Beitr a-dieresis ge zur Algebra und Geometrie \noindentContributionsB e i tot r Algebra $ \ddot and{ Geometrya} $ ge zur Algebra und Geometrie Volume 42 open parenthesis 2001 closing parenthesis comma No period .. 1 comma 1 hyphen 3 7 period \noindentBeitr a¨ Contributionsge zur Algebra und to Algebra Geometrie and Geometry PlatonicContributions .. Hypermaps to Algebra and Geometry Volume 42 ( 2001 ) , No . 1 , 1 - 3 7 . VolumeAntonio 42 J period ( 2001 .. Breda ) , Nod quoteright . \quad Azevedo1 , 1 to− the3 7power . of 1 .. Gareth A period Jones Departamento de Matem acute-aPlatonic tica comma .. Universidade Hypermaps de Aveiro \ centerline { P l a t o n i c \quad Hypermaps } 3800 Aveiro comma PortugalAntonio J . Breda d ’ Azevedo1 Gareth A . Jones Department of MathematicsDepartamento comma .. de University Matem of Southamptona´ tica , Universidade de Aveiro \ centerlineSouthampton{ Antonio SO 1 7 1 BJ J .comma\quad .. UnitedBreda3800 Kingdom d ’ Aveiro $ Azevedo , Portugal ˆ{ 1 }$ \quad Gareth A . Jones } Abstract period .. WeDepartment classify the regular of Mathematics hypermaps open , parenthesis University orientable of Southampton or non hyphen orientable closing parenthesis whose \ centerlinefull automorphism{Departamento group is isomorphicSouthampton de Matem to the $SO symmetry\acute 1 7 1{ BJ groupa} ,$ of t aUnited ic Platonic a , \ Kingdomquad solid periodUniversidade de Aveiro } ThereAbstract are 185 of . them commaWe classify of which the 93 regular are maps hypermaps period .. We ( also orientable classify the or nonregular - orientable hyper hyphen ) whose \ centerlinemaps withfull automorphism{3800 automorphism Aveiro group , group Portugal A sub is 5 isomorphic: ..} there are to19of the these symmetry comma all group non hyphen of a Platonic orientable solid comma . and 9 of themThere are maps are period 185 of .. them These , hypermaps of which are 93 constructed are maps . as combinatorial We also classify and topologi the regular hyphen hyper - \ centerline {Department of Mathematics , \quad University of Southampton } cal objmaps ects comma with automorphism many of them arising group asA coverings5 : there of Platonic are 19 so of lids these and , Kepler all non hyphen - orientable Poinsot , and 9 polyhedraof them open are parenthesis maps . viewed These as hypermaps hypermaps closing are parenthesis constructed comma as combinatorial or their associates and period topologi .. We - conclude by showing that \ centerlineany rotarycal obj P{Southampton latonic ects , hypermap many ofSO i them s 1 regular 7 arising1 BJ comma , as\quad so coverings thereUnited are of no Platonic chiral Kingdom Platonic so} lids hypermaps and Kepler period - Poinsot 1 periodpolyhedra .. Introduction ( viewed as hypermaps ) , or their associates . We conclude by showing that \ centerlineThe convexany rotary{ polyhedraAbstract P latonic in R. to\quad hypermapthe powerWe classify of i 3 s with regular the the most , so regular interesting there are hypermaps symmetry no chiral properties (Platonic orientable are hypermaps the orPlatonic non . − orientable ) whose } so1 lids . : ..Introduction these are the t etrahedron T comma the cube C comma the octahedron O comma the dodecahedron D comma and \ centerline { full automorphism group is isomorphic to the symmetry group of a Platonic solid . } theThe icosahedron convex polyhedra I comma described in R3 inwith Plato the quoteright most interesting s dialogue .. symmetry Timaeos open properties square bracket are the 25 closing Platonic square bracket period .. The rotation groupsso of lids T comma : these of are the t etrahedron T , the cube C , the octahedron O , the dodecahedron D , \ centerlineCand .. and the O commaicosahedron{There .. are and ofI 185 , D described and of I them are isomorphicin , Plato of which ’ to s dialogue the 93 alternating are mapsTimaeos and . symmetric\quad[ 25 ]We .groups also The A subclassify rotation 4 comma groups the S sub regular 4 hyper − } andof AT sub , of 5 C semicolon and these O , are and subgroups of D and of index I are 2 isomorphic in the i sometry to the groups alternating of these solids and comma symmetric i somorphic groups to \ centerline {maps with automorphism group $ A { 5 } : $ \quad there are 19 of these , all non − orientable , and 9 } SA sub4,S 4 comma4 S sub 4 times C sub 2 and A sub 5 times C sub 2 period Eachand PlatonicA5; these solid are P can subgroups be regarded of index as a map 2 in comma the i that sometry i s comma groups as a of graph these imbedded solids , in i asomorphic surface to \ centerlineopen parenthesis{ of them in this are case maps the sphere . \quad closingThese parenthesis hypermaps semicolon are .. constructed more generally as comma combinatorial P can be regarded and topologias a hypermap− } .. open parenthesis a hypergraph \ centerline { cal obj ects , many of them arising as coverings of Platonic so lids and Kepler − Poinsot } imbedded in a surface closing parenthesis commaS4,S ..4 and× C within2andA5 either× C2. category P i s a regular obj ect comma .. in Vince quoteright s sense hline \ centerline1 subEach The author{ Platonicpolyhedra is grateful solid ( Pto viewed canthe Research be as regarded hypermaps and Development as a map ) , or, Joint that their Research i s , associates as aCentre graph of imbedded the . \ Commissionquad We in a conclude of surface the by showing that } European( in this Communities case the sphere for financially ) ; more supporting generally this , research P can be period regarded as a hypermap ( a hypergraph \ centerline0imbedded 1 38 hyphen{any in 482 a rotary 1 surface slash 93 P ) dollar , latonic and2 period within hypermap 50 circlecopyrt-c either i s category regular 200 1Heldermann P , i s so a regularthere Verlag are obj no ect chiral , in Vince Platonic ’ s hypermaps . } sense \noindent 1 . \quad Introduction

\noindent The convex polyhedra in $ R ˆ{ 3 }$ with the most interesting symmetry properties are the Platonic

\noindent 1soThe author l i d s : is\ gratefulquad these to the Research are the and t Development etrahedron Joint T , Research the cube Centre Cof , the Commission octahedron of the O , the dodecahedron D , and theEuropean icosahedron Communities I , describedfor financially in supporting Plato this’ s research dialogue . \quad Timaeos [ 25 ] . \quad The rotation groups of T , of C \0138quad− 4821and/ O93$2 , .50\quadcirclecopyrtand of− c D2001 andHeldermann I are isomorphic Verlag to the alternating and symmetric groups $ A { 4 } , S { 4 }$

\noindent and $ A { 5 } ; $ these are subgroups of index 2 in the i sometry groups of these solids , i somorphic to

\ begin { a l i g n ∗} S { 4 } ,S { 4 }\times C { 2 } and A { 5 }\times C { 2 } . \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }Each Platonic solid P can be regarded as a map , that i s , as a graph imbedded in a surface

\noindent ( in this case the sphere ) ; \quad more generally , P can be regarded as a hypermap \quad ( a hypergraph imbedded in a surface ) , \quad and within either category P i s a regular obj ect , \quad in Vince ’ s sense

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ hspace ∗{\ f i l l } $ 1 { The }$ author is grateful to the Research and Development Joint Research Centre of the Commission of the

\noindent European Communities for financially supporting this research .

