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1998 Exceptional Objects John Stillwell University of San Francisco, [email protected]

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Recommended Citation John Stillwell. Exceptional Objects. The American Mathematical Monthly Vol. 105, No. 9 (Nov., 1998), pp. 850-858. DOI: 10.2307/ 2589218

This Article is brought to you for free and open access by the College of Arts and Sciences at USF Scholarship: a digital repository @ Gleeson Library | Geschke Center. It has been accepted for inclusion in Mathematics by an authorized administrator of USF Scholarship: a digital repository @ Gleeson Library | Geschke Center. For more information, please contact [email protected]. THE EVOLUTION OF... Edited byAbe Shenitzer Mathematics,York University, North York, Ontario M3JlP3, Canada

ExceptionalObjects

JohnStillwell

1. INTRODUCTION. In the mind of everymathematician, there is tension be- tween the general rule and exceptionalcases. Our conscience tells us we should strive for general theorems,yet we are fascinated and seduced by beautiful exceptions.We can't help lovingthe regularpolyhedra, for example, but arbitrary polyhedraleave us cold. (Apart fromthe Euler polyhedronformula, what theorem do you know about all polyhedra?)The dream solutionto thisdilemma would be to finda general theoryof exceptions-a complete descriptionof theirstructure and relations-but of course it is stillonly a dream.A more feasibleproject than mathematicalunification of the exceptional objects is historicalunification: a descriptionof some (convenientlychosen) objects, their evolution,and the way theyinfluenced each otherand the developmentof mathematicsas a whole. It so happens thatpatterns in the worldof exceptionalobjects have oftenbeen discoveredthrough an awareness of history,so a historicalperspective is worth- while even forexperts. For the restof us, it givesan easy armchairtour of a world that is otherwisehard to approach. A rigorousunderstanding of the exceptional Lie groups and the sporadic simple groups,for example, mighttake a lifetime. They are some of the least accessible objects in mathematics,and from most viewpointsthey are way over the horizon.The historicalperspective at least puts these objects in the picture,and it also shows lines that lead naturallyto them, thus paving the way for a deeper understanding.I hope to show that the exceptionalobjects do have a certainunity and generality,but at the same time theyare importantbecause theyare exceptional. Some of the ideas in this article arose fromdiscussions with David Young, in connectionwith his Monash honoursproject on octonions.I also receivedinspira- tionfrom some of the Internetpostings of JohnBaez, and help on technicalpoints fromTerry Gannon.

2. REGULAR POLYHEDRA. The firstexceptional objects to emerge in mathe- maticswere the five regularpolyhedra, known to the Greeks and the Etruscans around 500 BC. These are exceptionalbecause there are only fiveof them,whereas there are infinitelymany regular polygons. The Pythagoreansmay have knowna proof,by consideringangles, that only five regular polyhedra exist. Many more of their propertieswere worked out by Thaetetus, around 375 BC, and by 300 BC theywere integratedinto the general theoryof numbersand geometryin Euclid's Elements.The constructionof the regularpolyhedra in Book XIII, and the proofthat there are onlyfive of them,is

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Tetrahedron Octahedron Icosahedron Cube Dodecahedron

Figure1 the climaxof the Elements.And the elaborate theoryof irrationalmagnitudes in Book X is probablymotivated by the magnitudesarising from the regularpolyhe- dra, such as ((5 + V5)/2)1/2,the diagonal of the icosahedronwith unit edge. Thus it is probable that the regular polyhedrawere the inspirationfor the Elements,and hence formost of the later developmentof mathematics.If I wished to demonstratethe influenceof exceptionalobjects on mathematics,I could rest my case rightthere. But there is much more. The most interestingcases of influencehave been comparativelyrecent, but before discussingthem we should recall a famousexample from400 years ago. In Kepler's MysteriumCosmographicum of 1596 the regular polyhedramade a spectacular, though premature,appearance in mathematicalphysics. Kepler "explained" the distancesfrom the sun of the six knownplanets by a model of six spheres inscribingand circumscribingthe five regular polyhedra. Alas, while geometrycould not permitmore regular polyhedra,physics could permitmore planets,and the regularpolyhedra were blown out of the skyby the discoveryof Uranus in 1781. Kepler of course neverknew the fatal flawin his model, and to the end of his days it was his favouritecreation. Indeed it was not a complete waste of time, because it also led him to mathematicalresults of lasting importanceabout polyhedra.

