No title

ERIC C. R. HEHNER

From Booean Agebra to Unified Agebra

oolean algebra is simpler than number algebra, with applications in programming, circuit design, law, specifications, mathematical proof, and reasoning in any domain. So why is number algebra taught in primary school and used routinely by scientists, engineers, economists, and the general public, while boolean algebra is not taught until university, and not routinely used by anyone? A large part of the answer may be in the terminology and symbols used, and in the explanations of boolean algebra found in text­ books. This paper points out some of the problems delaying the acceptance and use of boolean algebra, and sug­ gests some solutions. Introduction

This paper is about the symbols and notations of boolean algebra, and about the way the subject is explained. It is about education, and about putting boolean algebra into general use and practice. To make the scope clear, by "boolean algebra" I mean the algebra whose expressions are of type boolean. I mean to include the expres­ sions of propositional calculus and predicate calculus. I shall say "boolean algebra" or "boolean calculus" inter­ changeably, and call the expressions of this algebra "boolean expressions". Analogously, I say "number algebra" or "number calculus" interchangeably, and call the expressions of that algebra "number expressions". Boolean algebra is the basic algebra for much of computer science. Other applications include digital circuit de­ sign, law, reasoning about any subject, and any kind of specifications, as well as providing a foundation for all of mathematics. Boolean algebra is inherently simpler than number algebra. There are only two boolean values and a few boolean operators, and they can be explained by a small table. There are infinitely many number values and number operators, and even the simplest, counting, is inductively defined. So why is number algebra taught in pri­ mary school, and boolean algebra in university? Why isn't boolean algebra better known, better accepted, and bet­ ter used? One reason may be that, although boolean algebra is just as useful as number algebra, it isn't as necessary. Informal methods of reckoning quantity became intolerable several thousand years ago, but we still get along

© 2004 SPRINGER-VERLAG NEW YORK, LLC, VOLUME 26, NUMBER 2, 2004

3

with informal methods of specification, design, and reasoning. Another reason may be just an accident of edu­ cational history, and still another may be our continuing mistreatment of boolean algebra. Historical Perspective To start to answer these questions, I'm going to look briefly at the history of number algebra. Long after the in­ vention of numbers and arithmetic, quantitative reasoning was still a matter of trial and error, and still conducted

3 goats and 20 chickens to be divided equally between his 2 sons, 8 chickens, the solution was determined by iterative approximations, prob­

in natural language. If a man died leaving his and it was agreed that a goat is worth

ably using the goats and chickens themselves in the calculation. The arithmetic needed for verification was well understood long before the algebra needed to find a solution. The advent of algebra provided a more effective way of finding solutions to such problems, but it was a dif­ ficult step up in abstraction. The step from constants to variables

is as large as the step from chickens to num­

bers. In English 500 years ago, constants were called "nombers denominate" [concrete numbers], and variables were called "nombers abstracte". One of the simplest, most general laws, sometimes called "substitution of equals for equals",

x=y�fx=fy seems to have been discovered a little at a time. Here is one special case [20]: In the firste there appeareth

2

nombers, that is

14x+15y

equalle to one nomber, whiche is

71y .

But if you

marke them well, you maie see one denomination, on bathe sides of the equation, which never ought to stand. Wherfore abating [subtracting] the lesser, that is that is, by reduction,

lx=4y .

15y

out of bathe the nombers, there will remain

Scholar: I see, you abate

15y

14x =56y

from them bathe. And then are thei equalle still,

seyng thei wer equalle before. According to the thirde common sentence, in the patthewaie: If you abate even [equal] portions, from thynges that bee equalle, the partes that remain shall be equall also. Master: You doe well remember the firste grounds of this arte. And then, a paragraph later, another special case: If you adde equalle portions, to thynges that bee equalle, what so amounteth of them shall be equalle. Each step in an abstract calculation was accompanied by a concrete justification. For example, we have the Commutative Law [0]: When the chekyns of two gentle menne are counted, we may count first the chekyns of the gentylman hav­ ing fewer chekyns, and after the chekyns of the gentylman having the greater portion. If the nomber of the greater portion be counted first, and then that of the lesser portion, the denomination so determined shall be the same. This version of the Commutative Law includes an unnecessary case analysis, and it has missed a case: when the two gentlemen have the same number of chickens, it does not say whether the order matters. The Associative Law [0]: When thynges to be counted are divided in two partes, and lately are found moare thynges to be counted in the same general! quantitie, it matters not whether the thynges lately added be counted together with the lesser parte or with the greater parte, or that there are severalle partes and the thynges lately added be counted together with any one of them. As you can imagine, the distance from

2x+ 3 =3x+ 2

to

x= l

was likely to be several pages. The reason for all

the discussion in between formulas was that algebra was not yet fully trusted. Algebra replaces meaning with symbol manipulation; the loss of meaning is not easy to accept. The author constantly had to reassure those readers who had not yet freed themselves from thinking about the objects represented by numbers and vari­ ables. Those who were skilled in the art of informal reasoning about quantity were convinced that thinking about the objects helps to calculate correctly, because that is how they did it. As with any technological advance, those who are most skilled in the old way are the most reluctant to see it replaced by the new. Today, of course, we expect a quantitative calculation to be conducted entirely in algebra, without reference to

thynges.

Although we justify each step in a calculation by reference to an algebraic law, we do not have to

keep justifying the laws. We can go farther, faster, more succinctly, and with much greater certainty. In a typi­ cal modem proof we see lines like

4

THE MATHEMATICAL INTELLIGENCER

Arar=(bab-1Y=barb-1=ar br=Arbr=(AbY=(a-1baY=a-1bra (a1-1b1)2=a1-1b1a1-1b1=a1-1Cb1a1 -1)b1=a1 -1(p,a1 -1b1)b1=JLa1-2b12 (a1-1b1y=JL1+2+... +(,·-1)a1-rb{=JL1+2+. . +(r-1)=JLr(r-1)12 These lines were taken from a proof of Wedderburn's Theorem (a finite division ring is a commutative field) in

[15] (the text used when I studied algebra). Before we start to feel pleased with ourselves at the improvement, let me point out that there is another kind of calculation, a boolean calculation, occurring in the English text be­ tween the formulas.In the example proof

[15] we find the words "consequently", "implying","there is/are", "how­

ever","thus", "hence", "since", "forces", "if ...then", "in consequence of which", "from which we get","whence", "would imply", "contrary to", "so that","contradicting"; all these words suggest boolean operators. We also find bookkeeping sentences like "We first remark ...","We must now rule out the case ...";these suggest the struc­ ture of a boolean expression.It will be quite a large expression, perhaps taking an entire page. If written in the usual unformatted fashion of proofs in current algebra texts, it will be quite unreadable. The same problem oc­ curs with computer programs, which can be thousands of pages long;to make them readable they must be care­ fully formatted, with indentation to indicate structure. We will have to do likewise with proofs. A formal proof is a boolean calculation using boolean algebra; when we learn to use it well, it will enable us

to go farther, faster, more succinctly, and with much greater certainty.But there is a great resistance in the math­ ematical community to formal proof, especially from those who are most expert at informal proof. They com­ plain that formal proof loses meaning, replacing it with symbol manipulation. The current state of boolean al­ gebra, not as an object of study but as a tool for use, is very much the same as number algebra was five centuries ago.

Boolean Calculation

Given an expression, it is often useful to find an equivalent but simpler expression. For example, in number al­ gebra

xx(z+1) - yX(z-1) - zX(x-y) (xxz + xX1) - (yxz - yX1) - (zxx- zxy) xxz + x - yxz + y - zxx + zxy x + y + (xXz - xxz) + (yXz - yxz) x+ y

distribute unity and double negation symmetry and associativity zero and identity

We might sometimes want to find an equivalent expression that isn't simpler;to remove the directionality I'll say "calculation" rather than "simplification". We can use operators other than

= down the left side of the calcula­

tion; we can even use a mixture of operators, as long as there is transitivity. For example, the calculation (for real

x)

2:

xx(x + 2) x2 + 2 Xx x2 + 2 Xx + 1 - 1 (x + 1)2 - 1

distribute add and subtract

1

factor a square is nonnegative

-1

tells us

xX (x + 2) 2: -1 Boolean calculation is similar. For example,

-

-

-

-

And so

{:=:

(a==>b) V (b==>a) -,a V b V --,b V a a V -,a V b V --,b ru t e V tru e

replace implications

V is symmetric excluded middle, twice

V is idempotent

tru e

(a==>b) V (b==>a) has been simplified to 3n· n + n2 = n�' instance 0 + Q2 = Q3 arithmetic

ru t e

,

which is to say it has been proven. Here is another example.

ru t e And so

(3n· n + n2 = n3)

f2, . . . } of C[a,b 1. Its proof, in fact, con­ tains the main ingredient of the Hahn­ Banach theorem: it is shown that if the inequality holds for a given n (and all f.Li) and if fn + 1 is an additional contin­ uous function on [a,b), then it is possi­ ble to find Cn + 1 such that n+1

I I f.LiCi I i�1

:5

n+1

M il I f.Lifi I I i�1

holds for all choices of f.Li· Both Hans Hahn and Stefan Banach were later to use this idea in their proofs of the Hahn-Banach theorem, without quot­ ing Helly. It would take decades before Helly got due credit for his achieve­ ments. By 1913 Helly's prospects looked good. He had secured a position as a

Preparing a Course

Indeed, Eduard Helly, born June 1, 1884, in Vienna, had done brilliantly as a student there under Ludwig Boltz­ mann, Franz Mertens, Wirtinger, and the young Dozent Hans Hahn. In 1907, he wrote his Ph.D. thesis on one of the hottest topics of that time: "Contribu­ tions to the theory of Fredholm's inte­ gral equation." Wirtinger was impressed and organised a stipend, from the Fund for the Training of Future University Teachers, which allowed Helly to spend his postdoc years where it mattered: in Gottingen, with David Hilbert, Felix

was nothing else but a norm. As Harro Heuser wrote later, "in a curious way the abstract norm existed before the abstract normed space." In the same paper, Helly extended Riesz's solution of the moment prob­ lem to prove the following theorem (in modem terminology): Let the func­ tions fi E C[a,b), the real numbers ci ,

Eduard Helly. There exist many photographs of Helly, but not one that shows him smiling. Yet despite his pensive, occasionally melancholic look, he is consistently described as a very warm and friendly person with a large circle of friends.

school teacher (just as Frigyes Riesz had done a few years earlier), he was highly visible in the Viennese Mathe­ matical Society, and he gave courses at the Viennese Volksbildungsheim-a kind of worker's university, quite ad­ vanced for its time. In addition, he spent a lot of his free time at the Vien­ nese Mathematical Seminar, which had

VOLUME 26, NUMBER 2, 2004

23

Seeking knowledge, eighteen-year-old Eduard enrolled at the University of Vienna, the Technical University, and (later) the University of Got­ tingen. Students had to keep a booklet to register the "testates" of their professors. Helly's book contains the signatures of Boltzmann, Wirtinger, Mertens, Hahn, and Hilbert.

just moved into an imposing new build­ ing not far from his flat. Helly had also just discovered a ba­ sic theorem of convex geometry. In­ deed, while handling the above system of inequalities, he had used a simple statement: if a family of compact in­ tervals has the property that aU pair­ wise intersections are non-empty, then the intersection of aU intervals is non­ empty. He now hit upon a splendid gen­ eralisation: if a family of compact con­ vex sets in R"' has the property that any n + 1 of them have non-empty inter­ section, then the intersection of aU the sets is non-empty. Helly gave a lecture on this result, which would lead later to a luxurious growth of "Helly-type" the24

THE MATHEMATICAL INTELLIGENCER

orems, but he did not publish it-there seemed no hurry, and he wanted first to work out some generalisations. Moreover, Helly was in love: a young mathematics student, Elise (Liese!) Bloch, who had the good sense to work on functions of bounded variation­ she tried to extend its definition to sev­ eral variables, no trivial task-was clearly sensitive to his attentions. Ed­ uard had dedicated a handwritten ver­ sion of his first paper to Fraulein Bloch as early as 191 1 , but it would be ten years before they would marry. The Far End of the Line

In 1914, when the war broke out, Helly volunteered on the spot, like many

young intellectuals. His military train­ ing took place close to Vienna, and he was even able, for a time, to keep lec­ turing at the Volksbildungsheim. But in 1915, his unit was sent to the Russian front, where things were getting out of hand. In September of that year he was shot through the lung, and captured. The shot had dislocated his heart, but Helly somehow survived the wound, and a series of Russian hospitals, in Kiev, Kursk, and Voronezh. In 1916, he was deemed fit enough for transporta­ tion to Siberia; his camp, Berezovka, was close to Tobolsk. By 1917, the war with Russia ground to an end, and POWs were supposed to be returned to their home countries. But there was a

"Fur Elise." In 1 91 1 , Eduard Helly dedicated two theorems to his beloved, Fraulein Bloch. It would take ten years before the two were married, and even longer before the theo­ rems were fully appreciated.

1)) .

fi!Ltl 03� �-�

9� aJt o( eo

-

%11

....._

•

�-- ·

"Siberiaks" would meet, years later, and reminisce about the Siberian clar­ ity of their seclusion. Camp authorities did not greatly worry about escape at­ tempts; where might escapees turn? Su­ pervision, while erratic, was usually light. In summer, posses of POWs would roam far and wide as lumberers: in win­ ter, most camps produced their own brand of cultural life, with makeshift universities, printing presses, and mu­ sical performances. Eduard Helly found some fellow-inmates interested in mathematical research. One, Paul Elbogen, wrote an "Introduction to ax­ iomatic methods" which was pub­ lished, ten years later, by his bereft mother. In the foreword, Helly wrote about "the terrible physical and psy­ chical pressure" under which the book­ let had been written. He himself re­ sumed the lecturing which he had been obliged to give up at the worker's uni­ versity, and found an exceedingly tal-

1 '1 -1 -1 .

snag: the Russian revolution had caused a fierce civil war, and indepen­ dent units of the Whites, the Reds, Eng­ lish and Japanese intervention troops, a free-roaming Czech unit, and sundry semi-military gangs led by desperate warlords ranged up and down the for­ mer Czarist empire and fought for chunks of the Trans-Siberian railway. In 1918, the prisoners finally left their camp, but only to end up in another camp, even further away, in Nikolsk­ Ussurisk, the last stop on the line be­ fore Vladivostok. Siberian internment camps, from well before Dostoevski to well after Solzhenitsin, were wellsprings of intellectual heroism. The fierce climate and the utter remoteness led to a solidarity among the inmates which often endured after their release: such old

Pop science. The municipality of Vienna strongly supported movements for spreading knowl­ edge to the masses. Mathematicians Eduard Helly, Hans Hahn, and Heinrich Tietze, physi­ cists Erwin Schrodinger and Friedrich Kohlrausch, and philosopher Karl Popper lectured to non-academic audiences. Helly would later use the skills acquired at this "workers' univer­ sity" to teach calculus to fellow prisoners of war, then to US Army officer candidates.

VOLUME 26, NUMBER 2, 2004

25

.-.­

-

---

A day in the life of Lt. Helly (POW}. Despite the terrible hardships of Siberian camps, some of the young men were later to view their years in trans-Baikalean remoteness as a formative experience of almost other-worldly quality.

ented young Hungarian inmate named Tibor Rad6, a student destined to go far. Incidentally, Rad6 went far in more than one sense; in 1920 he escaped north and walked home (a hike of sev­ eral thousand miles through the Arc­ tic), and in 1930-by then a professor at Ohio State University-he immor­ talised his name by solving the Plateau problem. In the late summer of 1920 things fi­ nally got moving for Helly. After stopovers in Manchuria, Japan, and Egypt (where he spent another few months in a British internment camp), he returned to Vienna in December 1920, ending a five-year odyssey. His Penelope was waiting for him: Elise Bloch, after finishing her PhD thesis under Wirtinger's supervision in 19 15, had found a job as a school teacher. 26

THE MATHEMATICAL INTELLIGENCER

They would not lose any more time; they would marry as soon as Eduard got this habilitation.

cepted by the Monatshefte, although there was no money to go to press. Functional analysis had come of age after the war, with three almost simul­ taneously appearing memoirs by Ba­ nach, Wiener, and Hahn. A few weeks after Helly's return, Hahn was recalled from Bonn as the third full professor at the Mathematical Seminar. Furtwan­ gler and Wirtinger were glad to see someone from the younger generation in charge, and Hahn took command with boundless energy. He immedi­ ately recognised the value of Helly's manuscript and wrote a glowing re­ port. Within a few months, Helly be­ came Dozent and got married. But his hope for an appointement at the Uni­ versity failed to materialise. Only one position was vacant, that of an extra­ ordinary professor for geometry. Hahn paved the way, not for Helly, but for a young German, Kurt Reidemeister. Elise Helly, much later, hinted at an anti-Semitic factor behind the decision. Anti-Semitism was indeed rampant in Austria, and particularly at the Univer­ sity. The current head of state, prelate Ignaz Seipel, went on record with his opinion that the percentage of Jewish students was too high and would have to be limited. (He coined the word Notwehrantisemitismus-Notwehr means "self-defense".) However, Hahn himself was Jewish. Some might have felt that the appointement of another Jewish professor would be undiplo­ matic, but Hahn was unlikely to be moved by such timorous considera­ tions. In any case, Reidemeister proved to be a brilliant choice, and functional

Intersections

The Mathematical Seminar had not changed much. Rooms were unheated, ofcourse, in that freezing winter, and stu­ dents looked undernourished. Wirtinger had gone a bit deafer, Furtwangler was lame, and cranky old Tauber lived like a recluse. Helly, for that matter, had also lost some of his health and hair, but none of his enthusiasm for mathe­ matics. At the age of thirty-seven, he still had only two publications to his credit, but he had not been idle in Siberia, and he submitted a superb new manuscript in long-hand, "On systems of linear equations with infinitely many unknowns." It was immediately ac-

I· '

'

" . '

¥'

:a.l

' '

--

To my Mama. Stranded in Siberia for five years, Helly could communicate with his mother only very rarely. An intrepid Countess Kinski, the "Angel of Siberia," managed to visit the camps and distribute some help-in Helly's case, a loan of forty rubles, to be paid back by Eduard's mother.