\noindent $ 0 1 38 − 482 1 / 93 \$ 2 . 50 circlecopyrt −c 200 1 $ Heldermann Verlag 2 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps \noindentopen square2 bracket\quad 27A closing . J . square Breda bracket d ’ Azevedothat the automorphism ; G . A . group Jones Aut : P\quad acts transitivelyPlatonic on Hypermaps the .. quotedblleft blades quotedblright .. out of2 which A .P J is . Breda d ’ Azevedo ; G . A . Jones : Platonic Hypermaps \noindentconstructed[ period 27 ] that the automorphism group Aut P acts transitively on the \quad ‘ ‘ blades ’ ’ \quad out of which P is constructed[ 27 ] that the . automorphism group Aut P acts transitively on the “ blades ” out of which P Ourisaim constructed in this paper . is to classify the regular Platonic hypermaps comma the regular hypermaps H .. open parenthesis orientable .. or .. non hyphen orientable closing parenthesis .. whose .. automorphism .. group .. Aut H .. is .. i somorphic Our aimOur in aim this in paper this paper is to is to classify classify theregular regular Platonic Platonic hypermaps hypermaps, the , the regular regular hypermaps hypermaps .. toH .. the ( orientable or non - orientable ) whose automorphism group Aut H is i H automorphism\quad ( orientable group .. G ..\quad = .. Autor P\ ..quad thicksimnon =− Sorientable sub 4 comma S ) sub\quad 4 timeswhose C sub\ 2quad .. or Aautomorphism sub 5 times C sub\quad 2 .. ofgroup some Platonic\quad solidAut H \quad i s \quad i somorphic \quad to \quad the somorphic to the automorphism group G = Aut P ∼= S ,S × C or A × C Pautomorphism semicolon .. for group \quad G \quad $ = $ \quad Aut P \quad $ \sim4 =4 S2 { 4 }5 ,S2 { 4 }\times C { 2 }$ of some Platonic solid P ; for t echnical reasons in dealing with A × C it is also useful for us \quadt echnicalor reasons$ A { in5 dealing}\times with A subC 5 times{ 2 C}$ sub\ 2quad it is alsoof usefulsome for Platonic us to include5 solid2 the P case ; G\quad = subf thicksim o r A sub 5 period to include the case G = A . The number of G - hypermaps ( regular hypermaps H tThe echnical .. number reasons .. of G hyphen in dealing∼ hypermaps5 with .. open $ Aparenthesis{ 5 }\ regulartimes .. hypermapsC { 2 .. H}$ .. with it is.. Aut also H .. useful thicksim for = .. Gus closing to include parenthesis the .. case G with Aut H ∼= G ) for each of these four groups G i s given in Table 1 , for$ = .. each{\ ..sim of these} A { 5 } . $ Thewhich\quad alsonumber indicates\quad howo f G many− hypermaps of these hypermaps\quad ( rH e g u are l a r orientable\quad hypermaps , are maps\quad ,H and\quad are with \quad Aut H \quad fourorientable groups .. maps G i s given ; inthe Table final .. column 1 comma shows .. which where also .. we indicates have how provided many of more these detailed hypermaps information H $ \aresim orientable= $ comma\quad ..G) are maps\quad commaf o r ..\ andquad areeach orientable\quad mapso f semicolon these .. the final column shows where we have fo urabout groups the individual\quad G i hypermaps s given in . Table \quad 1 , \quad which a l s o \quad indicates how many of these hypermaps H provided more detailedG information G - hypermaps about the individual orientable hypermaps maps period orientable maps areG .. orientable G hyphen hypermaps , \quad .. orientableare maps .. maps, \quad .. orientableand are maps orientable maps ; \quad the final column shows where we have providedTable ignored! more detailed information about the individual hypermaps . Table 1 period .. Number of G hyphen hypermapsTable ignored! \ centerlineIn the .. orientable{G \quad .. casesG − commahypermaps .. we ..\ findquad .. thato r i ..e n besides t a b l e ..\ thequad .. hypermapsmaps \quad .. oforientable genus .. 0 .. corresponding maps } to the Platonic solids themselves commaTable there 1 are . examples Number of generaof G - 1 hypermaps comma 3 comma 4 comma 5 comma 9 and 13 semicolon .. in the non hyphen orientableIn the cases orientable the possible cases genera , are we 1 comma find 4 comma that 5 comma besides 6 comma the 10 comma hypermaps 14 comma of 1 6 genus comma 20 0 comma 26 comma 30 comma 34 andcorresponding 38 period to the Platonic solids themselves , there are examples of genera 1 , 3 , 4 , 5 , 9 and \ vspaceThe13 first ;∗{3 step in ex the}\ incenterline the non argument - orientable{\ ..framebox[.5 open cases parenthesis the\ l possible i whichn e w i d can t h genera ]comma{\ t e x are ..t b in f 1{ principleTable , 4 , 5 , ignored! comma 6 , 10 be ,}}}\ 14 applied , 1vspace 6 to , any20∗{ , finite3 26 ex ,} group G30 closing , 34 parenthesisand 38 . i s to determine the algebraic G hyphen hypermaps H for each G semicolon as explained in open square bracket 4 closing square bracketThe first comma step the in i thesomor argument hyphen ( which can , in principle , be applied to any finite group phismG ) i classes s to determine of these correspond the algebraic bij ectively G - tohypermaps the orbits of H Aut for Geach on the G ; triples as explained r sub 0 comma in [ 4 r ] sub , the 1 comma i somor r sub 2 \ centerline { Table 1 . \quad Number o f G − hypermaps } of- involutions phism classes generating of these G comma correspond or equivalently bij ectively to the to normal the orbitssubgroups of Aut H in theG on free the product triples r0, r1, r2 Capitalof involutions Delta = C subgenerating 2 * C sub G 2 * , C or sub equivalently 2 with Capital to Delta the slash normal H = sub subgroups thicksim G H period in the .. Next free comma product the orders l sub i of the products \noindent In the \quad o r i e n t a b l e \quad c a s e s , \quad we \quad f i n d \quad that \quad b e s i d e s \quad the \quad hypermaps \quad o f genus \quad 0 \quad corresponding to r sub∆ j r= subC2 ∗ kC open2 ∗ C2 parenthesiswith ∆/ H open=∼ braceG . i comma Next j , comma the orders k closingli of brace theproducts = open bracerjr 0k( comma{ i , j , 1 k comma} = {0 2, 1 closing, 2}) brace closing parenthesis the Platonic solids themselves , there are examples of genera 1 , 3 , 4 , 5 , 9 and 13 ; \quad in the non − inin G G give give us theus the type type .... open( parenthesisl0, l1, l2) of l sub each 0 comma H , and l sub from 1 comma this we l sub can 2 closingcalculate parenthesis it s Euler .... characof each H - comma and from this we can calculateorientablet erist it s ic Euler ; cases H charac is orientable the hyphen possible i f and genera only if are H l 1 ies , in 4 the , 5 , “ 6 even , 10 subgroup , 14 , 1 ” 6 ,∆+ 20of , index 26 , 2 30 in , 34 and 38 . t∆ erist, and ic semicolon knowing H this is orientable we can i then f and calculate only if H l the ies in genus the .. of quotedblleft H . ( This even subgrouprather brief quotedblright outline ignores .. Capital Delta to the power of plus ofThe indexcertain first 2 in Capital step questions in Delta the involving comma argument and boundary\quad components( which can , since , \quad theyin donot principle exist for , our be chosen applied groups to any finite group Gknowing )G i . ) s The thisto determinewe final can stage then calculate thein the algebraic process the genus i sG of to− H useperiodhypermaps this .. openinformation H parenthesis for each to This construct G rather; as briefexplained the G outline - hypermaps ignores in [ 4certain H ] , the i somor − phismquestionsas combinatorial classes involving of boundary theseor topological correspond components obj comma ects bij . ectivelysince We they shall do to donot the this exist orbits using for our several chosen of Aut groupst echniquesG on G the period , triples including closing parenthesis $ r { 0 } , r { 1 } ,Thethe r final{ 2 operations stage}$ in the process on i s hypermaps to use this information of Mach to construct`ı [21] theand G hyphen James hypermaps [ 16 H ] , Walsh ’ s ascorrespondence combinatorial or topological [ 28 ] obj between ects period hypermaps .. We shall and dobipartite this using several maps t , echniques and the comma double including coverings of \noindentthehypermaps .. operationsof involutions introduced .. on .. hypermaps in generating [ 4 ] . .. of These Mach G , grave-dotlessi ideas or equivalently are explained open square to in general bracket the normal 2 in 1§ closing2 subgroups, and square are described bracket H in .. the and free .. James product .. open square bracket$ in\Delta greater16 closing detail= square C in bracket{ [ 42 ]} . comma ∗ ..C Walsh{ 2 quoteright} ∗ C s ..{ correspondence2 }$ with .. open $ \Delta square bracket/ $ 28 H closing $ = square{\sim bracket}$ G . \quad Next , the orders $ lbetween{ Ini § hypermaps}3$ we of introduce the and products bipartite the Platonic maps $ comma r so{ j lidsand} the Pr as double{ hypermapsk } coverings( \{ ,of and hypermaps$ describe i , j introduced , their k $ automorphism\} = \{ 0 , 1 , 2 \} ingroups open) $ square and bracket rotation 4 closing groups square ; bracket we alsoperiod .. describe These ideas the are explained great dodecahedron in general in S 2 comma GD , and a are described in greater detail in openregular square map bracket of type 4 closing{ square5 , 5 } bracketwhich period forms the basis for several later constructions . Having \noindentIndescribed S 3 we introducein our G give methods the usPlatonic the in so type general lids P\ h as f ihypermaps l l in $§ (2 comma , l we{ and0 apply} describe, them their l { in automorphism1 }§§4, – 7 l in the{ 2 } cases) $ \ h f i l l of each H , and from this we can calculate it s Euler charac − groupsG ..= and∼ rotationS4,A5,A groups5 × semicolonC2 and .. weS4 ..× alsoC2 ..respectively describe the .. . great In .. dodecahedron each of these .. GD sections comma we .. a first regular map \noindentofenumerate type .. opent erist , brace then ic5 describe comma ; H is5 , closing and orientable finally brace .. which i f andforms only the basis iffor H lseveral ies later in the constructions\quad ‘‘ period even .. Having subgroup described ’’ \ ourquad $ \Delta ˆ{ + }$ ofmethods indexconstruct 2.. in in the general $ relevant\Delta .. in .. hypermaps S 2, comma $ and .. , withwe .. apply tables them summarising in .. S S 4 endash their basic 7 in the properties .. cases .. G. .. = Many sub thicksim of S sub 4 comma A sub 5 commaknowingthem A sub turn this 5 times out we toC can sub be then2 ( ..or and to calculate be closely the related genus to ) of such H familiar. \quad obj( This ects as rather the regular brief polyhedra outline ignores certain questionsSdescribed sub 4 times involving by C sub Coxeter 2 respectively boundary in [ 8 ] period , or components the .. In regular each of , maps these since sectionsdescribed they we do firstby not Coxeter enumerate exist and forcomma Moser our then chosen in describe [ 1 0 groups] commaand Gand . finally ) Theconstructclassified final the stage ( relevant for low in hypermaps genus the process ) by comma Brahana i with s to tables[ 1 use ] , summarising Sherk this [ information 23 their] and basic Garbe properties to [ construct 1 2 ] period . .. the Many G − of hypermaps H asthem combinatorial turn out to be open or topological parenthesis or to obj be closely ects related. \quad to closingWe shall parenthesis do this such using familiar several obj ects as t the echniques regular polyhedra , including thedescribed\quad byo Coxeter p e r a t i o in n sopen\quad squareon bracket\quad 8 closinghypermaps square\ bracketquad commao f Mach or the $ regular\grave maps{\imath described} by[ Coxeter 2 1and Moser ] $ in\quad open squareand \quad James \quad [ 16 ] , \quad Walsh ’ s \quad correspondence \quad [ 28 ] bracketbetween 1 0 closing hypermaps square and bracket bipartite and maps , and the double coverings of hypermaps introduced inclassified [ 4 ] open . \quad parenthesisThese for ideas low genus are closing explained parenthesis in by general Brahana in open\S square2 , bracket and are 1 closing described square bracket in greater comma detail Sherk open in square [ 4 ] . bracket 23 closing square bracket and Garbe open square bracket 1 2 closing square bracket period In \S 3 we introduce the Platonic so lids P as hypermaps , and describe their automorphism groups \quad and rotation groups ; \quad we \quad a l s o \quad describe the \quad grea t \quad dodecahedron \quad GD , \quad a regular map o f type \quad \{ 5 , 5 \}\quad which forms the basis for several later constructions . \quad Having described our methods \quad in g e n e r a l \quad in \quad \S 2 , \quad we \quad apply them in \quad \S \S 4 −− 7 in the \quad c a s e s \quad G \quad $ = {\sim } S { 4 } ,A { 5 } ,A { 5 }\times C { 2 }$ \quad and $ S { 4 }\times C { 2 }$ respectively . \quad In each of these sections we first enumerate , then describe , and finally