3. REGULAR POLYTOPES. It is a measure of the slow progressof mathematics untilrecent times that the place of the regularpolyhedra did not essentiallychange until the 19th century.Around 1850, they suddenlybecame part of an infinite panorama of exceptionalobjects, the n-dimensionalanalogues of polyhedra,called polytopes.Among other things,this led to the realisationthat the dodecahedron and icosahedron are more exceptionalthan the other polyhedra,because the tetrahedron,cube, and octahedronhave analogues in all dimensions. The 4-dimensionalanalogues of the tetrahedron,cube, and octahedron are knownas the 5-cell (because it is bounded by 5 tetrahedra),the 8-cell(bounded by 8 cubes), and the 16-cell(bounded by 16 tetrahedra),respectively.

/

5-cell 8-cell 16-cell

Figure2

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This content downloaded from 138.202.189.11 on Thu, 11 Jun 2015 21:12:47 UTC All use subject to JSTOR Terms and Conditions The other4-dimensional regular polytopes were discovered by Schlafli in 1852:the 24-cell(bounded by 24 octahedra),120-cell (bounded by 120 dodecahedra),and 600-cell(bounded by 600 tetrahedra).

1~~~24cl 12-el 600-cell

Figure3

Schlaflialso discoveredthat in five or more dimensionsthe only regular polytopesare theanalogues of thetetrahedron, cube, and octahedron.Thus from the n-dimensionalperspective the dodecahedron,icosahedron, 24-cell, 120-cell, and 600-cellare the genuineexceptions. This also suggeststhat 3 and 4 are exceptionaldimensions, a fact that just might explain the dimension of thespace (or spacetime)in whichwe live,though after Kepler one shouldbe cautiousabout suchspeculations!

4. INFINITE FAMILIES PLUS EXCEPTIONS. Schlafli'sdiscovery may be sum- marisedby saying that the regularpolytopes may be classifiedinto three infinite familiesplus five exceptions. The classificationof other structures follows a similarpattern: * Regulartessellations of RD: one infinitefamily and fourexceptional tessella- tions.These were enumerated by Schlafli in 1852.The infinitefamily is the tessellationof R' by "n-cubes",which generalises the tessellationof the planeby squares. Two of theexceptions are thedual tessellationsof R2 by equilateraltriangles and regularhexagons. The othertwo, discovered by Schlafli,are dual tessellationsof R4, by16-cells and 24-cells. * SimpleLie groups:four infinite families An, Bn, Cn, Dn, plus five exceptional groupsG2, F4, E6, E7, E8. The infinitefamilies were discovered by Lie, and the exceptionsby Killingin 1888 and Cartanin 1894. The processof classificationresembles that of regularpolyhedra, because it reducesthese continuousgroups (in a farfrom obvious way) to discretearrangements of linesegments in Euclideanspace (rootsystems), with certain constraints on anglesand relativelengths. * Finitereflection groups: four infinite families plus seven exceptional groups, discoveredby Coxeterin 1934.These generalisethe symmetrygroups of polyhedra,and are linkedto the simpleLie groupsvia root systems.It emergedfrom Coxeter's work, and also thework of Weyl in 1925,Cartan in 1926,and Stiefelin 1942,that each simpleLie groupis determinedby a reflectiongroup, now called its Weyl group. * Finite simplegroups: 18 infinitefamilies plus 26 exceptional(sporadic) groups,discovered by the collective work of manymathematicians between 1830and 1980.These are also linkedto the simpleLie groupsby passage fromthe continuous to thediscrete. Galois, around 1830, first noticed that finitesimple groups arise fromgroups of transformationswhen complex

852 THE EVOLUTION OF ... [November

This content downloaded from 138.202.189.11 on Thu, 11 Jun 2015 21:12:47 UTC All use subject to JSTOR Terms and Conditions coefficientsare replacedby elements of a finitefield. The idea was extended byJordan and Dickson,and reachedfull generality with Chevalley in 1955. However,even the exceptional Lie groupsyield infinite families of finite simple groups,so the sporadicsimple groups appear to be the heightof exceptionality. Still,they are not completelyunrelated to the otherexceptional objects. Other linksbetween exceptions appear when we pickup otherthreads of the story.

