Emergence,ComplexityandComputation
Volume45
SeriesEditors
IvanZelinka,TechnicalUniversityofOstrava,Ostrava,CzechRepublic
AndrewAdamatzky,UniversityoftheWestofEngland,Bristol,UK
GuanrongChen,CityUniversityofHongKong,HongKong,China
EditorialBoard
AjithAbraham,MirLabs,USA
AnaLucia,UniversidadeFederaldoRioGrandedoSul,PortoAlegre,RioGrande doSul,Brazil
JuanC.Burguillo,UniversityofVigo,Spain
Sergej ˇ Celikovský,AcademyofSciencesoftheCzechRepublic,CzechRepublic
MohammedChadli,UniversityofJulesVerne,France
EmilioCorchado,UniversityofSalamanca,Spain
DonaldDavendra,TechnicalUniversityofOstrava,CzechRepublic
AndrewIlachinski,CenterforNavalAnalyses,USA
JouniLampinen,UniversityofVaasa,Finland
MartinMiddendorf,UniversityofLeipzig,Germany
EdwardOtt,UniversityofMaryland,USA
LinqiangPan,HuazhongUniversityofScienceandTechnology,Wuhan,China
GheorgheP˘aun,RomanianAcademy,Bucharest,Romania
HendrikRichter,HTWKLeipzigUniversityofAppliedSciences,Germany
JuanA.Rodriguez-Aguilar ,IIIA-CSIC,Spain
OttoRössler,InstituteofPhysicalandTheoreticalChemistry,Tübingen,Germany
VaclavSnasel,TechnicalUniversityofOstrava,CzechRepublic
IvoVondrák,TechnicalUniversityofOstrava,CzechRepublic
HectorZenil,KarolinskaInstitute,Sweden
TheEmergence,ComplexityandComputation(ECC)seriespublishesnewdevelopments,advancementsandselectedtopicsinthefieldsofcomplexity,computationandemergence.Theseriesfocusesonallaspectsofreality-basedcomputation approachesfromaninterdisciplinarypointofviewespeciallyfromappliedsciences, biology,physics,orchemistry.Itpresentsnewideasandinterdisciplinaryinsighton themutualintersectionofsubareasofcomputation,complexityandemergenceand itsimpactandlimitstoanycomputingbasedonphysicallimits(thermodynamicand quantumlimits,Bremermann’slimit,SethLloydlimits…)aswellasalgorithmic limits(Gödel’sproofanditsimpactoncalculation,algorithmiccomplexity,the Chaitin’sOmeganumberandKolmogorovcomplexity,non-traditionalcalculations likeTuringmachineprocessanditsconsequences,…)andlimitationsarisinginartificialintelligence.Thetopicsare(butnotlimitedto)membranecomputing,DNA computing,immunecomputing,quantumcomputing,swarmcomputing,analogic computing,chaoscomputingandcomputingontheedgeofchaos,computational aspectsofdynamicsofcomplexsystems(systemswithself-organization,multiagent systems,cellularautomata,artificiallife,…),emergenceofcomplexsystemsandits computationalaspects,andagentbasedcomputation.Themainaimofthisseriesis todiscusstheabovementionedtopicsfromaninterdisciplinarypointofviewand presentnewideascomingfrommutualintersectionofclassicalaswellasmodern methodsofcomputation.Withinthescopeoftheseriesaremonographs,lecture notes,selectedcontributionsfromspecializedconferencesandworkshops,special contributionfrominternationalexperts.
IndexedbyzbMATH.
Moreinformationaboutthisseriesat https://link.springer.com/bookseries/10624
· SouvikRoy · KamalikaBhattacharjee
Editors
TheMathematicalArtist
ATributeToJohnHortonConway
SukantaDas
Editors
SukantaDas DepartmentofInformationTechnology
IndianInstituteofEngineeringScience andTechnology
Howrah,India
KamalikaBhattacharjee DepartmentofComputerScience andEngineering NationalInstituteofTechnology Tiruchirappalli,India
SouvikRoy C3iHub IndianInstituteofTechnology Kanpur,India
ISSN2194-7287ISSN2194-7295(electronic) Emergence,ComplexityandComputation
ISBN978-3-031-03985-0ISBN978-3-031-03986-7(eBook) https://doi.org/10.1007/978-3-031-03986-7
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JohnHortonConway December26,1937–April11,2020
[ImageCourtesy:DeniseApplewhite,PrincetonOfficeofCommunications,PrincetonUniversity]
Preface
Thebeginningoftheyear2020hadseenthesuddenappearanceofanunknownvirus, namedas SARS-CoV-2, whichcausedCoronavirusdisease(COVID-19)andstarted wreakinghavocovertheglobewithinafewmonths,resultinginapandemic.The worldliterallystalledunderthiswrathofnature.Thiswasthetimewhenwelost JohnHortonConway.
WhoisJohnConwaytous?Wearethestudents,researchers,andpractitionersof CellularAutomata (CAs).Allofuswerepracticallyintroducedtocellularautomaton (CA)throughthe GameofLife byourteachers.This GameofLife, sometimessimply knownas Life,isazero-playergamedevelopedbyJohnConwayin1970.Although historicallyCAwasinitiatedbyJohnvonNeumannandStanislawUlaminthe1940s, itwasConway’s GameofLife whichpopularizedCAtothepeoplebeyondacademia. Thepowerofitssimpleruletocreateaninfiniteworldofpossiblecomplexpatterns isboundtomakeeveryonelikeuscharmedandintriguedtoknowmoreaboutCAs andadvertentlycometodoresearchonCAs.
However,JohnConwaywasnotlimitedtoonly GameofLife.Infact,itisa verysmallshadeofhislifefilledwithcontributionsindifferentareasforminga colorfulpalette.BeittheCombinatorialgametheory,Grouptheory,Numbertheory, Geometry,Geometrictopology,Algebra,orTheoreticalPhysics,whateverareahe hadtouched,aremarkablecontributiononhisnamehasremainedthere.Welostthe legacyofthismathluminaryonSaturday,April11,2020,inNewBrunswick,New Jersey,USA,fromcomplicationsrelatedtoCOVID-19.
Nevertheless,civilizationcannotbestoppedbyapandemic.Itcreatedanew normalsituation—theuseofonlinemeetingstocontinueworkfromhome.Classes, meetings,conferences,etc.werefinallyresumedinvirtualmode.Duringthistime (August2020),we,someoftheIndianresearchersoncellularautomata,tookan initiativetocreateavirtualplatformwherethepeopledoingresearchindifferent shadesofTheoreticalComputerScience,especiallyonCellularAutomata,cangetto knoweachother’swork,participateandinteractinlivelydiscussionsonacommon platform.Inthisdiscourse,westartedSeriesofWebinarsonCellularAutomata whereeminentscientistsovertheglobeinthefieldoftheoreticalcomputerscience, ingeneral,andcellularautomata,inparticular,joinedusinenlighteningdialogues.
Asanoutcomeofthisinitiative,aninformalresearchgroup CellularAutomataIndia wasbornwhichnowhasover130membersfromallovertheworld.
Thisvirtualpodiumalsogaveusachancetothinkaboutpayinghomagetoour belovedJohnConwayinthewaywedesired.So,westartedplanningfromDecember 2020toorganizealectureseriesinmemoryofConwayonhisfirstdeathanniversary. OurplanwastohaveweeklylecturesforthemonthofApril2021startingonApril 10,2021.WewantedtolistenaboutotheraspectsofConwayaswellason Gameof Life andcellularautomata.And,ourwishwasifsomeoneclosetoConwaycouldtalk tousabouthispersonalfeelings,experience,memories,andlearningsfromConway. Weapproachedafewofsuchandaftersomeunsuccessfulattempts,wegotareply fromProf.RobertWilson,aneminentgrouptheoristanddoctoralstudentofJohn Conway.Thatletterwasalmostlikeanextractofpagesfromhisdiarythroughwhich wewerevisualizingseveralscenesfromConway’slife.ProfessorWilsoninteracted withuson10thAprilanditwassuchafascinatingeventforus.Wearegratefulto Prof.RobertWilsonforsharinghisinvaluableexperiencewithus.
Twootherspeakersalsobenevolentlyagreedtogivelecturesthenextweek,Prof. R.RamanujamandProf.GenaroJ.Martinez.ProfessorRamanujam,arenowned theoreticalcomputerscientistandactivist,talkedtousaboutConway’scontributionindifferentfields,especiallyonCombinatorialGameTheory,whereasProf. Martinez,anoteworthyscientistinthefieldofCellularAutomata,gaveusamesmerizingoverviewof GameofLife,itshistoricalevolutionthroughtheeyesofanexcellentresearcher.Finally,inthelastweek,wehadamongusStephenWolfram,a pioneeringscientist,physicist,andauthorof ANewKindofScience whosework createdaparadigmshiftincellularautomataresearch.Wewereenlightenedbythe richnessofhisdialoguesandhisexperience.Weareindebtedtoallofthemfortheir time,effort,andindulgence.
