2 (xi- Xj)

•

•

fori=j fori =F j.

The diagonal elements are strictly negative, the other ele­ ments 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 func­ tion 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. Forg2m+ 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

SpringerMath(JJxpress Per onalized book announcement to your e-mail box. www.springer-ny.com/express

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Many Subjects to Choose From Chao e from an array of specialties within the di ciplines of Astronomy,

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Busi ness,

Engineering, Geo. cience, Life Science, Mathematics, Medicine,

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

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

Yo rk 1967) 2. W.

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

Phy ic , and Statistics.

MATHEMATICS:

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Algebra

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Combinatoric and Graph Theory

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Functional Analy is and

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General Mathematics

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Geometry and Topology

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Mathematical B i ology

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Mathematical Methods

•

Mathematical Physics

Cohomology, volume II (Academic Press, New York and London 1 973)

Scientific Computing

3. J. M . Hammersley, The distrb i ution 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

Dynamical Sy tem.

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)

Operator Theory

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umber Theory

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Optimization, Control Theory and

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Probability Theory

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Real and Complex Analysis

Operation

Research

Springer

VOLUME 24, NUMBER 4, 2002

11

Mathematically Bent

Colin Adams, Editor

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 contin­ Colin Adams uing student.) Sarge munched noncha­ lantly on a toaster pastry. Was she remember that day as if it were yes­ really that unconcerned or was that terday. I will never forget it. Often, the impression she wanted us to have? I wake up at night, in sweat-soaked I didn't know for sure, but the toaster sheets, screaming, "Look out, Sarge, tart sure looked good. As we spread out over the lecture look out!" Often my roommate is screaming, too. "Shut up, shut up!" But hall, hunkering down in our foxholes, I can't shut up. I have to tell the story, I felt queasy myself. This was it. The the story of that fateful day. A day that real thing. No more training sessions, with dummy problems whizzing over­ can never be forgotten. We were fresh out of boot camp, head and a solutions manual available Leftie and me. Hardly knew an integral for cover. This would be live ammuni­ from a derivative. We thought the tion exploding around us. Everyone power rule was complicated. Just a else looked as frightened as I felt. Now, pair of snot-nosed calc students. But we fmd out what you're made of, I they said we were ready for Calc II. thought to myself, as the hour struck How ridiculous that sounds now. and the general down front signaled We arrived in country and were as­ the beginning of the battle. I gulped signed to a unit of misfits. Sarge was once and turned over the cover page. A couple of partial derivatives whis­ the only one of us who had seen real combat before. She had fought in tled overhead, and I thought to myself, WWWI , a web-based trig course. And I can handle this. I started firing, then there was Pipsqueak, Pops, Leftie, plugged a couple quick. Hey, no worse and me. They called me Kodowski. I than an afternoon of video games, I wanted them to call me Tootsie. But said to myself. Then I came up over the next page they refused. Before we had even fmished un­ and swallowed hard as I found myself packing our gear, we heard a yell. "In­ face-to-face with an armored series di­ coming!" Grunts dove for cover. Sarge vision. I didn't even stop to think. I just just kept eating her granola bar. "Re­ peppered them with Ratio Tests. A few lax," she said. "It's just a quiz." I stayed went up in flames. The rest rolled for­ low anyway. It seemed dangerous ward. I switched to Root Test, spray­ enough to me. But it wouldn't be long ing them indiscriminantly. A couple more went down but the rest rumbled before I understood the difference. I remember that fateful morning as forward. So I lobbed in a couple of Ba­ if it were yesterday. I woke to some­ sic Comparison Tests and a Limit Com­ thing dripping on my forehead. Leftie parison Test or two. Then I let loose had wet the upper bunk again. He gave with the Alternating Series Test and fol­ new meaning to the words math anxi­ lowed up with half a ton of nth Term ety. I pulled him off his bunk and we Tests. That ought to do it, I thought, as had a quick shoving match..Then we I waited for the smoke to clear. But threw on our uniforms. No time to among the littered carcasses on the field brush teeth or comb hair. Ours or any- before me, there still stood one lone se-

The Red Badge of Courage

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 e-mail: [email protected]

12

I

THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

ries. At first I couldn't make it out. But as it lumbered forward, I suddenly real­ ized 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 se­ ries? I tried desperately to remember. We had talked about this in basic train­ ing. 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 har­ monic series . . . . " Yes, yes, I thought, as I waited for the voice to finish its ex­ planation. "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 se­ ries." 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 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 won­ dering 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 ordi­ nance. I should have known Leftie wouldn't have the guts for it. Ever"since that quiz problem on improper inte­ grals, he had had the shakes.

22

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 de­ rivatives. 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 singlehand­ edly turning the tide. Suddenly I real­ ized the battle was almost over. I tri­ umphantly flipped the last page and found myself face-to-face with the nas­ tiest 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 be­ low. 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 echo­ ing in my ears. "Don't forget. p2 sin 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 an­ swer 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, Pip­ squeak, 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 pro­ moted, too. They made me a grader. They wanted me to go to officer' s train­ ing school at Princeton or maybe Berkeley. And maybe someday I will. Maybe that would make it all worth­ while. But I have to get over the night­ mares first. I have to reconcile my vic­ tory 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.

VOLUME 24, NUMBER 4, 2002

13

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THE MATHEMATICAL INTELLIGENCER

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S O C I ETY F O R l N D U ST R I A L A N D A P P L I E D M A T H E M AT I C S

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

Com mentary on an U n pu bl ished Lectu re by G . N . Watson on Solvi ng the Qu i ntic

he following notes are from a lecture on solving quintic equations given by the late Professor George Neville Watson (1 886-1 965) at Cambridge University in 1 948. They were discovered by the first author in 1 995 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 sep­ arated 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 mathe­ matics which has been of enduring and widespread inter­ est 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 deter­ mine which quintic polynomials have roots expressible in terms of radicals, and in 1991 Dummit [24) gave for­ mulae 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 pub­ lish Professor Watson's notes on his lecture, with com­ mentary 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 pro­ cedure will be especially valuable. Watson's method ap­ plies 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. 1 999.

The third author thanks Carleton University for a travel grant which enabled him to travel to the University of Birmingham, England in December 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

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 determi­ nations on a firm foundation, and Zhang [61], [62] has used Kronecker's limit formula to verify Watson's calcu­ lations. Professor Watson held the Mason Chair of Mathemat­ ics at the University of Birmingham from 1918 to 1 951. He was educated at Cambridge University (1904-1908), where he was a student of Edmund Taylor Whittaker (1873-1 956). He became a Fellow of Trinity College, Cambridge, in 1 91 0. From 1 91 4 to 1918 he held aca­ demic positions at University College, London. Watson devoted a great deal of his research to extending and pro­ viding proofs for results contained in Ramanujan's Note­ books [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 Mod­ em Analysis, first published in 1 915, 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 com­ mentary 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 quan­ tities associated with a quintic equation, the second gives twenty-jour pentagrams used in showing that permuta­ tions of the suffixes of

(X1X2 + X2X3 + X3X4 + X4X5 + X5X1 - X1X3 - X3X5 - X5X2 - X2X4 - X4X1)2 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 num­ bers 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 polyno­ mial equations of degree exceeding 2 was often needed. In a series ojsixjurther papers [50]-[51], [53]-[56], he de­ veloped an empirical process for calculating class in­ variants, which also depended heavily on solving poly­ nomial 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 re­ mained unproven until recent times. The remaining ones

16

THE MATHEMATICAL INTELLIGENCER

yield six distinct expressions, and the third gives Wat­ son'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 ex­ tent of the possibility of solving equations of various de­ grees 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 nowa­ days 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 com­ pliment 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 in­ dulging in a few reminiscences. In order to make my lee-

SHEET 1 The denumerate form of the quintic equation is PoY5 + P1Y 4 + P2 Y3 + PaY 2 + P4 Y + 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 z 5 + 10Cz 3 + 10Dz2 + 5Ez + F

=

0.(z

= ax + b)

The sextic resolvent is where K = ae - 4bd + 3c2,

-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- 70abc 2de + 80abcd3 + 35ac4e- 40ac 3d 2 - 25b4e2 + 100b3cde -50b2c 3e - 100b2c2d 2 + 100bc 4d- 25c 6.