5. SUMS OF SQUARES. Entirelydifferent from the storyof polyhedra,but at leastas old,is thestory of sums of squares. Sums of two squares have been studied sincethe Babylonian discovery of "Pythagoreantriples" around 1800 BC. Around 200 AD, Diophantusmade thestriking discovery that sums of twosquares can be multiplied,in a certainsense. His Arithmetica,Book III, Problem19 says

65 is "naturally"divided into two squares in twoways ... due to the factthat 65 is the productof 13 and 5, each of whichnumbers is the sum of two squares.

In 950 AD, al-Khazininterpreted this as a referenceto the twosquare identity

(a2 + b2)(a2 + b2) = (ala2 blbT+ 2 + (bla2 )2, as did Fibonacciin his LiberQuadratorum of 1225.Fibonacci also gave a proof, whichis nottrivial in his algebra-it takesfive pages! Thereis no similaridentity for sums of threesquares, as Diophantusprobably realised.15 is nota sumof three integer squares, yet 15 = 5 x 3, and 3 = 12 + 12 + 12, 5 = 02 + 12 + 22. Thus therecan be no identity,with integer coefficients, expressingthe productof sumsof threesquares as a sumof threesquares. It is also truethat 15 is not the sumof threerational squares. Diophantus may have knownthis too. He statedthat 15 is notthe sum of tworational squares, and the prooffor three squares is similar(involving congruence mod 8 insteadof mod4). But claimsabout sums of foursquares are conspicuouslyabsent from the Arithmetica.This led Bachetto conjecture,in his 1621 editionof the book,that everynatural number is thesum offour (naturalnumber) squares. Fermatclaimed a proofof Bachet'sconjecture, but the firstdocumented step towardsa proofwas Euler'sfour square identity, given in a letterto Goldbach, 4 May 1748:

2+ b2 + c2 + d2)(p2 + q2 + r2 + S2)

= (ap + bq + cr + ds)2 + (aq - bp - cs + dr)2

+(ar + bs - cp - dq)2 + (as - br + cq - dp)2. Usingthis, Lagrange completed the proof of Bachet'sconjecture in 1770. In 1818,Degen discoveredan eightsquare identity, which turned out to be the last in the series,so sumsof 2, 4, and 8 squaresare exceptional.This was not proveduntil 1898, by Hurwitz. In fact,Degen's identity was virtually unknown until thediscovery of...

6. THE DIVISION ALGEBRAS R, C, H, 0. A divisionalgebra of dimensionn overR1 consists of n-tuplesof real numbersunder vector addition, together with a "multiplication"that distributes over addition and admits"division". The idea is to make n-tuplesadd and multiplyas "n-dimensionalnumbers". Addition is no

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This content downloaded from 138.202.189.11 on Thu, 11 Jun 2015 21:12:47 UTC All use subject to JSTOR Terms and Conditions problem,but a decentmultiplication is exceptional-apart from the obvious case n = 1 it existsonly in dimensions2, 4, and 8. The firstclue how to multiplyin each of thesedimensions was theidentity for sums of 2, 4, or 8 squares. Diophantusassociated a2 + b2 withthe pair (a, b), identifiedwith the right-an- gledtriangle with sides a and b. Fromtriangles (a1, b1) and (a2, b2) he formedthe "product"triangle (a1a2 - b1b2,b1a2 + a1b2). His identity

(al + b 2)(a + b2) = (a1a2 - blb 2 + (bla2 + alb)2 showsthat the hypotenuse of the"product" is theproduct of thehypotenuses. In modernnotation his identity is