Duringthislectureseries,Prof.MihirK.Chakrabortyproposedtothinkofcreating aneditedbookwiththetranscriptsoftheseexcellenttalksandsomeinvitedpapers. ThankstoSpringerwhichreadilyacceptedourproposaltopublishthebookunder theseries“Emergence,ComplexityandComputing”.SouvikRoy,aneditorofthis book,tookuponhimselfthegigantictaskoftranscriptionforthethreelecturesgiven byProf.Wilson,Prof.Ramanujam,andStephenWolfram.Wearethankfultothe threeofthemforcheckingandapprovingtheirowntranscripts.Thefirstpartofthe bookcontainsthesetranscripts.ProfessorMartinezagreedtowriteacompletearticle onhistalkwithtwoofhiscollaborators.Thisarticleisaddedjustafterthetranscripts aspartoftheinvitedarticles(PartII).
Further,weinvitedsomeeminentscientiststocontributetothisbook.Weare especiallygratefultoProf.KenichiMoritawhoagreedtowriteforusonsuchshort noticeandgaveusanexcellentresearcharticle.WealsowanttothankProf.Carter Bays,Prof.PedroPauloBalbideOliveiraandhiscollaborators,Prof.HectorZenil andhiscollaboratorforcontributingtoourbook.Oneofus,SukantaDas,hasalso contributedanarticle.EachoftheinvitedarticleshasgonethrougharigorouspeerreviewprocessaspertherequirementofSpringer.Weareindebtedtoallthereviewers forindulgingourrequestforpromptreviewsaswellastheirtimeandeffort.
WewouldliketotakethisopportunitytoexpressourdeepgratitudetoProf.Mihir K.Chakrabortywhofirstencouragedustotakethisinitiative.Wealsothankoursmall teamofcellularautomataresearchers—SupreetiKamilya,SumitAdak,Sukanya Mukherjee,RajuHazari,NazmaNaskar,DebopriyaBarman,andBiswanathSethi aswellasthe CellularAutomataIndia communityasawholewithoutwhosesupport thisbookwouldhaveremainedonlyindreams.
Kolkata,India February2022
SukantaDas SouvikRoy KamalikaBhattacharjee
JohnHortonConway
APlayfulMasterofGameswhotransformedMathematics (December26,1937–April11,2020)
JohnHortonConwaywasalegendarymathematicianwhowasdistinguishedforhis loveofgamesandforbringingmathematicstothecommonpeople.Hewasone ofthemostmultifacetedmathematiciansofthepastcentury,awizard,polymath, storyteller,andanaturalproblem-solverwhoseunaidedaccomplishmentsoftenmade hiscolleaguesspell-bound.
ConwaybecamerenownedduringhisstayatCambridgeUniversity(asastudent andprofessorfrom1957to1987).However,heclaimedtoneverhaveworkedasingle dayinhislife.Instead,“hepurportedtohavefritteredawayreamsandreamsoftime doingnothing,beinglazy,playinggames”.YethewasPrincetonUniversity’sprestigious JohnvonNeumannProfessor in AppliedandComputationalMathematics since1987whichpositionhehelduntil2013whenhewastransferredto emeritus status.
ConwayreceivedtheBerwickPrizein1971andthePólyaPrizein1987(first recipient).HewasawardedtheNemmersPrizeinMathematics(1998)andtheLeroy P.SteelePrizeforMathematicalExpositionoftheAmericanMathematicalSociety in2000.HewaselectedasaFellowoftheRoyalSociety(1981),andFellowofthe AmericanAcademyofArtsandSciences(1992).Forhiscontributions,hereceived honorarydegreesfromtheUniversityofLiverpoolin2001andAlexandruIoan CuzaUniversityin2014andhonorarymembershipoftheBritishMathematical Associationin2017.
Knownforhisplayfulapproach,swiftcomputation,andnaturalproblem-solving skills,JohnHortonConwayhasbeencalleda“magicalgenius”.Forseveralyears,he frettedthathisobsessionwithplayingsillygameswaswastinghiscareer—untilhe recognized,itcouldsteerastoundingdiscoveries!Hisfascinationwithgamesgave birthtoatheoryofpartisangames—combinatorialgametheory,whichheoriginated withElwynBerlekampandRichardGuy.Thefamousbookseries WinningWaysfor yourMathematicalPlays isanoutcomeofthiscollaboration.Infact,hisbook On NumbersandGames (1976)givesthemathematicalfoundationsofthistheory.He alsodevelopedandanalyzedmanyotherpuzzlesandgames,suchasConway’s soldiers,pegsolitaire,Somacube,andphilosopher’sfootball.
xiiJohnHortonConway
Hehadtheunparalleledtendencyofjumpingintoanareaofmathematicsand completelychangingit.OnthesuggestionofJohnMcKay,hetriedtoprovesomethingaboutthepropertiesofthe Leechlattice andcameupwiththesymmetrygroup of Leechlattice.Hecontributedto spherepacking—findingthemostefficientway topackasmanyspheresinaslittlespaceaspossible.HecollaboratedwithRobert CurtisandSimonP.Nortontocontrivethefirstconcreterepresentationsofsomeof the sporadicgroups,whichhavebeennamedasthe Conwaygroups.Conwayand Nortonformulatedthecomplexconjecture—monstrousmoonshine whichisbased onanobservation(1978)byJohnMcKay.ConwayalsocollaboratedwithCurtis, Norton,RichardParker,andRobertWilsoninwritingthebook ATLASofFinite Groups.
In knot theory,Conwaycameupwithanewvariantof Alexanderpolynomial and producedthe Conwaypolynomial.Heinventedasystemofnotationfortabulating knots,knownas Conwaynotation andalsodeveloped tangle theory.Inthemid1960s,alongwithMichaelGuy,Conwayprovedthattherearesixty-fourconvex uniformpolychora.
InspiredbyLewisCarroll’sperpetualcalendaralgorithm,Conwaydevisedthe algorithmformentalcalculationin1973.Hisfamous Doomsdayalgorithm canbe usedtocalculatethedayoftheweek.Inhisfinalyears,hewasworkingontheoretical physicsandalsocontributedinthatfield.HeandSimonB.Kochenprovedthe free willtheorem (2004),anastoundingversionofthe“nohiddenvariables”principleof quantummechanics.
Thoughhemadeinfluentialcontributionstonumbertheory,grouptheory,algebra, geometry,topology,analysis,andcombinatorialgametheory,perhapsheisbest knownforinventingthe GameofLife,anenthrallingcellularautomaton-based“noplayernever-ending”gamewhereacollectionof cells continuouslyevolvesinto newconfigurationdependingonafewverysimple rules.Sincethe1970s,this GameofLifehasbeeninspiringpeoplefromdifferentagestofurtherexplorethe differentversionsandvariationsofitandcellularautomataingeneral.Itsmassappeal hascreatedacommunityofengineers,mathematicians,andresearchers,popularly knownasthe Life community,whohasbeenactivelyworkingonGameofLifeand itscomplexbehavior.
Knownforhisboundlesscuriosityandzealforsubjectsmuchbeyondmathematics,ConwaywasanadoredfigureinthehallwaysofPrinceton’smathematics buildingandattheSmallWorldcoffeeshoponNassauStreet,whereheactively engrossedwithfaculties,students,aswellasanymathematicalenthusiastswith equalinterest.Hewasanactiveresearcherandwasattachedtothecommonroomof themathematicsdepartmentofPrincetonwellintohis70s.However,threeyearsago, amajorstrokeconfinedhimtoanursinghome.Eventhenhewasregularlyvisitedby hiscolleaguesuntilthepandemicofCOVID-19madesuchvisitsimpossiblewhich forcedhimtodiscusshisideasonlyoverthephone.OnSaturday,April11,2020,he succumbedtocomplicationsrelatedtoCOVID-19.Atthattime,hewas82.
WeconcludewithacopyofthefirstemailreceivedfromProf.RobertWilsonas aresponseoftheinvitationmailforinteractingwithusonConway:
“Conwaywasaremarkableman.Hemadehisownrules,andlivedbythem.Healmost neveransweredlettersoremails,andrarelyevenreadthem.Inthe35yearsbetweenthe timeheleftCambridge,andhisdeath,herepliedto*one*ofmyemails.Andthatwasonly because,afternearly30yearsofeffort,Ihadfinally“solved”aproblemhehadtoldmewas impossible.