L

=

ture effective, I must endeavour to picture to myself what is passing in the mind of John Brown who is sitting some­ where in the middle of this room and who came up to Trin­ ity 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 mathe­ matician 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 Chan­ cellor. 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 re­ garding me as equally elderly, but I hope that, for a num­ ber 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 Cam­ bridge in the Lent term of 1905, and so my attempt at an anal­ ogy rather breaks down. On the other hand a visit was paid to Cambridge at the end of that term by a much more emi­ nent personage, namely the Sultan of Zanzibar. For the ben­ efit 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 Sul­ tan was W. H. de Vere Cole, a third-year Trinity undergradu­ ate. It was the most successful practical joke of an age in which practical joking was more popular than it is to-day.

VOLUME 24, NUMBER 4, 2002

17

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 professor­ ial lectures which were not much attended by undergrad­ uates); 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 Trin­ ity 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 nine­ teenth century than it is to-day when mathematics is tend­ ing 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 Op­ tics, whose name is sometimes associated with mine, is liv­ ing 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 Hob­ son 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 lectur­ ing on curvature of surfaces and the problem of three bod­ ies, respectively. Two things you may have noticed, the large proportion of my teachers who are still alive, and the

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 Col­ lege tutor was responsible for the teaching of his own pupils and of nobody else; he was aided by one or two as­ sistant 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 ir­ relevancies, 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 audi­ ence 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 famil­ iar with methods of obtaining algebraic solutions of qua­ dratic, 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 de­ gree 5, then the quintic equation.f(x) = 0 is solvable by rad­ icals 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 sub­ group 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 solv­ able 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(2 71i/5), r assumes the values 0, 1 , 2, 3, 4, and u Y, 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 re­ sults, you will find proofs of them in the treatise by Burn­ side 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].

Bruce Berndt with Ramanujan's Slate.

18

THE MATHEMATICAL INTELLIGENCER

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 conve­ nient for me to assume that this state of affairs still per­ sists, 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 im­ portance in the theory of ellipticfunctions, apart from their intrinsic interest.

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

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 el­ liptic 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 equa­ tions, not only of the fifth degree, but also of the sixth, sev­ enth and other degrees higher still. It consequently became necessary for me to co-ordinate the work of previous writ­ ers 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].

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 ex­ pressions 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- {3 o, etc., or simpler still,

af3 + yo,

ay + {3o,

a o + 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 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 pos­ sess algebraic solutions (and it proves to be the reason) is that certain non-symmetric functions of the roots ex­ ist such that the values which certain powers of them can assume are fewer in number than the degree of the equa­ tion. Thus, in the case of the quadratic equation with roots a and {3, there are two values for the difference of the roots, namely

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 coeffi­ cients. 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 im­ mediately obtainable. The cubic equation, with roots a, {3, y, can be treated sim­ ilarly. Let €3=1 (E i=- 1).Then we can form six expressions

� + yE + ail',

f3 + y€2 + aE,

- (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

a- {3, f3- a.

a + {3E + y€2, a + {3€2 + yE,

a,X3 - 6a2cX2 + (16bd- 4ae)aX

y + aE + {3€2, y + ail' + {3E,

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 I=ae- 4bd + 3c 2, J

=

a b c b c d c d e

I have discussed the problem of solving the quartic equa­ tion 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 re­ ducing cubic of a quartic equation has no root which is ra­ tional in the field of its coefficients, and any expression for the roots of the quartic involves cube roots; on the other

VOLUME 24, NUMBER 4, 2002

19

SHEET 1A The discriminant Ll o f the quintic equation i n its standard form i s equal to the product o f 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 + 3456ac 5f 2 - 11520ac4def + 6400ac4e3 + 5120ac3d3f- 3200ac3d2e2 + 256b 5f 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 anal­ ogous 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 quar­ tic and biquadratic indifferently. (I once discussed this ques­ tion with my friend Professor Berwick, who in his lifetime was the leading authority in this countcy on algebraic equa­ tions, 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 or­ der 4, the cyclic group 7lJ47L of order 4, or the dihedral group D4 of order 8. Since D4 is not abelian, it is not ap­ propriate 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 Thomp­ son, 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 in­ stance, if my friend Mr. P. Hall of King's College is here this evening, he will probably be horrified at the ignorance which 20

THE MATHEMATICAL INTELLIGENCER

I shall show when I say anything derived from the theory of groups. On the other hand, while to the best of my knowl­ edge 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 equa­ tion, 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=10p , d = lOOp , 2 3 e=2000p4, f= l0 5p5. 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

4

1

2

3

1

2

5

4

5

2

2

3

5

3

2

4

5

4

3

2

5

2

4

3

3

3

1

1

1

3

4

5

2

5

5

4

3

1

1

4

2

3

2

2

5

3

1

3

4

VOLUME 24, NUMBER 4, 2002

21

SHEET

3

The roots of the quintic in its reduced form are

Zr+ l with

w

=

=

wrul + u.J.ru2 + w3ru3 + w4ru4

exp(277i/5), r = 1, 2, 3, 4, 0.

(1) U 1U4 + 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�u 1 - u�u4 + u�u2) =

New unknowns, (} and

(5) (6)

T,

=

-F.

defined by

U 1U4 - U2U3 = 2 (}, u1u3 + u�u2 - u�u 1 - 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�u l )/(UzUs), etc., U� = (uiusi(u�ul)/(UzUd, C(D2 - T 2) + (C2 - 02)(C2 + 302 - E) = R1R2 0. (D2 - T 2 )2 + 2C(D2 - T2)(C2 + 3 ()2) - 8Co2(D2 + T 2) +(C z o2) 2 (C 2 502) 2 + 1 6D03 T + E2 (C2 ()2) - 2CE(D 2 - T 2) - 2E(C2 - 02)(C 2 + 302) = 0. (DO + CT)(D 2 - T 2) + T(C 2 - 502) 2 - 2CDEO -ET(C2 + 02) + FO(C2 - 02) = 0. =

=

=

(7) (8) (9)

_

_

_

etc.

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 For this quintic,

- y 4 + y 3 - 2y 2 + 3y- 1

=

0.

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

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)

22

THE MATHEMATICAL INTELLIGENCER

=

0

x

79V5.

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

so that t = -10 and tjJ Watson.

=

500 in agreement with

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

ES

5 acts on this element by

4 5 2 3 u((x1 + WX"z + w x3 + w x4 + w x5) ) = (Xu( 1) + WX"u(2) + u?xu(3) + w3Xu(4) + w4Xu(5))5.

An easy calculation shows that a

=

Cx1 + WX"z

if and only if u = Hence

u

preserves

+ u?x3 + w3x4 + w4x5)5

(1 2 3 4 5)k for

stab85(a)

=

some k E

{0, 1, 2, 3, 4}.

(( 1 2 3 4 5)),

so that lstabs5(a)l = 5.

Thus, by the orbit-stabilizer theorem [27, p.

)I = ?

lorbs5(a

1" I

=

120 -5

=

139], we obtain

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 ex­ pressed as functions of the coefficients of the quintic; more­ o�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 ex­ pression 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 So­ ciety. 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 read­ ing from Cayley [6] where the names of Cockle and Harley are linked [6, p. 3 1 1 ] .