Elise Helly (1 892-1992), nee Bloch, in a nearly professional glamour shot by her husband. Miss Bloch got her PhD in mathematics un­ der the guidance of Wilhelm Wirtinger. In the early years of twentieth-century Europe, it be­ came almost fashionable for girls with brains to study mathematics {the wife of the writer Thomas Mann being

probably the best­

known example). During the First World War, they dominated the PhD lists.

met success. Inflation was overcome, and Vienna knew its share of the golden twenties. Working hours were long for Helly, but he found time to finish a note pre­ senting a proof of his theorem on in­ tersections of convex sets, which he had discovered ten years before. In the meantime, an Austrian and a Hungar­ ian, Johann Radon and Denes Konig, had each published his own proof of Helly's theorem, duly quoting him. Helly still was persuaded that the convexity assumption was too narrow, and he hunted for a topological extension. Topology was flourishing in Vienna during those years, and Helly had con­ tacts with Leopold Vietoris, Adolf Hur­ witz, and Karl Menger. He kept fre­ quenting the Mathematical Seminar. On Monday evenings he met his friends in the famous Cafe Central, where

analysis, as everyone agreed, was well covered by Hahn himself. Helly had to look for another job and luckily found one right away, as a bank clerk of the Bodencreditanstalt, a highly respected bank dealing mostly with landed property. Prospects looked bright enough for Liesel Helly, who was frail of health, to give up her teacher's job. A promise of prosperity hung in the air. Within a few years, the tough fi­ nancial measures of chancellor Seipel

Charming country. Eduard Helly was a highly gifted and competent photographer, and a pas­ sionate hiker. He sold this picture to the Austrian Tourist Office. In his later years of exile, the innocent text must have appealed to Helly's sense of humour.

VOLUME 26, NUMBER 2 , 2004

27

lively discussions and dead smoke filled the air. He was a charming, gre­ garious man, with a vast range of in­ terests-an avid chess player, a gifted photographer, and a passionate hiker, despite his dislocated heart. Although not a member of the Vienna Circle, he knew and associated with many of its members, and in particular two mem­ bers-at-large who often returned to their home town, Richard von Mises and Philipp Frank Von Mises headed the large Institute for Applied Mathe­ matics in Berlin, and Frank held a chair of physics in Prague. Helly must have thought of looking for a professorship abroad, but perhaps he was too timid to apply, or just too happy to be back in his native town. The Hellys kept an open house; Liesel enjoyed their con-

•

.. . . .. . .. .

• • . . " .

!

. I ea• •

Crash Course

Thus Helly resigned himself to staying in the Bodencreditanstalt. But in Oc­ tober 1929, within weeks of Wall Street's Black Friday, this bank, one of the largest in Austria, long patron­ ised by the imperial family and the landed gentcy, had to declare insolvency. Other Austrian banks were forced to

. · :,; - -:;

. . •.• .

•

was to be the fifth of Reily's mathemat­ ical publications. It came too late to help him get the job as Reidemeister's suc­ cessor, when the latter left for Konigs­ berg. In the meantime, a young shooting star named Karl Menger had written a dozen papers on dimension theocy, and Hahn preferred to appoint twenty-five­ year-old Menger rather than forty-one­ year-old Helly.

tacts with mathematicians. She lec­ tured at the Volksbildungsheim and translated some monographs. Helly met his topological mentor in Walther Mayer (of Mayer-Vietoris fame), a Dozent in mathematics who stood out, among the run of Viennese coffee-house intellectuals, for owning a coffee-house himself. (Albert Einstein would also look for Mayer's expertise: in 1932, he arranged to have him as his assistant in Berlin, and soon afterwards in Prince­ ton.) Eventually, Helly found how to ex­ tend his theorem beyond convex sets: if there exists, in R", a family of cells with the property that the intersection of any n + 1 of them is a cell, then the inter­ section of the whole family is a cell. (Helly defines a cell as a compact non­ empty set with all Betti numbers 0.) This

.• .

eibung - Signalemenk

�li

Frau - Fe

5elv. M iA J�� (A, U, L

j e d. ,);

;lit{.Me-

J

""

� o/ ,(f

lJro{� tA"" �oJ,.e, ttn, 1.tn. ole... Irrrie

li.L &e-.. G�e.. !lrM 'k-1 -JZ

fAfo..iv. f/1.

/1-111

e.'

e1n), m 1 ,2 = 1 + 3 (=n), m l, 3 = 1 + 2 (:Sn) . . . . This yields the configuration =

=

In the same way, the indication 3, 2, 1, 4 left of the rows and 4, 3, 1 , 2 above the columns, produces

4

3

1

2

3 2 1 4 To generate all of the configurations possible using the same column se­ quence, a diagram is made as follows. Starting with the row sequence 1, 2, 3, 4, the permutation of rows provides 24 4! possibilities of Figure 4. Per­ muting the column sequence, another 24 possibilities result in 24 tables, each containing 24 configurations. A block with all possibilities will thus contain 576 = 4! X 4! different configurations. In every table, some configurations are =

Figure 1. Map showing the Khant region.

Figure 2. A. Y. Filtchenko's photograph of a traditional Khant log-house (1 998).

Figure 3. A shirt with rich cross-stitched patterns. The arrows indi­ cate patterns explained in the text (Photo: Markku Haverinen).

VOLUME 26. NUMBER 2, 2004

35

September 29, 2003, the museum or­ ganized a special exposition titled: "Sibe­

ria Ufe on the Taiga and Tundra" (see

http://www.nba.fi/NEWS/LEHDISTO/ 2002/siperialehdistokuvat.htm). The mu­ seum is housed in a former indoor ten­ nis complex, at the "Etelilinen Rauta­ tiekatu

8 I Salomonkatu 15" in Helsinki.

The multiplex has been renovated to house the museum as well as the Helsinki City

Art Museum, Finnkino's

cinema centre, and a number of shops, restaurants, and cafes.

R&D&I- Research, Develop, Invent 2 Union Street Kew 3 1 0 1 Victoria, Australia e-mail: g . [email protected]

Figure 4.

Figure 6.

Figure 5.

the horizontal mirror images of each

these patterns in different regions of

other, whereas in each block the con­

Siberia (see Figure

figurations of some tables are vertical

plex example), or elsewhere in the

7 for a more com­

mirror images. The pattern of the con­

world, is an interesting ethno-mathe­

nections, valid inside the tables and in­

matical subject. The mathematical set­

side the blocks, is shown in Figure 5.

up here may help to classify different

6 is a magnified image of Fig­

stitched patterns and to compare con­

ure 3; it uses repeatedly the configura­

figurations made by cultures from dif­

tion 1, 3, 2, 4 (rows) and 3, 1, 4, 2

ferent regions or from different eras.

Figure

n = 4, and the configu­

A museum that attaches great im­

ration 1, 4, 2, 5, 3 (rows) and 5, 1, 3, 2,

portance to these fur coats with their

mirror images.

Helsinki

(columns) for

4 (columns) for n = 5, along with their To investigate the occurrence of

36

THE MATHEMATICAL INTELLIGENCER

intriguing geometrical patterns is the Museum

of Cultures.

Re­

cently, between May 14, 2002, and

Figure 7. More cross-stitched Khant pat­ terns.

1Mfijii§jr@ih££h§4£i 11 i.J i§.id ..

Autologlyphs

M i chael K l e b e r a n d Ravi Vak i l , Ed itors

For solutions, see page 411.

Henry Segerman and Paui-Oiivier Dehaye

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

G O +2 3 8lJ OODVOD

Please send all submissions to the Mathematical Entertainments Editor,

� �� � OO � ffi CW

Ravi Vakil, Stanford University,

Department of Mathematics, Bldg. 380, Stanford , CA 94305-2 1 25, USA e-mail: [email protected]

© 2004 SPRINGER-VERLAG NEW YORK. LLC. VOLUME 26, NUMBER 2. 2004

37

Il 11 � I 1 11 I

11 11 11 11

THE MATHEMATICAL INTELLIGENCER

I

11

I II

I

II

I

38

11 1 1 11

I I

\

• c::-. )

)

r (I

)

I

c

\. .

VOLUME 26, NUMBER 2, 2004

39

ip.i$�•ffli•i§[email protected]?Ji

Kolmogorov and Aleksandrov in Sevan Monastery, Armenia, 1 919 Vahe G . Gurzadyan

Does your hometown have any mathematical tourist 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?

L

D i rk H uy l e b ro u c k , Ed itor

ake Sevan is one of the most beau­

and princess Mariam. It includes two

tiful spots in Armenia. It has always

churches, the Arakelotz (the Church of

attracted poets and painters. Maxim

the Apostles) and Astvatsatsin (the

Gorky called it "a piece of sky which

Church

came down among the mountains. " In

panoramic view on the lake from the monastery. In the 1930s, the water flow from the lake via the river Hrazdan was

33, two mathematicians just starting

increased for irrigation and hydroelec­

their life-long friendship, visited Sevan.

tric power purposes. This lowered the

For about a month they enjoyed the

level of the lake, and transformed the

beauties of the lake, living in a cell in

island into a peninsula. However, in

the monastery on the island of Sevan.

1929 the island still existed, and Kol­

They visited other sites of Armenia, but

mogorov colorfully described the life

still actively continued their research.

on that isolated spot:

Both mathematicians mentioned Sevan in their memoirs, many decades later:

At that time, Sevan was in its full glory. We bathed several times a day in its cool limpid waters. We also did a lot of work; in particular, I worked on "Aleksandrov-Hopf' (the type­ writer was always with me) (Alek­ sandrov [ 1]). Naturally we were immediately at­ tracted by the rocky islet, which with the decrease in the level of the Lake Se­ van has now became a peninsula, and we wanted to settle there. This proved to be a simple matter. The cells of the monastery were empty and we occu­ pied one of them (Kolmogorov [2}). Sevan provides an unusual combi­

column a picture, a description of its

titude lake, 1900 m above sea level, 75

may follow in your tracks.

km in length and 55 km in breadth. The

water is fresh and transparent, and the people of the region drink it abun­ dantly. A number of rivers flow into the lake and only one, the Hrazdan, flows from it. The lake is famous for its trout and beautiful peninsula, which was once an island. During the Middle Ages various battles occurred

over that

Please send all submissions to

rocky island, where Armenians often

Mathematical Tourist Editor,

sheltered from their enemies.

Dirk Huylebrouck, Aartshertogstraat 42,

The monastery on Sevan island was

8400 Oostende, Belgium

founded in 874 AD by the Armenian

e-mail: [email protected]

king Ashot, of the Bagratuni dynasty,

40

There is a

N. Kol­

nation of natural conditions, a high-al­

a map or directions so that others

of the Virgin).

mogorov, 26, and Pavel S. Aleksandrov,

the summer of 1929, Andrei

If so, we invite you to submit to this mathematical significance, and either

j

THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC

The permanent population of the is­ land consisted then of the archiman­ drite of the monastery (who had a fairly big house), his wife (who looked after some cows), the head of the me­ teorological station with his small family, and finally the "Captain, " who did indeed command "the Sevan fleet" consisting of one motor boat and a few of the unusual pleasure boats. His picturesque figure has occasion­ ally been described in literature (for example, by Marietta Shaginyan). Every day the archimandrite opened the lower church (there were two abandoned temples on the top of the hill), lit candles and, in complete soli­ tude, recited the service. Obviously the head of the meteorological station car­ ried out his duties. At times, the Cap­ tain brought honored guests, such as Sar'yan or the then President of the All-Union Central Executive Commit­ tee of Armenia, but he was also ready to support humble tourists. On the island, we both set to work. With our manuscripts, a typewriter, and a folding table, we sought out the secluded bays. In the intervals be­ tween our studies, we bathed a lot. To study, I took refuge in the shade, while Aleksandrov lay for hours in full sun­ light wearing only dark glasses and a white panama. He kept this habit of working completely naked under the burning sun well into his old age.

to the monastery of Hayravank under the guidance of our captain). The Hayravank monastery with its IX-century church is situated on a high rock facing Lake Sevan. According his student Hrant Maranjian, Kolmogorov recalled tiny details about that stay at Sevan's island during his visit in Ar­ menia in 1973.

For Aleksandrov the day was ap­ proaching on which he had arranged to journey to Gagra, and we set off to­ gether for Yerevan (where we stayed for some days in a student hostel). The Pavel Aleksandrov and Andrei Kolmogorov (photo Uspekhi Mat. Nauk, 1 986). temperature was 40°C, the sky was a hazy blue, and only after sunset did there unexpectedly appear the peak of On Sevan, Aleksandrov worked on "On analytic methods in the theory of Ararat, suspended in this blue sky. We visited Echmiadzin (where we de­ various chapters of his joint mono­ probability. " Given its position, Lake Sevan cided not to visit Katholikos as we did graph with Hop!, "Topology" [3}, and he helped me to write the German text mostly enjoyed sunny weather, but not have the right clothes). From Ech­ of my article on the theory of inte­ sometimes clouds com'ing from the miadzin, we walked to Alagez (spend­ grals. Besides writing this paper, I East fiUed up from the mountains, ing the first night by the lake, where was busy with ideas about the ana­ dropped down to the water and then, physicists working on cosmic showers lytic description of Markov processes on contact with it, va,nished. We stayed very kindly put us up). After spending with continuous time, the end prod­ there for about 20 days without leav­ one night (stiU without suits, wearing uct of which later became the memo'ir ing the pla,ce (apart from excursions only shorts), we climbed the south

The Sevan monastery where Kolmogorov and Aleksan-

General view of the Sevan peninsula, in 1 929 still an island, where the monastery

drov lived in 1 929 (picture by the author, August 2003).

is situated.

VOLUME 26, NUMBER 2, 2004

41

The summit of Mount Aragatz, climbed by Kolmogorov (photo H. Badalian).

Alagez summit, which did not present any complications (4000m). From the top there opened up a view of the rocky northern summit (41 00m), separated from the south by a huge ridge of snow, at the very bottom of which could be seen a small lake, its shores frozen and covered with snow. Of course, Aleksandrov wanted to climb down there and bathe, but I preferred climbing the northern summit. At the time, reaching the top of Mt. Aragatz (Alagez) involved a 30 km walk over mountainous terrain, the roads not having been built until the 1940s. Aragatz has a spectacular 3km-wide crater with a glacier and 4 summits surrounding it. The northern summit is not only the highest but also the most difficult one for climb­ ing; its easiest route is classified "lB" in the mountaineering category of dif­ ficulty. The cosmic ray station of Yerevan Physics Institute is situated 3200 m above sea level. The nearby beautiful lake, mentioned by Kolmogorov, is an

42

THE MATHEMATICAL INTELLIGENCER

artificial reservoir that was con­ structed in the Second Millennium BC, as archaeological studies have found [5]. The surface of the lake is free of ice only 2 to 3 months a year, and Alek-

sandrov's desire to bathe in the icy wa­ ter is as characteristic as Kolmogorov's climbing of the most difficult northern summit of Aragatz. After the dissolution of the Soviet

The cosmic ray station, 3200 m, where Kolmogorov and Aleksandrov stayed the night, before climbing the Aragatz mountain the next day. Many scientists who went mountaineering later visited the station. The photo shows (from right to left), L. D. Faddeev (St. Petersburg), Alexan­ der Migdal (Princeton), R. Kallosh (Stanford), Arcady Migdal (Landau Institute, Moscow), A. M. Polyakov (Princeton), V. G. Gurzadyan (Yerevan Physics Institute), and A. G. Sedrakian (Yerevan Physics Institute) descending the Aragatz summits in 1 983. Walking behind was an­ other member of the group, A. Linde, now a famous cosmologist {his hat is visible in the back).