Iz1121z212 =Iz1z212 where z1 = a1 + ib1, Z2 = a2 + ib2, and itexpresses the multiplicativeproperty ofthe norm jzj2 = a2 + b2 of z = a + ib: thenorm of a productis theproduct of thenorms. A multiplicativenorm makes divisionpossible, because it guaranteesthat the productof nonzeroelements is nonzero. The same productof pairs (withoutthe identificationwith triangles) was rediscoveredby Hamiltonin 1835 as a definitionof the productof complex numbers: (a1, bj)(a2, b2) = (aja2 - bb2 bja2 + alb2). Hamiltonhad been tryingsince 1830 to definemultiplication of n-tuplesand retainthe basic properties of multiplicationon R and C: * multiplicationis commutative and associative * multiplicationis distributive over vector addition * thenorm x2 + x2 + *. +x2 of (x1,x2,..., xn) is multiplicative. He gotstuck on triplesfor 13 years,not knowing that a multiplicativenorm was ruledout by elementary results on sumsof threesquares. If he had knownthis earlier, would he havegiven up thewhole idea, without tryingquadruples? van der Waerden [7, p. 185]thought so, implyingthat Hamilton triedquadruples in October1843 onlyto salvagesomething from his long and fruitlesscommitment to triples. He had alreadygiven up on commutativemultipli- cation,but withquadruples he saved the otherproperties, in his algebraH of quatemions.It was lucky,in van derWaerden's opinion, that Hamilton didn't know aboutsums of three squares. But whatif he had knownabout sums of foursquares? He wouldthen have seen a multiplicativenorm of quadruples,and would perhapshave discovered quaternionsin 1830.This is not purespeculation. Gauss, who knewEuler's four squareidentity, discovered both a "quaternionform" of it and also a "quaternion representation"of rotationsof thesphere, the latter around 1819. However,Gauss did notpublish these discoveries, so Hamiltonwas luckyafter all-he was firstto see thefull structure of the quaternions,and he receivedall thecredit for them. JohnGraves, who had been in correspondencewith Hamilton for years on the problemof multiplyingn-tuples, was galvanisedby the discoveryof quaternions and the associatedfour square identity(which Hamilton and he at thattime believedto be new).In December1843 Graves rediscovered Degen's eight square identity,and immediatelyconstructed the 8-dimensionaldivision algebra 0 of octonions.

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This content downloaded from 138.202.189.11 on Thu, 11 Jun 2015 21:12:47 UTC All use subject to JSTOR Terms and Conditions Dickson[4, p. 159]condensed all theseresults into one formula,showing that each algebrain thesequence DR, C, 0X,0 comesfrom the one beforeby a simple generalisationof Diophantus'rule for multiplying pairs: (a1, b1)(a2, b2) = (a1a2 - b2b1,b2a1 + b1a2) where (a, b) = (a, -b). Fromthis it is easilyproved that the quaternions are associativebut not commuta- tive,and thatthe octonions are notassociative. The nextalgebra in thesequence, consistingof pairs (a, b) of octonions,is not a divisionalgebra. Theorems by Frobenius,Zorn, and others confirmthat DR,C, 0H, and 0 are indeed exceptional-theyare the onlyfinite-dimensional division algebras over R1.For furtherinformation on themsee [5].

7. LATTICES. The polyhedronthread of our storyintertwines with the division algebrathread when we reconsidertessellations of R8n. The two exceptional tessellationsof R82 = C, byequilateral triangles and hexagons,are bothbased on the latticeof Eisensteinintegers 1+ - m + n 2 for m, n E Z. The triangletessellation has latticepoints at vertices;the hexagon tessellation has themat facecentres. Similarly,the two exceptional tessellations of R' = H are based on theHurwitz integers 1 + i + j + k + + + sk for P 2 qi rj p,q,r,s EZ , calledthe "integer quaternions," and wereused byHurwitz in 1896to givea new proofthat every natural number is a sumof foursquares. The analogous"integer octonions" form a latticein R88,in which the neighbours of each latticepoint form a polytopediscovered by Gosset in 1897. It is not regular,but is neverthelesshighly symmetrical. Its symmetrygroup is none other thanthe Weylgroup of the exceptionalLie groupE8. Gosset'sstory, and the historyof regular polytopes in general,is toldin [3].