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 determi­ nation of signs is that terms associated with adjacent ver­ tices are assigned + signs, while those associated with op­ posite 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 de­ rived 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 co­ efficients 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 b c/>1 = c/>1, we have stabs5( c/>1) 2::

(a, b la5 = b2 = e, bab

=

a4)

=

D5 ,

so that

l

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

On the other hand, the first two columns of Watson's pen­ tagram 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, cl a5 = c2 = e, c- 1ac = a3) = Fzo, so that lstabs5(c/>I)I 2:: 20. From thefirst column of the pen­ tagram 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 equa­ tion of degree less than 6, but it is possible to construct a sextic resolvent equation in which two of the coefficients

VOLUME 24, NUMBER 4, 2002

23

are zero. We succeeded in constructing a sextic in ql- be­ cause 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 se­ lection 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 se­ lected also form a sestet in which the sum of the members is zero; one of these sestets is represented by the penta­ grams in the first column, the other by the pentagrams in the second column.

lfr is odd, the transposition (12) changes the above equa­ tion 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

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.

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. It

.

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 re­ marks 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

=

..

. , cr( 4>6) } � orbs54>I>

l orbs5 1>1 l

=

12,

so that Thus if T E s5 is a transposition, · · ·

+ 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]. Di­ viding Er by x1 - x2 , we obtain Er = (xl - x2)q(x2, . . . , X5)

where

24

THE MATHEMATICAL INTELLIGENCER

(Xm - Xn)

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 E1 and E3 must be identically zero, while E5 must be a con­ stant 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.

( 4> - 4>I)(4> - c/>2)

and

r(Er) = r(4> [ +

II

l�m !),

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

+ r(x2, . . . , X5),

•

.

.

( 4> - 4>6)

has coefficients in ()I or Q(VI)), where D is the discrim­ inant 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 inven­ tion), 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 equa­ tion (0)

a6 4>6 - 100Ka44>4

+ 2000La2 ql-

2 - 800a 4>-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 a8/3 125. 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 dis­ criminant D of the quintic is [26, p. 205]

by

a8(x1 - x2)2 (x1 - x3)2 · · · (x4 - x5) 2 = 3125Ll = 55d. There is no obvious way of constructing any simpler re­ solvent and so it is only natur.§ll to ask "Where do we go from here?" It seems fruitless to attempt to obtain an al­ gebraic 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 Wat­ son in [49]. It seems inappropriate to refer to this prob­ lem 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'IK was equal to v=79, and consequently it was an Abelian quin­ tic, 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 combi­ nation of the two.

On pages 263 and 300 in his second Notebook [39], Ra­ manujan indicates that 2 114G79 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 1)(Z)

=

e ;,.;z/12

00

II

m= l

(1 - e2 11'imz)

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 offind­ ing 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 prod­ ucts of theta functions. After this digression, let us return to the sextic resol­ vent·' 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 so­ lution 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 in­ terested in the somewhat laborious task of completing the details of the solution of the quintic after the determina­ tion 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 con­ temporary 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 pro­ fessional mathematician at all. The few details of his ca­ reer 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 subse­ quently 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 num­ ber of papers on the algebraic solution of equations, in­ cluding three in the American Journal of Mathematics

VOLUME 24, NUMBER 4, 2002

25

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

(2) u1u3 + u�u1 + u§u4 + u�u2 = - 2D, 3 3 3 2 2 + U2U32 - U1U U3U4 - U1U (3) U1U4 2 2 2 - 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 Ramanu­ jan 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

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 ural procedure is now to determine

penultimate stage of our journey by being confronted with two simultaneous equations in the unknowns From

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

(1) and (5)

0 and T.

we have

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,

while from (2) and

and not as a consequence of deliberate and systematic

u1u3 + u�u2

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

=

(6)

we have

-D + T, u�u1 + u§u4 = -D - T;

and hence it follows that

u1u3 - u�u2 = u�u1 - u§u4 =

Y(D - T? + 4(C - 0)2(C + 0) = : R1, say; ± 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(u lu4i(u2u3), (u�u1 - u§u4)2 = (u�u1 + u§u4)2 - 4(u2u3i (u lu4).

intelligible, systematic and symmetrical manner. Take the reduced form of the quintic equation

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

These last equations enable us to obtain simple expres­ sions for the various combinations of the u's which occur

and suppose that its roots are

in

(3) and (4).

Thus, in respect of

(3), we have

U2U3

where w

=

exp(2 17i/5),

r = 1, 2, 3, 4, 0.

Straightforward but somewhat tedious multiplication

with similar expressions for u�u4, u�u1, stitute these values in

u�u3. When we sub­ (3) and perform some quite straight­

forward reductions, we obtain the equation

shows that the quintic equation with these roots is This shows incidentally that, when

termined, the signs of

R1

and

R2

0 and T have been de­

cannot be assumed arbi­

trarily but have to be selected so that R1R2 has a uniquely de­

terminate value. The effect of changing the signs of both R1

and R2 is merely to interchange The result of rationalising

u1 with u4 and u2 with u3. (7) by squaring is the more

and a comparison of these two forms of the quintic yields

formidable equation

four equations from which

(D2 - T2 )2 + 2C(D2 - T2 )(C2 + 302 ) - 8C0 2(D2 + T2 ) (8) + (C 2 - 02 )(C2 - 502 )2 + 16D03 T + E 2(C2 - 0 2 ) -2CE(D2 - T 2 ) - 2E(C2 - 02 )(C2 + 302 ) = 0.

mined, namely

(1) 26

THE MATHEMATICAL INTELLIGENCER

u1, u2, u3, u4

are to be deter­

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

u15 _

(uiu3i (u�u l ) , etc., (U2U3)2

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 R1R rationally.

2

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 R1R the value which 2 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 -ET(C2 + (}2) + Fe(C2 - (}2) (9)

= 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, u , u3, u4 are immedi­ 2 ately 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 un­ knowns 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 in­ spection. He does, in fact, solve the equations by inspec­ tion 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 mis­ taken in your assumption. Consequently Young's investi­ gations 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 pub­ lished 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 theexpression on the left is equal to

5(u lu4 - 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 = wru l + w2ru2 + w3ru3 + w4ru4 (r E { 1, 2, 3, 4, 5}) MAPLE gives Z1Z2 + . . . -z1Z3 - . . .

= 5(u lu4 - u2u3)(w - w2 - w3 + w4)

so that z1z2 +

· · ·

- z1z3 -

· · · =

5(u lu4 - u2u3)V5

since Consequently, to obtaill a value of (} which satisfies Young's simultaneous equations, all that is necessary is to ob­ tain a root of Cayley's sextic resolvent; and the determina­ tion 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 con­ stant term of Cayley's sextic resolvent (0) is 40000M. When (} has been thus determined, Young's pair of equa­ tions contain one unknown T only, and there is no diffi­ culty 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.

VOLUME 24, NUMBER 4, 2002

27

where

Watson's metlwd of finding a real root of the solvable quintic equation: 4 ax5 + 5bx + 10c.i3 + 10dx2 + 5ex + f = 0

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

First transform the quintic into reduced form x5 + 10Cx3 + 10Dx2 + 5Ex + F = 0.

Step 7. Determine u4 from u1u4 = -c + 8.

Watson's step-by-step procedure gives a real root of the re­ duced 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 =

Step 8. Determine u2 from u�u2 = ( -D + T - R1)/2. Step 9. Determine usfrom

U2U3 =

exp(277i/5).

INPUT: C,D,E,F

OUTPUT· A real root of the quintic is x = u1

Us + U4.

Step 1. Find a positive integer k such that

kj 16 X 108 X JJ12,

- C - 8.

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 al­ ready implied, I have used it to solve rather more than 100 Abelian quintics. If any of you would like to attempt the so­ lution of an Abelian quintic, you will find enough informa­ tion about Ramantijan's quintic given at the foot of the third sheet to enable you to complete the solution. You may re­ member 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 some­ what unobtrusively in the values which I have quoted for the u's.

eVk/a is a root of (0) for E = 1 or - 1.