Union, the former communist con­ straints on religion have been re­ moved in the Republic of Armenia. The activity of the Sevan monastery is much increased, not only because of the larger number of priests but also because a seminary has been opened on the peninsula. The monastery and the peninsula are now attractive tourist areas. Few of the tourists, however, can guess that

the gloomy cells, with walls completely blackened by many centuries of candle smoke, once accommodated two lead­ ing mathematicians of the twentieth century.

[3] P. Alexandroff, H. Hopf, Topologie, Berlin, 1 935. [4] P. Cuneo, Architettura Armena , De Luca Edittore, Rome, 1 988. [5] A. Kalantar, Armenia: From the Stone Age to the Middle Ages, Civilisations du Proche­

Orient, Neuchatei-Paris, 1 994.

REFERENCES

[1 ] P. S. Aleksandrov, Russian Math. Surveys 35, 3 1 5 , 1 980.

Vahe G. Gurzadyan

[2] A. N . Kolmogorov, Russian Math. Surveys

Yerevan Physics Institute, Armenia

4 1 ' 225, 1 986.

e-mail: [email protected]

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For the group G = A4 defined in Example 4, and B = [6_10_ 1 1 ) . Then A - 1 = [2_3) = A B- 1 [5_9_12) AB = [6_6_7_7_10_ 1 1 ) BA = [6_7_7_10_ 1 1_ 1 1 )

Order Invariants

The main order invariants are the order of G, namely iG', and the order multiset of G defined by

j

ord(G) = [ord(.r) .r

In the example,

mA(x) > 0

[2_3)

=

More generally, we allow rational combinations like :3_A + B, which is a 1-multiset. We may also form the direct rod­ uct A X B by the rule

A XB =

lSf ·

This will be a convention: general multisets denoted by cap­ itals, and the associated 1-multisets denoted by the corre­ sponding small letter. Special cases of the product of multisets deserve men­ tion. For x E G and A a multiset from G,

2A

±

s

mB(x) > 0

where mA(x) is the multiplicity of the object x in the mul­ tiset A. If A and B are multisets from a group G, in other words if A � G and B � G, then define the multisets

A- 1 = [a- 1 I E A) AB = [ ab I [ a,b] E A X B) . a.

In this latter definition of the product AB (not to be con­ fused with the direct product A X B defined above), we must consider all possible products ab with repetitions included. Note that this generally yields a multiset even if A and B are subsets of G, so that AB has here a dif­ ferent meaning from that used in almost all texts in the subject! We see that if A and B are multisets from a group G, then

VlEI = f,4 , jBf . It follows that the product of any two 1-multisets from a group G is itself a 1-multiset. That means that 1-multisets

E G)

where ord(.r) is the least nonzero natural number n such that xn = e. Clearly if two groups are isomorphic then their orders and order multisets must agree. Exercise 3 Write a program which outputs the order mult·iset of a group G. There are secondary order invariants which one may

define, which I have not seen studied. Given x and y in a group G, an ordered pair of integers [n,m] such that xn = y"' may be called a period pair for [.r,y] . Period pairs for [.r,y] form an integral lattice and include the vec­ tors [ord(.x:),O] and [0, ord(y)] . One may associate to this some invariants, the simplest of which is the area a(x,y) of a fundamental domain. Define the area multiset of G by

a(G) = [a(x,y) i x,y E G) . Similarly, for a triple of elements .r, y, z we can consider period triples [n,m,l] satisfying xn = ym = z1, an associ­

ated volume multiset, and so on. For any element x in a group G, the subgroup {.x:" I k = 1, 2, ord(.r) ) is called the cyclic subgroup generated by x. Any subgroup of a cyclic group is also cyclic. The set of cyclic subgroups of a group G is a poset under inclusion, ·

·

·

,

VOLUME 26, NUMBER 2, 2004

49

which we call the cyclic subgroup poset of G. Clearly the period lattices for x and y are closely related to the cyclic subgroups generated by x and y. We must be attuned to the possibility of counting some­ thing at all times in this subject. To this effect, multisets play an important organizational role. For another exam­ ple, if the number of elements that commute with an ele­ ment x is c(x), then we may study the multiset

[c(x) I x E

G].

You will be able to define and study many more such or­ der invariants as you proceed with your investigations. Conjugacy Classes and the Class Hypergroup

Now we come to central analysis on a group G. Let C; be a conjugacy class of G. Then the associated 1-class is the 1-multiset - C;

C; -

lCJ .

A multiset S from G with the property that {x) S {x- 1 ) = S for all x E G will be called G-invariant, or central. De­ noting the conjugacy classes by C0 = { e), C1 , · · · , Cr, any central multiset is a linear combination of them in the mul­ tiset sense, so any central 1-multiset s is a combination n

s = I r;c; ;�o where r; 2: 0 and �?� o r; = 1 .

Since the product o f central multisets is central, for each

i and j there are positive rational numbers nt summing to 1 such that

n

c;ci = I n t ck. k�O

The coefficient nt is the probability that the product xy is in ck if X is in C; and y is in Cj. The multiplication c; ci = c;ci of 1-classes is associa­ tive, has identity c0, and has a partial notion of inverse: for every c; there is a unique c� = c;* with the property that c;c7 contains a non-zero multiple of c0; ci is the conjugacy class of inverse elements of c;. We require that the map c; ----> ci be an involution. This defines a (finite) hypergroup. In terms of the coefficients nri and the map i ----> i*, the definition amounts to the following conditions.

erally much smaller than G, especially if G is highly non­ commutative. The monster group M, largest of the sporadic simple groups, has roughly 8 X 1053 elements. Its class hy­ pergroup K(M) has 194 elements, and so its multiplication program contains at most 1943 rational numbers. Furthermore, K(G) is always commutative, meaning that c;cj = CjCi for all i,j. Thus classical tools of harmonic analy­ sis, such as characters, the Fourier transform, Plancherel the­ orems, Parseval identities, and so on are available. Commu­ tative hypergroups provide us with a powerful extension of abelian harmonic analysis with applications far beyond group theory; to physics, number theory, combinatorics, and func­ tional analysis (see [BH], [W2], and the references therein).

Let G = A4, with conjugacy classes Co = { [ 1234] ) c1 = { [2143]_[3412]_[432 1 ] ) c2 = { [2314]_[4132]_[3241]_[ 1423] ) c3 = { [3124]_[2431 ]_[42 13]_[1342] ). The multiplication program for the class hypergroup K(A4) = { co, c 1 , c2 , c3) is given by the table Example 6.

c1 · c1

Co

Co

Co

c1 c,

c,

c,

l eo + l c ,

c2

CJ

c2

c2

c2

c3

l eo + l c ,

CJ

c3

3

3

c2

c3

c2

c3

l eo + l c ,

CJ

4

4

4

4

c2

We can see that ci = c; for all i (in such a case the hy­ pergroup is called Hermitian). So determination of the 1-class structure constants nt is one of the first and most important tasks to be performed for any group G. Here is my favourite group-theory problem.

Problem 4. Find a good formula for the 1-class struc­ ture constants for the symmetric group Sn.

·

�f=o n�j = 1 Vi,j nt 2: 0 \:fi,j,k m l l m \:fi,j,k,m 3 . "'-l=O n;J n lk - .:-1� 0 n il nJk 4. n�; = nro = 8 ;k \:fi,k 5. n � > 0 � j = i* Vi,j 6. nt = nJ:i \:fi,j,k The set K(G) = { co,cb , c,) 1.

2.

"""

_

,.-n

*

·

·

·

with the multiplication and involution defined above is the class hypergroup of G. Why would we consider this seemingly complicated ob­ ject? There are two good reasons. First of all, K(G) is gen50

THE MATHEMATICAL INTELLIGENCER

Character Tables

Let's now establish some further central invariants, espe­ cially the character table of a group. This latter forms the foundation for harmonic analysis on non-abelian groups, with applications to chemistry, physics, and other branches of mathematics, as well as to further development of the subject. This presentation has its origins in the work of Frobenius [F] and Kawada [K]. Suppose that K(G)

= {co, c 1 ,

·

·

·

,

c,)

is the class hypergroup of a group G, with structure equa­ tions n

c;ci = I ntck. k�O This can be viewed as a system of (n + 1)2 rational equa­ tions in the variables c;, with the property that the right­

hand side of each equation is a linear expression, and the left-hand side is a quadratic product.

Perhaps thefundamental computational problem in the subject is the determination of all complex-valued solutions to such a system. That is, we want to replace the variables ci with complex numbers so that the equations still hold. Almost surely you already possess algorithms (well devel­ oped in MAPLE and MATHEMATICA) that will input a sys­ tem of quadratic/linear equations, and in certain cases, find all solutions. Let's outline the basic idea. First define a character of the commutative hypergroup above to be a complex-valued function x which satisfies n ci c1 = ) x( )x( I ntxCck) 'Vi,j. k�O

There is always at least one character, the constant func­ tion

'Vi ;

xo(ci) = 1

x (co) = 1. Example 7. There are exactly jour characters of the hy­ pergroup K(A4), given by the rows of the following table C (here a- = e27Ti13)

X1

X2

Co

- 1 /3

0

0

(T2

X3

(T

It turns out that there are always exactly (n + 1) char­ acters of a fmite commutative hypergroup K = {co, c1, · · · , Cn }. We can find them as follows: For each element ci E K, let Mi be the matrix of multi­ plication by ci with respect to the ordered basis { c0, c1, , cnl · The matrices Mi satisfy exactly the same multiplicative relations as the elements ci, and in particular they mutu­ ally commute. Furthermore, they are linearly independent, for if � i ziMi = 0, then multiplying � i ZiCi by cJ and con­ sidering the resulting coefficients of c0 yields z1 = 0. The matrices Mi can be simultaneously diagonalized by some invertible matrix Q. That is, 1 QMiQ- = Di •

·

·

for some diagonal matrices Di. The diagonal entries of the matrices Di are then the characters of K, that is, we define the hypergroup character table C(i , J) = Xi (c1) = Dj(i,i). In particular, there are exactly as many characters as con­ jugacy classes. It is worth pointing out that in general only one of the matrices Mi is required. Suppose that M1 happens to have distinct eigenvalues. If a1 is one of these, then replacing c1 with a1 in the n - 2 equations for c1c2, · · · , c1cn yields a , Cn which has a unique system of linear equations in c2 , solution up to a scalar. Substituting into the equation for c1 yields an absolutely unique solution, so this gives us one character, or row of the character table. As aj runs through the eigenvalues of M1, the rows so obtained constitute the full character table. ·

·

·

Example 8.

For K(A4), the matrices Mi are

[� � � �l [� � ! �] ] [! Mo

0 0 1 0

0 0 0 1

114 3/4 0 0

Mz

and simultaneous diagonalization ofthese yields the table of Example 7.

and for any character x we must have

Xo

So the other equations, not involving c1, can be viewed as providing additional information to cover the case of eigenvalues of M1 with multiplicity greater than one.

The hypergroup character table C contains much useful information. Its columns furnish us with a model of the hy­ pergroup K, in the sense that pointwise multiplication of columns obeys the same structure equations as does hy­ pergroup multiplication, while the rows, being the charac­ ters, can also be multiplied pointwise to obtain what turns out (in the case of the class hypergroup of a group) to be a hypergroup structure in complete duality with K, called the character hypergroup and denoted K(G"). In the particular case when G is itself a commutative group of order n, so that G = K(G), this procedure shows that both G and G" = K(G") are isomorphic to a subgroup of the n-fold product en. In fact it is not hard to see that in this case all the entries of the character table must be n­ th roots of unity, so G and G " are subgroups of products of cyclic groups. (From this one may deduce, with some more work (!), that both G and G" are themselves iso­ morphic to products of cyclic groups.) Weights and Orthogonality Relations

The character table for a finite commutative hypergroup exhibits some remarkable orthogonality. To describe this, first define the weight w(c;) of an element Ci of a hyper­ group K to be

which in the case of K(G) is just the size of the associated conjugacy class Ci. Define the weight w(K) of a hypergroup K = { co, c1, · · · , cn l to be n w(K) I w(ci) , =

i�O

so that in our case w(K(G)) ! G j . For any hypergroup K the rows of the character table, 1 viewed as elements in en + , are orthogonal with respect to the usual Hermitian inner product modified by the weights w(ci) , in other words by the inner product =

VOLUME 26, NUMBER 2, 2004

51

while the columns are similarly orthogonal with respect to the dual weights w(xi) of K(G"). The weight list

[

WK(G) = w(co) , · · · , w(cn)

J

Write multiplication programs for the class hypergroups and character hypergroups of all simple groups. Problem 6.

and the dual weight list WK(G A) = [w (xo) , · · · , w(xn) l

are implicit in the character table, but the orthogonality re­ lations suggest that it is useful to identify them explicitly as important invariants. Calculating the following five central invariants of a group G is thus a primary task

Write programs which determine the class hypergroup, the hypergroup character table, the charac­ ter hypergroup, the weight list, and the dual weight list of a given group G. Problem 5.

Example 9. You may check that the character hyper­ group K(A4) has multiplication table XI " XJ

Xo

X1

X2

Xa

xo

Xo

X1

X2

X3

X1

X1

X1

X1

X2

X2

+ �X1 + _1_X2 + _1_X3 _1_Xo 9 3 9 9 X1

X3

xo

X3

X3

X1

Xo

X2

and that the weight list is WKc.A4) weight list is wKc.A4) = [ 1 ,9, 1 , 1 ] .

=

while the dual

[ 1,3,4,4]

By multiplying each row Xi of the hypergroup character table of the class hypergroup K(G) by the square root of the dual weight w(xi), we obtain the group character table C'. In other words, we define Xi(Cj)

=

Xi(cj)

VwCXJ

and let C'(i,J) Xi(Cj) · The group character table has the advantage of often having many integral entries, but it has the disadvantage of obscuring the symmetry between the class and character hypergroups. Its definition is usually left to a second course in group theory and traditionally in­ volves the deeper notion of a representation of a group. One of the advantages of the approach sketched here is the introduction of character tables without representation theory, which was actually Frobenius's original point of view! Recall that the passage from the multiplication table to the character table as I have outlined it involves only counting and solving a quadratic/linear system of equations, or equivalently diagonalizing some matrices. The weight list WK(G) is the list of sizes of the conjugacy classes, while the dual weight list WK(G A) has also the in­ terpretation as the list of squares of the dimensions of the irreducible representations (which are not defined here). =

Example 10.

Here is the group character table C' ofA4.

X,(CJ )

Co

c1

c2

Ca

3

-1

0

0

Xa

x1

x2

1

1

u

u2

x3

52

1

THE MATHEMATICAL INTELLIGENCER

The next problem is in principle straightforward, since the character tables for simple groups are clearly laid out in the Atlas [ CCNPW].

1

u2 u

It is a consequence of the classification of simple groups that a simple group is determined by its group character table, and so by its hypergroup character table. The following fundamental problem probably requires some new insight for its resolution. (The pattern of cir­ cles on a simple group, introduced later, might help in this regard.) Problem 7. Write a program that constructs a simple group from its group or hypergroup character table. Remark 2. The dual of an arbitrary commutative hy­ pergroup is not always a hypergroup, as negative coeffi­ cients can occur. Nevertheless, by enlarging our view to include so-called signed hypergroups, there is a complete duality theory generalizing Pontryagin duality forfinite commutative groups. This fact was discovered in the slightly different context of C-algebras by Kawada [K]; see [W2] and [OW] for a modern treatment. Cosets and Subgroup Analysis

Let's now introduce subgroup analysis. Let H be a sub­ group of G. A multiset S from G with the property that { y } S = S (respectively, S{y } S) for all y E H will be called H left-invariant (respectively, H right-invariant). A mul­ tiset S from G which is both H left-invariant and H right­ invariant will be called H hi-invariant. The subgroup H is itself H hi-invariant. For any x E G, the sets {x) H and H {xj are H right-invariant and H left-in­ variant, respectively, and are called left cosets and right cosets of H, respectively. The set of all left cosets of H forms a partition of G, as does the set of all right cosets of H. These two partitions of G are equal precisely when H is a normal subgroup of G. Because l lxJHI IH(xJ I = IHI, we conclude =

=

IHI divides IGI for any subgroup H of a group G. For any x E G the multiset H{x}H [h1xh2 l h 1 . h2 E H] is a double coset of H. It is H hi-invariant and is an IHI2Theorem 2 (Lagrange's Theorem).

=

set. In contrast to left and right cosets, a double coset is generally not a set. Note carefully that at this point our notation diverges from the standard usage of HxH as a

set.

