Symmetry 2 : unifying human understanding

1049 Pages·1989·70.782 MB·English

by  István Hargittai

InternaSteiroiinnea sl
MODERN APPLMIAETDH EMATIACNSD COMPUTESRC IENCE
Volum1e8
Y.
SeriEedsi toErR:V IN RODINW,a shingUtnoinv ersity
(Vloumeisnt hsee rhiaevsae l sboe epnu bliasshS epde cIisasluo efts h jeo urnal
ComputaenrdMs a themawtiitAchps p lications)
RELATED PERGAMONT ITLES
OtherV olumesi nt heS eries
JOHNSON
FormTahle oroifPe osl itics
MITTNIK
System ThMeeotrheoitdnEis cc o noMmoidce llli ng
HARGITATI
CrysStyamlme tea tlS.r hiuebsn:i Ckeonvt ennial Papers
WITTEN
MathemaMtoidceaillnMs e dicVionleu me
2
LEE MACDANIEL
Oce&aA nc ousPtriocp agbaytF iionniD itffee reMnecteh ods
YAVIN PACHTER
&
Purts-uEivaDsiiffoeennr tGiaamle s
OtherB ookso f Interest
ENNALS
ArtifiIcnitaell ligence
JESSHOPE
ParaPlrloecle ssing
KRISHNA
ReceAndtv anicnXe -s rCahya racteorfiM zaatteirIoiIna ls
MAUNG
MeasurefmreonMmta sp s
RODIN AVULA
Mathe& maMtoidceallil niS ncgi encTee cahnndo logy
RelateJdo urnals*
ApplMiaetdh emaLteitctse rs
BulloeftM iant hemaBtiioclaolg y
ComputaenrdGs r aphics
ComputaenrdMs a themawtiitAchps p lications
MathemaatnidCc oamlp uMtoedre lling
Pattern Recognition
ProgrienCs rsy sGtraolw atnhd C haracterization
System
Technoilno gSyo ciety
*Sampcloep iaevsa iloanrb elqeu est
SYMMETRY 2
UnifyiHnugm anU nderstanding
Editbeyd
ISTVAHNA RGITTAI
HungarAicaand emoyfS cienacnedsE otvUonsi verBsuidtayp,e st
PERGAMONP RESS
OXFORD NEWY ORK BEIJINGF RANKFURT
· · ·
SAOP AULO SYDNEY TOKYO TORONTO
· · ·
U.K. PergaPmroepnsi sHe e,a dinHgitHloalnl l,
OxforOdX 3O BWE,n gland
U.S.A. PergaPmroenIs nscM .a,x wHeoluls Fea,i rvPiaerwk ,
ElmsfoNredwY, o r1k0 5U2.3S,. A.
PEOPLER'ESP UBLIC PergaPmroenRs oso,m 4Q0i3a7n,mH eont Beelji,i ng,
OFC HINA PeopRleep'usbo lfi cC hina
FEDERARLE PUBLIC PergaPmroeGnsm sb HH,a mmer6w,e g
OFG ERMANY D-6242 KronbReerpgu,bo lfGFi eecrd mearnayl
BRAZIL Pergamon LtEddRaiu,ta�o ard aeQ uei3r4o6s,,
CEP0 04l,Pl araSiasPooa ,u lo, Brazil
AUSTRALIA PergaPmroeAnsu ss trPatLlyti daP, . OB.o 5x4 4,
Potts NP.oSi.2nW0t1.1,A, u stralia
JAPAN PergaPmroen5s tsFh,l oMoart,s uCoeknatB ruaill ding,
1-17N -ishishSihnijnujkuuTk,ou k-y1ko06u ,J, a pan
CANADA PergaPmroenCs asn aLdtad , NSou2.i71 t,e
25C3o llSetgreTe oerto,n to,C aOnnatMdaSarTI i RoS,
Copyr©i 1g8h99tP ergaPmroepnsi se
RigthsR eserNvope adr.oft t hpiusb limcaaytb ieo n
rAlle prodsutcoeirdnea, dr etrsiyesvtaoelrtm r ansmitted
in faornmyo rb ya nmye anesl:e cterlotenrciocs,t atic,
mangettipacem ,e chanpihcoatploy,ci onrge,ci onrogdr
otherwwiitshepo,eu rtm isisni ownr firotmi tnhge
publisher.
ISB0N 0083 7263 7
Covielrl usdtersaitbgiynoM enad g doHlanrag ittai
Publiassah s epde ciaolft hijeso suurCeno amlp uters
andM athemawtiitAchps p licaVtoilounm1se,7,
Numbe1r-sa6 n dsu pptlois eudb scraispb aerortfs
theniorr msaulbi spctrion.