\noindent construct the relevant hypermaps , with tables summarising their basic properties . \quad Many o f them turn out to be ( or to be closely related to ) such familiar obj ects as the regular polyhedra described by Coxeter in [ 8 ] , or the regular maps described by Coxeter and Moser in [ 1 0 ] and classified ( for low genus ) by Brahana [ 1 ] , Sherk [ 23 ] and Garbe [ 1 2 ] . A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 3 \ hspaceOur methods∗{\ f i l of l } enumerationA . J . Breda are by d inspection ’ Azevedo comma ; Gdepending . A . heavily Jones on : specific\quad propertiesPlatonic Hypermaps \quad 3 of Capital Delta open parenthesis generation by involutions closing parenthesis .. and G .. open parenthesis faithful permutation representation Our methods of enumerationA . are J . Bredaby inspection d ’ Azevedo , depending ; G . A . Jones heavily : Platonic on specific Hypermaps properties 3 of low degreeOur commamethods of enumeration are by inspection , depending heavily on specific properties of o fdirect $ \ productDelta decomposition( $ generation comma etc by period involutions closing parenthesis ) \quad periodand .. G However\quad comma( faithful there are permutation situations where representation one needs to vary of low degree , direct∆ ( generation product decomposition by involutions ) , etc and . G ) . (\quad faithfulHowever permutation , there representation are situations of low where degree one , needs to vary Capitaldirect Delta product comma decomposition as in S 9 comma , etc for . instance) . However comma or , thereG comma are as situations in .. open where square one bracket needs 1 8 to comma vary .. 19 closing square bracket where$ \ GDelta i s a Ree, group $ asor a in Suzuki\S 9 group , for period instance .. In , orG , as in \quad [ 1 8 , \quad 19 ] where G i s a Ree group or a Suzuki group . \quad In such∆, as cases in §9 , , for direct instance methods , or G may , as not in be [ 1 feasible 8 , 19 ] where, so in G i\S s a8 Ree we group introduce or a Suzuki a much group more . general suchIn cases such comma cases ,direct direct methods methods may may not be not feasible be feasible comma so , so in inS 8§ we8 we introduce introduce a much a much more general more general enumerativeenumerative method method due due to P to period P . Hall Hall open [ square 1 3 ] bracket , which 1 3 closing in particular square bracket provides comma which a useful in particular numerical provides check a useful numerical forenumerative our direct method calculations due to P . . Hall [ 1 3 ] , which in particular provides a useful numerical check checkfor our direct calculations . for our direct calculations period We concludeWe conclude , in \ ,S in9§9 , , by by considering considering the therotary rotary Platonic Platonic hypermaps hypermaps. An . \ orientablequad An orientable hyper - hyper − Wemap conclude H without comma boundary in S 9 comma i s rotary by consideringi f it s rotationthe rotary group Platonic ( orientation hypermaps period - preserving .. An orientable automorphism hyper hyphen mapmap H H without without boundary boundary i s rotary i s rotary i f it s rotation i f it group s rotation open parenthesis group orientation ( orientation hyphen preserving− preserving automorphism automorphism group ) Aut+ H acts transitively on the “ brins ” of H . ( Traditionally , such hypermaps grouphave closing often parenthesis been called Autregular to the power[ 6 , of plus 7 ] H , butacts transitivelywe have used on the Wilson .. quotedblleft ’ s t erm brinsrotary quotedblright[ 30 ] .. to of H period .. open parenthesis Traditionally\noindent commagroup .. $ such ) hypermaps Autˆ{ have+ }$ H acts transitively on the \quad ‘ ‘ b r i n s ’ ’ \quad o f H . \quad ( Traditionally , \quad such hypermaps have oftenavoid been confu called - sion regular with Vince [ 6 ’ ,s concept\quad 7 of ] regularity , but we have [ 27 used ] used Wilson here ’ ; s t s ermimilarly rotary , our [ 30 ] \quad to avoid confu − oftenregular been hypermaps called regular are open t ermed square bracketreflexible 6 commain ..[ 6 7 , closing 7 ] .square ) Amongbracket comma the orientable but we have hypermaps used Wilson quoteright s t erm rotary opension square with bracket Vince 30 ’closing s concept square bracket of regularity .. to avoid confu\quad hyphen[ 27 ] \quad used here ; \quad s imilarly , our regular hypermaps are t ermedwithout reflexible boundary , in \ “quad regular[ 6 ” , implies\quad 7 “ ] rotary . ) \quad ” , butAmong the converse the orientable i s false ; hypermaps nevertheless without boundary , \quad ‘‘ regular ’’ sionwe with will Vince show quoteright , using results s concept from of regularity§8 , .. open square bracket 27 closing square bracket .. used here semicolon .. s imilarly comma our regulari m p l i hypermaps e s \quad are‘‘ rotary ’’ , but the converse i s false ; nevertheless we will show , using results from \S 8 , that if H i s rotary with Aut+ H = Aut+ P for some Platonic solid P , then H is regular and Aut t ermed reflexible in .. open square bracket∼ 6 comma .. 7 closing square bracket period closing parenthesis .. Among the orientable hypermaps H = Aut P ( so H i s one of the regular Platonic hypermaps classified in §§4 , 6 or 7 ) . without\noindent boundary∼ that comma if H .. i quotedblleft s rotary regular with quotedblright $ Aut ˆ{ + }$ H $ = {\sim } Aut ˆ{ + }$ P for some Platonic solid P , then H is regular and Aut2 H . $ = Classifying{\sim }$ regular Aut P hypermaps ( so H i s one of the regular Platonic hypermaps classified in \S \S 4 , 6 or 7 ) . impliesFirst .. , we quotedblleft briefly review rotary quotedblright the theory ofcomma regular but thehypermaps converse i ; s false see semicolon [ 4 ] nevertheless or [ 1 we 5 will ] show for a comma using results from S 8 commamore general account of hypermaps , and [ 6 ] for the orientable case . \noindentthat if H i2 s rotary . \quad with AutClassifying to the power regular of plus H =hypermaps sub thicksim Aut to the power of plus P for some Platonic solid P comma then H is regular If G is any group then a regular hypermap H with Aut H ∼= G is called a regular G- and hypermap . As explained in §2 of [ 4 ] , H corresponds to a generating \noindentAut H = subFirst thicksim , we Aut briefly P open parenthesis review the so H theory i s one of of the regular Platonic hypermaps hypermaps ; \ classifiedquad see in S\quad S 4 comma[ 4 6 ] or\ 7quad closingor parenthesis\quad [ 1 5 ] \quad for a more general accounttriple ofr0 hypermaps, r1, r2 of G , and [ 6 ] for the orientable case . period 2 satisfying ri = 1( i = 0, 1, 2), or equivalently to an epimorphism θ : ∆ → G ,Ri 7→ ri, where ∆ is 2the period free .. productClassifying regular hypermaps IfFirst G is comma any group we briefly then review a regular the theory hypermap of regular hypermaps H with Aut semicolon H \quad .. see$ ..\ opensim square= $ bracket\quad 4 closingG i s square c a l l e d bracket a \quad .. orr .. e g open u l a r G − squarehypermap bracket . 1\ 5quad closingAs square\quad bracketexplained .. for a more\quad generalin \quad \S 2 \quad o f \quad [ 4 ] , \quad H \quad corresponds \quad to \quad a generating triple \quad ∆ = hR ,R ,R | R2 = 1i = C ∗ C ∗ C ; $ raccount{ 0 of} hypermaps, r { comma1 } and, open r square{ 02 } bracket1$ 2 \quad 6i closingo f G∼ square2 bracket2 2 for the orientable case period Ifwe G is call any the group triple thenr a regular= (r0, r hypermap1, r2) a ∆ H− withbasis Autfor H G.. thicksim . The = set .. GΩ isof called blades a .. of regular H can G be hyphen identified \noindenthypermapwith G period,satisfying so that .. As G .. permutes explained $ r ˆ{ .. the2 in} .. blades{ S 2i ..} of by ..= right open 1 square - multiplication ($ bracket i $= 4 closing ; the 0 square i - , faces bracket 1 of comma H, ( i 2= .. H0 ), ..1, 2) corresponds, ,$ orequivalentlytoanepimorphism .. to .. a generating triple$ \thetathat .. r sub is , 0 the: comma hypervertices\Delta r sub 1 comma\rightarrow , hyperedges r sub 2 ..$ of and G G hyperfaces $ , R { ofi H}\ , aremapsto identifiedr with{ thei } cosets, $ gD where ( $ \Delta $ is the free product satisfyingg ∈ G )r subof the i to dihedral the power subgroups of 2 = 1 open D parenthesis= hr1, r2i, ihr =2, 0 r0 commai, hr0, r1 1i commaof G , 2 with closing incidence parenthesis given comma by or non equivalently to an epimorphism theta- : empty Capital intersection Delta right arrow . G The comma edge R sub- lab i arrowright-mapsto elled trivalent graph r sub i G comma associated where with H is the Cayley \ [ Capital\graphDelta Delta for= G is with the\ langle free respect product toR the{ 0∆}− basis,Rr . { 1 } ,R { 2 }\mid R ˆ{ 2 } { i } = 1 \rangle = {\sim } C Capital{ 2The} Delta automorphisms∗ =C angbracketleft{ 2 } of ∗ R H subC ( or 0{ commaequivalently2 } R; sub\ ] 1 of comma the edge R sub - lab 2 bar elled R sub graph i to the G power ) are inducedof 2 = 1 right by the angbracket = sub thicksim C sub 2 * Cleft sub - 2 translations * C sub 2 semicolon g 7→ x−1 g where x ∈ G , so Aut H ∼= Aut G ∼= G . Two regular G - wehypermaps call the triple H r and = openH0 parenthesisare isomorphic r sub 0 if comma and only r sub if 1 their comma corresponding r sub 2 closing edge parenthesis - lab elled a Capital graphs Delta G hyphen basis for G period .... The\noindent setand CapitalG0 arewe Omega call i somorphic of the blades triple , of that H can ri s be , $= identified i f and ( only r { i f0 their} ,∆− bases r { r1 }and,r0 are r equivalent{ 2 } ) under $ a $ \Delta − $ basis for G . \ h f i l l The s e t $ \withOmegaAut G G comma$ . of One so blades that can G permutes therefore of H can the classify be blades identified by the right regular hyphen G multiplication - hypermaps semicolon by finding the thei hyphen