Step 2. Determine 8 from 8=

eaVk

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 - T 2 ) + (C2 - 82)(C2 + 382 - E))/8.

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.

Step 6. Determine u1 from x2y 115 ul z2 ' _

+ u2 +

( )

Three Examples Illustrating Watson's Procedure - 5x + 12 = 0 The Galois group of x5 - 5x + 12 is D5. Here

Example 1. x5

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 2 1 2, YM = 520. Equation (0) is

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

Step 1

k = 10. Step 2 1

(} = V5 " Step 3

2 T = V5 "

28

THE MATHEMATICAL INTELLIGENCER

Continues on next page

Examples (continued) Step 4

Step 5

�

Rz = - Y5 - v5.

Step 6 X=

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

_

ul - -

( cv5 + V5 + v5)2 cv5 + V5 - v5) ) 115.

1

- v5 ,

25

Step 7

Step 8 _

Uz - -

Step 9

_

U3 - -

25

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

0 is x = u 1 + u2 + U3 Example 2. x5 + 15x + 12 = 0

A solution of x5

- 5x + 12

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

The Galois group of x5

=

+ u4

.

This agrees with [43, Example 1].

+ 15x + 12 is Fzo. Here

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.

Equation (0) is cf>6 -

3004>4 + 54000� - 230400v54> + 1080000

=

0.

Step 1 k = 180. Step 2

Step 3

Step 4 R1 =

12VIO . 25 Continues on next page

VOLUME 24, NUMBER 4, 2002

29

Examples (continued) Step 5

6Vlo R2 - 25 .

Step 6

X=

15

- 15 + 3vl0 z = -� 6Vlo Y= ' ' 5' 25 25 /5 - - 75 - 21Vlo l u1 125

+

Step 7

u4 Step 8

)

(

( - 75

·

)

21Vlo l/5 125

+

(

)

(

)

·

225 - 72Vlo 1/5 u2 125 Step 9

·

72Vlo 1/5. U3 -- 225 +125 This agrees with [43, Example 2]. Example 3. x5

- 2fii3 + 50.1? - 25 = 0

The Galois group of x5 - 2fii3

Equation (0) is

+ 50.1? - 25 is 7L/57L. Here

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 = 5 7 X 72, � = 54 X 7.

¢6 - 1875¢4 + 671875¢2 - 3500000¢ - 19140625 = 0. Step 1 k=

625.

Step 2 (} =

-v5 . 2-

Step 3

T = 0. Step 4 Step 5

R 1 = Y -25

+ IOV5.

R2 = Y - 25

-

IOV5. Concludes on next page

30

THE MATHEMATICAL INTELLIGENCER

Examples (continued) Step 6

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

_

ul -

-5

Y=

+ v -25 - 10V5 2

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

z2

, Z=

5 + V5 , 2

4

Step 7

25 + 15V5 - 5Y- 130 - 5sV5 4 Step 8

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

Step

9

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

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tions of the fifth order, Philosophical Transactions of the Royal So­

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

ciety of London CLI (1 861 ), 263-276. [1 5, Vol. IV, Paper 268, pp.

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

309-324.]

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

7. Arthur Cayley, Note on Mr. Jerrard's researches on the equation of the fifth order, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XXI (1 861), 2 1 0-21 4. [1 5, Vol. V, Paper 3 1 0, pp. 5Q-54.] 8. Arthur Cayley, On a theorem of Abel's relating to equations of the

(1 897), Vol. XIII (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

fifth order, The London, Edinburgh and Dublin Philosophical Mag­

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

azine and Journal of Science XXI (1 86 1 ) , 257-263. (1 5, Vol. V, Pa­

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

per 31 1 , pp. 55-61 .] 9. Arthur Cayley, Note on the solution of an equation of the fifth or-

London, Edinburgh and Dublin Philosophical Magazine and Jour­ nal of Science XX (1 860), 1 45-1 48.

VOLUME 24, NUMBER 4, 2002

31

A U THO R S

BRUCE C. BERNDT

BLAIR K. SPEARMAN

KENNETH S. WILLIAMS

Department of Mathematics

Department of Mathematics and Statistics

School of Mathematics and Statistics

University of Illinois

Okanagan University College

Carleton University

Urbana, Illinois

Kelowna, British Columbia V1V 1V7

Ottawa, Ontario K1 S 586

U.S.A.

Canada

Canada

e-mail: [email protected]

e-mail: [email protected]

e-mail: [email protected]

Bruce C. Bemdt became acquainted with

Blair K. Spearman completed his B.Sc. and

Kenneth S. Williams did his B.Sc. degree

Ramanujan's notebooks in February 1 974,

M.Sc. degrees at Carleton University in Ot­

at the University of Birmingham, England, attending lectures in the Watson Building.

while on a sabbatical year at the Institute for

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

Advanced Study. Since then he has devoted

gree in mathematics at Pennsylvania State

He completed his Ph.D. degree at the Uni­

almost all of his research efforts toward prov­

University underW. C. Waterhouse in 1 98 1 .

versity of Toronto in 1 965 under the su­

ing results from these notebooks and Ra­

He currently teaches at Okanagan Univer­

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

manujan's lost notebook. In 1 996 the Amer­

sity College, Kelowna, BC, Canada. His re­

the University of Manchester he came to

ican Mathematical Society awarded him the

search interests are in algebraic number

Carleton University in 1 966, where he has

Steele Prize for his five volumes on Ra­

theory.

been ever since. He served as chair of the

manujan's Notebooks. Similar volumes on

Mathematics Department from 1 980 to

the lost notebook, to be co-authored with

1 984

George Andrews, are in preparation. He is

currently on sabbatical leave working on a

and again from 1 997 to 2000. He is

most proud of his three biological children,

book on algebraic number theory with his

Kristin, Sonja, and Brooks, his seventeen

colleague Saban Alaca.

mathematical children, his five mathemati­ cal children in preparation, and his current postdoc.

1 9.

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

James Cockle, On the resolution of quintics, Quarterly Journal of

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

Pure and Applied Mathematics 4 (1 861 ), 5-7. 20.

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

30.

21 .

James Cockle, On transcendental and algebraic solution-supple­

31 .

matics 5 (1 862), 337-361 .

Magazine and Journal of Science XXIII (1 862), 1 35-139. Winifred A Cooke, George Neville Watson, Mathematc i al Gazette

32.

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

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

33.

Sigeru Kobayashi and Hiroshi Nakagawa, Resolution of solvable

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

34.

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

49 (1 965), 256-258. 23. 24.

25.

1 996.

quintic equation, Mathematc i a Japonicae 37 (1 992), 883-886.

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

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

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

3 1 5.

tice Hall, New Jersey, 1 99 1 . 26.

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

35.

Houghton Mifflin Co. , Boston MA, 1 998.

36.

27.

Joseph A Gallian, Contemporary Abstract Algebra, Fourth Edition,

28.

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

32

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

THE MATHEMATICAL INTELUGENCER

John Emory McClintock, Analysis of quintic equations, Amerc i an

Journal of Mathematics 8 (1 885), 45-84. John Emory McClintock, Further researches in the theory of quin­ tic equations, American Journal of Mathematics 20 (1 898),

matics 7 (1 885), 1 78-1 79. 29.

Robert Harley, On the theory of the transcendental solution of al­ gebraic equations, Quarterly Journal of Pure and Applied Mathe­

mentary paper, The London, Edinburgh and Dublin Philosophical 22.

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

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

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

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.

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

23-39.]

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

38. Srinivasa Ramanujan, Question 699, Journal of the Indian Mathe­ matical Society 7 (1 91 7), 1 99. (40: p. 331 .]

Birkhauser, Boston, 1 996, p p . 81 7-838. 62. Liang-Cheng Zhang, Ramanujan's class invariants, Kronecker's

39. Srinivasa Ramanujan, Notebooks, 2 vols., Tata Institute of Funda­ mental Research, Bombay, 1 957.

limit formula and modular equations (Ill), Acta Arithmetica 82 (1 997), 379-392.

