RegularPolyhedraandtheFootball

離l・碑紅レぺ（F，Hirzel）r11Ch）

TheregularpolyhedraarediscussedinEuclid sbook（300B・C・）・Ishowthemtoyouhere・て￣−￣二子＿二RegularPolyhedraandtheFootball（FJIirzebruch）

Theyoccurinnatureandinart．Herearetwoexamplesforthe
dodecahedron．

Coccospha′reVOnBraarudosphaerabielowi
Vergr6鮎rungetwa5000Ⅹ

AusdemMiozan（vorca・20Mill・Jahren）
S．A．Jafar，Ttibingen，1975

50周年記念講演会講演

Ledod6caedreenargent
trouviaSaint−Pierre de Geneve

LorsdelucLlmP乙IgrleLlrCh60loglquedefouiilesentreprlSe
ご1J oq？PSiondelaJernitrerestaurこ1tiondelacathedraZede
SLlinトPierretlGeneve，underomainen（ormededod6cat−Jre、datLlnt du4e sitcleこ1Pr色sJさsus−Christ．a6te mis Llu」Our．Les12rLICCS r）entLlgOnL11es en乙1rgen［POrtentles7三SignesJuzodiこ1que：ilest rempliJe pJomb（poids297g）・
Le／くJe〉）LIPrObこIbJemcnt serviご11LIPrさdiction dellこlVenir
P乙Ir tC）Cu・mこIislこ1Pr（）VenilnCe etl tltilisLltion restcnt
inconnues pourl′ i−−St乙In【・Ihavethisn・OmDr．G6tzeofSpringerMVerlag・RegularPolyhedraandtheFootball（F・Hirzebruch）

Thereg111arpolyhedrahavethefbllowlngprOPerties・Thereare
nat11ralnumbersn，mSuChthat：Eachfaceisaregularn−gOn．

Fromeachvertex（corner）medgesleave・Whichn，marePOS−
Sible？Hereis asimple argument．The sum ofthe anglesin an−POlygonequals（乃−2）・1800・Therebre乃−2m・＋・

1800＜3600

乃（thesum ofthe anglesin each cornermust belessthan3600
becauseofconvexity）・Thisisequivalenttol l lー＋一Jご＞・り？

Thesolutionofthisinequalityinnaturalnumbers≧3are（3，3），（4，3），（3，4），（5，3），（3，5）・Ishowyouthefollowingtransparency（inGerman）・〕。血h†虹r Fぞ占′乙ゼeル≠＝）q‰−‰−左脇声山船

旬＝旬一∈し良三山；¶息色々仙レ日収こ如匝一打イgOじ

最小化伽ノ′ルィgoo ＜ 3G00

1t

十●＞
・1／h／

50周年記念講演会講演e 盈p01yedermnbヱ竺0oblb2

64

戸主女色√

Tetraeder

ココ

4Hexaeder4

38 126ど†＆Oktaeder346 128ム ヂ仁

Dodekaeder

5320 30 12＝kosaeder

351230 20

灯ム抽。4
仏側か√

TheGreeknames壬brthepolyhedraaresimilarinEnglishandI
S11ppOSeinJapanese．Platonassociatesthebasicelementsofour
existencetothefiveregularpolyhedra．bo，bl，b2arethenumbers
Ofcorners（vertices），edges，facesofthepolyhedron．TheGreek
namecoIneSfromthenumberoffaces．InGermanoftene，k，fareused（Eeken，Kanten，Flachen）．Canwecalculateb。，bl，b2
from n and m．VVb have
わom ＝ 2♭1
わ2γ1＝ 2わ1

andweaddco11rageOuSlyE111er sformulaも0−ら1＋わ2＝2

and obtainl l l l−＋−−− ＝−乃 m 2 わ1◆RegularPolyhedraandtheF00tball（F・性行zebTllCh）

Now we haveた）rmulas for bo，bl，b2in terms ofn，m．Inter−
Changlngn，mleadstointerchanglngbo，b2・refγαんe（gγ0†l−−−ガe諾αんedγ抑（c現ゎ）−−−かodecαんe（ゴγ0れ一一−

refγαんedγ0†もOcねんedγ抑Jc（フβαんe（か071

Weshallseeinamomentthecloserelationshipbetweentheleft
and the right side.
Let）sfirst talk about E111er）sformula．E11lerwasbornin Basel（Switzerland）in1707，heworkedinthePiussianAcademy丘om
1741to1766underFredericktheGreatinBerlinwhotreatedhim
badly，thoughEulerwasageniusalsoin