orbits faces of of Aut H open G parenthesis i = 0 comma 1 commaon 2 the closing∆− parenthesisbases of commaG . \noindent with G , so that G permutes the blades by right − multiplication ; the i − faces ofH( i $= 0 that is commaMach the hypervertices`ı ’ s group comma S =∼ S hyperedges3 of hypermap and hyperfaces operations of H [ comma 4 , 2 1 are ] transforms identified with one the regular cosets gD G - ,openhypermap 1 parenthesis , 2 g ) in G , closing $ parenthesis of the dihedral subgroups D = angbracketleft r sub 1 comma r sub 2 right angbracket comma angbracketleftthatH is to , r the sub another hypervertices2 comma ( r called sub 0 right , hyperedgesan angbracketass ociate comma and hyperfacesH angbracketleftπ of H ) of r sub by H 0 , comma are renaming identified r sub 1 right hypervertices withangbracket the of cosets G comma gD with incidence ( g $ \ in $ G ) of the dihedral subgroups D $ = \ langle r { 1 } , r { 2 }\rangle , \ langle given, by hyperedges and hyperfaces of H , that is , by applying a permutation π ∈ S3 to the edge - r { 2 } , r { 0 }\rangle , \ langle r { 0 } , r { 1 }\rangleπ $ of G , with incidence given by nonlabels hyphen i = empty 0, 1, 2 intersectionof G , or period equivalently .. The edge to the hyphen generators lab elled trivalentri of G graph . S G ince associated H and withH Hdiffer is the only Cayley nongraphin− theirempty for G lab with intersection elling respect , it to the i s Capital . sufficient\quad DeltaThe for hyphen us edge to basis− findlab r one period elled representative trivalent from graph each G S associated - orbit on the with H is the Cayley graphTheregular automorphisms for G G - with hypermaps of respect H open . parenthesis to the or$ equivalently\Delta − of$ the edgeb a s i hyphen s r . lab elled graph G closing parenthesis are induced by the left hyphen translations g arrowright-mapsto x to the power of minus 1 g where x in G comma so Aut H thicksim = Aut G thicksim = G period ..\ hspace Two regular∗{\ f G i l hyphen l }The hypermaps automorphisms of H ( or equivalently of the edge − lab elled graph G ) are induced by the H and H to the power of prime are isomorphic if and only if their corresponding edge hyphen lab elled graphs G and G to the power of prime are \noindenti somorphicl e comma f t − thattranslations i s comma .. g i f and $ \ onlymapsto i f theirx Capital ˆ{ − Delta1 hyphen}$ g bases where r and x r $ to\ thein power$ G of , prime so Aut are equivalent H $ \sim under= Aut $ G periodAut G .. One $ \sim = $ G . \quad Two regular G − hypermaps Hcan and therefore $ H ˆ classify{\prime the regular}$ are G hyphen isomorphic hypermaps if by and finding only the if orbits their of Aut corresponding G on the Capital edge Delta− hyphenlab bases elled graphs G and $ G ˆ{\prime }$ areof G period iMach somorphic grave-dotlessi , that quoteright i s , s\ groupquad Si = fsub and thicksim only S isub f 3 their of hypermap $ \Delta operations− open$ square bases bracket r and 4 comma $ r ˆ 2{\ 1 closingprime square}$ bracket are equivalent under Aut G . \quad One transformscan therefore one regular classify G hyphen the hypermap regular G − hypermaps by finding the orbits of Aut G on the $ \Delta − $ bases o fH G .. to . .. another .. open parenthesis called .. an .. ass ociate H to the power of pi .. of H closing parenthesis .. by .. renaming .. hypervertices comma .. hyperedges .. and \ hspacehyperfaces∗{\ f of i l H l } commaMach that $ \ isgrave comma{\imath by applying} $ a ’sgroupS permutation pi in $= S sub{\ 3 tosim the edge} S hyphen{ 3 labels}$ ofi = hypermap0 comma 1 comma operations 2 of [ 4 , 2 1 ] transforms one regular G − hypermap G comma or equivalently to the generators r sub i of G period .. S ince H and H to the power of pi differ only in their lab elling comma it \noindenti s sufficientH for\quad us toto find\ onequad representativeanother \ fromquad each( Sc a hyphen l l e d \ orbitquad onan the\ regularquad ass G hyphen ociate hypermaps $Hˆ period{\ pi }$ \quad o f H ) \quad by \quad renaming \quad hypervertices , \quad hyperedges \quad and hyperfaces of H , that is , by applying a permutation $ \ pi \ in S { 3 }$ to the edge − l a b e l s i $ = 0 , 1 , 2 $ o f G , or equivalently to the generators $ r { i }$ o f G . \quad S ince Hand $Hˆ{\ pi }$ differ only in their lab elling , it i s sufficient for us to find one representative from each S − orbit on the regular G − hypermaps . 4 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps \noindentBeing regular4 \ commaquad A .. H. ..J is . without Breda .. d boundary ’ Azevedo i f and ; G .. only . A .. . i fJones each r sub: \quad i negationslash-equalPlatonic Hypermaps 1 comma .. a condition .. always satisfied4 A in . Jour . Breda case since d ’ none Azevedo of our ; groups G . A .. . G Jones can be : generated Platonic by fewer Hypermaps that .. three .. in hyphen \ hspacevolutions∗{\ periodf i l l } ..Being Having regular empty boundary , \quad commaH \quad .. H ..i is s orientable without i\ fquad and onlyboundary if it s hypermap i f and .. subgroup\quad only \quad i f each $ r { i } \not= 1 ,Being $ \quad regulara , c o n H d i t i o is n without\quad always boundary i f and only i f each ri 6= 1, a Hcondition = ker theta i always s contained in the even subgroup Capital Delta to the power of plus = angbracketleft R sub j R sub k open parenthesis j equal-negationslash k closing parenthesis right angbracket .. of index 2 in Capital Delta semicolon .. this is \noindentsatisfiedsatisfied in our case in since our none case of since our groups none of G our can groups be generated\quad G by can fewer be that generated three by in fewer that \quad thre e \quad in − equivalent- volutions to .... . G having Having a subgroup empty .... boundary G to the power , H of plus is .... orientable of index 2 containingi f and only none if of it the s generators hypermap r sub i comma .... in v owhich l u t i o case n s G . to\quad the powerHaving of plus empty = sub boundary thicksim Capital , \quad DeltaH to\quad the poweris orientable of plus slash H i thicksim f and = only Aut toif the it power s hypermap of plus H\ periodquad ..subgroup subgroup H = ker θ i s contained in the even subgroup ∆+ = hR R ( j 6= k )i of index 2 in TheH type $ = of $ H is ker open $ parenthesis\theta $ l sub i 0 s comma contained l sub 1 comma in the l sub even 2 closing subgroup parenthesisj $k \Delta comma whereˆ{ + l} sub= i i s\ thelangle order of rR sub{ j rj sub} kR { k } ∆; this is ( $and j open $ brace\ne $ i comma k $ j comma) \rangle k closing$ brace\quad = openof brace index 0 comma 2 in 1 $comma\Delta 2 closing; brace $ \quad semicolont h i.. s thus i s each i hyphen face i s a 2 l equivalent to G having a subgroup G+ of index 2 containing none of the generators r , in sub i hyphen gon comma .. with sides alternately lab elled j .. and i which case G+ = ∆+/ H ∼= Aut+ H . The type of H is (l , l , l ), where l i s the order of r r \noindentk period ..equivalent There are∼ N sub to i\ =h bar f i l Gl G bar having slash 2 l a sub subgroup i i hyphen\ facesh f i semicolonl0 l 1 $2 G .. ˆ{ since+ } Gi$ has\ h .. f bar i l l Gof bar index ..j verticesk 2 containing and 3 bar G bar none slash of 2 the generators and { i , j , k } = {0, 1, 2}; thus each i - face i s a 2l − gon , with sides alternately lab elled edges$ r comma{ i } H has, $ Euler\ h f i l l in i j and k . There are N = | G | /2l i - faces ; since G has | G | vertices and 3 | G where to the power of characteristici eta = 1 dividedi by 2 chi or = 1 sum as H to the power of N sub i s to the power of i to the power of minus \noindent| /2 edgeswhich , H has case Euler $ G ˆ{ + } = {\sim }\Delta ˆ{ + } / $ H $ \sim = Aut ˆ{ + }$ H . \quad The type of H is orientable 1 divided by 2 to the power of bar G bar =|G 1| divided= |G byP 2 sub or to the power of bar G sub not to the power of bar sub period to the $ (where l characteristic{ 0 } η,= 1 χ l ={ 11P}N i,−orientable l { 21 } )1 | ,( $l − wherei1 − 1). $ l { i }$ i s the order of $r { j } r { k }$ power of open parenthesis2 toor the powerasH ofis sum to the2 power of2 or l minusnot. i to the powerThe of 1 minusgenus 1 of closing H parenthesis is then g period .... The genus of H .... isand then= \{η g(2 ....−i =χ ,) eta, j open , k parenthesis $ \} = 2 minus\{ chi0 closing , parenthesis 1 , comma 2 \} ; $ \quad thus each i − f a c e i s a $ 2 l { i } − $These gonThese arguments , \ argumentsquad enablewith us enable sides to give us topologicalalternately to give topological descriptions lab elled descriptionsof the j regular\quad Gofand hyphen the regular hypermaps G - hypermaps comma , and kand .the\ thequad final finalThere task task is is are to to construct construct $ N them{ i them} as combinatorial= as combinatorial\mid $ obj G ects $ objperiod\mid ects .. Here ./ we 2 Here will lrely we{ on willi the}$ rely i on− f the a c e s t ; \quad s i n c e G has \quad $ \tmid echniques$ G described described $ \mid in$ open in\ [quad square4 ] , constructingvertices bracket 4 closing and each square $ H 3 out bracket\mid of some comma$ G Platonic constructing $ \mid solid each/ P ( H with2 out $ Autof edges some P Platonic∼ ,Hhas= solid Euler P open parenthesis withG Aut ) P thicksim in such = a symmetric way that each automorphism of P induces an automorphism of H ; \noindentGthus closing Aut parenthesis$ H where≥ Aut .. ˆ{ inP suchcharacteristic∼= aG symmetric , and by way comparing}\ thateta each orders automorphism= \ f we r a c can{ 1 of prove P}{ induces2 equality}\ anchi automorphism . { or } = of{ H1 semicolon}\sum { as H }ˆ{ N }ˆ{ i } { i s }thusˆ{ −A Aut }particularly Ho greater r i e n t equala b useful l e Aut\ f t P r aechnique thicksim c { 1 }{ = , G2 described comma}ˆ{\mid and in by greaterG comparing\mid detail orders in= §§}\ we1 canf and r