LibraorfCy o ngreCsast aloguing-inD-aPtuab lication
Symme2t:urn yi fhyuimnagun n derstandbiyn g/edited
IstHvaarng ittai.
p.c m.-(Intesrneartiiinmeo osnd aealrp np lied
mathemaantcdio cmsp ustceirev n.1c )8e ;
Publiassah s epde ciisasoluft e h jeo urnCaolm puters
anmda themwaittaihpc psl icva.1t ,7in oon1.s- ,6 .
1S.y mmeItH.ra yr.g iItsttavia,n .
IIT.i tSlyem:m ettwroIy.I S Ie.r ies.
Q1275..S951S99839 5080-91d-c612360
BritLiisbhr aCrayt alogiunPi unbgl icDaattiao n
SymmeUtnriyf:hy uimngau nn derstanding.
1S.y mmetry
I. HargittaCio,m puIatsnetMdrvas at nh emIaIt.i cs
witahp plicIaItSIie.or nise s
500
ISB0N- 08-037237-6
Prinitne dGB rreiatbtayB i PnC WCh eatLotnEdsx,e ter
ERRATA
Pag3e7 l9i,bn eefo trhele a (setq ua7t)si hoonub le(d 3 C4LC L6 0);.
Pag3e8 b3o,t tpoamrt thfo;er mufolrac ey clopecnytcylloehnaeenxd,dyi lmeenteh ine
unisthso bule(d > CH5-,() >CH6-a )n(d> CH-HCr<:e)s pectively.
Page e3q8u9a,(t 1ibo7)sn,e conds htoeubrle(md > C-HHC<:)qg.
Pag3e9 e0q,u a(t2i0osane) c,ot nedrs mh ouble(d > CH-HC)<:t g.
Equat(1i2)oas,ne cotnedro mft hfier lsitn e bseh( o>uClHd-H C)::qa::gn: td h e
secotnedro mft he sleicsnoheno dub le(d C >H -H::C::):t g.
Pag3e19 e,q ua(t2i5ofianr) ts,et ro mfs eclonidn e bse(h >oCuHl-dH:: C::):qa gn d
thsee cotnedro mft hteh ilridsn heo ubled C (H ->H C)::t::g:.
Pag3e8 5, loansl ti 4ns weho orurdel adud n iints toefua ndi te.
Pag3e8 7, 1Tc,ao bllu1emt, nh fo er mufolrah exadecaohnte hdsere acnoen d and
thilridsn heo rueladad sfo llows
(HC(3C( HH)CC3•H )
(HCC(H3 ))3 ( HCC)H.) 3
HARGITTSAyIm:m e2t.Ir SyB 0N0 8 0367 237
PREFACE
Thsiesc voonldu omfSe y mmewtarisyn itiirnae tsepdto otn hsgeeo od reocfteh fipert vsiotol nu me,
SymmetryU:ny ifinHgum anU nderst(aEndIdH..ia n gr giPtetragiaP)mr.oe (ns91 s8 6P)e.o pdlied
noste etmob eo verwhbeyl mlteahdrev g oel uomvee,r -a-ptahgobeuussct,aa nldl edfo ri nstead
itcso ntinuawtaitsoh mnea. i fanTc htiionosr u dre citsoei mobnau rpkot nhs ee cvonoudlm. eW e
werael psloe aasnegddr atbiyf igteohdoe rd e viaenwtdsh r ee cogtnhitathtie o nj ournal version
wansa me"dT hBee sSti nIgslseou fae J ourbnyat lh"e ProafensSdsc ihoonlPaaulrb llyi shing
Diviosfti hoAens socioafAt mieorniP cuabnl ifosrh1 8e9r6Ns.o twithsittrasen ldaithniigvg ehl y
pritchebe,o owka osu otfp riwnitt thwiyone aorfis tp su bli.c ation
Thpeu rpoofs est ehceo nd vtohleu mseat mhioeas ftt ahfsei .rO sutr jperwcoates n visioned
fromt hvee rbye ginansaic nogn tritbotu htbeir oindr goiolnfesg y mmetbrryi nbgyi ng together
thmeo sdti vefires.lAe dn so jbecwtaitsvop e r estehcneot n temsptoarotafsur ysym mesttruyd ies
asw ealslv arivoiueoswn ts h reo alneid m porotfat nhcceoe n cHeapvtit.nh ge psuarmpefoo rs e
thsee cond dvooenlsoum tme eaj nu msotr oeft hsea mbeu,it mt e aanlses xop antsofui rotnh er
fiseM.le ddisccaile anncedec so noamriec tsbw uootf t hnee awr eas wwehrnieoic tnh c liund ed
thfei vrosltu Imnse o.m oet hfieerlt dhsee x tenwsaiison n deptthh airnnba rtehaedrt h.