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 q uintics 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), 8 1 -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 1 4 . 5 9 . George P . Young, Solution of solvable irreducible quintic equations, without the aid of a resolvent sextic, American Journal of Mathe­

STATISTICIANS ffl ENTURIES C of t h e �- �- �- �

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33

ip.iM$$j:J§..@hl£ili.JIIQ?-Ji

Dirk H uylebro u c k ,

Editor

Mathematics in M the Hal l of Peace Norbert Schmitz

iinster is one of the few cities fa­ mous not for a bloody battle but for a fruitful peace-the Peace of West­ phalia. In 1648, the signing of the peace treaty in Miinster and Osnabriick marked the end to the dreadful Thirty Years' War, which had caused unimag­ inable suffering throughout central Eu­ rope-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 indepen­ dence of the Netherlands. Both peace treaties were ratified in the Hall of Peace, the old council cham­ ber of the Miinster town hall. Famous for its magnificent gable, this town hall is regarded as one of the finest exist­ ing examples of secular Gothic archi-

Does your lwmetown have any mathematical toumt attractions such as statues, plaques, graves, the cafe where thefamous 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 e-mail: [email protected]

34

THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

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 itself looks back upon an eventful history of more than 1200 years. Around 1577, the Hall of Peace was decorated with a rich array of Renais­ sance woodcarvings. In the window re­ cesses, one can see Moses the Legisla­ tor as well as the seven liberal arts (for the history of these arts and their "por­ traits" by sculptors and painters, see "The Liberal Arts" by B. Artmann, The Mathematical Intelligencer 20 (1998), no. 3, 40-41), in particular Ars Arith­ metica with tablet and stylus and Ars Geometria with tablet and compass. Like the other carvings, these fig­ ures are embellished with ornaments,

Figure 2. The Hall of Peace.

Figure 3. Arithmetica and Geometria.

VOLUME 24, NUMBER 4, 2002

35

arabesques, and pediments with an­ gels' 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 visi­ tors' 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 (Augs­ burg), on the authority of M. Koecher could not be verified-either by check­ ing hundreds of pages of the visitors'

36

THE MATHEMATICAL INTELLIGENCER

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 Uni­ versity (Westfillische Wilhelms-Univer­ sitat Munster) is one of the leading departments in Germany. Here, F. Hirzebruch, H. Grauert, and R. Rem­ mert 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 de­ partment again embraced Arithmetica and Geometria.

G. Faltings wrote his Ph.D. and Ha­ bilitation theses in this department. Since 1998 a lively Sonderforschungs­ bereich (Special Research Field) "Geometrische Strukturen in der Math­ ematik" (Geometric Structures in Mathematics) is supported by the Deutsche Forschungsgemeinschaft. Yet, the mathematics building is less in­ teresting than the town hall-architec­ turally, at least. lnstitut fOr Mathematische Statistik Universitat Munster Einsteinstr. 62 D-48149 Munster Germany

LEON GLASS

Loo ki n g 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 statistical 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 blur­ ring of each dot, so that each dot grew. On the other hand, local inhibitory fields introduced by the charge transfer un­ derlying 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 itera­ tions), there was a pretty stable pattern of dots. This analogue system mimicked lateral inhibitory fields that play a role in developmental biology and visual per­ ception, 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 auto­ correlation function of the dots. But when I did this, I made a surprising finding. Super­ imposing 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 de­ scribed this effect and proposed a way that the visual sys­ tem 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 un­ ravel 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 ran­ domly sprinkle dots on the plane. Next we plot the loca-

© 2002 SPRINGER-VER LAG NEW YORK, VOLUM E 24, NUMBER 4, 2002

37

_I

:

.

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.

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.

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 1 5th iterations.

. ·. ·· ·.·

.

. . ' ·' . .

.

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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

�:cF·l� �

�

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

. �1.,..,. < ' 2 and n > 1, perfect strongly covering codes d o 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 choos­ ing n large enough. The technique somehow intertwines the binary case with the q-nary case. A

litftll;ifi

n ___:_

_

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 vec­ tors [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 con­ struction, but we now alter it in an in­ teresting way: let L be the set of ele­ ments of Qn whose image under T is a k-tuple whose coordinates are all non­ zero. 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 ba­ sis 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 proba­ bility comes much closer to the asymp­ totic bound. The basic idea is to make a random choice and then alter it as necessary. This is a well-known situa­ tion in coding theory: the best known explicit constructions fall short of what can be achieved by suitably mod­ ified 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 oth­ erwise, here is a collection of (some­ what) easier hat problems. The first two can be found in Peter Winkler's.charm­ ing 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 contribu­ tions to recreational mathematics. 10. The TV game show host introduces the following more extreme varia­ tion of the game. The hats game is played as described originally ex­ cept that passes are not allowed, and players making false state­ ments are executed. What "worst­ case" 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 varia­ tion? The team members are lined up in a manner that allows play­ ers 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 state­ ments are executed. All players hear all of the statements, but not their consequences. 12. Same as the previous problem, ex­ cept 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 as­ sume that their teammates are highly rational, and the rules per­ mit 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 de­ lightful 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, 301-3 1 3 in Puzzlers' Tribute, edited by David Wolfe and Tom Rodgers, A. K. Peters, Ltd., 2002. Department of Mathematics Reed College Portland, OR 97202 USA e-mail: [email protected]

VOLUME 24, NUMBER 4, 2002

49

l@ffli • i§rr6hlf119.1rr1rr11!.1h14J

MASS Program at Penn State Anatole Katok, Svetlana Katok, and Serge Tabachnikov

Marjorie Senec hal ,

Editor

T

he 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 mathe­ matics programs for American under­ graduates, both based abroad: Bu­ dapest 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

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal, Department

of Mathematics, Smith College, Northampton, MA 01 063 USA e-mail: [email protected]

50

three founders of the MASS pro­ gram (the first two authors of this ar­ ticle and the first MASS director, A Kouchnirenko) are steeped in the Russian tradition where interested stu­ dents are exposed to a variety of math­ ematical endeavors, often of nonstan­ dard kind, at an early age. By their senior undergraduate years such stu­ dents are already budding profession­ als. We briefly describe this tradition in the Appendix. The US educational sys­ tem is built on completely different principles, and interested young stu­ dents are routinely encouraged to progress quickly through the required curriculum. Here a typical mathemati­ cally 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 inter­ esting and challenging mathematics from the core fields of algebra, geom­ etry, and analysis, going way beyond the usual curriculum. The second author's first exposure to an intensive program for US under­ graduates was at the Mills College

THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

I

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 pro­ gram, is rather limited, and we decided not to restrict the pool of potential par­ ticipants. The result was the SURI (Summer Undergraduate Research Ini­ tiative) program at Penn State in the summer of 1993, where all three future founders of the MASS program came together. During this program it be­ came clear that a semester-long format would be even more productive for an intensive program organized mostly around advanced learning with ele­ ments of research initiation. And so we envisioned a semester­ long program for undergraduate stu­ dents from across the country. We thought it crucial for the success of the program that the cost for the partici­ pants 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 pro­ gram could begin.

Program Description The main idea of the MASS Program, and its principal difference from vari­ ous honors programs, math clubs, and summer educational or research pro­ grams, is its comprehensive character. MASS participants are immersed in mathematical studies: since the pro­ gram is intensive, its full-time partici­ pants are not supposed to take other classes. All academic activities for a se­ mester are specially designed and co­ ordinated to enhance learning and in­ troduce the students to research in mathematics. This produces a quantum leap effect: the achievement and en­ thusiasm of MASS students increases

much more sharply than if they had

most importantly, talk mathematics

been exposed to a similar amount of

most of the time. Of course, this is ex­

material over a longer time in a more

actly how "mature" mathematicians

conventional environment.

operate in their professional life! A

oped independently according to the •

interests and abilities of the student. A weekly 2-hour

interdisciplinary seminar run by the director of the

A key feature of the MASS experi­

necessary condition for this environ­

MASS program (the third author of

ence is an intense and productive in­

ment is the gathering of a critical mass

this article), which helps to unify all

teraction among the students. The en­

of dedicated and talented students,

vironment is designed to encourage

which is one of the chief accom­

such interaction: a classroom is in full

plish ments of MASS.