8. PROJECTIVE CONFIGURATIONS. Other exceptionalstructures in geometry are the projectiveconfigurations discovered by Pappus,around 300 AD, and by Desarguesin 1639.

Theoremof Pappus. If the verticesof a hexagonlie altematelyon twostraight lines, thenthe intersections of theopposite sides of thehexagon lie in a straightline.

(12) (4) (3)(5)( 6)

Figure 4

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This content downloaded from 138.202.189.11 on Thu, 11 Jun 2015 21:12:47 UTC All use subject to JSTOR Terms and Conditions Theorem of Desargues. If two trianglesare in perspective,then the intersectionsof theircorresponding sides lie in a line.

B

Figure5

These statementsare "projective"because they involve only incidence: whether or notpoints meet lines. Moreover, Desargues' theorem in spacehas a prooffrom incidenceproperties-the properties that

* twoplanes meet in a line, * twolines in thesame plane meet in a point.

The diagramof the Desarguesconfiguration hints at this,by suggestingthat the triangleslie in threedimensions, even though the diagram itself necessarily lies in theplane. Yet, strangely,the Desargues and Pappustheorems in theplane do not haveproofs from obvious incidence properties; their proofs involve the concept of distanceor coordinates. Whyis thisso? In 1847von Staudtgave geometric constructions of + and X, thus"coordinatising" each projective plane by a divisionring. Then in 1899Hilbert madethe wonderful discovery that the geometry of the plane is tiedto thealgebra ofthe ring:

* Pappus' theoremholds : thedivision ring is commutative * Desargues'theorem holds thedivision ring is associative.

Conversely,any division ring R yieldsa projectiveplane RP2. So by Hilbert's theorem,

RD p2 and Cp2 satisfyPappus, * Hp2 satisfiesDesargues but not Pappus, and * op2 satisfiesneither.

In 1933Ruth Moufang completed Hilbert's results with a theoremsatisfied by 0P2, the"little Desargues' theorem", which states uniqueness of the construction of the fourthharmonic point D of pointsA, B, C. She showedthat "little

856 THE EVOLUTION OF ...... [November

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Figure6

Desargues" holds if and onlyif the divisionring is altemative,that is, forall a and b, a(ab) = (aa)b and b(aa) = (ba)a. Alternativityis a characteristicproperty of 0 by the followingresult of Zorn from1930: a finite-dimensionalalternative division algebra over DRis _DR, C, H or 0. Thus the octonionprojective plane OP2 is exceptional,simply because 0 is. However,0p2 is moreexceptional than R p2, Cp2, and H p2 , because each of the latterbelongs to an infinitefamily of projectivespaces. The methodof homogeneouscoordinates can be used to constructa projective space RP' of each dimensionn forR = DR,C, H. But thereis no OpP3, because

existenceof OP3 => Desargues' theoremholds => 0 is associative.

Thus OP' and 0p2 are exceptionalprojectivespaces, comparedwith the infinite familiesEERpn, P p , HDpn

9. 0: THE MOTHER OF ALL EXCEPTIONS?. We have already observed that manyobjects inherit their classification from that of the simpleLie groups An, Bn, Cn,Dn, G2,F4, E6, E7, E8. In particular,classifications based on the A, D, and E series arise so oftenthat explaining them has become a flourishingindustry. For a recentsurvey of thisfield of "ADE classifications"see [6]. To findthe source of such classifications,we shouldtry to understandwhere the simple Lie groups come from.The groups An, Bn, Cn, Dn are automorphism groupsof n-dimensionalprojective spaces (or subgroups),so the infinitefamilies of simple Lie groups arise fromthe infinitefamilies of projectivespaces Rfpn, Cipn, H pn Since thereis no Qpn for n > 2, one does not expectmany Lie groupsto come fromthe octonions,but in fact all fiveexceptional Lie groupsare related to 0. It seems that,in some sense, the exceptionalLie groups inherittheir exceptionality from0.1 This astonishingrelationship began to emerge when Cartan discoveredthat G2 - Aut(O). He mentionedthis casually in a 1908 article [2] on the historyof hypercomplexnumber systems, saying only that the automorphismgroup of the octonionsis a simple Lie group with 14 parameters.However, he knew fromhis