Henevertalkedaboutcellularautomata.Ifsomeonementionedthesubject,hedismissedit ashistory,nolongerinteresting.Hedelightedindoingtheoppositeofwhatwasexpected. Whenintroducedtosomeoneatapartywhowasamathematician,heissupposedtohave said“Iwasnogoodatmathematicsatschool”whichisprobablybothtrueandtheultimate put-down.
Idon’tknowwhatitisyouwantfromme,butIcanimaginethatitmaynotbequitewhat Icanprovide.IcantellyouverylittleaboutConway,butwhathetaughtmeabouthowto approachproblems,andhowtoensurethatyouareincharge,nottheproblem,hasneverleft me.”–RobertWilson
SukantaDas SouvikRoy KamalikaBhattacharjee
TranscriptfromConwayMemorialLectureSeries
JohnHortonConway:AMasterofAllTrades ........................3 RobertWilson
TwoDifferentDirections:JohnConwayandStephenWolfram ........21 StephenWolfram
ConwayMemorialSeries:TheMathematicalArtistofPlay ...........73 R.Ramanujam
InvitedArticles
SomeNotesAbouttheGameofLifeCellularAutomaton ..............93 GenaroJ.Martínez,AndrewAdamatzky,andJuanC.Seck-Tuoh-Mora GlidersintheGameofLifeandinaReversibleCellularAutomaton ...105 KenichiMorita
FromMultipletoSingleUpdatesPerCellinElementaryCellular AutomatawithNeighbourhoodBasedPriority .......................139 PedroPauloBalbi,ThiagodeMattos,andEuricoRuivo GameofLife,AthenianDemocracyandComputation ................159 SukantaDas
AlgorithmicInformationDynamicsofCellularAutomata .............171 HectorZenilandAlyssaAdams
TheGameofLifeinThreeDimensions,andOtherTessellations .......191 CarterBays
Contributors
AndrewAdamatzky UnconventionalComputingLab,UniversityoftheWestof England,Bristol,UnitedKingdom
AlyssaAdams AlgorithmicNatureGroup,LABORES,Paris,France; MorgridgeInstituteofResearchandDepartmentofBacteriology,Universityof Wisconsin-Madison,Madison,WI,USA
PedroPauloBalbi FaculdadedeComputação&InformáticaePós-Graduaçãoem EngenhariaElétricaeComputação,UniversidadePresbiterianaMackenzie,São Paulo,SP,Brazil
CarterBays DistinguishedProfessorEmeritus,ComputerScienceandEngineering,UniversityofSouthCarolina,Columbia,SC,USA
SukantaDas DepartmentofInformationTechnology,IndianInstituteofEngineeringScienceandTechnology,Shibpur,India
ThiagodeMattos Pós-GraduaçãoemEngenhariaElétricaeComputação,UniversidadePresbiterianaMackenzie,SãoPaulo,SP,Brazil
GenaroJ.Martínez ArtificialLifeRoboticsLab,EscuelaSuperiordeCómputo, InstitutoPolitécnicoNacional,MexicoCity,Mexico; UnconventionalComputingLab,UniversityoftheWestofEngland,Bristol,United Kingdom
KenichiMorita HiroshimaUniversity,Higashi-Hiroshima,Japan
R.Ramanujam TheInstituteofMathematicalSciences,Chennai,India
EuricoRuivo FaculdadedeComputaçãoeInformática,UniversidadePresbiteriana Mackenzie,SãoPaulo,SP,Brazil
JuanC.Seck-Tuoh-Mora AreaAcadémicadeIngenieríayArquitectura,ICBI, UniversidadAutónomadelEstadodeHidalgo,Hidalgo,Mexico
RobertWilson QueenMaryUniversityofLondon,London,UK
StephenWolfram WolframResearch,Champaign,IL,USA
HectorZenil OxfordImmuneAlgorithmics,Reading,UK; TheAlanTuringInstitute,BritishLibrary,London,UK; AlgorithmicDynamicsLab,KarolinskaInstitute,Stockholm,Sweden; AlgorithmicNatureGroup,LABORES,Paris,France
JohnHortonConway:AMasterofAll Trades
RobertWilson
1APersonalRemembrance
IreallyonlyknewConwayforabout11yearsfromwhenIwenttoCambridgeas astudentin1975towhenhe(Conway)leftCambridgetogotoPrincetonin1986. SoI’lljustsayafewwordsabouthowIinteractedwithhiminthose11years.Of courseasanundergraduatestudent,Iwenttohislectures,Ididn’ttalktohim,or interactdirectlywithhimbutIwenttoacoursehegaveinthefirstyear,whichwas anoptionalcourse,noexams,onformallogicandsettheory.Andtheninthesecond yearhegaveacoursecalledAlgebraIII,onthingslike—linearmaps1 andquadratic forms2 —thingslikethat.AndwhatImostrememberaboutit,wastherevisionnotes thatheprovidedforthiscourse.Attheendofthecourse,hehandedouttoevery memberoftheaudienceonesheetofpaper.Thatonesheetofpapercontainedthe entirecourseincludingallproofs,alltheexamples,everything,theentirecourse.It wastypedontwosidesofonesheetofpaperandhadeverythinginit.Heworkeda lotreallytoget24lecturesontoonepieceofpaper.AndIthinkwhatinspiredme mosttoworkwithhimismostlythefourthyearcoursehegaveonsporadicsimple
1 Inmathematics,alinearmap(alsocalledalinearmapping,lineartransformation,vectorspace homomorphism,orinsomecontextslinearfunction)isamapping V → W betweentwovector spacesthatpreservestheoperationsofvectoradditionandscalarmultiplication.
2 Inmathematics,aquadraticformisapolynomialwithtermsallofdegreetwo(“form”isanother nameforahomogeneouspolynomial).Forexample,4 x 2 + 2 xy 3 y 2 isaquadraticforminthe variables x and y
R.Wilson(B)
QueenMaryUniversityofLondon,London,UK
e-mail: r.a.wilson@qmul.ac.uk
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 S.Dasetal.(eds.), TheMathematicalArtist,Emergence, ComplexityandComputation45, https://doi.org/10.1007/978-3-031-03986-7_1
groups.3 WhichwasreallyexactlywhatIwantedtodo,asIfounditsofascinating thatIknew,Iwantedtodothat.
“Attheendofthecourse,hehandedouttoeverymemberoftheaudienceonesheetofpaper. Thatonesheetofpapercontainedtheentirecourseincludingallproofs,alltheexamples, everything,theentirecourse.Itwastypedontwosidesofonesheetofpaperandhad everythinginit.Heworkedalotreallytoget24lecturesontoonepieceofpaper.”
Thenattheendofthefourthyear,it’saverybrutalsysteminCambridge,they havetheexamwhichisabsolutelybrutal.Everyseparateexamhasthreequestions— thefirstquestionis‘writeouteverythinginthefirsteightlecturesofthecourse’;the secondquestionis‘writeouteverythinginthesecondeightlecturesofthecourse’; andthelastquestionis‘writeouteverythinginthelasteightlecturesofthecourse’. Andyougotthreehourstodothis.Andattheendofthisyouwereranked,andthe firsthowevermanypeopleitis,howevermanyplacesthey’vegot,thefirstsomany peoplecandoaPh.D.inCambridge.
IhadthoughtthereisnowayI’mgoingtohavedonewellenough,butIwas ‘just’luckyenough.Iwas,Ithink,intheverybottomofthelist.Thesamedayyou havetogoandfindasupervisor.SoIwenttoConwayandIsaidwouldyoubemy supervisor.Iwanttodosporadicgroups.Andhesaid,wellyouknowIamnota grouptheorist.Andthenhetriedotherwaysofputtingmeoffbysayingwellyou knowmanyofmystudentsneverfinishtheirPh.D.’s.Sowehadabitofadiscussion andeventuallyheagreedtotakemeonashisstudent.AndwhenIstarted,hegave meafewthingstoreadwhichwerenotaboutgrouptheoryatall,theywereall aboutlattices.AndIreadseveralrecentpapersonlattices.AndeventuallyIfound somegroups,thelatticestookmetowardsthegroups.Andtheytookmetowards thequaternionicreflectiongroups,4 inparticulartheHall–Jankogroup.5 Justoneof thesporadicgroupsanditiscontainedintheConwaygroup6 whichisrelatedtothe Leechlattice.7 Sothat’showIwenttowardsanotherversionofLeechlattice—the
3 Ingrouptheory,asporadicgroupisoneofthe26exceptionalgroupsfoundintheclassificationof finitesimplegroups.Asimplegroupisagroup G thatdoesnothaveanynormalsubgroupsexcept forthetrivialgroupand G itself.Theclassificationtheoremstatesthatthelistoffinitesimplegroups consistsof18countablyinfinitefamiliesplus26exceptionsthatdonotfollowsuchasystematic pattern.These26exceptionsarethesporadicgroups.Theyarealsoknownasthesporadicsimple groups,orthesporadicfinitegroups.BecauseitisnotstrictlyagroupofLietype,theTitsgroupis sometimesregardedasasporadicgroup,inwhichcasetherewouldbe27sporadicgroups.