a prove c { 6 of1 equality}{ [ 42 ] ,} periodi{ s toor take}ˆ{\mid G }ˆ{\mid } { not }ˆ{ ( ˆ{\sum }} { . }ˆ{ l } −{ i }ˆ{ 1 } − Aa1 particularly regular ) G . useful -$ map\ th echnique fM i l lofThe type comma genus{ m described , nof} H, that\ inh greaterf i is l l , ai sdetail regular then in S g G S\ -1h hypermap and f i l l 6 of$ open = of square type\eta bracket (( n , 4 2 2 closing , m− ) square \ chi bracket) comma , $ i s to ˆ take . If M i s not bipartite then it does not cover the hypermap B0 ( defined in §2 of [ 4 ] ) which These arguments enable us to give topological descriptions of the regular G − hypermaps , ahas regular two G bladeshyphen map, transposed M of type by openR brace0 and m fixed comma by nR closing1 and braceR2; we comma can that therefore is comma form a regular the disj G hyphenoint hypermap of type .. open parenthesisandproduct the n final comma [ 4 , task 2 comma§ 3] isM mto0ˆ closing construct= M parenthesis×B them0ˆ, perioda as regular combinatorial.. If M map of type obj ects{m0, .n\}quadwhereHere wem0 will= rely on the ti echniques slcm not ( bipartite m , 2 described ) then . it This does in not is [ cover 4 a ] the bipartite , hypermap constructing double B to the covering each power H of ofout circumflex-zero M of , unbranched some Platonic open i parenthesis f m i solid s even defined P , ( and within S 2 ofAut open P square $ \sim bracket= 4 $ G) \quad in such a symmetric way that each automorphism of P induces an automorphism of H ; closingbranched square bracket at the closingN = parenthesis| G | /2 m which face has - centres two of M i f m is odd , so χ(M0ˆ) = 2χ( M thus Aut H $ \geq 2$ Aut P $ \sim = $ G , and by comparing orders we can prove equality . blades comma transposed by R sub 0 and fixed by R sub 1 and R sub 2 semicolon we can0ˆ therefore form the disj oint product ) or 2χ( M ) − N2 respectively ; if M is orientable then so is M ( but not conversely open square bracket 4 comma .. S0ˆ 3 closing square bracket M to the power of circumflex-zero = .. M .. times B to the power of zero-circumflex A particularly) . Being usefulbipartite t echnique, M i s the , Walsh described map inW (greater H ) [ 4 detail , §6 in ;\S 28\S ]1and6 of a hypermap of [ 4 ] , i s to take comma .. a regular−1 map0ˆ .. of type .. open brace m to the0ˆ power of prime comma n closing brace .. where .. m to the power of prime = .. lcm open parenthesisa rH e g u= l aW m r comma G(M− map) 2 closingon M theo f parenthesis type same\{ surfacem period , n as ..\} ThisM, ..: that isthe .. a is hyperfaces , a regular of H G correspond− hypermap to of the type faces\ ofquad ( n , 2 , m ) . \quad I f M 0ˆ ibipartite sM not, while bipartitedouble the covering hypervertices then of M it comma does and unbranched hyperedges not cover i fcorrespondm the i s hypermapeven comma to the and $ Btwo branched ˆ{\ monochromehat at{0 the}} N sub sets( 2$ = of bardefined vertices G bar slash in \S 2 m2 of [ 4 ] ) which has two ˆ bladesfaceof hyphenM0 ,, transposed centresone vertex of M byof .. i each f$ m R is colour odd{ 0 comma covering}$ ..and so each .. fixed chi vertex open by parenthesis of $R M .{ M1 to}$ the and power $ of R circumflex-zero{ 2 } ; $ closing we parenthesis can therefore = 2 chi form open the disj oint product [ 4 , \quad \S $ 3 ] M00 ˆ{\hat{000}}1 =0 $ 1 \quad M \quad $ \times B ˆ{\hat{0}} , $ \quad a regular map \quad o f type \quad parenthesisThus HM hasclosing type parenthesis ( n , n .., m or) 2, where chi openm parenthesis= 2 m = M2 closingm or m parenthesis as m i s minuseven or N sub odd 2 . .. respectively Moreover , semicolon H .. if M .. is $ \{ m ˆ{\prime } , $ n \}\quad where \quad $ m ˆ{\prime } = $ \quad lcm ( m , 2 ) . \quad This \quad i s \quad a orientablei s regular then , withso is M Aut to the H power=∼ Aut of circumflex-zero M =∼ G . open Many parenthesis regular G but - not hypermaps conversely arise closing in parenthesis this way . period .. Being bipartite comma Mbipartite to theWe power double will of zero-circumflex apply covering the i s of the M above Walsh , unbranched map process W open iparenthesis to f m our i s H even closing chosen , parenthesis and groups branched G at, the in each $ N { 2 } = \mid $ G $ \openmidcase square/ enumerating bracket 2 $ 4 m comma , describing .. S 6 semicolon and finally .. 28 closing constructing square bracket the regular .. of a hypermap G - hypermaps H = W to H the . power In the of minus 1 open parenthesis M f a c e − c e n t r e s o f M \quad i f m i s odd , \quad so \quad $ \ chi ( M ˆ{\hat{0}} ) = 2 \ chi ( $ to thecases power G of circumflex-zero= S4 and closingA5 it parenthesis i s convenient .. on the same to regard surface elementsas M to the of power G as of permutations zero-circumflex :in .. the the hyperfaces of H M)correspondnatural\quad or representation to the $ faces 2 of\ chi M of to degreethe( power $ M n of= $ circumflex-zero 4 )or 5− on {N1 comma,{ . .2 .} , while$ n }\quad. the hypervertices Inrespectively particular and we hyperedges will ; \quad represent correspondi f M \quad to thei s two orientable then so is $Mˆ{\hat{0}} ( $ but not conversely ) . \quad Being bipartite $ , Mˆ{\hat{0}}$ monochromeeach involution sets of verticesri in ofr M to the power of circumflex-zero comma .. one vertex of each colour covering each vertex of M period i s the Walsh mapW( H ) Thusas a H disj has ointtype openset of parenthesis edges n ( comma corresponding n comma to m to the the 2 power - cycles of prime of ri prime) in closing an n - parenthesis vertex graph commaG where: m to the power of prime prime[ 4this = , 1\ idividedquad s the\ permutationbyS 6 2 m ; to\quad the power graph28 of] for\ primequad G on =of 1{ divided1 a , hypermapH . . by . , 2 n m} or, mor $= as equivalently m i Wˆs even{ or− the odd1 Schreier period} ( .... coset MMoreover ˆ{\ graphhat comma{0}} H ) $ \quad on the same surface as $ Mi sfor ˆ regular{\ a hat comma{0}} with: Aut $ \ Hquad = subthe thicksim hyperfaces Aut M = sub of H thicksim G period .. Many regular G hyphen hypermaps arise in this way period correspond to the faces of $M ˆ{\hat{0}} , $ while the hypervertices and hyperedges correspond to the two Wepoint .. will - stabiliser .. apply .. theGα ..of above index .. n process in G , .. with to .. respectour .. chosen to a ..∆ groups− basis .. Gr commafor G ; .. thus in .. eachG is .. the case quotient .. enumerating comma monochrome sets of vertices of $ M ˆ{\hat{0}} , $ \quad one vertex of each colour covering each vertex of M . describingG /Gα of and the finally Cayley constructing graph G the of regular G corresponding G hyphen hypermaps to r , factored H period out .. In by the the cases action G .. = of S subGα on 4 and G . A3 sub . 5 .. Theit .. i sPlatonic convenient solids to regard as elements hypermaps of G as permutations in the natural representation \noindentofThe degree n PlatonicThusHhas = 4 or 5 on solids open type brace P (n , 1 comma or ,n more $period precisely, period mˆ{\ their periodprime comma 2 - skeletons\prime n closing} , brace) are period ,$ all .. where examplesIn particular $mˆ we of{\ willprime represent each\prime involution} = r\ subf r amaps c i{ .. in1 r}{ (2 and} hencem ˆ{\ hypermapsprime } = ) on\ surfacesf r a c { 1 homeomorphic}{ 2 }$ mormasmi to the sphere s . even or odd . \ h f i l l Moreover , H as a disjThe oint properties set of edges of ..the open t etrahedron parenthesis corresponding T , cube C , to o the ctahedron 2 hyphen O cycles , dodecahedron of r sub i closing D parenthesisand icosahe .. in an n hyphen vertex graph \noindent i s regular , with AutH $= {\sim }$ Aut M $ = {\sim }$ G . \quad Many regular G − hypermaps arise in this way . overbar- dron G : .. I this are well - known [ 8 , 1 7 , 22 ] , and are summarised in Table 2 , where N0,N1 and N2 idenote s the permutation graph for G on open brace 1 comma period period period comma n closing brace comma or equivalently the Schreier coset Wegraph\quad for a w i l l \quad apply \quad the \quad above \quad p r o c e s s \quad to \quad our \quad chosen \quad groups \quad G, \quad in \quad each \quad case \quad enumerating , describingpoint hyphen and stabiliser finally G sub constructing alpha of index n in the G comma regular with G respect− hypermaps to a Capital H Delta . \quad hyphenIn basis the r cases for G semicolon G \quad thus$ overbar = S G is{ the4 }$ quotientand $G A slash{ G5 sub}$ alpha\quad of thei t Cayley\quad graphi s G convenient of G corresponding to regard to r comma elements factored of out G by as the permutations action of G sub alpha in the on G natural period representation of3 period degreen .. The $=Platonic 4$ solids or5onas hypermaps\{ 1 , . . . , n \} . \quad In particular we will represent each involution $ rThe{ ..i Platonic}$ \quad .. solidsin P comma r .. or more precisely their .. 2 hyphen skeletons comma .. are all .. examples .. of maps .. open parenthesis and hence hypermaps closing parenthesis on surfaces homeomorphic to the sphere period \noindentThe propertiesas a of disj the t etrahedron oint set T of comma edges cube\quad C comma( corresponding o ctahedron O comma to the dodecahedron 2 − c y c lD e s and o f icosahe $ r hyphen{ i } ) $ \quad in an n − vertex graph $\ overlinedron I are{\}{ well hyphenG } known: $ open\quad squaret h bracket i s 8 comma 1 7 comma 22 closing square bracket comma and are summarised in Table 2 comma wherei s N the sub permutation 0 comma N sub graph 1 and N for sub G 2 denote on \{ 1 , . . . , n \} , or equivalently the Schreier coset graph for a \noindent point − stabiliser $ G {\alpha }$ of index n in G , with respect to a $ \Delta − $ basis r for G ; thus $\ overline {\}{ G }$ is the quotient G $ / G {\alpha }$ of the Cayley graph G of G corresponding to r , factored out by the action of $ G {\alpha }$ on G .