Suppose that the distinct left cosets of H are H = H0, H1, . . . , Hz. Define the associated left 1-cosets by the rule

with h be

H· = hj 1�1

=

h0. Now defme the GIH =

right quotient

{ho, h1, · · ·

, hzJ.

of G by H to

Note that G/H is not defined to be a set of cosets, but rather a set of 1-cosets! Because H is a subgroup, we have

h2 = h h- 1 = h. Furthermore, the product of H right-invariant 1-multisets is also an H right-invariant 1-multiset, so there exist ratio­ nal numbers Pt such that l

hihj = I Pt hk. k�O This is a product on G/H which shares some of the prop­

erties of a hypergroup; for example, it is associative and for fixed i andj the right quotient structure constants Pt are probabilities. However in general ho is a right identity, that is, hiho = hi for all i, but not necessarily a left identity. There are inverses but they are not unique, and the product is gen­ erally not commutative. Nevertheless, the structure constants Pt are computable invariants of the quotient G!H. So any

right quotient G/H carries an algebraic structure.

The situation is entirely symmetrical when we interchange lefts and rights in the above discussion and consider right cosets of H in G and the associated left-quotient H\G. In the special case when H is a normal subgroup of G, the left-invariant and right-invariant 1-cosets coincide, and the above product structures coincide and indeed form a group called the quotient group and denoted by either G/H or H\G. In general the connection between the left­ quotients and right-quotients is given by the inverse map­ ping x � x- 1 , which transforms left cosets to right cosets and vice-versa. So the right-quotient and left-quotient struc­ ture constants are essentially the same. In fact there is a simpler and more fundamental invari­ ant associated to a subgroup H of a group G which is a hy­ pergroup. A 1-multiset of G of the form all

H {x } H a= � will be called a double 1-coset of H in G. The double 1cosets of H in G are either identical or disjoint, and we let

ffiGIH = {ao, a 1

arJ denote the set of all double 1-cosets of H in G, with a0 = ho = H!ll:ij. Then there are rational numbers q�1, which again ,

·

·

·

,

have an interpretation as probabilities, such that

The multiplication table is a3

a1 · a1

ao

a1

a2

ao

ao

a,

a2

a3

a,

a,

ao

a2

a3

a2

a2

a2

a3

l ao + l a ,

a3

a3

a3

l ao + l a, 2

2

2

a2

2

This is obviously commutative, so (G,K) is a Gelfand pair. The character table is ao

x;lail xo

a1

a2

-1

X1 X2

0 a a2

X3

a3

0

r? a

The similarity with the group character table in this case is largely accidental. Given a group G and a subgroup H, (write a program to) find the structure constants p�1 and q t above and determine whether (G,H) is a Gelfand pair. Exercise 8.

Circles: Translates of Conjugacy Classes

A translate of a conjugacy class in G will be called a cir­ cle (this is not a usual definition). More particularly, a translate of the i-th conjugacy class ci will be called an i-circle. Since {x}Ci = Ci{x) for any x E G and any conju­ gacy class Ci, it is unnecessary to specify left or right trans­ late. We will say that the circle {x)Ci has color i and cen­ ter x, both of which may fail to be unique. Example 12.

Let

G = s3 = { [ 123] , [213] , [132] , [321] , [231] , [312] } with conjugacy classes Co = { [ 123] } c1 = { [213]_[132]_[321]} Cz = { [231]_[312] } . Then of course every element is a 0-circle trivially; fur­ thermore,

{[123] } c1 = { [231J J c1 = { [312]} c1 = c1 and ! [213]} c1 = { [ 132] } c1 = { [321]} c1 { [123]_[231]_[312]}, so there are only two !-circles and their centers are not unique. There are six distinct 2 -circles, each with a unique center, namely =

r

aiaJ = I q�J ak k�O which makes ffiGJH into a bona fide hypergroup which we call the double coset hypergroup of H in G. In general ffiGIH is not commutative; when it is, (G,H) is known as a Gelfand pair.

Let G = A4 and K = { 1_2} . Then K\GIK = {ao_a1-a2-a3) where ao = 21 [1_2] a 1 = 21 [3_4] az = ± [5_8_9_12] a3 = ± [6_7_1 0_1 1]

Example 1 1 .

( [123] } Cz = {[231]_[312]} { [231]} Cz = {[123]_[312]} { [312] } Cz = {[123]_[231]} ( [213] } c2 = ( [132]_[321] ) { [321]} Cz = { [213]_[ 132] } {[132] } c2 = ( [213]_[321]} . Example 13. For G = A4 the class structure equations reveal that a circle may have more than one color: C2 and VOLUME 26. NUMBER 2, 2004

53

C3 are translates of each other, so each is both a 2 -circle

and a 3-circle.

Restricting to simple groups, the situation is much more regular than the above examples might lead us to suspect. Theorem 3. Let G be a finite non-commutative simple group. Then every circle in G has a unique color and a unique center. Every point x E G lies on precisely lci l i-circles, and so lies on a total of I G I circles.

Let Ci,Ci be two distinct conjugacy classes of G. We first show that no translate of Ci can also be a trans­ late of Ci. If Hi (z E G I (z}Ci = Gil then Hi is a subgroup of G and is normal. It cannot be all of G unless G is trivial, in which case there are not two distinct conjugacy classes. Thus Hi = (e) since G is simple. Now suppose that (x}Ci = (y)Cj for some x,y E G, or equivalently that (z}Ci = Ci for some z. Conjugating by w E G, we see that (wzw- 1 }Ci = Ci. But then (z- 1wzw- 1 }Ci = Ci and so by the previous re­ mark z- 1wzw- 1 = e. Since w is arbitrary, z is in the cen­ ter of G and so z = e. But this is impossible because Ci and cj are distinct. For the second claim, note that the statement is true for x = e, for the centers of the i-circles on which e lies are ex­ actly the elements of the class c; of inverses of the ele­ ments of Ci, which has I Gil elements. Since the family of i­ circles is invariant under translation, every point x E G lies on precisely I Gi l i-circles, and so on a total of I GI circles. • Returning to general groups, the circles of G are related to the structure constants n �i of the class hypergroup K(G) = ( co, C1 , , Cn } . Theorem 4. For any conjugacy classes C;, Ci, Ck of G and z E Ci, Proof.

=

•

•

•

I (z} ci n ckl = lci l n�i· Proof. Suppose that the product CiCi contains Ck a total of N �i times, necessarily an integer. Then since ci = C/ I Ci I , Nfi l ck l _ - nkij· I ci I I cj I Of course I (z} cj n ck I = I (xzx- 1} cj n Ck l for any x, so that counting in different ways yields l ci l l lz} Cj n ck l = N�j l ck l· • Combining, we get the required equality. In particular we see that ICj I n�j is always a positive in­ teger, which in tum allows further deductions, such as the following. Corollary 1 . If Ci and Ci are conjugacy classes of G whose orders are relatively prime, then CiCi is a multi­

could recover the circle pattern of a simple group from the character table, we could hope to recover G from its action, on the left and right, as symmetries of the circle pattern. Relations and Isomorphism Theorems

So far we have been focusing on individual groups, their conjugacy classes, and their subgroups. Now let us con­ sider possible relations between two or more groups. The following treatment is more general than usual, but I leave most of the proofs to you. If G and H are groups, then a subgroup R of the prod­ uct group G X H will here be called a group relation be­ tween G and H. Note that the order of G and H matters; interchanging the order of all ordered pairs in R yields the transpose group relation RT between H and G. A homomorphism from a group G to a group H is a map if! : G � H satisfying for all x, y E G

cp(xy) = cp(x) cp(y). Perhaps you have already been pre-programmed to think of homomorphisms, not relations, as the natural objects of study in algebra. In that case, note that, given a homomor­ phism cp from G to H, we may define a group relation R be­ tween G and H by the rule R = ( [x, cp(x) J i x E G). If R is a group relation between G and H, then define the image and the transpose image of R, respectively, by im(R) imT(R)

=

=

(y E H I [x,y] E R for some x E G) (x E G I [x,y] E R for some y E H).

These are subgroups of H and G, respectively. If both imT(R) = G and im(R) = H then we define R to be full. The next problem is deep, and generalizes aspects of repre­ sentation theory.

Write a program that inputs an ordered pair ofgroups and finds all full group relations between them.

Problem 9.

If R is a group relation between G and H, then define the kernel and the transpose kernel of R, respectively, by ker(R) = (x E G I [x,eH] E R} kerT(R) = (y E H I [ea,Yl E R},

where eH and ea are the identities in H and G, respectively. More generally, for any x E G and y E H define RY = (x E G I [x,y] E R} Rx = (y E H I [x,y] E R}

ple of a single conjugacy class. Proof. In this case n�i is both an integer and a probabil­

so that

The pattern of circles on a group is an in­ teresting geometrical object, closely connected to the sub­ ject of association schemes (see for example fBI]). If we

Rea· Then Rx is both a left coset and a right coset of kerT(R) for all x E imT(R), and RY is both a left coset and a right

ity for each k, and so must be either 0 or 1 .

Remark 3.

54

THE MATHEMATICAL INTELLIGENCER

•

ker(R) kerT(R)

= Ren

=

coset of ker(R) for all y E im(R), so that ker(R) and kerT(R) are normal subgroups of imT(R) and im(R), respectively.

A U T H O R

r�

If R is a group relation between groups G

Theorem 5.

and H, then

.

imT(R)Iker(R) = im(R)IkerT(R).

j\

Given a group relation R between groups G and H, we may associate to each x E imT(R) the 1-multiset r(x) = rx = Rxl I R:r I in H. In that case, the homomorphism equa­ tion applies, showing that a group relation can be viewed as a (partially defined) homomorphism that is allowed to have 1-multisets as values. If R is a group relation between G and H satisfying imT(R) = G and kerT(R) eg, then R determines a homo­ morphism q; from G to H by the rule

..

•

:

·

N . .J. WILDBERGER

UNSW

Sydney, NSW 2052 Australia

e-mail: [email protected] A native Canadian, N. J. Wildberger has settled 1n the alto­ gether sunnier environment of Australia, where he enjoys bush­

walking, sw1mm1ng, and plaYJng GO. He studied at the Uni­

In this case the theorem reduces to

versity of Toronto and Yale, and then taught at Stanford and Toronto before heading down under.

G!ker(R) = im(R). Then R and RT both determine homomorphisms if and only if R determines an isomorphism. Consider a useful example. Suppose G is a group with normal subgroups K and L, which have associated 1-mul­ tisets k and l, respectively. Define a group relation R be­ tween the groups G/K and GIL by the rule that [ la} k, lb}l] E R if and only if I a} k n I b} l is non-empty. Let's first check that this makes sense. Because K and L are normal, la} k = k laJ and lb} l = l lbJ for any a, b E G. Thus if X 1 E lai } k n lbi } l and X2 E la2 } k n lb2}l, then X1X2 E lai} k la2 J k = la1a2} kk la 1a2 } k. Similarly X1X2 E lb 1b2 } l; thus R is indeed a group relation. Now kl lk because both are G-invariant, and so (kl) 2 kl. This implies that kl is the associated 1-multiset to a nor­ mal subgroup M (which is usually denoted by KL in the lit­ erature) containing both K and L. It is not hard to see that ker(R) = MIK and kerT(R) MIL, so the above theorem yields the equation =

=

=

GIK = GIL . MIK MIL In the special case when K is a subgroup of L (and so a normal subgroup of L), we get M = L and the following iso­ morphism theorem.

Acknowledgments

I thank Peter Donovan, Hendrik Grundling, and David Har­ vey for useful comments and discussions. REFERENCES

[BI] E. Bannai and T. Ito, Algebraic combinatorics /-Association Schemes, Benjamin and Cummings, Menlo Park, 1 984.

[BH] W. R . Bloom and H. Heyer, Harmonic analysis of probability mea­ sures on hypergroups , de Gruyter Studies in Mathematics, Vol. 20,

Walter de Gruyter, Berlin, 1 995. [CCNPW] J. H. Conway, R. T. Curtis, S. P. Norton, R. A Parker and R. A Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1 985. [F] G. Frobenius, Ober Gruppencharacktere, Gesammelte Abhandlun­ gen, Vo l . I l l , pp. 1 -37, Springer-Verlag (1 968), Berlin. [K] Y. Kawada, Ober den Dualitatssatz der Charaktere nichtcommuta­ tiver Gruppen , Proc. Phys. Math. Soc. Japan 24(3) (1 942), 97-1 09.

[OW] N. Obata and N. J . Wildberger, Generalized hypergroups and or­ thogonal polynomials , Nagoya Math. J . 1 42 (1 996), 67-93.

[W1 ] N . J . Wildberger, A new look at multisets, preprint, 2003.

Theorem 6.

[W2] N. J. Wildberger, Duality and Entropy for finite commutative hy­

GIK = GIL . LIK

Show that this theorem extends to general (non-normal) subgroups by developing a theory of quo­ tients for the "hypergroup-like" objects GIH.

Exercise 10.

.

School of Mathematics

=

[x,q;(x)] E R.

�

.1. .

r(xy) = r(x)r(y)

=

.

. .

pergroups and fusion rule algebras, J . London Math. Soc. 56(2)

(1 997), 275-291 . [W3] N. J . Wildberger, Finite commutative hypergroups and applica­ tions from group theory to conformal field theory, In Applications of

Hypergroups and Related Measure Algebras, Proc. Seattle 1 993 Conference, Contemporary Math. 1 83 (1 995), 41 3-434.

VOLUME 26, NUMBER 2 , 2004

55

Mathemati c a l ly Bent

Col i n Ad a m s , Ed itor

This Theorem Is Big Colin Adams The proof i s i n the pudding.

Opening a copy of The

Mathematical

Intelligencer you may ask yourself uneasily, "lthat is this anyway-a mathematical journal, or what?" Or you may ask, "lthere am /?" Or even "ltho am !?" This sense of disorienta­ tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01 267 USA e-mail : [email protected]

56

H had to call you, because I have a ey, Barry. It's me, Sid. Listen, I just

theorem like no other theorem you ever saw. You are going to go apeship over this. This one is going to be big. And not just a little big. We are talking another Atiyah-Patodi-Singer Index Theorem. Think Mostow Rigidity or Reine-Borel. This is mega blockbuster material. I'm shopping it around, and let me tell you, there's a lot of interest. But I wanted you to have a shot at it. I know you guys haven't had a big hit in a while, not since Riemann-Roch. So I know you're hungry. This one has the potential to be a blockbuster. Get ready, because here it is. The Mandelbrot­ Thurston-Wiles Pi-Orbifold Syzygy The­ orem. How about that, huh? You don't get more star power than Mandelbrot, Thurston, and Wiles. These are box office heavies. And if you're picking a real number to put in the title of a theorem, you can't get hot­ ter than 7T. And, hell, we have a word in there whose only vowel is "y" AND it appears three times. This word is so end-of-the­ alphabet-heavy, it screams brilliance. Now you are probably wondering how I got these three big names to agree to this project. And the honest truth is that I don't yet have them all on board. But this is where you come in. The only reason they are hesitating is the need for a sponsoring institution, a place of your caliber. They aren't go­ ing to sign on with some community college with a 4-4 teaching load. No, they need to know that you're going to provide the resources necessary for this theorem to get the attention it de-

THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC

serves. Let me tell you what I have in mind. I want you to get invitations for all three to speak at the Institute for Ad­ vanced Study, the Fields Institute, and the International Congress. I'm betting you can make that happen. That's why I called you. And I'm thinking there should be some awards from some wealthy Middle Eastern countries. Or maybe Japan. You may have to pull some strings, but I'm betting you can do that. I'm thinking big gold ingots, with a picture of Gauss stamped on them, or maybe a big belt, like they give out at the World Wrestling Federation. On the PR front, we have lots of ideas. We're thinking of spicing things up a bit, having a shoving match be­ tween Thurston and Wiles at the AMS meetings. All faked, of course; but imagine the PR potential. Stars not get­ ting along. Disagreement over the credit for the corollaries. This is hot stuff. We'll get the front page of the tabloids. And of course, there will be the love interest. Get this. Thurston, dyed-in­ the-wool topologist, right? He falls for analytic number theory. See? So that's what he contributes to the theorem. But then, there is some lemma trouble. ANT's pretty rigid stuff. Thurston's frustrated. Thinking maybe he's made a mistake. Misses his old sweetheart topology. She was supple, forgiving. But in the end, the lemmas work out, and love transcends all. I'm telling you, there won't be a dry eye in the house. You're probably wondering what the theorem will say. Me, too. Quite frankly, I haven't got a clue. But I haven't brought the writers on board yet. I'm thinking maybe it will have to do with the structure of singularities. Yeah, that has a ring to it. But there won't be three or four singularities, there will be 7T of them. Yeah, an irra­ tional number of them. We'll general­ ize the very definition of cardinality of sets. And we'll have a sequence. No, a lot of sequences. Wait, wait, make

those spectral sequences. Bockstein this. The co-authors don't care. They spectral sequences. And they are con­ prove it anyway. Wow, this is good. But verging to other spectral sequences. of course, we're still in the early de­ Grothendieck spectral sequences. And velopment stages. We'll need Mandel­ the indices on the sequences are them­ brot, Thurston, and Wiles to fill in some selves sequences. And there's a tower of the details. of these sequences reaching way up Oh, and the theorem will have im­ into the sky. plications. Oh yes, will it ever have im­ And there will be some incomplete­ plications. It will certainly imply the ness. Say a Cauchy sequence that Riemann Hypothesis at the very least. doesn't converge. Or a statement that Maybe Poincare, and Goldbach, too. is true but unprovable. Yeah, but get And it will show the airlines how to dis-

tribute their planes to the airports to maximize their profit. Maybe then the President could thank the three au­ thors in a special televised address to the nation, with a whole lot of happy pilots standing behind him. Oh, this is good. Anyway, let me know what you think We have to move on this fast, be­ fore someone else does it. Who knows, if we play this theorem right, maybe Ron Howard will make it into a movie. Sorry for the long message. Call me.