4 Itisagroupoflineartransformationsinaquaternionicvectorspaceofdimension n < ∞ generated byelementsthatfixan(n 1)-dimensionalsubspacepointwise.
5 Intheareaofmodernalgebraknownasgrouptheory,theJankogroup J2 ortheHall-Jankogroup HJ isasporadicsimplegroupoforder27 · 33 · 52 · 7 = 604800 ≈ 6 × 105
6 Intheareaofmodernalgebraknownasgrouptheory,theConwaygroupsarethethreesporadic simplegroups Co1 , Co2 and Co3 alongwiththerelatedfinitegroup Co0 introducedby(Conway 1968,1969).ThelargestoftheConwaygroups, Co0 ,isthegroupofautomorphismsoftheLeech lattice ∧ withrespecttoadditionandinnerproduct.Ithasorder8, 315, 553, 613, 086, 720, 000,but itisnotasimplegroup.Thesimplegroup Co1 oforder4, 157, 776, 806, 543, 360, 000isdefined asthequotientof Co0 byitscenter,whichconsistsofthescalarmatrices ±1.
7 Inmathematics,theLeechlatticeisanevenunimodularlattice ∧24 in24-dimensionalEuclidean space,whichisoneofthebestmodelsforthekissingnumberproblem.ItwasdiscoveredbyJohn
JohnHortonConway:AMasterofAllTrades5
quaternionicversionoftheLeechlattice.AndsoIstudiedthat.Thatwaswhatmy firstpaper8 wasabout,thegrouptheoryassociatedtothisquaternionicLeechlattice. ThenIthought,wellifthereisarealLeechlatticeandacomplexLeechlattice,and twodifferentquaternionicLeechlattices,whatabouttheoctonionicLeechlattice? whataboutCayleynumbers?9 Ididlotsofcalculationsandnothingworked.And Conwaysaid“it’simpossible”.Youcan’tdooctonioniclattices,becausetheyarenot associative,sothereisnolinearalgebra,thereisnolattice,nothingyoucando,it’s impossible.
Well,letmetellyouanotherthingaboutConway’sattitudetowardsproblems. Manysupervisorswillsayyoushouldworkonaneasyproblemandonahard problem.Soifyoucan’tsolvethehardproblem,atleastyou’vegottheeasyproblem forbackup.Conwaysaidyoushouldhavefourproblems—youshouldhaveoneeasy problemandonehardproblem.Youshouldalsohaveatrivialproblem.Soifyou can’tdotheeasyproblem,youcanstilldothetrivialproblem.Andyoushouldhave oneimpossibleproblem.Becauseifyougetreallyluckyandyousolvetheimpossible problemthenyou’vereallymadeit.Soyoushouldhavethefourproblems—from trivialtoimpossible.Infact,Ithink,thatwastheversionoffewyearsbefore,I perhapsgotalaterversionthathehasexpandedthistosixproblems—youshould have,aswellastrivial—easy—hard—impossibleproblems,youshouldalsohave moderatelyeasyandmoderatelyhardinbetween.Andhereallyputthisintopractice. Hewasalwaysworkingonsixdifferentthingsatthesametime.Andmakingwork, yes,makingprogressonmostofthem—notnecessarilyalwaystheimpossibleones.
“Conwaysaidyoushouldhavefourproblems-youshouldhaveoneeasyproblemandone hardproblem.Youshouldalsohaveatrivialproblem.Soifyoucan’tdotheeasyproblem, youcanstilldothetrivialproblem.Andyoushouldhaveoneimpossibleproblem.Because ifyougetreallyluckyandyousolvetheimpossibleproblemthenyou’vereallymadeit.”
Soanyway,whatImeanishowversatilehewas.Ididn’tmakeprogressonthe impossibleproblemofanoctonionicLeechlattice.Andittookme20years,then eventuallyIsucceeded,workingouthowithadtowork.AtthatpointIthought,that wasIthinkin2009,andsoIthoughtI’dbettercontactConwayandtellhimabout this.Nowofcourse,Conwaydidn’thaveanemailaddressthatheread.Imean,he gotthousandsandthousandsofemails,thathecouldn’tpossiblykeepacount.So,
Leech(1967).Itmayalsohavebeendiscovered(butnotpublished)byErnstWittin1940.The Leechlattice ∧24 istheuniquelatticein24-dimensionalEuclideanspace, E24 ,withthefollowing listofproperties:
• Itisunimodular;i.e.,itcanbegeneratedbythecolumnsofacertain24 × 24matrixwith determinant1.
• Itiseven;i.e.,thesquareofthelengthofeachvectorin ∧24 isaneveninteger.
• Thelengthofeverynon-zerovectorin ∧24 isatleast2.
8 Thequaternioniclatticefor2 G 2 (4)anditsmaximalsubgroups,J.Algebra,77(1982),449–466.
9 TherearetwocompletelydifferentdefinitionsofCayleynumbers.Thefirstandmostcommonly encounteredtypeofCayleynumberistheeightelementsinaCayleyalgebra,morecommonly knownasoctonions.AquantitywhichdescribesaDelPezzosurfaceissometimesalsocalleda Cayleynumber.
anyhow,hehadasecretaccount.SowhatIdidwasthis—Iemailedtheofficeinthe Princetonmathematicsdepartmenttosay—‘I’vegotsomethingreallyexcitingtotell Conway,Ithink,hewillreallybeinterested,canyouforwardthisemailhim’.So, theydid.Andthenextday,IgotreplyfromConwaysayingincapitalletters—“I thinkthatisFANTASTIC”.SoittookmetwentyyearsbutIgotthereintheend. So,anyway,let’sperhapsgobacktomyPh.D.andhowIwasworkingwith ConwayforthePh.D.Itoldyou,hegavemethesepapersandsaidreadthem. Henevergavemeaproblem.HemighthavedoneeventuallybutIfoundmyown problems.IsaidaboutthisquaternionicLeechlattice.IsaidIworkedonthemaximal subgroupproblem.Thebreakthroughcamewhen,atsomepointinmythirdyearI think,Conwaywentonsabbaticalforamonthortwo.Hedidn’ttellmebeforehe wenthewasgoingaway.SoIhadamonthwithoutasupervisor.And,that’sthe timeIstartedhavingtalktootherpeopleandItalkedtoSimonNortonandJohn Thompson.Andtheygavemesomeideas,thatIwasmissing.Sotheyenabledmeto actuallystudytheLeechlatticeinadeeperwaythanIhadbeforeandtounderstand themaximalsubgroups10 probleminadeeperwayandmakeprogress.AndthenI eventuallymanagedtogetapostdocaftermyPh.D.Isupposetheotherthingthat reallyhelpedmetogetajobwasthatIworkedonthe‘Atlasoffinitegroups’which wasaprojectthatConwaystartedprobablyabout10yearsbeforewithRobertCurtis. Andgettingmynameonthefrontofthatbook11 wasveryimportantformycareer. WhatelseshouldIsay,probablyIshouldnotsaysomuchaboutmyselfmoreabout Conway.
AsIsaidIwenttomanyofhislecturecourses.Andheoftengavetheimpression, thelectureswerenotpreparedbecausehetalkedasthoughhewasjusttalkingtoyou inacafeorsomething.Hewouldn’ttalkasifhewasreadingapreparedscriptof thelectures.Andoftenhewouldsaywell—wecoulddoitlikethis,wecoulddoit likethat,whichdoyouwant—thingslikethis.Andlateronwhenhegaveresearch seminarstoaudienceshewouldsayI’vegotsixtalkswhichonedoyouwantmeto give.AndeventuallyIrealizedthatitwasn’tthatthetalkswerenotprepared.They wereverywellprepared.Buttheyweren’twrittendown.Theywerepreparedinhis head.Andso,becausetheywereinhishead,theyvaryslightlyfromoneperformance toanotherjustlikeapieceofmusicmight.Sometimeshewouldgooffinadifferent directionordothingsdifferentlyandsoon.He’dplaythemusicinadifferentway. Thatwaswhatgavetheimpressionofitbeingspontaneous.Whenitreallywasn’t. Ijustmadeamistakeatonepointofthinkingthatitispossibletogivelectureslike thiswithoutpreparation.Iveryquicklyfoundthatitwasnot.Itisquitepossibleto
10 Ingrouptheory,amaximalsubgroup H ofagroup G isapropersubgroup,suchthatnoproper subgroup K contains H strictly.Inotherwords, H isamaximalelementofthepartiallyordered setofsubgroupsof G thatarenotequalto G .Maximalsubgroupsareofinterestbecauseoftheir directconnectionwithprimitivepermutationrepresentationsof G .Theyarealsomuchstudiedfor thepurposesoffinitegrouptheory.