\noindent 3 . \quad The Platonic solids as hypermaps

\noindent The \quad P l a t o n i c \quad s o l i d s P , \quad or more precisely their \quad 2 − s k e l e t o n s , \quad are a l l \quad examples \quad o f maps \quad ( and hence hypermaps ) on surfaces homeomorphic to the sphere .

The properties of the t etrahedron T , cube C , o ctahedron O , dodecahedron D and icosahe − dron I are well − known [ 8 , 1 7 , 22 ] , and are summarised in Table 2 , where $ N { 0 } ,N { 1 }$ and $ N { 2 }$ denote A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 5 \ hspacethe numbers∗{\ f i of l l vertices}A . J comma . Breda edges d and ’ facesAzevedo period ; .... G A . S A chl . dieresis-a Jones : fl\ iquad symbolPlatonic open brace Hypermapsm comma n closing\quad brace5 means that the faces are A . J . Breda d ’ Azevedo ; G . A . Jones : Platonic Hypermaps 5 \noindentm hyphenthe gons numbers comma n meeting of vertices at each vertex , edges comma and so faces as a hypermap . \ h f i lP l hasA S type c h open l $ parenthesis\ddot{a} l sub$ 0 f l comma i symbol l sub 1\{ commam , l n sub\} 2means that the faces are the numbers of vertices , edges and faces . A S chl a¨ fl i symbol { m , n } means that the faces closingare parenthesis = open parenthesis n comma 2 comma m closing parenthesis comma s ince \noindenteach edgem meets− gons two vertices , n meeting and two faces at each period vertex , so as a hypermap P has type $ ( l { 0 } , l { 1 } , l {m2 - gons} ) , n = meeting ($ at n,2,m),since each vertex , so as a hypermap P has type (l0, l1, l2) = ( n , 2 , m ) , s ince Peach .. Schl edge a-dieresis meets fl twoi .. N vertices sub 0 N andsub 1 two N sub faces 2 .. . Aut P .. Aut to the power of plus P Table ignored! P Schl a¨ fl i N N N Aut P Aut+ P \noindentTable 2 periodeach .. edge The five meets P latonic two solids vertices P as hypermaps and0 two1 faces2 . Figure 3 period 1 period .. The five Platonic solids Table ignored! \ centerlineThe Platonic{P so\ lidsquad P areSchl i llustrated $ \ddot ....{ opena} $ parenthesis f l i \quad as convex$ N polyhedra{ 0 } closingN { parenthesis1 } N ....{ 2 in} Figure$ \quad 3 periodAut 1 period P \quad .... In$ each Aut ˆ{ + }$ Pcase} we can regard P .. as .. a sphericalTable map 2 . comma The .. five and P we latonic .. obtain solids a representation P as hypermaps of P .. as a .. open parenthesis topological closing parenthesis hypermap .. by .. quotedblleft thickeningFigure 3quotedblright . 1 . The .. the five .. Platonic vertices .. solids and .. edges to .. give .. 2 n hyphen .. and 4 hyphen sided .. regions .. onThe .. the Platonic so lids P are i llustrated ( as convex polyhedra ) in Figure 3 . 1 . In each case we \ vspacespherecan∗{ regard comma3 ex }\ ..Pcenterline representing as a{\ the sphericalframebox[.5 hypervertices map\ .. ,l i and n eand w hyperedges i d t we h ] {\ t obtainsemicolone x t b f { Table a .. representation the ..ignored! complementary}}}\ of Pvspace regions as∗{ a comma3 ex (} .. which havetopological 2 m sides ) comma hypermap .. are the byhyperfaces “ thickening period .. This ” i the s i llustrated vertices in Figure and 3 period edges 2 comma to give .. where 2 for n - convenience theand sphere 4 - sided has been projregions ected stereographically on the sphere onto , the representingplane R to the the power hypervertices of 2 period and hyperedges Figure; the 3 period complementary 2 period .. T as regionsa map and , as which a hypermap have 2 m sides , are the hyperfaces . This i s i \ centerlineInllustrated Figure 3{ periodTable in Figure 2 open 2 . 3 parenthesis\ .quad 2 ,The where b closing five for P parenthesis convenience latonic the solids black comma P as grey hypermaps and white} regions are the i hyphen faces of T for i = 0 comma 1 andthe 2 comma sphere that has been proj ected stereographically onto the plane R2. \ centerlinei s comma the{ Figure hypervertices 3 . 1 commaFigure . \quad hyperedges 3 . 2The . five and T as hyperfaces Platonic a map and of Tsolids as comma a hypermap} while the 24 vertices are the blades ofIn T Figureperiod .. 3 Figure . 2 ( 3 b period ) the 3 black shows , the grey underlying and white trivalent regions graph are G ofthe T i comma - faces where of T each for edge i = j 0, oining1 and two 2 , \noindenti hyphenthat facesThe i Platonic s labelled i period so lids P are i llustrated \ h f i l l ( as convex polyhedra ) \ h f i l l in Figure 3 . 1 . \ h f i l l In each case we i s , the hypervertices , hyperedges and hyperfaces of T , while the 24 vertices are the blades of \noindentT . Figurecan regard 3 . 3 shows P \quad the underlyingas \quad a trivalent spherical graph map G ,of\ Tquad , whereand each we \quad edge jobtain oining two a representation i of P \quad as a \quad ( topological ) hypermap- faces i\ squad labelledby \ iquad . ‘‘ thickening ’’ \quad the \quad v e r t i c e s \quad and \quad edges to \quad g i v e \quad 2 n − \quad and 4 − sided \quad r e g i o n s \quad on \quad the sphere , \quad representing the hypervertices \quad and hyperedges ; \quad the \quad complementary regions , \quad which have 2 m sides , \quad are the hyperfaces . \quad This i s i llustrated in Figure 3 . 2 , \quad where for convenience

\noindent the sphere has been proj ected stereographically onto the plane $ R ˆ{ 2 } . $

\ centerline { Figure 3 . 2 . \quad T as a map and as a hypermap }

\noindent In Figure 3 . 2 ( b ) the black , grey and white regions are the i − facesofTfor i $= 0 , 1 $ and 2 , that

\noindent i s , the hypervertices , hyperedges and hyperfaces of T , while the 24 vertices are the blades o f T . \quad Figure 3 . 3 shows the underlying trivalent graph G of T , where each edge j oining two i − faces i s labelled i . 6 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps \noindentFigure 3 period6 \quad 3 periodA . .. J The . underlyingBreda d graph’ Azevedo G of T ; G . A . Jones : \quad Platonic Hypermaps For6 each A regular . J . Breda hypermap d ’ Azevedo H comma ; the G corresponding . A . Jones : trivalent Platonic graph Hypermaps G i s the Cayley graph for \noindentG = Aut HFigure with respect 3 . to 3 a . Capital\quad DeltaThe hyphen underlying basis r satisfying graph G of T ForFigure each 3 regular . 3 . hypermapThe underlying H , the graph corresponding G of T For each trivalent regular hypermap graph G i H s , the corresponding Cayley graph for rtrivalent sub i to the graph power G ofi 2 s = the open Cayley parenthesis graph r sub for 1 r sub 2 closing parenthesis to the power of l sub 0 = open parenthesis r sub 2 r sub 0 closing parenthesis to the power of l sub 1 = open parenthesis r sub 0 r sub 1 closing parenthesis to the power of l sub 2 = 1 semicolon G = Aut H with respect to a ∆− basis r satisfying \noindentwhen H isG a Platonic $=$ solid AutHwith P these are defining respect relations to a for $ G\Delta comma ..− corresponding$ basis to r the satisfying fact that P is the universal map open square bracket 10 comma Ch period 8 closing square bracket or hypermap open square bracket 7 closing square r2 = (r r )l0 = (r r )l1 = (r r )l2 = 1; bracket\ [ r ˆ of{ type2 } open{ i parenthesis} = ( l sub r 0 comma{ i1 } l1 subr2 1{ comma2 2} 0 l) sub ˆ{ 20 closingl 1 { 0 parenthesis}} = period ( r { 2 } r { 0 } ) ˆ{ l { 1 }} = (Thewhen r ..{ automorphism H0 is} ar Platonic{ group1 } solid ..) Aut ˆ P{ Tthesel of{ T are ..2 open}} defining parenthesis= relations1 as; \ a] map for G comma , corresponding .. hypermap .. or to convex the fact polyhedron that closing parenthesis .. is i somorphicP is the universal to S sub 4 map : .. the [ 10 automorphisms , Ch . 8 ] or induce hypermap all possible [ 7 ] permutations of type (l0, of l1, the l2). four vertices comma withThe the rotations automorphism inducing the group even permutations Aut T of commaT ( as so a Aut map to ,the power hypermap of plus T or thicksim convex = polyhedron A sub 4 period ) .. The dual pair C and O \noindent when H is a Platonic solid P these are defining relations for G , \quad corresponding to the fact haveis i automorphism somorphic to groupsS4 : Autthe C automorphisms = sub thicksim Aut induce O = sub all thicksimpossible S permutations sub 4 times C sub of the 2 : .. four the vertices first factor , is the rotation group comma that P is the universal map [ 10 , Ch . 8 ] or hypermap+ [ 7 ] of type $ ( l { 0 } , l { 1 } , l { 2 } correspondingwith the rotations to the permutations inducing the of the even four permutations pairs of opposite , so verticesAut ofT C open∼= A parenthesis4. The dual or faces pair of C O and closing O parenthesis comma ) . $ andhave the automorphism second factor i s generatedgroups Aut by the C antipodal=∼ Aut O symmetry=∼ S4 × periodC2 : ..the S imilarly first factor D and itis s the dual rotation I group have, corresponding automorphism group to the A permutations sub 5 times C sub of 2 the : .. four the 20 pairs vertices of opposite of D can be vertices partitioned of C into ( or five faces sets of of O ) , Thefourand\quad comma theautomorphism second each set factor the vertices i group s generated of an\quad inscrib byAut ed the tT etrahedron antipodal o f T \quad comma symmetry( asand a the . map rotation S , imilarly\quad grouphypermap A D sub and 5 induces it s\quad dual the Ior convex polyhedron ) \quad i s i somorphic to $ S { 4 } : $ \quad the automorphisms induce all possible permutations of the four vertices , evenhave permutations automorphism of these group t etrahedraA5 × C semicolon2 : the the 20 vertices antipodal of symmetry D can be generates partitioned the factor into C five sub sets 2 period of four Another, each hypermapset the vertices we shall of need an i inscrib s the great ed tdodecahedron etrahedron GD , and period the .. rotation This i s Poinsot group quoterightA5 induces s star the hyphen even \noindent with the rotations inducing the even permutations , so $ Aut ˆ{ + }$ T $ \sim = A { 4 } . $ polyhedronpermutations denoted of by these open t brace etrahedra 5 comma ; the 5 divided antipodal by 2 symmetry closing brace generates in open square the factor bracketC 82. comma Ch period VI closing square bracket comma\quad whichTheAnother dual contains hypermap pair a detailed C and we description shall O need of i this s the andgreat dodecahedron GD . This i s Poinsot ’ s star - haveotherpolyhedron automorphism star hyphen denoted polyhedra groups by semicolon{5, Aut5 } in C [ .. 8 $GD , = Ch can{\ . VI besim ]identified , which}$ Aut with contains O the $ regular =a detailed{\ mapsim open description} braceS { 54 comma of}\ thistimes 5 and bar 3 closingC { brace2 } of: type $ open\quad the first factor is the rotation group , corresponding to the permutations2 of the four pairs of opposite vertices of C ( or faces of O ) , braceother 5 comma star 5 - closing polyhedra brace and ; GD can be identified with the regular map {5, 5 | 3} of type { 5 , 5 } andgenusand the 4 genus in second open 4 square in factor [ 1 bracket 0 , § i8 s1 . 0 5 generated comma ] ( see S 8 alsoperiod by the [ 25 closing , antipodal p . 14 square ; 26 bracket ,symmetry pp . .. open 1 . 9\ parenthesis–quad 2 2 ]S ) . imilarly see alsoIt has open D the square and 1 it bracket s dual 2 comma I p period 14have semicolon2 vertices automorphism 26 comma and 30 pp edges group period of .. $ 1 A 9 endash{ 5 }\ 2 2 closingtimes squareC bracket{ 2 } closing: $ parenthesis\quad the period 20 .. vertices It has the 1 of 2 vertices D can and be 30 partitioned edges of into five sets of fourthethe icosahedron , icosahedron each set I semicolon the I ; for vertices each for each vertex vertex of anv v of of inscrib I I the the 5 5 neighbouring ed t etrahedron vertices vertices lie , onlie and a on circuit the a circuit rotation open parenthesis ( abcde group ) inabcde I $ Aclosing{ 5 parenthesis}$ induces in the even permutations of these t etrahedra ; the antipodal symmetry generates the factor $ C { 2 } . $ I, comma to which to which we attach we attach a pentagonal a pentagonal face face CapitalΦv of GD Phi sub . v This of GD i speriod i llustrated .. This i in s iFigure llustrated 3 in . 4 Figure , where 3 period 4 comma where the visiblethe visible part of one part of of the one 1 2 of faces the of 1 GD 2 faces i s shaded of GD semicolon i s shaded note ; that note in that this picture in this comma picture faces , faces intersect intersect comma \ hspace ∗{\ f i l l }Another hypermap we shall need i s the great dodecahedron GD . \quad This i s Poinsot ’ s star − so, it so does it notdoes represent not represent an imbedding an imbedding of GD in R of to theGD power in R of3, 3 commathough .. thoughit does it il does lustrate il lustrate the the fact fact that that AutAut GD GD = sub=∼ thicksimAut I Aut=∼ A I5 =× subC2. thicksim A sub 5 times C sub 2 period \noindentFigure 3 periodpolyhedron 4 period .. denoted TheFigure great by dodecahedron 3 . $ 4\{ . The5 GD great , \ f dodecahedron r a c { 5 }{ 2 GD}\} $ in [ 8 , Ch . VI ] , which contains a detailed description of this and \noindent other s t a r − polyhedra ; \quad GD can be identified with the regular map $ \{ 5 , 5 \mid 3 \} $ o f type \{ 5 , 5 \} and genus 4 in [ 1 0 , \S 8 . 5 ] \quad ( see also [ 2 ,p . 14 ; 26 ,pp . \quad 1 9 −− 2 2 ] ) . \quad It has the 1 2 vertices and 30 edges of