AppliedMathematics

alld helped the Kinglnlllany teChnicalproblenlS・Froll11766

untilhisdeathin1783hewasintheRussianAcademyinSanktPetersburgunderKatherinaIIwhotreatedhimbetter．E111er

pict11reisonaSwissbanknotes50周年記念講演会講演

IshowyoualsoCarlFriedrichGauJ3（1777−1855）onaGermanl）ank notewhoseworkoncllrVatllreiscloselyrehtedtoElller sformula．HowtoproveEuler

sform11la？

Weshowthatitistrueforan arbitraryconvexpolyhedron．

The surfaceofthe polyhedron can be mapped onto the plane．
Thefacesbecomecountries．Onecountrybeingabigocean．We

imagineapointatinfinity（intheocean）tomaketheplanetoasphere．Theideaisexplainedinthefbllowlngpicture．

Fig．48

Fig・49

RegularPolyhedraandtheFootball（F・HirzebruCh）（FromthefamousbookbyRademacherandToeplitz VonZahlen
undFig11ren）．Itshowsthediagramofanicosahedronandanoctahedron．Openadam（removeanedge）toletwaterin．An
edge disappears and two countries are united. Open again if

thereis wateronly on oneside．Each time blisreduced byl
andalsob2．Thenbo−bl＋b2StaySthesameuntilyoureacha

COnneCtedstringofedges（withoutacycle）andonecountry（theOCean has taken over）．Obviously，at the end ofthis process，

bo−bl＝1andb2＝1．Thusbo，bl＋b2＝2attheendandat
thebeginnlngOftheprocess・

The relation between the icosahedron and the dodecahedron
becomes clear by studying the group ofsymmetries．A sym−
111etryis a rotn・tion arollnd an axisl）y a Certain angle which

Carriesthepolyhedrontoitself．Icosahedronanddodecahedronhavesixsymmetryaxis（throughavertexanditsoppositefbr
theicosahedron，throughthe center ofaface anditsopposite
fbrthedodecahedron）．Wecanrotatebyk・72O（k＝1，2，3，4，fivefo1dsymmetry）．Similarly wehavelOaxiswiththreefold

and15with two払1dsymmetry．The number ofsymmetriesis
6・4＋10・2＋15・1＝59．ThegroupofsymmetrieshasN，＝60

elementsbecausewehavetoincludetheidentity．Forallregularpolyhedra N＝2bl．From the point ofview ofthe symmetrygrouplCOSahedronanddodecahedron，Cubeandoctahedroncan−

notbedistinguished．Thefacecentersofonecorrespondtothe
verticesofthe other．

Let us consider theicosahedron．Take a point onit whichisnotspecial（not avertex，nOtthecenterofanedge，nOtthecenterofaface）．Applyallsymmetriestoit．Thenweget60

points．Thisistheorbitofthegivenpoint．Forexample，takeaface（triangle）oftheicosahedronandapointontheboundary，butnotavertexandnotthecenterofanedge．Thentheorbit
has60points，tWOOneaChedge．

50周年記念講演会講演■−−−t−
Now study a football．It usua11y has12black pentagons
with60vertices．Thisissuch an orbit．Theusualfbotballcorre−

SPOndstothecasewhentheedgeofthetriangleisdividedintotIlree equalparts．Inside a face we obtain a regular hexagon．

We callCut Offtlle COrllerS Oftlleicosalledrollby aplallar Cut
tIlrOughthe5pointsoftheorbitnearthecorner．Thenweob−

tain the truncatedicosahedron with12regular pentagons and20regular hexagons．Therefore b2＝12＋20＝32．We havebl＝30＋60＝90andbo＝60．Euler

sequation60L90＋32＝2

clleCks．

Next wecometothe applicationofthetruncatedicosahe−droninchemistry．

t‡egularl olyhedraand the Football（F．IiirzebruCh）

Abb・1・PerspektivischeDarstellungdesfu名ball−

f6rmlgen C60MolekBIs・Die Kohlenstoffitome
befindensichandenEckeneinesPolyeders，das