11 “Atlasoffinitegroups”byJHConway,RTCurtis,SPNorton,RAParkerandRAWilson;Oxford UniversityPress,1985.Reprintwithcorrections,2004.
goupandtalkforanhourwithoutnotesandgothroughthesubjectmatterbutit isabsolutelyessentialtoprepareverythoroughly.Ok,Ithinkthat’sprobablyabout everythingthatIprepared.So,whatwouldyouliketoknowaboutConway?
2InteractionwithRobertWilson
KamalikaBhattacharjee:Youweretalkingaboutexperienceofworkingunder Prof.Conway.Hewasalreadyveryfamouswhileyouwenttohimforyour doctoralresearch.Sodidyoufeelsomekindofextrapressure?Yousaidthathe hadalsotriedtodiscourageyoubeforehandthatyoushouldn’ttakehimasa superviser.HowmuchpressureyoufeltatthattimetomakeConwaysatisfied aboutyourwork?
That’sagoodquestion.Idon’trememberfeelinganyextrapressureIthinkit wasjustthesamewithanysupervisor.AlsoIdidn’tactuallyknowverymuchabout Conway.Ididn’tknowhowfamoushewasforallthesedifferentthings.Idon’t thinkIknewabouttheGameofLife.Ididn’tknowabouthisworkonfoundationsin numbersandgames.Ididn’tknowanythingabouthimapartfromtheConwaygroup. Hewasn’taprofessoratthatstage.HewasjustDr.Conway.HebecameaFellowof theRoyalSocietywhileIwasdoingmyPh.D.Andthenhebecameaprofessor.By thatstageitbecameabitmoreobviousthathewasareallypowerfulmathematician. It’sthebreadthofhismathematicsthatissoimpressive.Notjustgrouptheory,but cellularautomataandphysicsaswell.Hewasinterestedineverything.Andheread anenormousamountofstuff.Oneofthewaysheusedtowork,hewouldgoto thelocalcoffeeshop,inCambridgeandhewouldgothereeverymorningforhis breakfast,andworkinthesenoisysurroundings.Anddependingonwhathewas workingon,hewouldbeusingnapkins,pieceofpaper,orreadingbooks.Andthese bookswouldbeaboutthehistoryofimportantproblemsinmathematics,orphysics, orsomethingelse.Sohewaslearningaroundthesubjectsallthetime.Verywide reading,soheknewanenormousamountaboutanenormousamountofdifferent things.Andifyoucaughthimintherightmoodwhichwasnotdifficultactuallyhe wouldtellyouaboutanyofthesethings.Ifyoushowedinterestaboutanyofthese things,hewouldtellyouaboutit.AndIlearnedalotthatway,justfromlisteningto himtalk,inthecommonroom,inthebar,intheevening,hehadanenormousnumber ofstories.Mostlymathematical,notallofit.Andhetaughtmathematicsnotina lectureroombutinthepub.Icouldn’tkeepupbecause,wewouldgotothepubafter theseminar,drinkbeer.Conwaydrankalotofstrongbeer.Aftertwobeers,Icouldn’t keepupanymore.Itwasimpossible.Butstill,Ilearntahugeamountofbackground materialthatwaythatIprobablywouldn’thavelearntotherwise.Conwayneversaid, goandreadthatbooktolearnit.Insteadhesaid,Iwilltellyouaboutit.Hereishow
youdoit.Andhesatdownforanhourandexplained.Itwouldtakeprobablyaweek toreadabook.Fromthatpointofview,hewasanidealsupervisor.
“Andifyoucaughthimintherightmoodwhichwasnotdifficultactuallyhewouldtellyou aboutanyofthesethings.Ifyoushowedinterestaboutanyofthesethings,hewouldtellyou aboutit.AndIlearnedalotthatway,justfromlisteningtohimtalk,inthecommonroom, inthebar,intheevening,hehadanenormousnumberofstories. Conwayneversaid,go andreadthatbooktolearnit.Insteadhesaid,Iwilltellyouaboutit.Hereishowyoudoit.”
KamalikaBhattacharjee:Yousaidthathegaveyouonepagecontentofthe wholecourseinyourgraduationyear.Hehadthisfascinationaboutmaking thingsveryprecise.
–Exactly.Hewasneversatisfiedwithanyversionofanythinguntilhehadmade itassimpleaspossible.Ithadtobeassimpleaspossible,exactlywhatEinstein said.Itmustbeassimpleaspossiblebutnotmoreso.ThatwasConway’sattitude ineverything.Sowhenhewasworkingonaresearchproblem,itwasverydifficult fortherestofustokeepupbecauseeverydayhewouldhaveanewsimplerversion thanhehadthedaybefore.Sowehadtostartalloveragainfromscratchbecause everythingcouldchange.Andintheendifyoucouldn’tfititononesheetofpaperit wasn’tworthdoing.Irememberoneparticularexample,Iwasquiteinvolvedinthis, oneofthesporadicgroups,theJankothirdgroup( J3 ).12 Itwasreallytheworkof RichardParker,whowasdoingsomecomputingworkwiththisparticulargroup.And hediscoveredanewrepresentation.Soinsteadofhaving18dimensionalcomplex representations,hefounda9dimensionalrepresentationoverthefieldoforder4.And soRichardParkerprovedthattheJankogroupisasubgroupoftheunitarygroup,13 indimension9.AssoonasConwaygotholdofthatfact,hestartedscribblingon bitsofpaper.NowIknowIcandoitwith9by9matrices.Iwanttofindthesenine byninematricesbyhand,onpiecesofpaperandtellyouwhattheyare.Andhedid thatandeverydayhemadeachangefromthedaybefore,itgotsimpler.Untilat theendoftheday,attheendofafewdaysofthishehadonepieceofpaperwith completelynewnotation.Conwayalwaysisinventingnewnotationsforeverything. Completelynewnotationwhichhadononepieceofpaper,theentireconstruction of J3 .Idon’tknowifheeveractuallypublishedthat.ItappearedintheatlasbutI don’tthinkhepublishedapaperonit.Butthattaughtmeawayofworkingthathas beenveryusefultomelateron.Forexample,ItoldyouaboutthisoctonionicLeech lattice.Ididthesamething.IsaidtheremustbeanoctonionicLeechlattice.Let’s tryandwriteitdown.Ittookmeoveramonthandseveralhundredspiecesofpaper,
12 Intheareaofmodernalgebraknownasgrouptheory,theJankogroup J3 ortheHigman–Janko–McKaygroup HJM isasporadicsimplegroupoforder27 · 35 · 5 · 17 · 19 = 50232960. J3 isone ofthe26SporadicgroupsandwaspredictedbyZvonimirJankoin1969asoneoftwonewsimple groupshaving21+4 : A 5 asacentralizerofaninvolution(theotheristheJankogroup J2 ). J3 was showntoexistbyGrahamHigmanandJohnMcKay(1969).
13 Inmathematics,theunitarygroupofdegree n ,denoted U (n ),isthegroupof n × n unitary matrices,withthegroupoperationofmatrixmultiplication.Theunitarygroupisasubgroupofthe generallineargroup GL (n , C).Hyperorthogonalgroupisanarchaicnamefortheunitarygroup, especiallyoverfinitefields.