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

Let us describe the main compo­

Three

other activities. The MASS

colloquium, a weekly

lecture series by distinguished math­

nents of MASS: •

•

ematicians, visitors, or Penn State research facuity.

core courses on topics cho­

All elements of MASS

(3 courses,

class times. Each student has a key and

sen from the areas of Analysis, Al­

can enter the room 24 hours a day. The

gebra/Number Theory, and Geome­

16 credit hours, all listed as Honors

students live together in a contiguous

try!ropology. Each course features

classes that are transferable to MASS participants' home universities. Addi­

the seminar, and the colloquium) total

block of dormitory rooms and they

three 1-hour lectures per week, a

pursue various social activities to­

weekly meeting conducted by

a

tional recognition is provided through

gether. The effect is dramatic: the stu­

MASS Teaching Assistant, weekly

prizes for best projects and merit fel­

dents find themselves members of a

homework assignments, a written

lowships. Each student is issued a Sup­

cohesive group of like-minded people

midterm exam, and an oral final ex­ amination/presentation.

plement to the MASS Certificate, which includes the list of MASS courses with

Individual student research proj­ ects ranging from theoretical math­

special achievements. It also includes

study together, attack problems to­

ematics to computer implementa­

the descriptions of MASS courses, the

gether, debug programs together, col­

tion. Most of the projects are related

list of MASS colloquia, and the de­

laborate on research projects, and,

to the core courses; some are devel-

scription of MASS program exams.

sharing a special formative experience. They quickly bond, and often remain friends after the program is over. They

•

credits, grades, final presentations, and

photo © S. Katok

VOLUME 24, NUMBER 4, 2002

51

These supplements are useful for the

fundamental facts about finite groups

student's

and

question from the course and a prob­

pro­

lem. Then the student has an hour to

lences and enhance the student's ap­

ceeded to what is often referred to as

prepare the answers, with no access to

plications to graduate schools.

"quantum

home

institution

equiva­

their representations

and

of

literature or lecture notes during this

The core courses are custom de­

knots and 3-dimensional manifolds as­

hour. The answers to the ticket ques­

signed for the program and are avail­

sociated with statistical physics and

able only to its participants. Each

usual undergraduate

(and, in many

cases, even graduate) curriculum. For example, the core courses offered at MASS

2001 were

invariants

"The MASS

course addresses a fundamental topic which is not likely to be covered in the

topology":

p rog ram has been the best semester of

Mathematical Analysis of Fluid Flow by A. Belmonte, Theory of Parti­ tions by G. Andrews, and Geometry and Relativity: An Introduction by N.

my l ife. " the Yang-Baxter equation. The course

Higson.

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

a

was received by the students with great

ence: a speaker cannot assume much

course, an instructor is challenged to

enthusiasm and is likely to direct some

background

reach a delicate balance between cov­

of them toward this active area of re­

makes the speaker's task harder, we

ering the basics, with which the stu­

search.

Designing

and

teaching

such

dents might be unfamiliar, and intro­

material.

Although

this

find that the quality of the talks usually

The final exams (three, in total) have

benefits from this restriction. To quote

a unique format. It is quite unusual for

the opening sentences of an inspiring

a US university and represents a cre­

article by

Consider, for example, a MASS 2000

ative development of a European tradi­

course Finite Groups, Symmetry, and Elements of Group Representations by

good colloquium" (see at www. math.

A student draws a random "ticket"

"Most colloquia are bad. They are too

A. Ocneanu. This class started with

which typically contains a theoretical

technical and aimed at too specialized

ducing

advanced

material

typically

taught in topics courses.

tion where examinations are often oral.

J. McCarthy "How to give a

psu. edu/colloquium/go odcoll. pdf) :

photo © S. Katok

52

THE MATHEMATICAL INTELLIGENCER

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 as­ signed to take notes and prepare a readable exposition of the talk. We also experiment with videotaping the talks. Choosing the speakers, we always in­ vite mathematicians known for their ex­ pository skills. We also try to represent as broad a spectrum of mathematical re­ search as possible. We find it beneficial to combine very well-known mathe­ maticians with those in the early stage

photo © S. Katok

of their careers. A complete list of MASS colloquium talks can be found on the

portant function of the seminar is to

web site www.math.psu.edu/mass.

bring out elements of unity of modem

The MASS seminar plays many roles

mathematics. Often identical or similar

in the program. One of them is to in­

notions appear in different courses in

troduce the students to the topics that,

various guises, and the seminar is the

otherwise, are likely to "fall between

place to explore, develop, and clarify

cracks in the floor." For example, one

these connections.

of the seminar topics in 2001 was the

ber of guest speakers, mostly Penn

State ..faculty, give expository talks at

the conference.

Here are two examples of REU stu­ dents' research projects.

·

"Simplices with only one integer point" (2 students; faculty mentor A. Borisov) . The students found an effective proce­

jective geometry: Pappus, Desargues,

The Summer Program: REU and MASS Fest

Pascal, Brianchon, and Poncelet. Once

The Penn State Summer RED (Re­

wojective geometry was a core subject

search Experiences for Undergradu­

in the university curriculum, but nowa­

ates) program started in 1999 as an ex­

days it is perfectly possible to obtain a

tension of MASS. Unlike MASS, this

doctoral degree in mathematics without

program is not unique: currently, there

a single encounter with these facts. An­

are about 50 RED programs in mathe­

"New congruences for the partition

other example: the theory of evolutes

matics available to undergraduate stu­

function" (1 student; faculty mentor

and involutes was a crowning achieve­

dents in the USA. The Penn State RED

Ono). This project started before the

classical configuration theorems of pro­

dure that allows them to describe all classes of simplices with vertices that have only integer coordinates and only one point with integer coordinates in­ side. Using computers they found all classes in dimensions 3 and

4. K.

ment of Calculus to be included into

is closely related to MASS: about half

RED program began. Using the theory

textbooks. Alas, a contemporary stu­

of its participants stay for the MASS se­

of Heeke operators for modular forms

dent is not likely to see these things any

mester in the fall. This makes it possi­

of half�integral weight, the student found

more. The MASS seminar is a natural

ble to offer research projects that re­

an algorithm for primes 13 :::;;

place to learn such a topic.

quire more than 7 weeks (the length of

which reveals 70,266 new congruences

Another purpose of the seminar is to prepare the students for the up­

RED program) for completion.

of the form p(An

Mathematical research usually in­

+ B) == 0

m :::;; 31

(mod

m),

where p(n) denotes the number of un­

coming MASS colloquium talks. A col­

cludes three components: study of the

restricted partitions of a non-negative in­

loquium speaker is asked whether cer­

subject, solving of a problem, and pres­

teger n. As an example, she proved that

tain material should be covered in

entation of the result. These three com­

p(3828498973n

advance so that the students get the

ponents are present in the RED pro­

for every integer n. The first three con­

+ 1217716)

==

0 (mod 13)

most from the talk. For example, as

gram: in addition to the traditional

gruences were found in 1 9 19 by Ra­

preparation for A. Kirillov's talk on

individual/small group research proj­

manujan, and after that finding new

Family Algebras in 2001, a 2-houJ sem­

ects supervised by faculty members,

ones was considered a very difficult

inar was devoted to the basics of Lie

the program includes two short courses,

problem. The paper written by this stu­

groups and Lie algebras. Still another

a weekly seminar, and the

dent has been accepted for publication.