1The fiveexceptional Lie groups could be of trulycosmic importance.Recent work in theoretical physicsseeks to reconcilegeneral relativity with quantum theory by means of "stringtheories" in which atomicparticles arise as modes of vibration.The possible stringtheories correspond to the exceptional Lie groups,hence stringtheory allows five"possible worlds".This pointwas raised by Ed Wittenin his Gibbs Lecture at the AMS-MAA JointMeetings in Baltimore1998, along withthe question: if there are fivepossible worlds,who lives in the otherfour?

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This content downloaded from 138.202.189.11 on Thu, 11 Jun 2015 21:12:47 UTC All use subject to JSTOR Terms and Conditions 1894 classificationof the simpleLie groupsthat G2 is the onlycompact simple Lie group of dimension14. In the 1950s several constructionsof F4, E6, E7, and E8 from 0 were discoveredby Freudenthal,Tits, and Rosenfeld.For example,they found that F4 is the isometrygroup of Op2, and E6 is its collineationgroup. Their success in findinglinks between these exceptional objects raises an interestingquestion, thoughperhaps one that will never be completelyanswered: How many excep- tional objects inherittheir exceptionality from 0? Some connectionsbetween 0 and sporadic simple groups are known,but the lattergroups remain the most mysteriousexceptions to date. Naturally,the sheer mysteryof these groups has only intensifiedthe search for a broad theoryof exceptionalobjects. This has led to some surprisingdevelopments, among thema revivalof the near-dead subject of "foundationsof geometry".Foundations have now growninto "buildings"(a concept due to Tits), whichare just part of a huge field of "incidence geometry"spanning most of the resultswe have discussed.A comprehensivesurvey of thisnew fieldwith ancient roots maybe foundin [1].

REFERENCES

1. F. Buekenhout,Handbook of IncidenceGeometry, Elsevier 1995. 2. E. Cartan,Nombres Complexes, Euvres Completes II, 1, pp. 107-246. 3. H. S. M. Coxeter,Regular Polytopes, Macmillan, 1963. 4.. L. E. Dickson, On quaternions and their generalisationand the historyof the eight square theorem,Ann. of Math. 20 (1919), 155-171. 5. H.-D. Ebbinghauset al., Numbers,Springer-Verlag, 1991. 6. I. Reiten: Dynkindiagrams and the representationtheory of algebras,Notices Amer. Math. Soc. 44 (1997), 546-556. 7. B. L. van der Waerden, A Histotyof Algebra, Springer-Verlag, 1985.

Departmentof Mathematicsand Statistics,,Clayton, Victoria 3168, [email protected]

The ChauvenetPrize forMathematical Exposition has been awarded to ProfessorP. R. Halmos of the Institutefor Advanced Studyfor his paper entitled"The Foundationsof Probability,"published in thisMONTH1LY for November,1944. This most recentaward of the Prize "for a noteworthy expositorypaper published in English by a memberof the Association" coversthe three-yearperiod, 1944-'46. The Associationfirst established the ChauvenetPrize in 1925. At that time it was specifiedthat the award was to be made everyfive years for the best articleof an expositorycharacter dealing with some mathematical topic,written by a memberof the Associationand publishedin English duringthe fivecalendar years precedingthe award. The Prize was not to be awardedfor books. Originallythe amountof the awardwas fixedat one hundreddollars. At a later date it was decidcd to award the Prize cverythrcc years, and the amountwas changed to fiftydollars. In 1942, it was furtherspecified that onlysuch papers would be consideredas "came withinthe range of profitablereading of Associationmembers." MONTiLY55 (1948) 151

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