JohnHortonConway:AMasterofAllTrades9 buteachtimeIgotaversionwhichwassimplerthanthepreviousversion.Andthen Iwentbacktothebeginningandsaidthatthisisn’tsimpleenough,let’ssimplify itsomemore.AndIkeptgoingaroundthatloopasConwaytoldandeventuallyit wassimpleenoughthatIcouldnotsimplifyitanymore.Itwasobviouslyatthat stageofEinstein’spoint“ofbeingassimpleaspossiblebutnotsimpler”.Thesetwo waysofworking,firstofallofhavingacompleterangeofproblemsfromtrivial toimpossible.Andsecondly,simplify,simplify,simplify.Nevermakethingsmore complicated,alwayssimplify.I’vebeentrying,sinceIretired,I’vebeentryingtodo thisforthegrouptheorythatexistsinphysics.Ithinkthismayberelatedtocellular automataatsomepoint.BecauseagainIwentbacktoConway’smethods,Iwent backtoEinstein’sthoughtsaswell.AndoneofthingsEinsteinsaidaboutphysics was,quantummechanicsinparticular,thereshouldbeadiscretetheoryofquantum mechanics.Itshouldn’tbeacontinuoustheoryofLiegroups14 andsoon.Itshould bediscrete,itshouldbeliketheGameofLife.Itshouldbecompletelydiscrete andcompletelydeterministic,acellularautomatoninfact.Sothat’smyimpossible problemtocreateafiniteanddiscretetheoryofquantummechanics.ThoughEinstein wasn’tabletocreateit,soperhapsitdoesn’texist.IfEinsteincouldn’tdoit,that’s clearlyanimpossibleproblem,wouldn’tyousay?ButjustbecauseEinsteincouldn’t doit,I’mnotgoingtogiveup.Iknowit’sanimpossibleproblem.AndIthinkI havebeenpreparingforitfor10years.I’vebeenworkingonitspecifically.Ihaven’t succeeded,butIhavegoneroundConway’ssimplificationloopabout20timesnow. Andit’sgettingsimplerandsimplereverytime.Notonlythat,eachsimplification loophaswholeloadsofsimplificationstepsonit.Soit’sgettingsimplerandsimpler. Beforetoolong,Imaybeabletowritedownacellularautomatonthatdoesquantum mechanicsinthewaythatEinsteinwouldhavewantedit.Wellnoonewillbelieve meuntilI’vefinishedit,thewholethingisdone.
“Hewasneversatisfiedwithanyversionofanythinguntilhehadmadeitassimpleas possible.Ithadtobeassimpleaspossible,exactlywhatEinsteinsaid.Itmustbeassimple aspossiblebutnotmoreso.ThatwasConway’sattitudeineverything.Sowhenhewas workingonaresearchproblem,itwasverydifficultfortherestofustokeepupbecause everydayhewouldhaveanewsimplerversionthanhehadthedaybefore.”
It’snogoodgoingroundthissimplificationloops19timesifitneeds20.Andit’s animpossibleproblemsoit’sstilloneofmy6problemsandIdivideddowninto smallerproblems.Iamlookingatsomeeasyproblemsandsomehardproblemsand soon,andhopingthattheywillcometogetherattheendofthedaytomakethis newtheoryofquantummechanics.Well,itmightnotwork.But,itmightwork.I thinkitmighthavebeenanothersayingofConway’sormightcomefromsomewhere elsewhichRichardParkersaidtome,neverdiscountthepossibilityofsuccess.In
14 Inmathematics,aLiegroupisagroupthatisalsoadifferentiablemanifold.Amanifoldisa spacethatlocallyresemblesEuclideanspace,whereasgroupsdefinetheabstract,genericconcept ofmultiplicationandthetakingofinverses(division).Combiningthesetwoideas,oneobtainsa continuousgroupwherepointscanbemultipliedtogether,andtheirinversecanbetaken.If,in addition,themultiplicationandtakingofinversesaredefinedtobesmooth(differentiable),one obtainsaLiegroup.
anotherwords,eventhoughyourproblemappearstobeimpossible,don’tdismiss thepossibilitythatyoumightactuallybeabletosolveit.AnywayI’mtalkingtoo muchaboutmyownworknow.
SukantaDas:AsIhaveunderstood,whenyouwereatCambridge,thenyou wereontouchwithConway.Andduringthattime,Conwaywasdevelopingthe GameofLifewhichisverybeautifulbutverysimple.Sotoreachtothissimple ruleforGameofLife,didheuseanykindofcomputersimulation?
Nono,hedidallofitonaGoboard.Andheexperimentedwithlotsofdifferent versionsoftherule.Beforehecameupwiththeonethatactuallyworksintheend hetriedallsortsofpossibilities.
AndIdidn’tseehimdothisbutpeopleIknowdidseehimdoitandhewouldspend somedaysononeparticularversionusinglotsofGostonestoseewhathappensand whennothinginterestinghappenshetriedtoanalyzewhyandmodifytherulestosee andtrysomethingelseandhetriedthesepossibilitiesuntileventually,suddenlythe rightonewasthere.Obviouslythisistherightonebecausehetriedallthepossibilities andit’stheonlypossibility.It’stheonlyonethatdoesanythinginteresting.Inthe physicalworld,inthe3dimensionalversionoftheGameofLife,andinquantum mechanics,itisalittlebitmorecomplicatedthansayingthisisjustaliveordead (states).Itisnotmuchmorecomplicatedthanthat,itislittlebitmorecomplicated. Irecognizedthateachindividualcellhas48states.I’dliketosimplifyitbelow48. Wellactuallyit’sreallyjustacube.Justimaginedividingthewholespaceintoa cubiclattice.Nowlookateachindividualcube.What’sthesymmetrygroup15 ofthe cube.Wellweshouldonlybeabletorotateit,weshouldn’tbeabletoreflectit.In realspaceyoucan’treflect.Wellthat’sagroupoforder24.Nowyoujustneedto doubleittogetthespin.spinup—spindown.So48iswhatyouneed.Butyoudo need48.Mypreviousversionhad24.Itdidn’twork.Whydoesn’titwork?Because it’stoosimple.Isimplifiedittoomuch,doesn’twork.The48Ithinkdoes.Sonow you’vejustgottolookatthesecubes.ColourthefaceslikeaRubik’scube16 so youcantellwhichone’swhich,andspinthem.Sospinup—spindown,insideand outside,whateveryouwanttocallit.Anditseemstomethatissimplifiedasfaras itpossiblycouldbe.Butnotmore.Soifthereisacellularautomatonthatworksto describequantummechanics,thismustessentiallybeit.You’vegottodividespace intocubes.Havealookattheindividualcubeswiththese48differentstatesthat eachcubehas.So,maybenextweekI’llbeabletotellyouwhatthe48statesare andhowtodoitbutthisweekIcan’t.
15 Ingrouptheory,thesymmetrygroupofageometricobjectisthegroupofalltransformations underwhichtheobjectisinvariant,endowedwiththegroupoperationofcomposition.Sucha transformationisaninvertiblemappingoftheambientspacewhichtakestheobjecttoitself,and whichpreservesalltherelevantstructureoftheobject.Afrequentnotationforthesymmetrygroup ofanobject X is G = Sym ( X ).
16 TheRubik’sCubeisa3-Dcombinationpuzzleinventedin1974byHungariansculptorand professorofarchitectureErn˝oRubik.OriginallycalledtheMagicCube.
“hewouldspendsomedaysononeparticularversionusinglotsofGostonestoseewhat happensandwhennothinginterestinghappenshetriedtoanalyzewhyandmodifytherules toseeandtrysomethingelseandhetriedthesepossibilitiesuntileventually,suddenlythe rightonewasthere.Obviouslythisistherightonebecausehetriedallthepossibilitiesand it’stheonlypossibility.”
VilleSalo:SoI’mguessingthattheanswerisasimple‘No’.Anyway,cellular automataareofcoursethemodelsofphysicalsystemsandotherdynamicalsystems.Thesesystemshaveconnectionwithgrouptheory.SoIwouldbeinterested toknowwhetherConwayevertriedtomakeanykindofconnectionsbetween grouptheoryandcellularautomaton,giventhatheobviouslyknewboth.
Idon’tthinkthathedid.Imean,hedidstudyautomataofverydifferenttypes, notjustcellularautomata.ButhecreatedaTuringmachineoutofanythinghecould. HeproducedalistofelevenrationalnumberswhichwasaTuringmachine.Ithink healsomayhavedonethesamethingwiththeGameofLife.Butmaybesomeone elsedidit.
KamalikaBhattacharjee:YouweresayingthatwheneverConwaywasasked aboutcellularautomata,hediscouragedthatdiscussionsayingitashistory. Whyheusedtosaylikethat?
ThepointisthatConwaywasinterestedineverything.Butmostpeopleonlyknow abouttheGameofLife.SowhenarandompersonwantstotalktoConway,they wanttotalkaboutGameofLife.Andhejustgotboredofit.
SupreetiKamilya:IsthereanyimpactofLifeinpuremathematics?