Thes eemirnagnldoyor md eorfit nhcgeo ntriobfut thfieio rvnsostl uwmaems e atnota dd
emphtaos iiisnt tse rdisTchiaippslp irnwoaaarcsih etn nyot.tia rbealnydi onn eptdrh eesv eonltu me.
Weo nltyr iteold et th e pflaopwgee rnse rfraolmml ayt hemaantsdic cisea nttc hebese ginning
towatrhdaesr itnsl atpearrM tasn.oy ft hpea pewrosun lodlt e ntdh emsteoal nvryei sg orous
classiefivceanwt eihi oafwnd i shfoerid tw ,h iwcehc ertdaiindno lty.
Myh opiensse ndSiymnmget r2yo ni twsaa yr tew o-Ifow lodu.ll idk see et os yemxmeertcriys ing
itasb itlohi ethlyup m an cporgoncietasinIsvdw ee os u;ll idk see sety om meltirnyvk airnigao ruesa s
ofh umaenn deavor.
Whecno mplSeymtmietnrg1y I w adso utbfwuhle tihhtea rad c compallitlsh hhaeatdbd e en
calfolrea dti tcso nceApttt ihcoeon m.p loefSty mimoetnr 2yi itcs l etaor t mheia mtte amnesr ely
anotshteiernpt hoer igpirnoajolen wc hti wcehe mbarkedfi vayele maaorgssoTt .h fiee lodfs
relevaantndhc reee levoafvn acreif oiuessle detsmoe xparnadtt hheatrno c loTshece.o ncoefp t
symmekterepypr so viidmipnefogtr un se tw houagnhsdte sa rches.
Ia md elibgyh tgetrdho ew iinntge rnaotfti hoseny amlmiepttryro yjT ehctetw .vo o luhmaevse
noiwn voslcvieedan ntadir sttfrisos mt2 sc3 o untSriinsecysem. m esttruyda eiri enss trumental in
getttiokn ngo w mpaansa'tnsc d u lttuhrieen ,t ernaotfti hoeinnsad leiahtvaypos rr o fessional
imporbteaynocgneod o d-awnifderl inld ship.
Iw oullidtk oee x prmeyas psp rectioaa lctloi notnr itbotu htvioosrl suf moterh edierd ication
anhda rwdo raknp da tioefntcieenn,v orlevviinasgni odot nhesex rtw roarw kh ichn ogdtei nde rate
thsel igchotmeps.lt a aiImn m tost gtroaa tlelf ruetlfeh rewe heoss e ruvnesde lafinswdhi ltyh
enthsumsW.ie ha avael bseon efifrtotmce odnr sutctcirvieta incsdiu sgmg estions from numerous
ferindasnc do lleTahgivusoe lsu.im osen caeg atihnre e soufwl otn fdueclro operbaymt ainoyn.
IH.A RGITTAI
ix
SYMMETRY 2
UnifyHiunmgaU nn derstanding
(Pa1r)t
Liqudiedn dforritmPi.hc o tog(r91a49pc)ho urotfGe ysoyr Kgeyp es.
ComputMeartsh A.pp licV.o l1.,7 N1o3-.,p p.1 -,11 2899 0097-494$33/.+80 090. 00
PrinitneG dr eaBtr itAalirlni .g hrtess erved Copyri©g 1h89t9 P ergamPornep sise
SYMMETRYR EVISITEDt
M.S ENECHAL
DepartomfMe antth emSamtiiCtcohsl ,l Neogret,h amMpAt0 o16n03,U, . S.A.
Abstraceta-dRiHenergr maWneny ln'osw -cl15a92sm soinco grSaympmhte royn,ie ss t rbuoctkh i tbsy
beauatnyd ibtys limMiatnoayftt himeoo nsistn. t erpersotbilnecgmo sn tienm psoyrmamretyth rey·o ry
concleorccnao ln figuraantgdir oontushp,e moaryyn obtet hoen loyrt, h bee tsotfo olsr t udtyhienmg.