MIS either Confirms nor Denies Allegations Concerning Alan Turing Andrew INine d ·p culation t hat 115 may

cr

b

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World War I I.

syst ms in

ybernetics is the cientific ·t ucly of control

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lr Turing died e.·actly fifty years ago,

in 1954, at the ha\·e found it u pici ou t hat lr Turing also was involv d in r ru·ch on method t o distinguish human b ings fr m non­ human, artificial rt>ason ing machines. In his artid om­ pul ing fa ·hinery and I nt el li gence Turing advallc d t he

guishablc from those

h

gramme of countemwasure. to substitution of artificial ertain associates of tr Tming have su gge te

s d ind(>ed, o f e:xllerimental surgery at­ tempting the first hum an cyborg, t hat is, a hybrid of or­ gani m and computing machine. They ugge t that hi '"Turing t t" may have been de ·igned for u on hinlS lf, \\it h the aim of allowing hi research team to discern th human or rutificial nature of h is re pon. e . Th pre,iou ly top secret documents rei as d today confurn that Tming underwent a serie of tate-dir ct d proc dur · nl'ar the end of his life. It had b .n report d in th ' past that Turing was atTe ted in 1 952 for gros in­ d cency, and after c onviction under the British Ob c ni ty Act was subjN:ted to treatments designed to wcure" his ho­ mo ·e.·uality, and that thi had led to h is suicid in 1954. Many people, in clud ing fonner 115 and , 116 ag nts, have long regarded this as a mere �cover tory." "Which tory i more piau ible'?" ask on fonner ag nt 110\ work­ ·

ing in the pri,·ate enee , or ity'?

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Vancouver V6Y 1 Z2 Ganada

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advance copy of an artiCle to appear n a lead ng British tabloid .

VOLUME 26, NUMBER 2, 2004

57

l'i¥1(9·{·1.1

Dav i d E.

Rowe ,

Editor

I

There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had com­ paratively little, general relativity the­ ory very considerable, influence, above aU upon the development of a general scheme for differential geometry. -Hermann Weyl, ''Relativity as a Stimulus to Mathematical Research, " pp. 536-537

The Mathematicians' Happy Hunting Ground: Einstein's N General Theory of Relativity David E. Rowe

o one was more familiar with the impact of Einstein's general the­ ory of relativity (GRT) on mathemat­ ics and mathematicians than Hermann Weyl, who threw himself headlong into this new field shortly after the appear­ ance of Einstein's classic paper [Ein­ stein 1916]. In the summer semester of 1917 Weyl taught a three-hour course at the ETH in Zurich on "Raum, Zeit, Materie." The idea to publish a book based on these lectures came from Ein­ stein's close friend, Michele Besso [CPAE, vol. 8B, 663]. One year later the first edition of Weyl's classic Raum­ Zeit-Materie [Weyl 1918] was already in print. Einstein praised it to the skies:

I am reading with genuine delight the page proofs of your book, which I am receiving piece by piece. It is like a symphonic masterpiece. Every word has its relation to the whole, and the design of the work is grand. What a magnificent method the infinitesimal parallel displacement of vectors is for deriving the Riemann tensor! How naturally it all comes out. And now you have even given birth to the child I absolutely could not muster: the con­ struction of the Maxwell equations out of the g11/s! (CPAE, val. 8B, 669-670}.

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

58

Weyl, for his part, always held Ein­ stein's theory of general relativity in the highest esteem, and Raum-Zeit­ Materie did much to promote Ein­ stein's fame as the "New Copernicus." Thirty years later, when Weyl reassessed

THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER·VERLAG NEW YORK. LLC

its impact in "Relativity as a Stimulus to Mathematical Research" [Weyl 1949], he spoke far more soberly, but still with decided enthusiasm. That lecture and the passage from it cited above came on an auspicious occasion. On March 19, 1949, more than three hundred scientists-including such eminent figures as J. R. Oppenheimer, Eugene Wigner, I. I. Rabi, and H. P. Robertson-gathered in Princeton to celebrate Albert Einstein's seventieth birthday, which fell five days earlier. The celebrant's young assistant, John Kemeny, later recalled the excitement in Princeton during the days leading up to this stellar event, held as a tribute to Einstein's scientific achievements. "People fought over tickets like mad," Kemeny remembered. "I had nothing to do with the tickets, but people some­ how thought that being Einstein's as­ sistant I had some pull, and more big shots came to me begging for an extra ticket. They were absolutely dying to get in, and Einstein just had no sense at all about what absolute reverence there was for him" [Sayen 1985, 227] . Einstein in Berlin and GoHingen

For Einstein, Weyl's address must have brought back some fond memories of the enthusiastic response his general theory of relativity received from lead­ ing mathematicians, especially after November 1915 when he succeeded in finding an elegant set of generally co­ variant gravitational field equations:

R11v

=

-K(

TJLV -

i ) g}Lvr

(*)

Most physicists, by contrast, found Einstein's whole approach to gravita­ tion a daring speculation, at best. Max von Laue, who later became one of the strongest advocates of GRT in Ger­ many, was originally dismayed even by Einstein's original theory based on the equivalence principle alone. Laue re­ lated his misgivings to Einstein in a let­ ter dated 27 December 1911: "I have now carefully studied your paper on gravitation and have also lectured

about it in our colloquium [Arnold Sommerfeld's colloquium in Munich). I do not believe in this theory because I cannot concede the full equivalence of your systems K and K'. After all, a body causing the gravitational field must be present for the gravitational field in system K, but not for the accelerated system K ' " [CPAE, vol. 5, 384] . Laue's criticism came before Ein­ stein had wandered into the thickets of the Ricci calculus, an adventure made possible through his collaboration with the Zurich mathematician Marcel Gross­ mann [Pais 1982, 208-227]. After Ein­ stein left the ETH in Zurich to become a member of the Prussian Academy in Berlin, he found virtually no sympa­ thy in his new surroundings for the Einstein-Grossmann approach, often called the Entwurf theory. Indeed, the only encouragement he received came from an Italian mathematician who happened to be the world's leading au­ thority on Ricci's absolute differential calculus, Tullio Levi-Civita. As Einstein mentioned to his friend Heinrich Zang­ ger in April 1915, corresponding with Levi-Civita gave him great pleasure: The theory of gravitation will not find its way into my colleagues' heads for a long time yet, no doubt. Only one, Levi­ Civita in Padua, has probably grasped the main point completely, because he is familiar with the mathematics used. But he is seeking to tamper with one of the most important proofs in an in­ cessant exchange of correspondence. Corresponding with him is unusually interesting; it is currently my favorite pastime [CPAE, val. 8A, 1 77-1 18}.

Meanwhile Berlin's two leading the­ oreticians, Laue and Max Planck, re­ mained skeptical. Einstein's fortunes in Germany began to change, however, when he visited Gottingen to deliver six lectures on general relativity during the summer of 1915. Afterward, Arnold Sommerfeld and David Hilbert both be­ gan to take a keen interest in Einstein's daring approach to gravitation and in­ ertia. The three of them corresponded 1 Einstein identified the term

nostrifizieren

fairly regularly throughout the autumn, by which time Hilbert had developed a strategy for combining Einstein's gravitational theory and Mie's electro­ magnetic theory of matter within the framework of a generally covariant mathematical formalism [Rowe 2001]. Einstein was less than enthusiastic but saw that he had to abandon the re­ stricted covariance of the Einstein­ Grossmann theory in order to make progress. This led to his four famous notes on general relativity from No­ vember 1915, the last of which con­ tained the equations (*). Writing to Sommerfeld three days later, he de­ scribed his dramatic struggle: "you must not be cross with me that I am answering your kind and interesting letter only today. But in the last month I had one of the most stimulating and exhausting times of my life, and indeed one of the most successful as well" [CPAE, vol. 8A, 206]. His unhappiness with Hilbert at this time can be seen most clearly from another letter to Zangger in which Einstein wrote:

The theory is beautiful beyond com­ parison. However, only one colleague has really understood it, and he is seeking to "partake" [nostrifizieren]l in it . . . in a clever way. In my personal experience I have hardly come to know the wretchedness of mankind better than as a result of this theory and everything connected to it. But it does not bother me [CPAE, val. 8A, 205}. It bothered him enough, however, that he wrote to Hilbert suggesting that they forget about this momentary distur­ bance to their friendship [CPAE, vol. 8A, 222]. Afterward they remained on excellent terms, in large part because of their mutual dedication to the ideal of an international scientific commu­ nity free from political pressures.

physical problems of greatest interest. He applied these methods in deriving first results for the three classic tests of GRT: the perihelion motion of Mer­ cury, gravitational redshift, and the bending of light rays in the sun's grav­ itational field. All of these findings faced challenges from the scientific community during the decade follow­ ing Einstein's breakthrough in 1915, but leading proponents of GRT found a series of improvements that helped strengthen Einstein's case. The first important result was con­ veyed to Einstein in December 1915 by the astronomer Karl Schwarzschild, who was then stationed at the Russian front. Einstein replied in late Decem­ ber and then in a longer letter dated 9 January 1916, which began: "I exam­ ined your paper with great interest. I would not have expected that the ex­ act solution to the problem could be formulated so simply. The mathemati­ cal treatment of the subject appeals to me greatly" [CPAE, vol. 8A, Nr. 181]. Schwarzschild gave the first exact solution of Einstein's equation (*) in the vacuum case, where the right-hand side is zero, by showing how to calcu­ late the exterior gravitational field of a static massive body that was spheri­ cally symmetric, like the sun, but treated as a mass point [Schwarzschild 1916). The resulting metric thus gave a precise means of calculating the tiny deviations from Newton's theory that were of such crucial importance for the success of general relativity. An even more useful derivation of this result was obtained by Johannes Droste, a student of H. A Lorentz, in May 1916, the very month in which Schwarzschild died after contracting a terminal skin disease in Russia. Jean Eisenstaedt has pointed out in [Eisen­ staedt 1989, 216] that the famous Schwarzschild solution found in all standard textbooks on general relativity

The Schwarzschild Solution

The Einstein equations (*) are notori­ ously difficult to solve, a circumstance that led Einstein to devise methods of approximation for handling the special

( �)

- 1 -

+

dt2

+

( � rl 1 -

r(drf!- + sin2

e

dr

dcf)

(**)

with the name Max Abraham, who was famous in Gbttingen for his back-stabbing behavior. But, if Einstein learned of this ex­

pression from Abraham, it was, in fact, a familiar one in Gbttingen mathematical circles, where visitors were often warned that good ideas had a way of getting "nos­ trified" once they got into the local atmosphere. See [Reid 1 976, 1 2 1 ] .

VOLUME 26. NUMBER 2, 2004

59

was actually first presented in modem form in Droste's dissertation [Droste 1916]. In Droste coordinates, one sees immediately that the metric breaks down at r 0 and r 2M. Converting from geometrized units, the condition r 2M becomes =

=

=

r

=

2 M G c2

=

3

( l!_ )

km,

Ms

where Ms is the solar mass. Clearly, for any conventional astronomical object, the Schwarzschild radius r lies well in­ side the body, so that the vacuum equa­ tions no longer apply. Indeed, for sev­ eral decades all leading experts agreed that the Schwarzschild radius had no physical relevance whatsoever (see [Eisenstaedt 1989]). The kind of cata­ strophic gravitational collapse that leads to black holes was simply un­ thinkable in those days. Arthur Stanley Eddington was em­ phatic about this point in Space, Time and Gravitation, where he described what would happen if an observer were to approach the surface r 2M with a measuring rod. "We can go on shifting the measuring rod through its own length time after time," he wrote, "but dr is zero; that is to say we do not re­ duce r. There is a magic circle which no measurement can bring us inside" [Ed­ dington 1920, 98]. Hermann Weyl, who derived the Schwarzschild solution in Droste coordinates in Raum-Zeit-Ma­ terie, noted that the gravitational radius of the earth's mass was a mere 5 mm. David Hilbert claimed that the Schwarz­ schild solution contained two singular­ ities, the obvious one at r 0 and the more mysterious one located on the two-sphere with radius r 2M. Follow­ ing the lead of Ludwig Flamm, Hilbert calculated the trajectories of non-radial light rays that pass near this imaginary sphere, concluding that those which ap­ proach it can never penetrate its surface because their velocity will approach zero before arrival (see Figure 1). Hilbert presented these findings in an unpublished lecture course offered in Gottingen [Hilbert 19 16-17], but Max von Laue borrowed a copy of the course notes when he was writing Die Relativitatstheorie [Laue 1921], the first textbook on general relativity in the German language. Laue repeated Hil=

=

=

60

THE MATHEMATICAL INTELLIGENCER

-"'--�--------------

--

-------- -

---·---- · -A_,_

Figure 1 . Hilbert's visualization in [Hilbert 1 916-17] of the trajectories of light rays in the neigh­ borhood of a strong gravitational field. Note how the rays fail to penetrate the "magic circle" (Eddington's term) determined by the Schwarzschild radius.

bert's pronouncement regarding the singular nature of the two-sphere with r 2M, and he even replicated Hil­ bert's diagram showing how light rays fail to penetrate the sphere (Figure 2). Like the Gottingen mathematician, he was convinced of the essential nature of this singular region, which "cannot be eliminated by using any other coor­ dinates; it is essential to the nature of the thing" [Laue 192 1 , 215]. Physically, this seemed patently obvious, since "every mass m . . . has a radius greater than . in fact we do not know as yet any counter-example even in the nucleus of atoms." (For a modem treat­ ment of the trajectories associated with the Schwarzschild solution, see [Wald 1984, 136-148].) Johannes Droste had actually pre­ sented solutions to the Einstein vac­ uum equations for a mass point in three different coordinate systems, and he was well aware that the r coordinate had no direct physical significance [Eisenstaedt 1989, 216-2 17]. Einstein, too, was completely clear about this cir­ cumstance. His friend, Paul Painleve, had been troubled by the plethora of different representations of the Schwarzschild solution, and passed his misgivings on to Einstein. Noting that one could obtain an infmite number of different formulas for the metric, he suggested this "gave a clear indication of the hazardous character of such pre­ dictions," concluding that "it is pure imagination to claim that such conse­ quences can be derived from the ds2" =

2C::

..