That’sadifficultquestion.IthinktheGameofLifehashadsuchabigimpacton somanythingsthattheanswermustbeyes.Insomeway,itmaybehardtopoint toanyparticulardirectapplication.ButthewayIseeit,thewayofthinkingthat createdthisgamehasinfluencedsomanypeople.Andthefactthatit’s,wellthisis goingbacktophysicsagainwiththefactthatcellularautomatahaveinspiredpeople tocreatephysicaltheoriesbasedonit.Withinpuremathematics,itcertainlyhasan impactinlogic,inTuringmachinesandsoon.Inalgebra,probablynot.No,Ican’t thinkofanythingspecific.
SupreetiKamilya:DoestheGameofLifemotivateyousomewayinyour researchfield?
Itinspiredmeinphysicsverymuch.Physicsmustbediscrete.Itmustbeacellular automaton.Thequestioniswhichone,andhow.
KamalikaBhattacharjee:Hehastouchedsomanyfieldsandmaderemarkable contributioneverywhere.Accordingtoyou,whatisConway’sbiggestcontributioninmathematics,andinphysics?
Idon’tknowthatIcanactuallychooseonetobethebiggest.Becausehehadmajor impactsinsomanydifferentareasinmathematics.AndtomewhenIwashisPh.D. studentobviouslyitwastheConwaygroupandhisworkontheLeechlattice,thatto mewasmostimportant.Buttootherpeople,otherthingsweremoreimportant.And hisworkonknottheory17 forexamplewasveryinspirational.Numbersandgames, thefoundationofnumbers,howtocreatenumbersoutofnothing.Itwasdifferent fromwhatanyoneelsehasdone.AndI’mnotsurethatpeoplewilluseitbutit’s whatyoucandoifyouthinkreallydifferentlyfromanybodyelseaboutaproblem completelyfromscratchwithoutbeingboundbyanyoneelse’srules.Andthat’sthe sameineveryareathatheworkedon.SoIwasjustgoingtosayagainIthinkhis workisimportantinlotsofareas.Youcan’tsaywhichonesaremostimportant.It dependsonyourownpriorities,notonhim.
“Idon’tknowthatIcanactuallychooseonetobethebiggest.Becausehehadmajorimpacts insomanydifferentareasinmathematics.AndtomewhenIwashisPh.D.studentobviously itwastheConwaygroupandhisworkontheLeechlattice,thattomewasmostimportant. Buttootherpeople,otherthingsweremoreimportant.”
KamalikaBhattacharjee:Whathasbeenthegreatestimpactofhimonyou. Youhavebeentellingthathetaughtyouhowtosolveproblems,or,howto approachtosolveproblems.Whetherthatistheonlyimpacthehadonyou, orhetouchedyousomehowinpersonallevelonthesimplicityofhislifestyle, oronthewayheapproachedproblemsthathasimpactedyouinyourfuture researchwhichwascontinuedafterleavingCambridge?Youwereintouchwith himwhilehewasinCambridge.Afterthat,howthatinfluencecarriedonyou, canyoutellus?
Wellofcourse.IworkedcloselywithhimforthreeorfouryearsduringmyPh.D. andlessafterthat.AftermyPh.D.Iworkedmoreorlessonmyown.Butofcoursehe wasalwaysaroundandthewaythatheworkedwasveryclear.Toseehimworking andhowhewasdoingitandhowhewasthinking.Ikindofmodelledmyworkonthe waythathedidthings.Doeverythingforyourself.Don’ttakeanyoneelse’swordfor it.IfsomeonecomesalongandsaystheJankogroupisasubgroupof U9 (2),don’t askwhichwaytheydidit.Doityourself.Startagainfromscratch.Thiswashis method,startagainfromscratch.Andsothat’smymethod.Itdoesn’tworkaswellit didwithConwaybutthereisacertainamountthatyoucanlearnfromthewaythat otherpeoplehaveapproachedproblems.Butattheendofthedayyouhavetothink
17 Intopology,knottheoryisthestudyofmathematicalknots.Whileinspiredbyknotswhich appearindailylife,suchasthoseinshoelacesandrope,amathematicalknotdiffersinthattheends arejoinedtogethersothatitcannotbeundone,thesimplestknotbeingaring(or“unknot”).In mathematicallanguage,aknotisanembeddingofacirclein3-dimensionalEuclideanspace, R3 (in topology,acircleisn’tboundtotheclassicalgeometricconcept,buttoallofitshomeomorphisms).
foryourself.Andifthereisanunsolvedproblem,it’sbecausethewaythatother peoplethoughtaboutitisn’tgoodenoughtosolvethatproblem.Youhavetothink aboutitinadifferentwayinordertosolveit.Thatmeansyoushouldnottaketoo muchnoticeabouthowotherpeopleapproachedtheproblem.Yes,learnabitabout abouthowpeopleapproachedtheproblem.Butdon’tthinkhowcanIgofurtherwith thiswayofthinking,butwhatiswrongwiththatwayofthinking.HowcanIthink aboutthewholeprobleminabetterway.Whatassumptionstheyaremakingthat arenotnecessarytomake,ortheremightevenbewrongassumptions.Andrightat thebeginningstartthinkingabouttheassumptions.Aretheassumptionscorrect.If not,howcanwechangethem.ThesamewiththeGameofLife,westartwithsome assumptions.Youworkonit,nothinghappens.Yougobacktotheassumptions, assumesomethingelse,tryagain,thatdoesn’twork,gobacktotheassumptionsand changetheassumptions,startagain.
“Toseehimworkingandhowhewasdoingitandhowhewasthinking.Ikindofmodelled myworkonthewaythathedidthings.Doeverythingforyourself.Don’ttakeanyoneelse’s wordforit. Andifthereisanunsolvedproblem,it’sbecausethewayotherpeoplethought aboutitisn’tgoodenoughtosolvethatproblem.Youhavetothinkaboutitinadifferentway inordertosolveit.”
VilleSalo:Youmentionedthatyouhaveacellularautomatonthatyou’re cookingup.Isthisareversiblecellularautomatonorisitinthestandardcellular automatonmodelsunderlocalrules?Iknowyoudon’twanttotalktopeople, maybeit’snotfinishedbutI’mjustinterestedinwhetheryouaretalkingabout thesamemodelthatGameofLifefixedorit’sgeneralizedmodel.
Ihavenoidea.Idon’tknowanythingaboutcellularautomata.
WellasIsayI’velearnedfromConwaythatIshouldlearnasmuchofother people’sworkasIfeelisnecessarytolearn,whichisusuallymuchlessthanother peoplethinkit’snecessarytolearn.AndthenIstartthinkingfromscratch.Makemy ownassumptions.Tryitandseewhatworksandwhatdoesn’t.Thatwayyoureally understandwhycertainthingsthatwork.Thatsaid,youknowit’salessonoflife isn’tit?Youlearnsomuchbetterfromyourownmistakesthanyoudofromother people’smistakes.Youcanspendyourlifereadingaboutotherpeople’smistakesand tryingtolearnfromthem.Doesn’twork.Youmakeyourownmistakesyouknow exactlywhyitdoesn’twork.Youknowexactlyhowtoputitright.Youknowwhere themistakeis,wheretoconcentrateon.
Andthisparticularmodel,youcanreadasmuchasyoulikeonmyblog.Apaper Iamwritingthisweek,butitwon’tbetheretillnextweek.Somebodysaidtoone oftheirstudents,don’tworryaboutotherpeoplestealingyourideas.Ifyourideas areanygood,youhavetoramthemdownotherpeople’sthroats.Inotherwords,if yourideasreallysolvetheproblem,thenyoushouldnotbeinthemodethatother peoplewilldoit,otherpeoplewon’tbeinterestedinthem,unlessyouramthem downpeople’sthroats.
SukantaDas:Iwasreadingyourrecentblog.YoubeganonMarch15named amateurgrouptheoryanditwasinterestingandwhatyouhavementionedand Ihaveseeninarecentpaper.Youweresayingthatgrouptheoryisextremely powerfulandwecanunderstandmanythingsofthisuniverseofphysicswith grouptheory.Somysimplequerytoyouasanexpertofmathematicsand physics.Whatweunderstandaboutgrouptheoryisastudyofsymmetrical systems.Whatdoyouthink,thatthenatureanduniverseisverysymmetrical oneandcanweabletounderstandmanythingsusinggrouptheory?