\noindent the icosahedron I ; for each vertex v of I the 5 neighbouring vertices lie on a circuit ( abcde ) in I , to which we attach a pentagonal face $ \Phi { v }$ o f GD . \quad This i s i llustrated in Figure 3 . 4 , where the visible part of one of the 1 2 faces of GD i s shaded ; note that in this picture , faces intersect , so it does not represent an imbedding of GD in $ R ˆ{ 3 } , $ \quad though it does il lustrate the fact that Aut GD $ = {\sim }$ Aut I $ = {\sim } A { 5 }\times C { 2 } . $

\ centerline { Figure 3 . 4 . \quad The great dodecahedron GD } A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 7 \ hspaceAt v comma∗{\ f i.. l l 5}A pentagons . J . meetBreda semicolon d ’ Azevedo .. a rotation ; G on . GD A . once Jones around : v\quad crossesPlatonic edges in the Hypermaps cyclic order \quad 7 open parenthesis va comma vcA comma . J . Breda ve comma d ’ vb Azevedo comma ; vd G closing . A . Jones parenthesis : commaPlatonic so Hypermapsit corresponds to 7 a rotation twice around v on I period\noindent .. openAt parenthesis v , \quad Thus5 GD pentagons i s the map meet ; \quad a rotation on GD once around v crosses edges in the cyclic order ( vaAt ,v ,vc , 5 vepentagons , vb , meet vd ) ; , so a rotation it corresponds on GD once to around a rotation v crosses twice edges around in the v cyclic on I order . \quad ( Thus GD i s the map H( sub va 2 , openvc , veparenthesis , vb , vd I closing ) , so parenthesis it corresponds .. obtained to a by rotation .. applying twice Wilson around quoteright v on I s . rotation ( Thus hyphen GD doubling i s operator .. H sub 2 .. to I ..$ open H square{ 2 } bracket( $ 30 I comma ) \quad 20 closingobtained square by bracket\quad periodapplying closing parenthesis Wilson ’ .. s Using rotation − doubling operator \quad $ H { 2 }$ \quadtheto map I \quadH2( I )[ 30 obtained , 20 ] by . ) \ applyingquad Using Wilson ’ s rotation - doubling operator H2 to I radial[ 30 proj , 20 ection ] . ) from Using the centre radial of I proj comma ection we see from that the GD centrei s a 3 hyphen of I , we sheeted see thatcovering GD of i the s a sphere 3 - sheeted comma radialwith branch proj hyphen ection points from of order the .. centre 1 at the of .. 1 I 2 vertices, we see v comma that .. GD where i s two a 3sheets− sheeted meet .. open covering parenthesis of the the third sphere sheet , withcovering branch of− thepoints sphere of , with order branch\quad - points1 at the of order\quad 1 1 2at vertices the 1 2 v vertices , \quad v ,where where two two sheets meet \quad ( the third sheet containssheets Capital meet Phi ( the sub vthird closing sheet parenthesis period .... The vertex hyphen figure at v i s a pentagram open brace 5 divided by 2 closing brace comma which explains Coxeter quoteright s symbol contains Φ ). The vertex - figure at v i s a pentagram { 5 }, which explains Coxeter ’ s symbol \noindentopen bracec5 o n commav t a i n s 5 divided $ \Phi by 2{ closingv } brace) for. $ GD\ inh f open i l l squareThe2 vertex bracket− 8 closingfigure square at v bracket i s aperiod pentagram .... Having $ 1\{\ 2 verticesf r a comma c { 5 }{ 2 } \} {5,,5 } $for which GD in explains [ 8 ] . Having Coxeter 1 2vertices ’ s symbol , 30 edges and 1 2 faces , GD has Euler characteristic 30 edges2 and 1 2 faces comma GD has Euler characteristic χ = 12 − 30 + 12 = −6; being a branched covering of the sphere GD i s orientable , so it chi = 1 2 minus 30 plus 1 2 = minus 6 semicolonχ .... being a branched covering of the sphere .... GD .... i s orientable comma .... so .... it \noindenthas $ \{genus5 g , \ f= r1 a c {5,−55}}{1 2= }\}4. Asamap$, GD forGDinisisomorphicremaining [ 8 ] . \ h f i l l Havingto 1 2 it vertices s , 30 edges and 1 2 faces , GD has Euler characteristic has subdodecahedron dodecahedron genus g .... ={ sub2 open2 [8 brace,Ch.6]. subThe overbar2 2 to the power of 1 5 commaKepler− toPoinsot the power of minus 5 closing brace 1 divided by 2 to thedual power?−polyhedra of chi, sub open square bracket 8 commathe to the power of = sub Ch sub period 6 small closingstellated square bracketare the to the power of 4 period sub \noindent $ \ chi = 1 2 5 − 30 + 1 2 = − 6 ; $ \ h f i5 l l being a branched covering of the sphere \ h f i l l GD \ h f i l l i s orientable , \ h f i l l so \ h f i l l i t periodgreat to the stellated power of dodecahedron As sub The to the{ 2 power,3} and of a it map s reciprocal sub 2 to the , the power great of comma icosahedron GD i s i somorphic{3, 2 }; these remaining are sub Kepler hyphen Poinsot to .... it7 s - dual sheeted big star coverings hyphen polyhedra of the sphere comma , .... and the as .... maps small they stellated are are i somorphic the to D and I respectively ( \noindentgreatsee Figurestellated$ 3 has dodecahedron . 5 ,{ wheredodecahedron open a face brace i s shaded5}$ divided genus in by each 2 g sub\ caseh comma f i l )l . 3$ closing = {\{} brace andˆ{ 1 it s} reciprocal{\ overline comma{\}{ the great2 }} icosahedron5 , ˆ open{ − brace} 5 3\}\ commaf r a 5 c divided{ 1 }{ by2 2 closing}ˆ{\ chi brace} semicolon{ Figure[ 8 these 3 . 5, are .}ˆ{ Star= } { - polyhedraCh }ˆ{ 4 . } { . 6 ] }ˆ{ As } { . } { The }ˆ{ a map }ˆ{ , } { 2 } GD i s i somorphic { remaining } { Kepler − Poinsot }$ to \ h f i l l i t s $ dual {\ star − polyhedra } 74 hyphen . sheeted Regular coverings hypermaps of the sphere with comma automorphism and as maps they group are i somorphicS4 to D and I respectively , $open4\ .h parenthesis 1 f i . l l the Enumeration see\ h Figure f i l l 3small period of 5the $ comma stellated hypermaps where a{ faceare i s shaded the in}$ each case closing parenthesis period FigureAs explained 3 period 5 at period the ..end Star of hyphen§2 , it polyhedra will be convenient for us to use the natural representation \noindent great stellated dodecahedron $ \{\ f r a c { 5 }{ 2 } { , } 3 \} $ and it s reciprocal , the great icosahedron 4of period G = ..S Regular4 on { 1 hypermaps , 2 , 3 , 4 with} , automorphism first determining group the .. S possible sub 4 4 - vertex permutation graphs G for $ \{ 3 , \ f r a c { 5 }{ 2 }\} ; $ these are 0 4G period with 1 period respect .. Enumeration to any ∆− ofbasis the hypermapsr = (ri) for G . Two ∆− bases r and r give i somorphic Ashypermaps explained at Hthe= endH of0 Si 2 f comma and only it will if theybe convenient are equivalent for us to useunder the naturalAut G representation ; s ince Aut S = Inn \noindent 7 − sheeted∼ coverings of the sphere , and as maps they are i somorphic4 to D and I respectively ofS G4 ∼ == SS sub4 this 4 on is open brace 1 comma 2 comma 3 comma 4 closing brace comma first determining the possible 4 hyphen vertex permutation graphs( see overbar Figure G for 3 G . 5 , where a face i s shaded in each0 case ) . equivalent to the corresponding graphs G and G differing only in their vertex - labels , so withby respect omitting to any vertex Capital - labels Delta wehyphen obtain basis the r = i open somorphism parenthesis classes r sub i of closing regular parenthesisS4− hypermaps for G period . .. Two S Capital Delta hyphen bases r\ centerline andimilarly r to the power ,{ Figure of prime 3 . give 5 . i somorphic\quad Star hypermaps− polyhedra } H = sub thicksim H to the power of prime i f and only if they are equivalent under Aut G semicolon .. s ince Aut S sub 4 = Inn S sub 4 thicksim =\noindent S sub 4 this4 is . \quad Regular hypermaps with automorphism group \quad $ S { 4 }$ equivalent to the corresponding graphs overbar G and hline sub G to the power of prime differing only in their vertex hyphen labels comma so by \noindentomittingby regarding vertex4 . 1 hyphen the . \ edgesquad labels ofEnumeration we G obtain as 3 the - coloured i ofsomorphism the , hypermaps with classes the of colours regular Sto sub be 4 replaced hyphen hypermaps with the period labels .. S i imilarly comma hline= 0, 1, 2 in any of the 3 ! bij ective ways , we obtain the orbits of S on these regular hypermaps . \noindentbyThere regarding areAs the two explained edges conjugacy of G atas classes3 the hyphen end of coloured involutions of \S comma2 , in it withS will4 the: the colours be six convenient to transpositions be replaced for with us , the to which labels use are the natural representation i = 0 comma 1 comma 2 in any of the 3 ! bij ective ways comma we obtain the orbits of S on these regular hypermaps period \noindentThere are twoo f G conjugacy $ = classes S { of4 involutions}$ on \{ in S1 sub , 24 : , .. 3 the , six 4 transpositions\} , first comma determining .. which are the possible 4 − vertex permutation graphs $\ overlinehlineodd ,{\}{ are representedG }$ f o r by G single edges in G , while the three double - transpositions , which are withoddeven comma respect , ..are are to represented represented any $ \ byDelta by single disj edges− o$ int in Gbasisr comma pairs while $= of the edges three( . rdouble{ Thei hyphen} double) transpositions $ f o - r transpositions G . comma\quad whichTwo are $ \Delta − $ bases r and $ r ˆ{\prime }$ give i somorphic hypermaps evenl ie comma in .. aare Klein represented 4 - group by .. disjV o intS , ..and pairs s ..ince of edges| S : periodV |> ..2 it The follows .. double that hyphen in order transpositions to generate .. l ie .. in .. a Klein H $ = {\sim } H ˆ{\prime }4 $C 4 i f and only if4 they4 are equivalent under Aut G ; \quad s i n c e Aut $ S { 4 } 4S hyphen4 at least group two V sub of 4 vartriangleleft S sub 4 comma and s ince bar S sub 4 : V sub 4 bar greater 2 it follows that in order to generate S sub 4 =at least$ Inn two of $ S { 4 }\sim = S { 4 }$ t h i s i s hline \noindentthe involutionsequivalent r sub i in rto must the be corresponding transpositions period graphs .. S ince $\ Goverline must be connected{\}{ G comma}$ and it is easily $ \ r seen u l e {3em}{0.4 pt } { G }ˆ{\prime }$ differingthatthe comma involutions only up to in graphri theirin hyphenr must vertex isomorphisms be− transpositionslabels and , permutations so . by S ince of edge G must hyphen be colours connected comma , itthe is only easily possibilities seen for omittingEquation:that , up vertexhline to graph .. hline− labels - hline isomorphisms we obtain and the permutations i somorphism of edge classes - colours of , regular the only possibilities $ S { 4 } for − $ hypermaps . \quad S i m i l a r l y , G are among the graphs G sub 1 comma period period period comma G sub 5 in Figure 4 period 1 period \ [ \ r u l e {3em}{0.4 pt }\ ]