[Eisenstaedt 1989, 222]. Einstein's re­ sponse illuminated the mathematical and physical issues that bothered Painleve and others:

When in the ds2 of the static solution with central symmetry you introduce any junction of r instead of r, you do not obtain a new solution because the quantity r in itself has no physical meaning . . . only conclusions reached after the elimination of coordinates may pretend to an objective signifi­ cance. Furthermore, the metrical in­ terpretation of the quantity ds is not ''pure imagination" but rather the in­ ner-most core of the theory itself (quoted in [Eisenstaedt 1989, 223}). Einstein's original treatment of the Schwarzchild problem was based on a specialized class of coordinate sys­ tems. He also postulated that the met­ ric tensor be spherically symmetric, independent of time, stationary, and asymptotically flat at spatial infmity. These conditions hold, of course, for the Schwarzschild-Droste metric (**) , and Hilbert showed that the last con­ dition was not required to deduce this solution. Then, in 1923, G. D. Birkhoff proved a far stronger result: Birkhoffs Theorem: Any sphericaUy symmetric solution of Einstein's vac­ uumjield equations is necessarily sta­ tic and has a Schwarzschild metric. In Relativity and Modern Physics [Birk­ hoff 1923], the Harvard mathematician

F

!b

sideration of systems of geodesics, but

E

another vast terrain was opened by the

Blj

study of general path spaces, a specialty

0

C

of the Princeton geometers. Weyl de­ scribed this research activity in the fol­

3 }'3

2a

����----�

I

3 l'3 2

--

Ct

Figure 2. Max von Laue's diagram of the same phenomena in [Laue 1 921] was clearly taken from [Hilbert 1916-17].

showed that the assumption of spheri­

One ofthose who followed Einstein's

cal symmetry alone was sufficient to en­

lectures closely was the differential

sure the existence of a coordinate sys­

geometer

tem in which the solution

Princeton's senior mathematician. In

(**) holds for

Luther

Pfahler

Eisenhart,

the vacuum case. The Schwarzschild so­

fact, Eisenhart instigated the invitation

lution thus bears a strong analogy with

that led Einstein to visit Princeton that

the Coloumb field associated with a sta­

spring, although he originally hoped to

tic charged particle in electrodynamics:

coax him into coming as a guest lecturer

in both cases the fields are unique, show­

1920-21 (Eisenhart to Einstein, 20 October 1920, [CPAE, vol. 7, 231]). Einstein declined this offer, but in February 1920 Kurt Blu­

ing that a monopole cannot emit radia­ tion. (For a modem treatment of Birk­ hoffs Theorem, see [Hawking and Ellis

1973, Appendix B].)

during the winter semester of

menfeld

persuaded the now-famous

physicist to undertake a trip to the United States to support the Zionist

Relativity and Differential Geometry

movement and raise funds for Hebrew

Einstein had met Birkhoff on his very

University [CPAE, vol.

first visit trip to the United States in

1921,

a whirlwind tour that ended with a stop

7, 231: note 48),

a circumstance that led him to recon­ sider Eisenhart's invitation.

in Princeton. There, in May, Einstein de­

Einstein surely recalled Eisenhart's

livered five Stafford Little Lectures on

avid interest in general relativity when

the theory of relativity. The first two of

he heard Hermann Weyl's lecture on

these were of an introductory nature,

"Relativity as a Stimulus to Mathemati­

whereas the latter three entered into the

cal Research." Weyl noted that a major

technicalities of the theory. Later that

part of this stimulation began with in­

year, Einstein worked on a revision of

vestigations of affinely connected man­

his

last three talks and these were pub­

ifolds, which generalized Riemannian

lished in booklet form under the title The

manifolds but still admitted infinitesimal

1922],

parallel transport of vectors. Since co­

Tran­

variant differentiation and the whole ap­

scripts of his first two talks, on the other

paratus of tensor analysis carried over

hand, were only recently published in

to this more general setting, it was not

7 of the Collected Papers of Al­ bert Einstein [CPAE, vol. 7, App. C ] .

necessary to assume the existence of a

Meaning of Relativity [Einstein one of his best known works.

volume

cused on the conformal structure of Rie­ mannian spaces, which led to the con­

metric tensor. One area of research fo-

lowing words:

. . . here is clearly rich food for math­ ematical research and ample oppor­ tunityforgeneralizations. Thus schools of differential geometers sprang up in the wake of general relativity. Here in Princeton Eisenhart and Veblen took the lead, Schouten in Holland. In France, E. Cartan's fertile geometric imagination disclosed many new as­ pects of the subject. Some of their out­ standing pupils are Tracy Thomas and J. M. Thomas in Princeton, van Dantzig in Holland and Shiing Shen Chern of the Paris school. A lone wolf in Zurich, Hermann Weyl, also busied himself in this field; unfortunately he was aU too prone to mix up his math­ ematics with physical and philosophi­ cal speculations [Weyl 1949, 538}. The work of Elie Cartan and Weyl eventually opened up the whole vast theory of fiber spaces of differential manifolds.

Cartan's

investigations

dealt with geometries possessing tran­ sitive transformation groups that act on their tangent spaces, an approach that drew much inspiration from Lie theory and Klein's Erlangen Program (see [Cartan

1974]). Weyl developed an

axiomatic treatment of maps of tan­ gent spaces by utilizing parallelism along smooth curves. The latter idea was first developed by Levi-Civita in

1917 for Riemannian spaces, but it was soon generalized by Weyl to manifolds with an affme connection. Using these constructs, Weyl sought to develop a purely

infinitesimal

geometry

that

could serve as a basis for a unified field theory for gravity and electromagnet­ ism [Scholz

2001].

Eisenhart took his inspiration from the generalization to general affine spaces of geodesics in Riemannian geometry, namely the paths satisfying the differential equations

d2xi dt2

--

+ f �.

]k

d:x} dx" dt dt

-

-

0.

(***)

VOLUME 26, NUMBER 2, 2004

61

In Riemannian geometry one uses the In an article in Science (December space caused by the Sun's gravitational metric tensor to define the Christoffel 21, 1923) the Princeton geometer came field. Eisenhart's article appeared in symbols rjk from which one then to Einstein's defense after Philipp the wake of a similar defense of Ein­ proves that the solutions of (***) are Lenard republished portions of an stein's priority published by David geodesics, the shortest paths joining older theory of gravitation presented Hilbert and Max Born in the widely any two points within an appropriate by Georg Soldner. Lenard claimed that read Frankfurter Zeitung (see the ac­ neighborhood of the manifold. In an Soldner had derived precisely the same companying box). Hermann Weyl also affine geometry, the rjk are defmed in­ result for the deflection of light in the countered the criticisms of anti-rela­ dependent of a metric, and the solu­ vicinity of the sun as that found by Ein­ tivists, but this only hardened the tions of (***) determine a congruence stein in 1911. Einstein later revised his views of experimental physicists like of paths with special properties. The figure to 1. 75" of arc in 1915, twice the "Lenard and Co.," who regarded gen­ geometry of paths in affine spaces can original value, due to the curvature of eral relativity as a theory devoid of also be regarded as a generalization of methods introduced by Riemann in his famous Habilitationsschrift of 1854, Soldner und Einstein in particular his systems of normal co­ ordinates. Eisenhart worked directly Von Prof. Dr. Hilbert und Prof. Dr. Born Gottingen with objects of the manifold, in con­ Der Aufsatz von L. Baumgardt veranlaBt uns zu folgender sachlicher trast to the methods of Cartan and Berichtigung: Weyl, which dealt with the associated tangent spaces. Unfortunately, the for­ 1. Die von Soldner 1801 abgeleitete Formel fur die Lichtablenkung stimmt mer approach leads to complications, mit der iiberein, die Einstein im Jahre 1911 als Resultat einer vorHiufigen Uber­ because the intrinsic geometric objects legung tiber den EinfluB der Schwere auf das Licht veroffentlicht hat. Diese associated with paths do not satisfy the Arbeit von 1911 entha.It aber noch nicht diejenigen Grundgesetze der Physik, transformation law for generalized die man als, "allgemeine Relativitatstheorie" bezeichnet. Christoffel symbols, a central feature 2. Die richtigen Grundgleichungen dieser allgemeinen Relativitatstheo­ in the affine theories of Weyl and Car­ rie wurden erst Ende 1915 von Einstein gefunden. Dabei ergab sich, daB tan. Eisenhart summarized his contri­ der von Einstein vorher 1911 vermutete Wert, der mit der Soldnerschen butions to the geometry of paths as Angabe iibereinstimmt, vom Standpunkt der Relativitatstheorie falsch ist; well as those of his students in Non­ vielmehr stellte sich die Lichtablenkung doppelt so groB heraus. DaB Ein­ Riemannian Geometry [ �isenhart stein bei seinem Suchen nach den richtigen Gesetzen der Schwere zunachst 1927], whereas the related theory of durch vorlaufige Abschatzungen zu dem falschen Wert gelangt war, ist le­ differential invariants was sketched by icht begreiflich; das Genie sieht lange, ehe ein Gedanke begrifflich bis in Oswald Veblen in "Invariants of Qua­ alle Einzelheiten geklart und formuliert ist, die wichtigsten Zusammen­ dratic Differential Forms" [Veblen hange voraus und schatzt die experimentell kontrollierbaren Wirkungen 1927, Chapter VI]. ihrer GroBe nach zunachst mit rohen Mitteln ab. 3. Der endgtiltige Wert der Lichtablenkung von Einstein folgt vollig ein­ Einstein's Enemies: deutig aus der allgemeinen Relativitatstheorie. Es ist ein lrrtum zu glauben, The Anti-Relativists daB Soldner eine Folge der Relativitatstheorie richtig vorausgesehen habe; Not surprisingly, these mathematical vielmehr ist es umgekehrt: wenn die von Soldner auf Grund der Newton­ developments were largely ignored by schen Theorie berechnete GroBe der Lichtablenkung (die mit Einsteins physicists, few of whom had sufficient vorlaufiger Schatzung von 191 1 zufa.Ilig tibereinstimmt) durch die Beobach­ knowledge of differential geometry to tungen als richtig erwiesen wiirde, so ware damit eine vollgiiltige Wider­ make any sense of them. Opponents of legung der Einsteinschen Relativitatstheorie erzielt. relativity, on the other hand, had long Die englischen Astronomen, die bei der Sonnenfinstemis des Jahres claimed that Einstein's theory was of 1920 die Messungen der Lichtablenkung ausgefiihrt haben, sind der Mein­ purely mathematical interest, pointing ung, daB nicht der auf der Newtonschen Attraktionstheorie fuBende Wert to Minkowski's four-space formalism von Soldner, sondem der von der Einsteinschen Relativitatstheorie vo­ to make their case. Aside from Weyl, rausgesagte Wert tatsachlich gilt. Neue Untersuchungen werden bei der most prominent mathematicians chose Sonnenfinstemis des Jahres 1922 stattfinden. to avoid engaging in polemics with vi­ tuperative anti-relativists. Still, just two years after Einstein's visit to Text of an article by Hilbert and Born in the Frankfurter Zeitung defending the originality of Princeton in 1921, the soft-spoken Einstein's prediction for the deflection of light in the sun's gravitational field. The authors em­ Eisenhart became embroiled in con­ phasize that Einstein's original prediction from 1 91 1 , which agreed with the value earlier ob­ troversies surrounding Einstein's the­ tained by Georg Soldner, was incorrect, and that Einstein's revised figure from 1 91 5 repre­ ory, which by then had spilled over sented the true relativistic result. Strangely, Hilbert and Born state that the British confirmed from Germany into the United States. Einstein's result during the solar eclipse of 1 920; the eclipse took place in May 1 91 9. 62

THE MATHEMATICAL INTELLIGENCER

12

March

ZS, 1922

Reuterdahl vs. Einstein : Nailing a Fallacy Former Insists Latter's· Theory Wrong and Questions Its Originality S THE Einstein enthusiasm is waning, we may well ask what thoughts and influences have 1 caused this slowing down of . the meteorical rise ·and advance of Einstein in giving the study of relativity its tremendous and unexpectedly sudden impetus. The History of Relativity shows us that in 1889 Rudolf Mewes, in an article entitied, "Das Wesen der Materie und des Naturerkennens/' gave a brief outline of his theory of relativity. Mewes held that the es­ sence of matter consists in its action as manifest ip continuous transformation in a.ccordance with cau­ sality. He found in his basic concept .a sow-ce of the relativity of space, time and matter. For him all reality depends upon the combination of space and time. From 1892 to 1894 M ewes made a cardul analysis of the derivation of Webn's Ltlw from the Do[ltler Principle. This resulted in the tvafuation by M�wcs of the transformation formula which later made · Lo­ rentz famous through its use by Einstein. Mewes clearly antedates 4rmor, Lorentz and Einstein in this development. In addition, Mewes grounds his system in sound philosophy. Arvid Reuterdahl presented a brief · outline of his Space-Time Potential and Theory of Interdependence · at the inaug'ural meeting of the American Electrochem­ ical Society, held ih · Philadelphi2., April 5, 1902, unde United states to de. no·un�.e- Einsteinism .as a- fallacy. Dean Reuterdabl ·was., born February lS,'!\1876, at Karlstad, Sweden, the son of .Christ ina. Reuterdabl. :Jn 1882 the- family came to the United rs�ates, eventually locating in Providence. The son�s early education c onsist ed in private instruction given by his. father, a former officer in th e Swedish army. and an es­ c cllent mathematician and philosopher.

J�·-�

between 41space" and "time" is a mere coincic:ilmce. In a statement given to the New York World (J une 12, 1921 ) , he again disclaims all knowledge of Reuterdahl. This disclaimer is irreconcilable with his own previous statement to the Afteuposten. The question naturally "Why was Einstein s o anxious to efface the arises ; name of Reuterdahl in connection w ith the Theory of Re-lativity? Does the five years' sojourn in Europe of Reuterdahl's manuscript answer th is question ?

to an infinitesimal portion of the great frame are the same as hold for a part conceived as great as the imagination may permit in conformity with the frame. The Einsteinean uJorld r�ference frame is Gaussian and four-dimr:nsional in type, the co-ordinate syste'm being .an attemvt at the amalgamation of space and time. Any reference section arbitrarify cut out of.."the fram� ·is a veritable co-ordinate fabric. Such section has been called a "R•fer•>�ce-Mollusk" by Einstein. •We may conceive that a large reference section or '"Mollusk" shrivels or contracts into a small section or vice ver sa. During such diminutions or augmentati()ns the laws of the relatKmal factors remain the same, and the re­ sulting section is either a reduced or - increased replica of the original. The changes are similar to the · effect produced by looking at an original,. first ·with a reducing glass, and then with a magnifying glass. All the parts of the co-ordinate fabric will remain in the same· rela · tive juxtaposition irrespective of the type of th�· ·gtass through which it is observed. In �uch a system force regarded as an independent entity does not exist.;. Action at a distance no longer becomes a problem because the world frame is a uni­ tary interacting system. Einstein holds that a further consequence of his worldsystem is the disappearance of the mass M (which pertains to a particular point) from the law of motion. He reduces the motion of a particle under the so-called inftuen.:e of gravitation to a mere "Following" of a geodetic line in his four-dimensional world� Nowhere has Einstein stated his own case as clearly as it has been- resented. in the above an�Jysi�. w�ich· is p due lo Reuterdahl. It IS even doubtful If E10stem has grasped the significance of his own vague and stumbling attemp ts at formulating this great world synthesis. His p itiable inability to express the essentials of the system m a clear and intelligent mann�r can readily be ac­ counted for on the supposition which grow! upon us­ that Einstein has taken his entire thought of this syn­ thesis from the work of Reuterdahl. Not grasping dearlr the intent of Reuterdahl's Space-Time Potential, Emstein camouflaged it in a dis­ torted manner into his own Space- Time Continuum­ by transferring the conception from Euclidean to non­ Euclidean. Rcutcrdahl's Space-Time Potential is a syn­ thesis,· not only of the same intent as Einstein's, but of a vastly more comprehensive order. Reut�rdahl's com­ plete cosmology proceeds direct from physical facts, wh_ereas Einstem presents us with a skeleton, incapable of harmonious functioninJ�. btcause - it is constructed fro� the fibers of incons1stent mathematical specula­ tion, which predict coming events with the hope that they will conform-and if they do not, they arc quietly thrown out of court. Rcutcrdahl has distinctly stated, that force docs not exist in the physical universe . as . all independent and distinct entity, and action-at-a � x

- k�� l { }, 1 L n

n k

-

,

where {x} denotes the fractional part of Surprising, indeed. Results like this don't come for free. Sacrifices are re­ quired. (I would not be surprised to learn of the mysterious disappearance of animals, even children, in the valley of chicks a century ago.) The above re­ sult has a probabilistic interpretation: if a large integer n is divided by each integer k = 1,2,3, . . . , n, then the av­ erage fraction by which the quotient nlk falls short of the succeeding inte­ ger is not 1/2, the value intuition sug­ gests, but y. Section 1 . 1 1 on Chaitin's constant is mind-blowing. This material is at the nexus of probability, computability, logic, and decidability, and it has its ba­ sis in Matiyasevic's proof that Hilbert's Tenth problem is unsolvable, i.e., the result that there exists no algorithm for determining whether a general Dio­ phantine equation has positive integer solutions. It is possible to construct a x.