MASS

Fest.

function of the seminar is to rehearse

MASS Fest is a 3-day conference at

the students' presentations of the re­

the end of the REU period at which the

search projects on the

participants present

,final exam. This

their

We would like to emphasize a unique

research.

role played by the RED coordinator, M.

usually occupies the last quarter of the

This is also a MASS alumni reunion.

Guysinsky, who has been coming to

semester. Probably an even more im-

Along with the RED students, a num-

Penn State for the summer since 1999

VOLUME 24, NUMBER 4, 2002

53

as a visiting Assistant Professor sup­ ported 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 re­ quires an unusual combination of math­ ematical and pedagogical talents, and we are very fortunate to have found this combination in Guysinsky.

Participants MASS participants are selected from ap­ plicants currently enrolled in US col­ leges or universities who are juniors, se­ niors, 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 cal­ culus sequence, basic linear algebra, and advanced calculus or basic real analysis. The search for participants is nation­ wide. Participants are selected based on academic record, recommendation let­ ters from faculty, and an essay. The number of MASS participants varies from year to year, with an aver­ age 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 cate­ gories: (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 cat­ egory, 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 pro­ gram). Another statistic: women rep­ resented 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 math­ ematics (one should keep in mind that some recent participants are still con­ tinuing their undergraduate studies). The distribution of the graduate schools is very wide. Without provid­ ing a comprehensive listing, we men­ tion some: Harvard, Cornell, Stanford, Princeton, Yale, University of Chicago, University of Michigan, University of California at Berkeley, University of Wisconsin, Indiana University, Univer­ sity 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 moti­ vated students and professors and challenged as never before. I was pushed by instructors, fellow-stu­ dents, and something deep inside myself to work and learn about math­ ematics, and my place in the mathe­ matical 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 in­ strumental 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. TAs are chosen from among the most accomplished Ph.D. students of the Penn State Department of Math­ ematics. 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 meet­ ing 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 them­ selves MASS graduates.

Student Research During the semester, each MASS par­ ticipant works on three individual projects. Usually a project consists in learning a certain topic in depth, work­ ing on problems (ranging from routine exercises to research problems, usu­ ally related to the subject of the re­ spective course), and making a pres­ entation 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 pro­ duced a significant piece of mathemati­ cal 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 ap­ pearance 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

+ Y4 + 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 ofthe form x2 y2 + 7z2 where x, y, and z are in­ tegers? 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 mod­ ular 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, "Ka­ plansky'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 biol­ ogy 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.

54

THE MATHEMATICAL INTELLIGENCER

matical analysis of fluid flow" has a re­ search project "A mathematical analysis of fluid flow through the urinary system." MASS students present their re­ search projects at the Undergraduate Student Poster Sessions at thi!.January AMSIMAA joint meetings. For exam­ ple, 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 out­ of-state students that reduce their tu­ ition to the in-state level. Further sup­ port 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 in­ dividually based on merit and need. In particular, several merit fellowships are awarded at the end of the MASS semes­ ter. The VlGRE grant also supports the MASS colloquium series by covering the speakers' travel expenses.

Perspectives We are confident that MASS will con­ tinue 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 sup­ port of the program. There is a con­ siderable interest in mathematics among private and corporate aonors, and the contribution of the MASS program to undergraduate mathe­ matics education is substantial. Ide­ ally, 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 for­ eign 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 lec­ ture notes style, detailed enough to serve as guidelines for a qualified in­ structor to design a similar course. As a first step, we are preparing a MASS presentation volume that will be published by the American Math­ ematical Society. This book will pre­ sent 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 on­ line, 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 im­ proving the mathematical education in the USA.

Appendix: On the Russian Tradition of Mathematical Education Russian mathematics constitutes one of the most vital and brilliant mathe­ matical traditions of the 20th century. Mathematicians trained in Russia are very well represented in the top eche­ lon of the world mathematical com­ munity. 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 all­ Union one (the first Mathematical Olympiad in the Soviet Union was held in Leningrad in 1934, and Moscow fol­ lowed suit the next year; the first all­ Union 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 usu­ ally met at the university once a week in the evening and were run by dedi­ eatenundergraduate or graduate stu­ dents with a tremendous enthusiasm for mathematics, very ·often them­ selves alumni of a kruzhok-a good example of "vertical integration"! The material discussed usually went well beyond the secondary school curricu­ lum and included challenging prob­ lems and nonstandard topics from ele­ mentary 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 lo­ cal 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 mathe­ matics in universities across the globe. Another well-known high school, the Boarding School for Mathematics No. 18 at Moscow State University, was estab­ lished 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 af­ filiated with the university and specially funded by the state. Still other cele­ brated Moscow schools for mathemat­ ics 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 cur­ riculum of a special school was more in-

VOLUME 24, NUMBER 4, 2002

55

Kvant, there was a

tensive and systematic than that of the

fession. Along with

kruzhki,

and this influenced our think­

rich popular literature; nwnerous col­

mix of undergraduates, graduates, and

ing about the structure of the MASS pro­

lections of problems for all ages, and

established

books on various topics in "serious"

from the third year of the university,

mathematics. We would like to mention

every student had an advisor and was

gram. An essential part of the tradition was

seminars were usually attended by a mathematicians.

Starting

the participation of prominent mathe­

some people who made a very substan­

considered a member of a research

maticians of various ages in teaching

tial contribution to popularization of

community in his or her field. It was not

and popularizing mathematics. A typical

mathematics: N.

B. Vasiliev, N. Ya

unusual for the best undergraduate stu­

Kvant (mean­

Vilenkin, I. M. Yaglom. The third author

dents at major universities to have pa­

ing "Quantum") on physics and mathe­

of this article was for a nwnber of years

pers published in first-rate research

example is the magazine

matics for school students published

Kvant had 12

since 1970.

issues a year

the Head of

Kvant's

Mathematics De­

partment.

journals by the end of their

5 years of

undergraduate studies. This system had multiple effects. On

and, at the peak of its popularity in the

At the university level, the emphasis

mid-1970s, boasted more than 300,000

on creative thinking continued, some­

the one hand, it stimulated early de­

subscribers. Among the authors were

times to the detriment of systematic

velopment of research interests and

well-known

D.

leaining. For example, the standard

mathematical precocity. On the other

B. Fuchs, I.

mandatory courses often did not fully

hand, it often led to inflated standards

M. Gelfand, S. G. Gindikin, A. A. Kirillov,

reflect the most current thinking in their

and expectations, and eventually to a

subjects, and were looked down on by

great waste of talent. A student with

mathematicians

Alexandrov, V. I. Arnold, D.

A.

A. N. Kolmogorov, M. G. Krem, Yu. V.

L. S.

the top students. A very important role

considerable talent but not very high

Pontryagin, among many others. For

was played by topics courses, offered in

self-esteem might be crushed by the system. Still, it succeeded spectacu­

Matiyasevich, S. P. Novikov, and

Kvant

a wide variety of subjects and attended

opened new horizons and determined

by a mixture of undergraduate and grad­

larly in producing creative and techni­

their choice of mathematics as a pro-

uate

cally powerful mathematicians.

many generations of students,

students.