Yes,yesIdo.Ialmosttoldyouwhatthegroupisbytakingasymmetricgroup ofacubeandtakingdoublecovertogetthespin.Wellthereareactually2double coverssoIhavetotellyouwhichone.Butifyouhavewatchedmyblogyoualready knowwhichistherightone.Sothatisthegroupoforder48thatIthinkisthekeyto thesymmetryoftheentireuniverse.SoI’mjusttryingtodevelopthemathematical modelstomakeitwork,andespeciallycomparingthisparticulargrouptowhat’sin thestandardmodel.Sothat’swhat’sinthepaperIamwriting,Iamtryingtodo. Thisgroupoforder48,howdoesitcomparetothegroupoforder64,thegroup generatedbytheDiracmatricesorgammamatrices.18 That’sagroupoforder64 whichunderliesthewholeofthestandardmodelofparticlephysics.Itdoesn’tdo everythingbutitdoesalot.SoIwanttocomparemygroupoforder48withDirac’s groupoforder64,wheretheydiffer,whytheydiffer,whichoneisright.Welllast nightIstartedthinkingaboutthis,howdoImatchmygroupoforder48tothe Diracgroup.AndbythetimeIwokeupthismorning,Icouldseetheanswer,I couldseewhat’swrongwiththeDiracgroupandhowthisgroup(mygroupoforder 48)correctsit.It’sjustpuremathematics,itisjustgrouptheory,justlookingatthe relationsbetweenthegammamatrices.Wehavetosquare,someofthemsquareto plusone,someofthemsquaretominusone,theyhavetoanti-commutewitheach other.Inonerelationwhichiswrong,youhavetoswaptwoofthegammamatrices inoneoftherelations.Andthenitworks,thenyouget3generationsofelectrons. Thenyougetthesymmetrythatactuallyexistsintherealworld.
SukantaDas:Isthenatureverysymmetrical?Inthenature,weseesomany asymmetries.So,canwemodelallthesethingstothesymmetry?
Ibelieveso.I’mstartingatthebottomandI’mtryingtoworkmywayup.
SupreetiKamilya:Youhaveworkedwithmonstergroupsthatdealswithso manydata.Socanyourelatemonstergroupwithcellularautomatasomehow?
18 Inmathematicalphysics,thegammamatrices,{ γ 0 ,γ 1 ,γ 2 ,γ 3 },alsoknownastheDirac matrices,areasetofconventionalmatriceswithspecificanticommutationrelationsthatensure theygenerateamatrixrepresentationoftheCliffordalgebra Cl 1,3 ( R ).Itisalsopossibletodefine higher-dimensionalgammamatrices.
Idon’tthinkso.Themonster19 isofcourseanextremelyinterestinggroupandhas connectionstoallsortsofpartsofmathematics.Somepeoplesayithasconnections tophysics.Idon’tthinkithasanyconnectionstophysics.Butoneoftheimportant thingsthatConwaydidofcoursewastheworkonmonstrousmoonshine(moonshine theory)20 whichwashappeningatthetimewhenIwasaPh.D.student,andwhich wasconnectingthemonstergroupanditscharactertable,anditsrepresentation theorywithsomeanalyticnumbertheory,whichisacompletelydifferentsubject andshouldn’thavebeenconnectedatall.Andyet,theyareconnected.Sothisisan extremelyimportantconnectionbetweentwodifferentpartsofmathematics.
Whatitmeansforphysics,Ithinkitmeansnothingtophysics.Buttherearestill peoplewhothinkitdoeshavesomeimpactonphysics.AndIthinkit’sthecellular automatathatdescribephysics,notthemonster.Andthatthereisnoconnection betweenthemonsterandcellularautomata.Andthereforethereisnoconnection betweenthemonstersandphysics.That’smyopinion.
SouvikRoy:SomyquestiontoProfWilson,bothasamathematicianand amusicianalso.So,Iamspecificallyaresearcherofcellularautomata,I’m verymuchinterestedinthebeautyofcellularautomataandpatterngenerated byit,asaartform.Also,manypeopleworkonmusicgenerationbycellular automata.So,otherfieldsofmathematicshavealsobeenusedinmusicand otherartformation.And,alsoatopicofresearch,algorithmicarthasbeen developedinlastdecade.So,asamathematicianandmusician,Iwanttoknow youropinionontheimpactofmathematicsinmusicandart.
Well,whenIwasyoungerIusedtolookatsomeofthesethingsandtryanduse mathematics,toanalyzemusicortowritemusic,somethinglikethat.Ithinkthat’s thewrongwaytolookatit.Now,Ifindthat,asawesternclassicalmusician,my trainingisverymathematical.Trainingtoplaythenotes.Youhavetoplayonenote afteranother,youhaveplayitintherightorder,youhaveplayintherightspeed. Veryverymathematicaltechnique.
Butyoucompletelyforgetthatthereismusic.Oneofmyfriendswhoplaysthe doublebass,intheorchestra,saidtomeonedaywhenhestartedtoplayinajazz band.Ithinkbeingtrainedasaclassicalmusician,theotherpeopleofthebandsaid tohim,youplaythenotes,butyou’renotplayingthemusic.Ithinkthatartormusic, it’saboutthesoul,it’snotaboutmathematics.SoIplaymusicratherdifferentlyfrom howIdidwhenIwasyounger.Anditalsopartlyrelatestolearningthiskora,Ifound ateacherinBirmingham,anAfricanwhoplaysthiswholearntitwhenhewasa
19 Intheareaofabstractalgebraknownasgrouptheory,themonstergroupM(also knownastheFischer-Griessmonster,orthefriendlygiant)isthelargestsporadicsimplegroup,havingorder246 320 59 76 112 133 17 19 23 29 31 41 47 59 71 =808, 017, 424, 794, 512, 875, 886, 459, 904, 961, 710, 757, 005, 754, 368, 000, 000, 000 ≈ 8 × 1053
20 Inmathematics,monstrousmoonshine,ormoonshinetheory,istheunexpectedconnection betweenthemonstergroup M andmodularfunctions,inparticular,the j function.Thetermwas coinedbyJohnConwayandSimonP.Nortonin1979.
childandhetaughtmesomeofit.Andagain,I’mtryingtoanalyzeitmathematically anditdoesn’twork.Doesn’tworkmathematically.Ithastobeplayedasmusic.I thinkthat’swhyhe’sfrustratedwithmeasaplayerbecauseIwanthimtoteachme thenotes.Forhimit’snotaboutthenotes.It’saboutthemusic.Andinthattradition ofmusicactually,itdoesn’treallymatterwhatnotesyouplay.Apartfromafew basicrules.It’smoreaboutyourimaginationandwhatyoufeelandhowyouwant toexpressyourself.Wellthiskindofmathematicalmusicseemstobefashionable thesedays.ButIdon’tfindanymusicatall,justmathematics.That’smyview.
“Ithinkthatartormusic,it’saboutthesoul,it’snotaboutmathematics.”
MihirK.Chakraborty:IhadaqueryaboutProfConway’sphilosophicalattitudetowardsmathematics.Actuallyweallknowthathegavedifferentinterpretationtonumbers.Ingametheoryinterpretationsandsoon,that’sfantastic. AndProfWilson,whatdoyousayabouthisphilosophicalpositionaboutthe ontologyofmathematics.Becauseifnumbersareinterpreteddifferentlythen whatremainsthen?Washereally“realist”inmathematics?Iftherearedifferentinterpretationspossible,thenhowyouwillaccountforthat?Overall,what washisattitudetowardsthephilosophyofmathematics?
Iactuallydon’tknow.Hewasdoingmathematics,orratherhewasplayingmathematics.Itwasagametohim.
MihirK.Chakraborty:So,ifoneplaysmathematics,hedoesn’tlookfortruth.
Isupposethat’sright,yes.Itisaninterestinggame,ausefulgame,that’sthepoint.
MihirK.Chakraborty:Maybeuseful,maynotbeusefulsometimesbutit isagameafterall.Imeanifsomebodyacceptsthatitisagame.Iwouldlike toknowwhatwasProfConway’sattitudetowardsthat.ForinstanceGodel’s incompletenesstheorem.Didheevercommentonthisincompletenessorthings likethat.
Nottome.Idon’trememberhimevertalkingaboutthingslikethat.Idon’tthink hereallyfoundthemthatinteresting.Itwasn’ttherightsortofgameforhim.Itwas somekindofrestrictiononwhatsortofgamesyoucanplay.ButConwaywasjust interestedinplayingthegames.
MihirK.Chakraborty:Butyoumentionedatthebeginningofthelecturethat youattendedoneofhislogiccourses.Whatwashisattitudetowardslogicwhen hewasintroducinglogic.Doyouremember?
WellIwasveryyoungatthetime,afirstyearstudent. Imeanhetaughtasyllabus.Therewerethingsthatyouaresupposedtoteach andhetaughtwhathewassupposedtoteachandsohetaughttheformalsystemof