Fiinf・undSechseckealsbegrenzendeFl五chenht（？Og．gekapptesIkosaeder）・ImUnterschiedzum

idealenPolyedermitgleichenKantenlingensind

im realen C60MolekiildieAbst五ndebenachbar−
ter Kohlenstoffatome nicht alleidenti5Ch．Ein−

schlie瓜1ichseinerElektronemvolkehatdasMole−
kiiletwalnm Durchmesserl

Dr・WblfglngKd【S⊂hm亡r，MlX−PllnCk−In5【l【u【fdr‰rnphγSik，H亡id亡1b亡柑
kいl〜J∴じゝFcぅ⊥ニチアhゼ0上

DohJよ R・一山桝叫′T しCi∂M′月ト；㌢州れ50周年記念講演会講演

Thereis thecarbon molecule C60With60atomssittingln theVerticesofthetruncatedicosahedron．Asexplained beforethisis an orbit of theicosahedralgroup．The Frankfurter Allge−
meine Zeitung of OctoberlO，1996，has an article

im Fussballfieber

Forscherand reports that the NobelPrize fbr chem−

istrywasgiventoHaroldW・Croto（GreatBritain），RobertF．
Curl，Jr．andRichardE．Sma11ey（USA）forthediscoveryofC60．
Congratulations！Maybe，Wolfgang Kratschmer of the Maxq
PlanckqInstitutfiirKernphysikinHeidelberg，WhoproducedC60

in macroscopIC quantities，COuld alsohave been acandidate．I
tlnderstand that aJapanese scientistis alsoinvoIved．But，inanycase，EuclidandArchimedesalsodeservepartofthisprlZe．
Al）01yhedrolliscalledanarcllillledeanpolyhedrollifallits
facesareregularn−gOnS（butnmayvary，fbrexamplen＝5，6

fbrthetruncatedicosahedron），ifa11edges haveequa11ength，

andifevery cornercan be moved by asymmetryto anyother
COrner．Thereare13interestingarchimedeanpolyhedra．Kepler

listedandinvestigatedtheminhis

Harmonicesmundiin1619．

Regular・Polyhedra and the Football（F．Hirzebruch）

d4 D￡FIGuRARuふl打A蝕0Ⅳ：よ．Trun．

V〝竺∫甥∂肋呼政財甜卯血古材挽ズ雫ク〝∫一彦吻j 呵5rわ′繭j♂

中。点．〝懲モむ○れ．

押〝什j〟〝，虎曾〟〟 タヱr励∫ご〝ピー
jⅩl．Rh畑沌
i亡○丘dod亡．
鵬む○皿．

V棚な肋r乎女♂〝ノ亡甜∫〟〝血β甜

り加郭密夕亡や〟竺〝茄子′雫ク〝上空J♂ん〃伽轡ガノ√そ仲ど柑〝りr′釣パ
郷少rr∫ タタカj■窄〟ム〟あ狗′ど Jふ・巧何据＿∂諺餌・子中G′膠β〝タグ〟〝〝ふ♪や桝ゆ如誓壷
釘む彷〝〝這労豆￠〝如〟一〟〝ふ窮地g読んr〟♂

50周年記念講演会講演

1 DE FIGl】RARu鵬HA腋肌ON：

卑jβ

叫勅ガ霊㌘器惣環ニノ加ゐ〝血僻炉材加r㌣す脚r〝β′

Ⅱ．息L亡uL●Jb亡血○●．

‥

lTt■h●■．

bidCub0．
t▲一心01ゝl

倣加町田坤卯品醐刑′ク・少年〟〝○鱒〝占哲〃壷♂，車〝ヤ卯〝′イrr彦彦，♂r∂雫作〟？

竃J〃 7讐励

鮪−′

Regular Polyhedra and the Football（F．Hirzebruch）

Number4isthetruncatedicosahedron（football）：．‥ et COngruunt duodecim Pentagonicum vlg−

intiHexagonisin unum Triacontakedyhedron，quOdappelloTruncumIcosihedron．Formamhabesslgnatam
numero4．H