Similarly,

specialized

A UTH O R S

ANATOLE KATOK

Department

SVETLANA KATOK

of Mathematics

Department

Department

of Mathematics

Pennsylvania State University

Pennsylvania State University

Pennsylvania State University

University Park, PA 1 6802

University Park, PA 1 6802

University Park, PA 1 6802

USA

USA

USA

e·mail: [email protected]

e-mail: [email protected]

e-mail: [email protected]

Serge Tabachnikov wrote his thesis (1 987)

Anatole

in the

Svetlana Katok (daughter of B.A. Rosen­

"Moscow mathematical school," as were

Katok was educated

feld, a "grand old man" of Moscow geom­

at Moscow State University on differential

his co-authors; A.N. Kolmogorov was

etry) immigrated to the United States in

topology and homological algebra. Later his

ref­

eree of his doctoral thesis. After immigrat­

1 978

ing in

got her Ph.D. in

1 978

to the United States (the coun­

with her husband 1 983

and

children, and

at the University of

interests shifted to symplectic geometry and Hamiltonian dynamics,

as

reflected in his

try of his birth), he taught at Maryland and

Maryland. Her research is on automorphic

book Billiards. Before coming to Penn State

Caltech before coming in

forms, dynamical systems, and hyperbolic

he taught at the University of Arkansas.

1 990

to Penn

State. Among his numerous publications

geometry. She is author of

are two books with his former student Boris

Groups

Hasselblatt:

ticle "Women in Soviet Mathematics,"

Introduction to the Modern

Theory of Dynamical Systems

forthcoming The

and the

Rrst Course in Dynamics

with a Panorama of Recent Developments.

56

SERGE TABACHNIKOV

of Mathematics

THE MATHEMATICAL INTELLIGENCER

Fuchsian

and Oointly with A. Katok) of the ar­ No­

tices of the American Mathematical Soci­ ety 40 (1 993), 1 08-1 1 6.

l]¥1f9·i.(.j

David E . Rowe , Editor

Einstein's Gravitational Field Equations and the Bianchi Identities David E. Rowe

j

I

n 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 gener­ ally covariant equations for gravitation ([Pais], pp. 250-261 ). This story includes the famous competition between Hilbert and Einstein, an episode that has re­ cently been revisited by several histori­ ans in the wake of newly discovered documentary evidence, first presented in [Corry, Renn, Stachel 1997]. In his earlier account, Pais empha­ sized that "Einstein did not know the [contracted] Bianchi identities

(

)

RJJ-V - .!gJJ-V R 0 (1) 2 ;v when he wrote his work with Gross­ mann." (The symbol ';' denotes covari­ ant differentiation, which here is used as the generalized divergence operator.) In 1913 Einstein and the mathe­ matician Marcel Grossmann presented their Entwurjfor a new general theory of relativity. Guided by hopes for a generally covariant theory, they never­ theless resolved to use a set of differ­ ential equations for the gravitational field that were covariant only with re­ spect to a more restricted group of transformations. However, when Ein­ stein 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)

1 RJJ-V - - gJJ-V R 2

(3)

where GJJ-V

Send submissions to David E. Rowe, Fachbereich 1 7 - Mathematik, Johannes Gutenberg University, 055099 Mainz, Germany.

==

is the Einstein tensor. (Here TJJ-v is the energy-momentum tensor and gJJ-v the metric tensor that determines the prop­ erties of the space-time geometry. The contravariant Ricci tensor RJJ-v is ob­ tained by contracting the Riemann­ Christoffel tensor; contracting again R. yields the curvature scalar RJJ-vg JJ-V =

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 indepen­ dent, 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 lat­ ter, 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 trans­ formations. Choosing a specific coordi­ nate system thus singles out a unique so­ lution among this family. Einstein for a long time resisted drawing this seemingly obvious conclu­ sion. Instead he concocted a thought experiment-his infamous hole argu­ ment-that purported to show how gen­ erally covariant field equations will lead to multiple solutions within one and the same coordinate system (see [NorJ989] and [Sta 1989]). His initial efforts there­ fore aimed to circumvent this paradox of his own making, for, on the one hand, physics demanded that generally co­ variant gravitational equations must ex­ ist, whereas logic (mixed with a little physics) told him that no such equations can be found (see his remarks in [Ein­ stein 1914], p. 574). Luckily, Einstein had the ability to suppress unpleasant con­ ceptual problems with relative ease. And so in November 1915 he plunged ahead in search of generally covariant equa­ tions, 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 ex­ plained in [Nor 1989] and [Sta 1989]). By 1916 Einstein was also quite aware that his field equations led di­ rectly to the conservation laws for mat­ 0. Nevertheless, he was ter T:Vv rather vague about the nature of this =

=

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

57

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 indepen­ dent 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 pro­ nouncement 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 tu­ multuous events of November 1915:

alize that the energy-momentum conservation laws

Einstein stiU did not know [the con­ tracted Bianchi identities] on No­ vember 25 and therefore did not re-

Pais's 20-20 hindsight no doubt identifies this particular source of Einstein's diffi­ culties, but it hardly helps to explain

Tf;," = 0

follow automatically from (1) and (2). Instead, he used these conser­ vation laws [ (5)] as a constraint on the theory! ([Pais 1982], p. 256).

Mathematische Gesel lschaft

'ula.

Hansen.

BIWDeuthal.

c. Miller.

SdUilinr.

JWIIert.

Dane7.

H1111el.

&bMidt.

1 90 2 .

H. lflllln. St:hwantelalhl.

(5)

r�hiYf.

R�t�n.

1-,l'i:�l-' ·

The identity of Bianchi seems, ho·wever, to be :published for the first time by E. Padova (Rendiconti .A ce. Lincei, v. (4) pp. 174-178, 1889) , who obtained it from G. Ricci. Com­ pare for this the book of Struik ( Grundzuge der mehrdimen ­ sionalen Dijfe1·entialg eometrie : B erlin, ,J. Springer, 1!j22, p. 141:l) , where the theorem is proved in a similar way as Mr. Harward do es. A similar p roof 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 gen eraliz ed geometries of Schouten (.1.1-latltem. Zeitschr. xiii. pp. 56-8 1 , 1922), o f 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, 1 923. 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.

64

THE MATHEMATICAL INTELLIGENCER

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 Ency­ clopii.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 so­ licited 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 Gross­ mann, Ricci's absolute differential cal­ culus 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 pseu­ donym 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 lat­ ter "relations appear to be still com­ pletely unknown" (ibid.). Forster con­ templated 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 as­ tronomer Willem De Sitter introduced the British scientific community to Ein­ stein's mature theory. This helped spark Arthur Stanley Eddington's in­ terest in GRT and the publication of [Eddington 1918], which contains the contracted Bianchi identities. Two years later, in [Eddington 1920], he ex­ pressed doubt that anyone had ever verified these identities by straightfor­ ward calculation, and so he went ahead and carried this out himself for the the­ oretical supplement in the French edi­ tion of [Eddington 1920]. In 1922 Eddington's calculations were simplified by G. B. Jeffery, and al­ most immediately afterward the Eng­ lish physicist A. E. Harward reproved Bianchi's identities (9) and used them to derive the conservation of energy­ momentum 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 Iden­ tity.' " More importantly, they empha­ sized that similar identities hold in affme spaces (those that do not ad­ mit a Riemannian line element ds 2 = gJJ- v dxJJ-dx v). Regarding these, they re­ ferred explicitly to [Bach 192 1 ] 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 Ricci­ Kalkiil. 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 im­ portance 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 consider­ able expertise in the use of sophis­ ticated 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 ten­ sor analysis had at least one important payoff. It gave the aged Felix Klein an inducement to explore the mathemati­ cal foundations of general relativity theory. He did so by drawing on ideas familiar from his youth, most impor­ tantly Sophus Lie's work on the con-

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 re­ marks on an earlier version of this col­ umn. The editor deserves a note of thanks, too, for posing questions that helped clarify some obscure points. Re­ maining errors and misjudgments are, of course, my own, and may even reflect a failure to heed wise counsel. REFERENCES

[Acz 1 999] Amir D. Aczel, God's Equation. Ein­ stein, Relativity and the Expanding Universe,

Mathematicians often prefer to fig u re out some­ t h i ng on thei r own rather than read someone else ' s work.

New York: Delta, 1 999. [Bach 1 92 1 ] Rudolf Bach, "Zur Weylschen Rela­ tivitatsthearie und der Weylschen Erweiterung .

des l