72

THE MATHEMATICAL INTELLIGENCER

universal Diophantine equation with this property whose solutions DN de­ pend on the parameter N. Now define a real number A by its binary expan­ sion O.A 1A2A3 . . . where

AN =

{

�

1 f DN =I= 0 0 If DN 0 =

There is no algorithm for deciding for arbitrary N whether AN is 1 or 0, so A is a perfectly well-defined but incom­ putable real number. This assertion has

"There i s a spigot

ample can be found in Section 3.12 on Du Bois Reymond's constants. Define the mth Du Bois Reymond constant by

Much is known about the Cm. Watson proved that c2k is a polynomial of de­ gree k in e2 with rational coefficients. No such information is known about the c2k+ 1, but there is an expression as an infmite series for all Cm· Denote by �1 , �2, �3 . . . the positive solutions of x = tanx. Then

algorith m for calculating

1r"

disquieting philosophical implications. There are many open problems, some philosophical, some mathematical, in this area. Section 2. 13, on Mills's Constant, gives rise to perplexing thoughts, too. Mills showed that there exists a posi­ tive constant C such that I C3N I yields prime numbers for all integers N. ( I . I denotes the largest-integer func­ tion.) Actually, there are many Cs. It can be shown that one way of con­ structing one is to define q0 2 and let qn+ 1 be the least prime exceeding q� for each n 2': 0. Then such a C is given by

Constants arising in the approxima­ tion of functions-including the topics of Fourier series, approximation by generalized orthogonal polynomials, and best rational and polynomial ap­ proximation-exert a powerful fasci­ nation. I love Lebesgue constants, which originally arose in Fourier se­ ries, but can be defined for expansions in any set of orthogonal polynomials. If Sn is the nth partial sum of the Fourier series for a functionfwith ::::::; 1, then

I! I

=

It is easy to compute that C 1.3063778838. . . . However, this ap­ proach is unsatisfying. First, it requires the prime numbers themselves, and second, one would require C to many places to compute just a few primes. Mills's formula would be of use as a prime-generator only if a way could be found to generate an exact value of some C. Scattered through the book are se­ ductive kickshaws which could lead one to fritter away hours that could more productively be spent in grading final exams, or engaging in legitimate (at Drexel this means funded) re­ search. Many of these originally ap­ peared as problems in the American Mathematical Monthly. A typical ex=

and Ln is called the nth Lebesgue con­ stant. Lo = 1, L1 1 .43599 . . . , L2 = 1.642 18 . . . , L3 = 1.7783. . . . Is Ln monotone increasing? Szego showed it was, by demonstrating the series =

Ln =

16 _9 7r

x (2n+l)k

Ll y�l ? k�

1 4k2 - 1

1 2'(}· - 1 ·

Less easily decidable issues arise from Lebesgue constants occurring in more sophisticated contexts, the theory of orthogonal polynomials, for instance. I want to mention an example from the theory of best approximation. Let's restrict ourselves to functions bounded on [ - 1,1] with norm ! = sup IJCx) l .

ll ll - l:s "'>1 x

Let Hn denote the norm o f the error of the best polynomial approximation of degree n to the function lx /. Bernstein

showed that the following limit ex­ isted, lim nHn n--.x

= {3,

=

=

and that {3 < .286. He conjectured that {3 1/(2 y;) .2821, but many years later, Varga and Carpenter showed this cor\iecture to be incorrect by comput­ ing {3 to 50 decimals. A closed-form ex­ pression for {3 is still unavailable. Con­ trast this with the error, call it En, of the best approximation to f by the ra­ tio of two polynomials each of degree n. In 1992 Herbert Stahl proved that lim

n--.ac

en-Yn En

=

o(1). = Bv� � B = 2 VZb = 2.9904703993 . ln(Mn)

Here where b

=r 0

+

ln(l - ln(1

. .

'

- e-x))dx.

Discrete structu res (I

8.

This example, which shows that some­ times the complex is more amenable to analysis than the simple, is one of those results that aren't to be had for free. However, I know Herbert, and he is an honorable man, incapable of commit­ ting enormities for the sake of mathe­ matics. I believe the fauna of Berlin are safe. Discrete structures (I prefer the term "combinatorics") yields a plethora of in­ triguing constants. I mention an exam­ ple which the reader should find ap­ pealing, since probably most readers of The Intelligencer are drivers. Cars of unit length are parked at random on the street. If a new car overlaps a car, our convention allows it to slide off to the front or rear, whichever is closer. It is then parked if there is a space for it. If not, it is discarded. The mean lim­ iting density of the cars that can fit is

m=f

of all members of the group? Bill Goh and Eric Schmutz, through a redoubtable feat of asymptotic analysis, were able to find the asymptotic expression

p refer the term "combi natorics") yields a plethora of i ntri g u i ng constants . Stong later gave a simpler formula for b: _

b-

J

x

0

ln(x + eX -

1) 1 dx.

Combinatorics yields another pot­ pourri of constants, with applications to manifold areas of applied science. Typically these constants occur as the lead constant in the asymptotic ex­ pression for some important combina­ torial quantity. One example: denote by rn the number of constitutional iso­ mers of the alkane series (methane,

ethane, propane, butane, etc.) Then if

(2x + 1)exp[ -2(x +

{3(x)

m=

=

exp

e-x - 1) ] {3(x)dx,

( -2 L 1 -/-( dt) , X

and .8086525183 . . . This seems reasonable, but in real life it is proba­ bly lessened by the occurrence of those cyclopean SUVs. Group theory has contributed its share of constants. Given a permutation 1T of n symbols, defme its order, ec 7r), to be the least positive integer such that 7f"' identity. Obviously, 1 ::s ec7r) ::s n!. What is its asymptotic mean value, call it Mn, taken over the orders

=

m

K

= .3551817423 lim �rnn512

n--.x

. . . . one has

= .6563186958 . . .

The theory of trees is responsible for these results, and for many other re­ sults about chemical isomers. All of this material is fertile with un­ solved problems, which the author doles out generously. A polyomino of order n is a connected set of n adja­ cent squares, i.e., squares containing a common side. Two polyominoes are distinct only if they have different shapes or different orientations. Let A(n) denote the number of polyomi­ noes of order n.

We have

A(1)

=

=

= =

1, A(2) 2, A(3) 6, A(4) 19, A(5) 63, A(6) = 216, A(7) 760 . . . =

=

Almost nothing is known about A(n). No one has found a generating function for it, although Augean computations by various researchers have provided values up to A(47). It is suspected, but not known, that A(n) is not com­ putable in polynomial time. Though it is known that the limit

a=

lim [A(n) ] 11n n--.oo

exists, its value is not known. Various bounds exist. Slight changes in the model can pro­ duce problems that are more tractable. A polyomino is row-convex if every row consists of an unbroken line of squares. Denote by B(n) the number of possible row-convex polyominoes consisting of n squares.

=

B( I ) 1, B(2) = 2, B(3) 6, B(4) 19, B(5) 61, B(6) 196, B(7) 629 . . .

=

=

=

=

=

A generating function is known for [1] :

B(n) f(x) =

x(1 - x? 1 - 5x + 7x2 - 4x3

=

I B(n)xn,

n=O

and this tells us everything we need to know. It leads to an explicit represen­ tation for B(n), even to one in terms of radicals, and to the asymptotic formula

B(n) - ccin,

a = 3.205569431 · c=

. 1908155020

B(n) has very curious properties. B(n) grows very rapidly, too rapidly, one would think, to have a very robust di­ visibility profile. Not so. Let b(n) B(n) - 2. It can be shown that every positive integer divides an infinite number of the b(n). Whether A(n) above er\ioys similar properties is not known. I close with one last example from this bounteous book, one which ex­ hibits the resourcefulness, the impres­ sive ingenuity of the mathematical imagination in its fullest flowering, namely, the material on Conway's Con-

=

VOLUME 26, NUMBER 2, 2004

73

stant (Section 6. 12). We start with a string of digits, say, " 13." Describing this sequence as "one one, one three" leads to the derived sequence, " 1 1 13." This we describe as "three ones, one three" to obtain "31 13." The next num­ ber is "132 1 13," and so forth. What can be said about the length L(k) of the kth string? At first glance, this seems to be precisely the sort of problem one should flee from. Conway, however, did not. Defying all expectations, he proved that L(k) �CAk, k � oo, A 1.3035772690. . . .

page in the book is an adventure, and I am grateful that he could enlist all of us as his fellow explorers. REFERENCE

[1 ] Odlyzko, A M . , "Asymptotic Enumeration Methods," p. 1 1 04, Handbook of Combina­ torics, v.ll, ed. R. L. Graham, M. Gr6tschel,

and L. Lovasz, North-Holland, 1 995. Department of Mathematics Drexel University Philadelphia, PA 1 91 04 e-mail: [email protected]

=

An amazing feature of A is that it is a universal constant, in the sense that the above asymptotic estimate prevails (with the trivial exceptions of the initial strings "0" and "22") no matter what

I th i n k al l mathematicians shou l d own t h is book . one starts with. Doron Zeilberger and Shalosh B. Ekhad (about whom I have warned lnteUi­ gencer readers in the strongest possible terms), both figure in this saga, as does the motif of utilizing mathematical soft­ ware to prove theorems. There is a con­ nection with chemistry. A is the unique largest (in modulus) eigenvalue of the 92 X 92 transition matrix whose (i,J)th element is the number of atoms of ele­ ment j resulting from the decay of one atom of element i. Occasionally one en­ counters cynics who claim there is lit­ tle really conceptually new to be dis­ covered in mathematics, that future knowledge will be a mere elaboration of the known. They are deceived. I think all mathematicians should own this book. To say that it is a tri­ umph of substance over style is too dis­ missive, and even inaccurate. There are sections where the author conveys admirably his excitement over some unexpected and beautiful sequence of ideas. I pay the author the earnest com­ pliment of stating that nearly every his cimmerian cohort,

74

THE MATHEMATICAL INTELLIGENCER

Hypercomplex Iterations by Yumei Dang, Louis H. Kauffman and Daniel Sandin WORLD SCIENTIFIC, 2002 SERIES ON KNOTS AND EVERYTHING, VOL. 1 7 1 64 pp. US$33 ISBN 981 -02-3296-9

REVIEWED BY KEITH BRIGGS

T

he prototype complex discrete­ time dynamical system is the qua­ dratic polynomial z � z2 + c; z,c E C. Thanks to pioneering work in the 1980s by Douady, Hubbard, Peitgen, and oth­ ers, we now have a good understand­ ing of both the structure of orbits for fixed c, and the structure of parameter space, that is, the dependence of the dynamics on the constant c. The for­ mer is described by Julia sets, the boundaries of which separate bounded orbits from those which diverge to in­ finity; and the latter is described by the Mandelbrot set: a value of c is in the in­ terior of this set if the orbit starting at 0 converges to a cycle. The Mandelbrot set consists of components, each topo­ logically a circle, on which the period of the cycle is constant; components meet tangentially at bifurcation points. Furthermore, it is easily shown that any complex quadratic polynomial is affine conjugate to z2 + c for some c, so that we have essentially a complete understanding of this family of dynam­ ical systems. It is therefore natural to ask whether such a description is available for dy­ namics in other spaces, and the present

book is an attempt to produce a paral­ lel theory for quatemion polynomials. The quatemions IHl form a 4-dimen­ sional non-commutative division alge­ bra, in which the three imaginary units i, j, k satisfy i 2 j2 = k2 = ijk - 1. S o how to d o maps q � q 2 + c ; q,c E IHl behave? Unfortunately (or fortunately, depending on our point of view), we get nothing new here, since there is al­ ways a rotation of IHl which puts this into the form z � z2 + c for z,c E C. In other words, the dynamics take place on a 2-real-dimensional subspace, iso­ morphic to C, in IHI. Thus, we can say that the quatemion quadratic q2 + c is already fully understood. The authors note this fact on page 66, but then they go on to ignore this raison de ne pas etre. Additionally, this particular qua­ dratic is no longer a generic represen­ tative of the set of all quatemion qua­ dratics, so it is not clear why it should be the focus of study. This book suffers from several de­ fects: a large number of factual and typographical errors; a curious imbal­ ance in the level of rigor, giving rigor­ ous theorems (though sometimes with faulty proofs) that are not used, and re­ jecting rigor when it is really needed; as well as uncertainty as to the intended audience. The ostensible main aim is to produce estimates of distances in IHl to Julia sets in order to produce computer graphics of these. However, we have al­ ready seen that these distances could easily be found by rotating to C, so it is hard to see any justification for devot­ ing a whole book to this aim. Let us survey the book to note some specific examples of these defects. On page 20 we read that "The Mandelbrot set was discovered by Mandelbrot in 1980." In fact, it was discovered by Brooks and Matelski in 1978. The proof of Lemma 3. 1 makes use of the "fact" that cos(m - n)d O for integers m,n, m =I= n. The proof of Lemma 3.3 is by contradiction: it begins, "Suppose to the contrary that f is a constant" and concludes, "Then f(z) is a constant, a contradiction!" The exclamation mark is indeed ironic. We have circular def­ initions such as that of Julia set on page 44: "Let us denote by Ap(oo) the basin of attraction of the Julia set generated =

J51T

=

by the polynomial p(z). Then the Julia set of p(z) is defined as Jp = aAp(oo)." Even definitions that are not meaning­ less are confused, such as the immedi­ ately following definition of Mandel­ brot set: "M = {q E IHI I 0 E Kp forp(z) z2 + c) where p(z) = zn + c. " A long but irrelevant section on analyticity of quaternion functions based on the work of Sudbery is then followed by an in­ correct statement of the maximum­ modulus theorem of complex analysis, which puts a condition onf ' (z). Section 4.8 is taken word-for-word from a paper of Bedding and Briggs, though the cita­ tion in this section incorrectly gives the author as Briggs. In copying equation (4.27) from that paper, five typographi­ cal errors were introduced. In addition, in the sentence "Second, let us see whether anything more interesting is possible with quadratic quaternion functions," the crucial word "regular" has been omitted before "quadratic." Eventually we reach section 7.3, where the distance estimation results be­ gin to appear. Here the authors work with maps F : fRN � fRN for general N, without bothering to define what multi­ plication might mean in such a space. Section 7. 1 has defined a restricted form of multiplication on the unit sphere, but this again appears to be reducible to the complex case by rotation. The authors assume without proof that the set of points not attracted to infinity under it­ eration of F is compact. Everything de­ pends on this assumption; we are not told for which F this might be true, and it is not checked for any explicit F. Page 75 uses the chain rule for a de­ gree k polynomial incorrectly: "npn + 1 = k(Fn)D(Fn)." In any case, the "correct" rule npn + 1 k(Fn)k- 1D(Fn) is proba­ bly meaningless in view of the lack of a definition of multiplication already mentioned. More defects of this kind continue throughout the book Because of this, it is difficult to have confidence that the computer graphics shown are cor­ rect. Unfortunately, this book does not add to knowledge, but will confuse a naive reader and infuriate a knowl­ edgeable reader. My opinion differs from that of Peter Ha'issinsky in Math­ ematical Reviews: "This book is well =

=

written and self-contained. A CD-Rom comes with the book which provides striking animated illustrations of such fractals." The attached CD indeed con­ tains striking images, but these are es­ sentially meaningless when based on dubious mathematics. BTExact Adastral Park Polaris 1 34 Martlesham Heath, Suffolk IP5 3RE, UK e-mail: [email protected]

On Ouaternions and Octonions: Their Geometry, Arithmetic, and Symmetry By John H. Conway and Derek

A.

Smith

NATICK, MASSACHUSETTS: AK PETERS 2003 1 50 pp. US $29.00 ISBN 1 -56881 - 1 34-9

REVIEWED BY GEOFFREY DIXON

A

resonant spike above background noise in one parameter as another parameter is varied is a frequent indi­ cator in experimental science that one has found something significant. Reso­ nance is good. It justifies the expendi­ ture of our time and can lead to the kinds of rewards that prompted us to run the experiment in the first place. In a broad sense there is a reso­ nance in mathematics detectable in much the same way. Using as the input parameter the set of positive integers, and as the output some measure of the number and richness of mathematical structures associated with the inputs, one could easily argue that the integers with the highest "resonant" spikes above the background noise are 1, 2, 4, 8, and 24. Associated with the first four of these integers are the parallelizable spheres, the n-square identities, lat­ tices of particular beauty and elegance, and the composition algebras, R, C, H, and 0 (real numbers, complex num­ bers, quatemions, and octonions). There is a holographic aspect to much of this. Starting with the parallelizable spheres, for example, we can derive

the existence of the division algebras, and from there the n-square identities, and so forth. The whole in that case is implicit in a part. For all that, the eas­ iest bits to play with are the so-called division algebras, R, C, H, and 0. And by the way, I include 24 in the list be­ cause that is the dimension of the Leech lattice. These mathematical objects have from the beginning inspired a great deal of research in both mathematics and physics (and beyond: the most ef­ ficient way to build a 3-d engine for computer games and modeling is using the quatemions). During the latter part of the nineteenth century, throughout much of the twentieth and on into the twenty-first, there have been those, like myself, convinced that this notion of mathematical resonance is more than metaphor. It seems reasonable that it should be able to help us answer some of the big questions of physics. In particular, why is the universe struc­ tured as it is? The father and patron saint of divi­ sion algebra obsession is surely W. R. Hamilton, the discoverer in 1843 of the quaternion algebra. By all accounts he harbored few if any doubts that our 3dimensional geometry could be most naturally described in H, and perhaps even ascribed to the existence of H. Small wonder then that he was galled by, and spent much of his life in resis­ tance to, the ultimately dominant de­ scription of vectors in 3-space in terms of ordered triples, along with the dot and cross products, both of which are inherent in quatemion multiplication. Hamilton's friend John Graves was impressed by the discovery of H, and by its connection to 3-space. But Graves, more of a mathematician, first and foremost saw a potentially ex­ tendable sequence of algebras. Very soon after learning about the quater­ nions he found the next algebra in the sequence, the octonions. He little real­ ized at the time that it was the last of its sort, that the sequence of division algebras is finite. One can imagine that Hamilton might have been disap­ pointed at some level by the existence of 0 beyond his beloved H, lessening the likelihood that H was a mathemat-

VOLUME 26, NUMBER 2, 2004

75

ical pinnacle linked inexorably to our

ence. Together with what many theo­

physical reality. Ultimately, however,

retical physicists feel is an insurmount­

theory, which many feel has a higher

his obsession may prove not to have

able

unification

been wrong or misplaced, but simply

nonassociativity of

too limited.

obstacle

to

application-the

0-this failure en­

space-time arises in the mysterious M­ potential

than

ordinary

String theory. What's more, this Jordan

In the arena of mathematical physics

hanced the perception in the majority

algebra also has spinors, which are

that these algebras were suspect, and

fermionic, a term in physics jargon re­

Hamilton's efforts to promote H as a

that association with them could only

ferring to how they transform with re­

description of 3-space over the more

hurt one's reputation. To

this day one

spect to Lorentz transformations. And

popular vector triplets tainted the for­

still finds a hint of apology in most ar­

the remarkable SOs triality transfor­

mer idea. Being on the losing side of

ticles relating the octonions to physics.

mation, mixing the two 8-dimensional

such a battle leads to a general per­

I myself was warned several times that

halves of these spinors with the 8 trans­

ception that the legitimacy of the

work in this area was a potential career

verse space dimensions (the bosonic

loser's cause is questionable. Fortu­

killer-and in fact it was.

part, in a similar piece of physics jar­

nately this perception never entirely killed off curiosity about the

two

And yet the articles continue. For

gon), is a kind of supersymmetry. And

many it seems just too unlikely that al­

then there are the links of this algebra

higher-dimensional division algebras

gebras linked to so much elegant math­

to exceptional groups (the exceptional

among mathematicians and physicists.

ematics are not also linked to central

group Es in particular has been promi­

To this day one

from the octonions. It all sounds just too

sti l l fi nds a h i nt of

alone who see the tendrils of division al­

In the first half of the twentieth cen­ tury such prominent figures as Pascual Jordan, John von Neumann, and Eu­ gene Wigner attempted to apply the oc­ tonions to quantum mechanics. There is a description of quantum theory in terms of Jordan algebras, and of all the infmitely many of these algebras there is one that is truly exceptional, stand­ ing apart from the rest: the Jordan al­ gebra of 3

X

3 hermitian

matrices over

nent in String theory), all of which arise

idea that the tendrils will be there what­

articles relating

ever direction is taken, so I threw them

all into the mix and ended up linking the

the octon ions

division algebras to the Standard Model of quarks and leptons, with its U(1)

to physics .

correlation of this algebra to any viable

gebra theory in their work My own work, for example, is founded on the

apology i n most

0. But they never found a satisfying

delicious. And it is not String theorists

SU(2)

X

X

SU(3) gauge symmetry.