The chemists areinterestedin polyhedra such that threeedgesleaveeveryvertexandthatwehavepentagonsandhexagons

Only．Theytrytofindcarbonmoleculeswhoseatomssitinthe
Vertices．Welearninclassicalchemistrythatcarbonhasvalence
4．NowuslngSuChapolyhedron，eaChatomisconnectedwith30thersbyedges．Therefbre，tOOneedgeshouldbeglVenmulti−

Plicity2・S11ChaKekll16strllCt11reCanbeglVentOthebotball：
The30edgesbetween2hexagonsget multiplicity2．In chem−

istry polyhedrawith the above properties arecalled Fullerenes
afterthefamousarchitectBuckminsterFu11er（1895−1983）．

It fbllowsfromEuler）sform111athat fbraFullerenethenum−

berofpentagonsis alwqys equalto12・Tb showthis wefirstprove same generalformulas valid董br any convex polyhedron．

Letbo（r）bethenumberofverticesfromwhichredgesleaveandb2（r）thenumberoffaceswithredgesintheirboundary（r≧3）・Tllell∑わ0（γ）＝わ0，∑わ2（γ）＝ら2∑γ♭0（γ）＝2わ1，∑γわ2（γ）＝2わ1・ThenthefollowlngequationsareconsequencesofEuler sformula
12＋∑（γ−6）ら0（γ）＋∑（2γ一6）わ2（γ）＝0（＊）

12＋∑（γ一6）わ2（γ）＋∑（2γ−6）わ0（γ）＝0

Thefirstequationisequivalentto
12＋2わ1−6わ0＋4ゐ1【6わ2＝O
andthisisEuler

shrmula．Thesameproofworks董brthesecond

equation（interchangeOand2）．Thesecondequationgivesfora

50周年記念講演会講演

Fullereneb2（5）＝12．Thefirstequationimplies
占0 ＝ 20＋2占2（6）わ1＝ 30＋3占2（6）わ2 ＝12＋占2（6）．
Forb2（6）＝Owehavetheregulardodecahedron，forb2（6）＝20
thefootball．Onecanshowthatthenumberb2（6）ofhexagons
Can have any value≧2・In the chemicalliterature（Iforgot

theprecisereference）therearecomputercalculationstofindthe
COmbinatorialtypesofallFullereneswithgivenbo．Thenumber

Oftypesbrbo＝60equals1760．Butthereisonlyonetypewith
disjoint pentagons，namelythefootball．Thereare21822types
hrbu＝78・But tllereareOnly5typeswitlldisjollltpentagOllS．

Thesecondequation（＊）impliesた）ranyCOnVeXPOlyhedron
3わ2（3）＋2む2（4）＋わ2（5）≧12・

Equalityholdsifandonlyifbo（r）＝0た）rr≠3andb2（r）＝O
for r≧7．Thisis satisfied fbrthe cube，the tetrahedron，thedodecahedron and allF1111erenes．

Keplermentionsthatinacorneroneregularpentagonandtworegularheptagons（7−gOnS）areimpossiblebecausethesum
Oftheangle

sisgreaterthan3600．
1080＋2・

5・1800＞3600．

NamunusPentagonuscum duobus Heptagonicis
JamSuPerat4rectos・‖

Weintroduce壬breachcornerthedeficit6as3600minusthesum
Ofallangles comlngtOgetherin this corner．Itis positive．A

res111tofDescartesisequlValentto（＊＊）∑6i＝720O（sumoverallcorners）
Leibniz（thegreatcontemporaryofSekiTakakasu）copiedDescartes，
resultinaParislibrary．Descartes