1.I NTRODUNC TIO
In1 091a,ta meeotfti hnBegr itAisssho cifoartt hiAeod nv anceomfSe cnite tnhcee ,c rys­
tallogWrialplhBieaarrmsla onwHd e nMriye prosi notuettdh tahthe i stooftr hdyee velopment
of tthheeoo frt yhs et ruocftc urryesw thailccslh,o psaerlaytl hldeee lvse loopftm hetenh te ory of
symmetry:
". .is. htihseto ofar ny atttoee xmpprtge esosm ettrhipech aylslpiyrc oaple orfct riyessat naadlte s a,c h
staogfte h per ocaenas psp etaotl h eir mkonropwhno lporgoipcearltd ireistv heghenea osm ettroi cian
widtehnse c ope ionfqy ua hinirtdso enhlidaser figne iotfhi oomno geneity."
Barlhoawod n lrye cecnotmlpyl heitdsee dr ivatthiteoh nr eoef- dicmreynsstiaolnlaolg raphic
grouTphse.as rete h e groruipgmsio dt oifoo rns sy,m mettrhiametas pa, t hree-dimensional
repeaptaitntge ranb-stthmreoa dcoetfl t hset rucotfau nri dee carly stali-t.osT nehtliofs
enumerfiartcsiator nro,iu betydt hGee rmmaant hemaAtritchSiucarhn o eannfltdih eResu ssian
crystalEl.o grSa.p ihFne1e rd8 o9mr1aor,vk t ehcdeo ncloufasl ioonsngt rutgodg elfewi hnaet
imse abnytc rystoarldleirn.e
Wea rseoa ccusttootm heidno kfci rnygs atspa elrsi mooddiucl,a r stthiractuto cmtaeussr es
as urptroli esaterh npa eto opnlcete h ouogthhte rHwoiwseeva.esl r aa,ts te h eei ghtceeennttuhr y
somree spsecctieewdne trsietsa itrlsgl u tihncagrt y stals g[rlolI w]nt hleviiikrete wh ,ep lvaenitnss
ancda viitnci reyssw thailwcsehn, o w raesgs atrrdu cdteeufcrtwasel,rc eh anfonre ltishn et ernal
distriobfau ptpiroonpj ruiiactees .
RobeHroto ktehs,ee vencteenetnutrhy Enwghlioifs sa hm ofosurchs iii esnn vteissttwi igtaht ions
thmei croswcaoosnp oeeft , h fier tsotc onstihdee riw nwa hyi ccrhy ssttarlum citguahrctec ount
focrr yfosrtma( lF iIg.)B. y 1 282t,h Fer enscchi eRnetnJieus Hstat i hiaywd o rkoeudath ighly
develtohpeeoodrfc y r yssttarlu citnwu hriec h tinbyu,i libddlieonnwcgtek irsace ra rla nigne d
periaordri(acFy is2g )t.;h ese arbreat yesr mciionnuav ltader diw oauyss , acfocrto huen ting
variaitnfoi romn asos fi ncgrlyess ptaelcD ieebsao.tn te hsre e aolfHi atyi ibyu'isl dinlge db locks
byt hmei ddolfte hn ei netceeenntttuohr yct ohnec oefapp to int (lFai3tg)At..fi tcteehrl e a ttices
wercel as(saibfio1e5u8d0t t) h e French mathJeomradutasinec dni eattwnho eoC glar mooiuflp l e
theotroey n latrhgdeee finoifht oimoong etnoie nictlyru edpee aptaitntwgei rtnrhso tational
symmeSturryp.r iisttio nomgkal nyy, ymeoarforersa c onsetnoes muesrt gherafl etce titoonos
shobuela dd dteotd h pea ttuenrdncesor n sideInr1 a98t1iiw toa ns. feisntaalbltlyhi tashthe erde
ar2e3 0 gorfmo outpisio ntn hsr ee-dismpeanacsneaid, o lniatltt lheta htnel; aa r1te7se yrm,tm rey
grouopfrs e peaptaitntigent r hnpesl ane.
Itwa tsh faem ouGse rmmaant hemaFteilKcilixea inn h awsdhu og getsoSt cehdo etnhflei es
probolfee nmu mertahcteri ynsgt alglroogurpaspw.ha itsKch fol ere eimneo xspto onfetn hvtei ew
thatth set uodfyg eomesthroyub led tshteuo dfys ymmetry orgm roorgueep nse,r ally
automorgprhoiuaspnmshd ,e p robarbelaylt ihzatethdg e e omeotfcr ryy swtoaulblsed nai ce
examtpoil lel uhsitisrd aeTtahesee .ffi caocfgy r oup fotrhm eaotrhye matical wcarsy stallography
tThpiasp iebsra soendl ectpurreesset nott heNede wY orAkc adeomfSy c ie6nN coevse,m 1b89e6ra, n tdo t hSep ecial
Sessoin"o Tneh M athaetmiSccaile onfHc eer mann A WCeeyn!t:eT nrairbyuA tmee"r,iM caatnh teimcaSaolc iety,
AmherMsats,s2 6.O ,c to1b8e95r.