After three decades of immersion in

quantum theory, and their efforts were

ideas in physics. Why would physics

mathematical physics, it dawned on me

soon overshadowed by developments

rest on less elegant structures? And

that the

in other areas, like quantum field the­

then there are the hints from String

sundry theories naturally get excited

Although

a

adherents

of various

and

ory, and the construction of devices

theory.

10-dimensional

when the application of some mathe­

that make very big booming noises.

Lorentzian space-time arises most fre­

matical idea or structure leads to a

The need for bigger and better booms

quently in String theory, the dimen­

sudden lessening of resistance to ad­

dried up many esoteric research ideas,

sions 3, 4, 6, 10, and 26 have all been

vancement. However, it could very

and the quatemions and octonions lan­

shown to be the only dimensions in

well be that this lessening of resistance

guished for a time.

which certain sets of conditions are

is more attributable to this mathemat­

In the 1970s they arose again, most

met. The associated transverse sub­

ical resonance than to any particular

prominently at Yale University. In that

spaces of these space-times have the

theory's merits, and that any theory,

familiar dimensions 1 , 2, 4, 8, and 24.

however absurd, will seem most right

decade I attended a talk at Harvard by someone from Yale about the relation­

The Lorentzian metrics of the 3-, 4-, 6-,

ship of the octonions to the increasingly

and 10-dimensional spaces arise from

prominent gauge Lie group, SU(3). A

the determinants of 2

short time before I had found an elegant

matrices over R, C, H, and

X

and exciting when resonating in tune with this mathematics.

2 hermitian

Finally, to segue away from physics,

0. These

it was the String theorist Martin Ced­

expression of all four of Maxwell's equa­

collections of matrices are also Jordan

erwall who introduced the octonion x­

tions as a single equation over the

algebras. But it is the last of these,

product ((Ax) (x*B), where x is a unit

quaternions, so I was ripe to be influ­

linked to the more important 10-di­

octonion, and x* its conjugate. I sub­

enced by news of a related algebra of

mensional space-time, which has re­

sequently generalized this to the xy­

potentially

ceived the most attention.

product ((Ax)(y*B)), and showed that

greater

significance

to

physics. All this spawned in me an ob­

But wait! The Jordan algebra of 3

X

0 has also

for general unit octonions x and y the algebra resulting from this new prod­

session for these algebras that may have

3 hermitian matrices over

rivaled or exceeded Hamilton's. But the

been the focus of much attention in

uct is still the octonion algebra (with a

research at Yale ultimately failed to ful­

physics. This contains not only a 10-di­

new identity, yx*). Then, inspired by

mensional space-time, but an extra

Conway and Sloane's Sphere Packings, Lattices and Groups, I showed that for

fill the hopes of its participants, and at

least one was embittered by the experi-

76

THE MATHEMATICAL INTELLIGENCER

space dimension. And 1 1-dimensional

any given orthogonal octonion basis­ with the product of any pair of basis units being another basis unit (plus or minus)-the set of unit octonions x, or pairs of unit octonions x and y, for which the product of any pair of basis units is still a basis unit, generates, in the former case, a pair of 8-dimen­ sional laminated lattices (E8), and in the latter case a pair of 16-dimensional laminated lattices. Lattices of the for­ mer sort appear prominently in Con­ way and Smith's new book, On Quater­ nions and Octonions. For the pure mathematician the study of these algebras is a friendlier pursuit. Not required in general to make big booms, nor to link the results of their re­ search to a tantalizingly out-of-reach physical reality, they are free to let the mathematics lead them where it will. For most of us with any familiarity with these algebras, they are associ­ ated with continuum structures: Lie al­ gebras; Lie groups; Euclidean spaces; spheres; Jordan algebras, and so forth. In On Quaternions and Octonions there is a fair amount of attention paid to rotation groups on spaces of di­ mension 2,3,4, 7, and 8, the odd num­ bers arising from the dimensions of the imaginary parts of H and 0. But one of the authors of this new volume is co­ author of Sphere Packings, Lattices and Groups, devoted to discrete struc­ tures in all dimensions. Not surpris­ ingly much of this new volume is de­ voted to discrete structures (lattices of integers) in dimensions 1, 2, 4, and 8, and to finite groups associated with the division algebras. That one can even define notions of integer and prime integer in C, H, and 0 is a consequence of their being com­ position algebras. If x and y are in one of these algebras, and N(x) is the Eu­ clidean norm of x, then

N(xy)

=

N(x)N(y).

This property has endless conse­ quences and lies at the heart of why these algebras are so important. In di­ mensions 2, 4, and 8 it allows us to define integers, primes, and unique prime factorization, up to certain conditions. There are three parts to the book,

one for each of C, H, and 0. The first is the shortest, and most readers will find the material familiar, save perhaps the sections on Gaussian and Kleinian integers over C. Chapters 3 to 5 are devoted to the quaternions. The first two focus on 3-di­ mensional and 4-dimensional groups, both finite and Lie (the dimensions re­ ferring to the spaces upon which the groups act). The noncommutativity ofH allows us to rotate both the 3-dimen­ sional purely imaginary part ofH (uAu*, u a unit quaternion), and the full 4-di­ mensional algebra (uAv*, u and v unit quaternions). Chapter 5 introduces Lip­ schitz and Hurwitz integral quaternions,

"Seven Rig hts Can Make a Left . " primes, units, and factorization of inte­ gers. The latter system of integers is connected to the elegant D4 lattice. All of this is mostly precursor to the octonion chapters, 6 to 1 2 . These chapters begin with a l o o k at octonion multiplication, and at a proof of the Hurwitz theorem: R, C , H, and 0 are the only composi­ tion algebras. The importance of this theorem cannot be overstated. There is more than one way to gen­ eralize this sequence of algebras (Dickson's is one, and Dixon's an­ other), but the higher-dimensional algebras that result are-to con­ tinue the resonance metaphor a bit further-little more than back­ ground noise. There is no associ­ ated spike of interesting mathe­ matical structures and ideas. The octonion algebra is noncom­ mutative and nonassociative. Because of nonassociativity the set of left mul­ tiplication maps, Lx: A � xA , is not closed. That is, for x and y in 0, there is in general no z such that LxLy = Lz. To close the set one must include nested actions of the form, A � x(y ( . . . (zA) . . . ) ) . Curiously, despite non­ commutativity any left (or right) action can be expressed as a combination of right (or left) actions. In the quaternion

algebra this is certainly not the case: left actions and right actions are dis­ tinct and commute with each other. In the octonion case, left actions do not in general commute with right actions, and the algebra of all left actions is equal to the algebra of right actions. This is complicated further by two­ sided actions, and the whole is inti­ mately tied to triality. Chapters 6 to 8 beautifully discuss many aspects of these multiplicative complexities. My favorite section title is "Seven Rights Can Make a Left." An integer system on 0, linked to the E8 lattice as the Hurwitz integers on H are linked to D4, is defined in chapter 9, and elaborated on in chap­ ters 10 and 11. The notion of integral­ ity on 0 is more difficult than on C or H and certainly deserves the extra at­ tention. Just a note: the lattices E8 and D4 are associated with the Lie groups E8 and SOs, both of which crop up prominently from time to time in many unification models. On Quaternions and Octonions concludes in chapter 12 with an origi­ nal look at the octonion projective plane. Projective spaces of all dimen­ sions can be constructed over R, C, and H, but the nonassociativity of 0 prevents anything beyond the plane. Similarly, R, C, and H are associated with infinite sequences of classical Lie groups, while 0 is linked primarily to just the five exceptional groups. This projective plane, by the way, is very closely linked to that previously men­ tioned exceptional structure, the Jor­ dan algebra of 3 X 3 hermitian matri­ ces over 0. Although Conway and Smith's book is pure mathematics, this last chapter highlights the fact that when it comes to the division algebras, pure and applied and seldom far apart. REFERENCE

Conway, J. H . , and Sloane, N. J. A : Sphere Packings, Lattices and Groups. Springer-Ver­

lag, 1 993. Department of Physics University of New Hampshire Durham, NH 03824

USA

e-mail: [email protected]

VOLUME 26, NUMBER 2, 2004

77

Stamping Through Mathematics by Robin J. Wilson SPRINGER-VERLAG, NEW YORK INC., 2001 1 26 pages US $29.95 ISBN 0-387-98949-8

REVIEWED BY VAGN LUNDSGAARD HANSEN

T

his cleverly designed book is based on the idea to tell the history of mathematics through motifs on stamps. The author, Robin Wilson, has a dis­ tinguished record in mathematics and has published extensively in graph the­ ory and combinatorics. Outside math­ ematics he holds a long-time passion for philately. He is well known to the readers of The Mathematical Intelli­ gencer for enlightening short stories about mathematicians, or mathemat­ ics, represented on stamps in the col­ umn "Stamp Comer," which he has edited for many years. It is commend­ able that Wilson by the publication of a book has shared his comprehensive knowledge about mathematics on stamps with a more general public. That the title of the book is witty is no surprise to those who have attended the author's entertaining lectures. The book is organized so that every right-hand page contains a number of stamps and the associated left-hand page tells an episode of mathematics illustrated by the stamps. The stamps are slightly enlarged and uncancelled to show each motif in complete detail, and the text is clear and informative. It is amazing how much mathematics Wilson has been able to extract from stamps. There are almost 400 different stamps in the book The reviewer counted 392, not counting the rare

78

THE MATHEMATICAL INTELLIGENCER

stamp on the back cover picturing the author himself, which still awaits for­ mal approval by the postal services. A stamp is a small piece of art, and the postal services in the various coun­ tries spend considerable time on preparing their graphics and texts. It is highly non-trivial for a person, an event, or a subject to get on a stamp. It therefore came as a surprise to this reviewer that enough mathematics-re­ lated topics have been selected for stamps to write a valuable short history of mathematics with stamps as the

A stamp is a smal l p iece of art . guide. Wilson has succeeded very well in this undertaking. He writes in the preface, "This is not a history of math­ ematics book in the conventional sense of the word. Several important mathe­ maticians or topics are omitted, due to the absence of suitable stamps featur­ ing them, whereas others may have as­ sumed undue prominence because of the abundance of attractive images. Where appropriate, I have felt free to let the stamps dictate the story." Wil­ son's book can be forgiven for not be­ ing the complete history of mathemat­ ics; in the opinion of the reviewer it is certainly a splendid first introduction. The great variety of topics touched upon is illustrated by some of the sec­ tion heads: Greek Geometry, Plato's Academy, Euclid and Archimedes, Early Islamic Mathematics, Go and Chess, Map-Making, The French Revo­ lution, The Geometry of Space, Mathe­ matical Education, and many more. In the section on The Nature of Light, I

missed a Danish stamp featuring the astronomer Ole R0mer, who discov­ ered that light has fmite speed. A sec­ tion on Mathematical Shapes is illus­ trated by stamps of different shapes, including a beautiful hexagonal stamp in the form of a honeycomb tessella­ tion and depicting various aspects of bee-keeping, issued by the Pitcairn Is­ lands in 1999. You can learn a lot just by studying such a stamp with an open mind, including geography! The oldest stamp I noticed in the book was issued by Colombia in 1869 and has the form of a scalene triangle. Another old stamp appears in the sec­ tion on Map-Making. It shows the world in Mercator projection and was issued by Canada in 1898. Among mathemati­ cians and mathematical physicists de­ picted on stamps, not surprisingly New­ ton and Einstein take the lead. Robin Wilson's book is attractively written, and even on a short train trip you can cover several episodes in mathematics. It should appeal to al­ most any kind of reader and it will be extremely useful for teachers and stu­ dents looking for a quick introduction to the history of mathematics. The book provides convincing evidence that mathematics can be found every­ where. It will be a good point of de­ parture for many interesting mathe­ matical conversations with philatelists and mathematicians alike. The design of the book is inviting and every bib­ liophile will find pleasure in paging through it. This magnificent book de­ serves to be widely read. Department of Mathematics Technical University of Denmark Building 303 DK-2800 Lyngby, Denmark e-mail: V.L. [email protected]

k1flrri .M$·h•i§i

Robin Wilson

The Philamath' s Alphabet- D

I

nite series and attempted to formalise

determine the distances between two

the idea of a limit so as to put the cal­

places. It dates from around 300 AD.

culus on a firm basis. He was also the

1898), better known as Lewis Carroll,

describes the motion of a vibrating

was the author of

string. In his later years he wrote many

Alice's Adventures in Wonderland and other children's

of the mathematical and scientific arti­

books. A mathematics lecturer at Christ

Encyclopedie,

Church, Oxford University, he wrote

which attempted to classify the knowl­

books on symbolic logic and the alge­

edge of the time.

bra of determinants, and he was the

cles for Denis Diderot's

D

'Alembert: Jean le Rond d'Alem­ bert ( 1 7 1 7-1 783) was a leading

Enlightenment figure. He stated the ra­ tio test for the convergence of an infi-

Dodgson: Charles Dodgson (1832-

first to obtain the wave equation that

Dedekind: Richard Dedekind (1831-

creator of many ingenious mathemati­

1916) invented the concept of an ideal to

cal puzzles. A great enthusiast for the

explain why certain types of "number"

teaching of Euclid's

factorise in more than one way-for ex­

dents, he produced several treatises

ample, 10

=2 X5=

(4 + v6)(4 - v6).

Elements to stu­

(both serious and humorous) on it.

The stamp below shows the unique fac­ torisation of an ideal as a product of powers of ''prime ideals." He also intro­ duced the idea of a "Dedekind cut" in order to provide a rigorous defmition of a real number.

Descartes: D'Alembert

In

1637 Rene Descartes

(1596-- 1650) wrote Discours de la meth­

ode, a philosophical treatise with a 100-page appendix, La geometrie, con­ taining fundamental contributions to an­ alytic geometry. Here he solved an an­ cient problem of Pappus on the path traced out by a point moving in a par­

De Witt

ticular way, by naming two lengths x and

y and calculating all the other lengths in terms of them, thereby obtaining a conic as the required path. Thus Descartes in­ Dedekind

troduced algebraic methods into geom­ etry, but he did not initiate the "Carte­ sian coordinates" (with orthogonal axes) named after him.

De Witt: Johan de Witt (1625-1672) was a talented mathematician and po­ litical leader whose concern with Hol-

Distance-measuring

land's financial problems led to his writing a treatise on the calculation of life annuity payments, an early attempt Descartes

to apply probability theory to econom­ ics. His important Elementa curvarum linearum was one of the earliest ac­ counts of analytic geometry. He was

Please send all submissions to

murdered by an angry mob for politi­

the Stamp Corner Editor,

cal reasons.

Robin Wilson, Faculty of Mathematics,

Distance-measuring

The Open University, Milton Keynes,

measuring instruments have survived

cart:

Various

MK7 6AA, England

from ancient China, including a distance­

e-mail: [email protected]

measuring cart that one pulls along to

80

THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER� VERLAG NEW YORK, LLC

Dodgson

cart