COPyOfLeibnizexists．

manuscriptislost，but theRegularPolyhedraandtheFootball（F・Hirzebruch）

n胱㍍ふム作別−イGど00t
叫色．tfj6…払亡サG「も〔v才ゼ・p・G5＋q鮎ゼ イ？gり1．r
β皮1♂ん色ザC｛こ㌣帥ム山仙仙

Fぞqlこノ叩…色十0仙hl払込l

一■？

こ叩頭ムつ絶丈士

3 1巳一字ヱ00三イげ∑せ−カキ
γ三三e∑ち＿ ヱ．

＿省一ヂ

二子ユ00

㍍〔ヂご‥八色ゝよヤ…如こ）考
うら0●−Sし叫仙そよ叶は、才㌦ 、乙50周年記念講演会講演The transparencyisin Latin and German（k＝bl，f＝b2，f，＝b2（r））・Weseethat Descartes
Euler

resultis equivalent tosformula．TheIbrmula（＊＊）isadiscreteversionofthefamousformulaofGauJ3thattheintegralIpcdFoverthecurva−
tureofaconvexsurfaceequals47T（＝7200）．Thereshouldbea
new Frenchbanknotewith Descartes．Proposal：

Howto constr11Ct F1111erenes？Therearemanymethods．I

do not have a complete survey．Recently，1n COnneCtion withmy work on Hilbert modular surfaces，Istudied an article byBertramKostant TheGraphoftheTruncatedIcosahedronandthe Last Letter ofGalois（Notices ofthe American Mathe−

maticalSociety，September1995）．Italked about thisin theKyoto colloqulum．Kostant mentions P．W．Fowler and D．E．
Manolopoulos，An AtlasofFu11erenes，0Ⅹford，1995．Idid notSeethisbookyet．Isha11explainhereaconstructionglVenbyM．

Goldbergin1936．Itispossibletotriangulateanicosahedronin20（a2＋ab＋Lb2）trianglesasexplainedbythefollowingdiagrams：

Regular Polyhedra and the F00tball（F．Hirzebruch）

ノ「・仁0ぞよ庖丁き乱ニュ7

恩ニユ

△ゼぞ．

イ？ちら50周年記念講演会講演■−−ノRegularPolyhedra and theFootball（F．HirzebruCh）

ThisglVeSthecombinatorialstructureofaconvexpolyhe−dronwhichhasonlytriangles：
わ2（γ）＝Oforγ≠3，ゐ2＝20（α2＋αわ＋わ2）．
Wehavebo（5）＝12andbo（r）＝Oforr≠5，6and

占。＝10（α2＋αゐ＋わ2）＋2
わ1＝ 30（α2＋αあ＋わ2）
わ2 ＝ 20（α2＋αわ＋わ2）

AccordingtoD・L・D・CasparandA．Kl11g1962（NobelPrize）
SOmeVirusseshaveprotectiveba111ikestruCtureS（capsides）con−
SistingofcapsornereSSittinginthecornersofsllChapolyhedron．

We can pass to the dualpolyhedra．The centers ofthe
facescorrespondtothevertices（云orners）ofthedualpolyhedron．

Thenboandb2areinterchanged．WeobtainFullereneswith
わ0 ＝ 20（α2＋αわ＋占2）
占1＝ 30（α2＋αわ＋占2）

わ2 ＝10（α2＋αわ＋わ2）＋2
占2（6）＝10（α2＋αわ＋占2−1）

Fora＝b＝1wegetthefbotball・Fora＝bwehavebo＝60a2・IIerearepicturesofC60a2．

50周年記念講演会講演

G4

G。

C58

Fullerenes
byRobertF．CurlandRichardE．Smalley，

ScientificAmerican，
October，1991．

Regular Polyhedra and the Football（F．Hirzebruch）

The dualfootball（a＝b＝1，bo＝32）wasdrawn byJ

LeonardodaVinci．

示ふ中耳小一伸

Leonardo da Vinci
in Luca Pacioli

DeDivinaProportione
Milano，Biblio〔heca Ambroslana1509

50周年記念講演会講演

It hasicosahedralsymmetry andistheprotectivecoverof
the Picorna virus with32capsomere．The centers ofthe60
triangles glVe the vertices ofthe fbotball．Irefbr to H．S．M．
Coxeter

Virusmacromoleculesandgeodesicdomes

1971．

［Thislectureisaveryshortenedandmuchmodifiedversion
oflect11reSin the ETH Zurich and the Siemens Foundationin
Munich・］（フリードリッヒ ヒルツェブルッフ）