2 M.S ENECHAL
FigI..R obeHroto kder'asw ionfcg rsy ssttarlu cfrtoumMr iec,r ogr(a16p65h)i.a
•
1,.,
� ·, "
' ' ,.., .
Fi2g..R .J .H aily'so fcc roynsscttearplut ca tnufodrr emf ,r oTmr adietC er istall(o8g12r2ap)h.i e
Symmertervyi sited 3
Fig3.. poAi lnatt tice.
stronglyb yts hudepi psocrootvfee ddtr ihyffe r aocfXt -irobanyyc sr ysitn1a 9l,1os n2 oeft hger eat
scieannttdie ficch lnaincdamlia nor ukcrse ntSuirnytc.he eg nr,o tuhpe hoarpsyl ayace edn tral
rloei nc rystallogrtahpcehl ya,s sbioefifccca arutysisbeoty na lssy mimste hter ys fitrespt
in dtehtee rmionfta htesiitorrn u cBtuuetrv eea.nc asiunaslp eocfVt oilo1.on fS ymmetry:
UnyiifnHgum anU nderstasnhdoitwnhsgna etpw r oblaerimems p licchiatlllgyer nogtuihpne go ry's
hegemony.
Thutso d8a7yy e,a arfstB earr laonMwdi eprrse setnhtepeiadrpw eera ,r e caognafrionn ted with
the nweieddte hntse oc oopfoe u irn quainredyn laorudgree finiotfhi oomno geanswe eia trye,
confrwoinpttahet dt wehrintcshh o,uo grhd edronl oyst,a tyti hsseft rriecqtu iroefcmr eynsttsa lline
ordIeitrns .o otn ltyhf ea moquusa sicwrhyissctuhag lgtseh saitls t,h tohuecgyehr tdaoiT.nh ley
probilmseu mc mho rgee nearnaidlit ,cs l etahrwa etm usgtob eyognrdo tuhpe ionro yr dteor
deawli tihtI .ti lse cslse wahra tn ettwh oeo wlistl ulro nut to be.
2.C LASSICAELT RSYY MM
A symmeotpreyr aitsai roing miodt iao mno,t itohnal te adviesst abnectewpseo einn ts
unchaTnhgeemsdoe.t iionncsl ude rroetflaettcritaoinnos,nal ,nav dta iroionoft u hsec iorm bina­
tioAn ssy mmeotpreyr aetffiteoscon r derrelptyei toainn,od r derrelpye tciathnia ovsnet rong
aestahpepteNiaoclst .u rprissyimnmgeiltosyrf n,ytf e uonda riatnn, ad r mtu seaurmweso ndfuelr
sourocfee xsa mpofls eysm meFtoreryx .a mlpelutesc , o nstiwdaoer rt irsetnidceo rfpi aniogrfss
hands.
Fig4u,br yeS anGtraa ziiesan ntii "tCalte"cd.hI nt hpiasio rnh,ea nidas r ighhatna dn tdh e
othael re wfhti,ic nsho stu rprBiusltio nocgka. r le:yft uhlehsaen adrsne o otn lrye lattoee adc h
othbeyar m irror rtehfleaeyrac elt vsiiooe nwb,ey ud sa si f tharm oiuorg!rrh
InF igw.e s e5e btehaueuls t ciuflp"tCautrhee bdyrR aold"iI nit.as t fi rass tup rritsofie n d
thahte re hbaontadhrsl e e hfatn drse,la ntobetyd r eflecbtubityo r no taIt iotnhti.hn atkth e
surparnitdsh ere o tatsiyomnmaealtr trehy e soofut rhscece usl 'pcsto umrpeeblelaiuntgy .
Mirrsoyrm meitfrsre yq ueenntcloyu nitnae rartne iddns cieInitcas pe r.o foucnodn c,we iptth
deeepc hionel si teraanptdhu irleo Tshoeparhreyse .t mialnly myosfrt iegarhnitlde te sf, tohne
larsgceao lfte h set ruocfttu hurene i voenrt shseem, a slclao lfee l emepnatratraiync adlta e lsl,
levienbl est wReoetna.t siyomnmaeiltfos ru yna dl meovsetr ywohfetceronem ,b ihnaerdm oniously
witrhec flteiWoenfi .n idt s nionw flakesfl,o wedsrotsma,eig snl,eaw dsi sn daonwpdso ttTehriys.