\noindentG are amongby regarding the graphs theG1 edges, ..., G5 in of Figure G as 43 .− 1coloured . , with the colours to be replaced with the labels i $= 0 , 1 , 2 $ in any of the 3 ! bij ective ways , we obtain the orbits of S on these regular hypermaps . There are two conjugacy classes of involutions in $ S { 4 } : $ \quad the six transpositions , \quad which are

\ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent odd , \quad are represented by single edges in G , while the three double − transpositions , which are even , \quad are represented by \quad d i s j o i n t \quad p a i r s \quad o f edges . \quad The \quad double − transpositions \quad l i e \quad in \quad a Klein 4 − group $ V { 4 }\ vartriangleleft S { 4 } ,$ ands ince $ \mid S { 4 } :V { 4 }\mid > 2 $ it follows that in order to generate $ S { 4 }$ at least two of

\ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent the involutions $ r { i }$ in r must be transpositions . \quad S ince G must be connected , it is easily seen that , up to graph − isomorphisms and permutations of edge − colours , the only possibilities for

\ begin { a l i g n ∗} \ tag ∗{$ \ r u l e {3em}{0.4 pt } $}\ r u l e {3em}{0.4 pt }\ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\noindent G are among the graphs $ G { 1 } ,...,G { 5 }$ in Figure 4 . 1 . 8 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps \noindenthline 8 \quad A . J . Breda d ’ Azevedo ; G . A . Jones : \quad Platonic Hypermaps Figure8 A 4 period . J . Breda 1 period d ..’ Azevedo Possibilities ; G for . G A when . Jones G = : sub Platonicthicksim S Hypermaps sub 4 \ [ In\ r all u l fivee {3em cases}{0.4 comma pt }\ the] subgroup angbracketleft r right angbracket of S sub 4 generated by r sub 0 comma r sub 1 and r sub 2 is transitive and contains a transposition semicolon .... in .... the .... cases overbar G sub 1 comma period period period comma overbar G sub 4 .... it .... also .... contains \noindent Figure 4 . 1 . \quad Possibilities for G when G $ = {\sim } S { 4 }$ .... aFigure .... 3 hyphen 4 . 1 cycle . .... Possibilities open parenthesis for G ab when closing G parenthesis=∼ S4 In period all five open cases parenthesis , the subgroup bc closingh parenthesisr i of S =4 open parenthesis acb closing In all five cases , the subgroup $ \ langle $ r $ \rangle $ o f $ S { 4 }$ generated by $ r { 0 } , parenthesisgenerated comma by ....r0 so, r1 ....and it r2 is transitive and contains r { 1 }$ and $ r { 2 }$ is transitive and contains hlinea transposition ; in the cases G1,..., G4 it also contains a 3 - cycle ( ab ) . ( bc ) = ( acb has) , order divisible by 4 period 3 period 2 and i s therefore so equal to S sub 4 period .. In the case of G sub 5comma it however comma r sub 0 comma\noindent r sub 1a transposition ; \ h f i l l in \ h f i l l the \ h f i l l c a s e s $\ overline {\}{ G } { 1 , } ..., \ overlineand r sub{\}{ 2 visiblyG generate} { 4 } an$ imprimitive\ h f i l l i t proper\ h f i subgroup l l a l s o ..\ openh f i l lparenthesisc o n t a i n D s sub\ h f 4 i lcomma l a \ h .. f in i l lfact3 closing− c y c parenthesisl e \ h f i l l period( ab .. ) Thus . ( bc there$ ) are = four ( $ acb ) , \ h f i l l so \ h f i l l i t has order divisible by 4 . 3 . 2 and i s therefore equal to S . In the case of G , however , r , r hline hline hline hline 4 5 0 1 \ [ \andr u l e {r23emvisibly}{0.4 generate pt }\ ] an imprimitive proper subgroup (D4, in fact ) . Thus there are Sfour hyphen orbits on the regular S sub 4 hyphen hypermaps comma .. corresponding to G = G sub 1 comma period period period comma G sub 4 period .. In the case of G sub 1 comma this orbit has length sigma = 3 : .. by assigning a lab el i = 0 comma 1 or 2 to the right hyphen hand edge we get \noindentthree non hyphenhas order i somorphic divisible hypermaps by 4 comma . 3 since . 2 andtransposing i s therefore the lab els on equal the other to two $ edges S { induces4 } . $ \quad In the case of $ G { 5 } ,$ however $ , r { 0 } , r { 1 }$ hlineS - hlineorbits hline on the regular S4− hypermaps , corresponding to G = G1, ..., G4. In the case of G1, andanthis isomorphism $ orbit r { has2 period} length$ visibly .... Sσ imilarly= 3 generate: Gby sub assigning 2 ancomma imprimitive a G lab sub el 3 .... i = and proper 0, 1 Gor sub 2 subgroup 4 to correspond the right\quad to - S hand hyphen$ ( edge orbits D we of{ get lengths4 } sigma, $ =\quad 1 commain 6 f a and c t ) . \quad Thus there are four 3 three non - i somorphic hypermaps , since transposing the lab els on the other two edges induces \ [ respectively\ r u l e {3em period}{0.4 ..pt }\ Thus therer u l e are{3em comma}{0.4 up pt }\ to i somorphismr u l e {3em}{ comma0.4 pt 3}\ plus 1r plus u l e { 63em plus}{ 30.4 = pt 13}\ regular] S sub 4 hyphen hypermaps H semicolon this agrees with P period Hall quoteright s result open parenthesis open square bracket 13 closing square bracket comma see also S 8 closing an isomorphism . S imilarly G , G and G correspond to S - orbits of lengths σ = 1, 6 and 3 parenthesis\noindent thatS − theorbits number ond sub the Capital regular2 Delta3 open $ S parenthesis4{ 4 } −S sub$ 4 hypermaps closing parenthesis , \quad of normalcorresponding subgroups to G $ = G { 1 } respectively . Thus there are , up to i somorphism , 3 + 1 + 6 + 3 = 13 regular S − hypermaps ,...,GH vartriangleleft Capital Delta{ 4 with} Capital. $ \ Deltaquad slashIn H the thicksim case = of S sub $G 4 i s{ equal1 } to 13,period $ 4 H ; this agrees with P . Hall ’ s result ( [ 13 ] , see also §8 ) that the number d (S ) of normal this4 period orbit 2 period has .. length Description $ \ ofsigma the hypermaps= 3 : $ \quad by assigning a lab el∆ 4 i $= 0 , 1$ or2to the right − hand edge we get thresubgroups e non − i H somorphicC ∆ with ∆ hypermaps/ H ∼= S4 ,i s since equal transposing to 13 . the lab els on the other two edges induces By4 using . 2 . Figure Description 4 period 1 to find of the the orders hypermaps l sub i of the permutations r sub j r sub k open parenthesis open brace i comma j comma k closing brace = open brace 0 comma 1 comma 2 closing brace closing parenthesis comma we see By using Figure 4 . 1 to find the orders l of the permutations r r ({ i , j , k } = {0, 1, 2}), we see \ [ that\ r u commal e {3em up}{0.4 to a pt reordering}\ r u l by e { S3em comma}{0.4 H pt has}\i typer u open l e {3em parenthesis}{0.4 pt l}\j subk ] 0 comma l sub 1 comma l sub 2 closing parenthesis = open that , up to a reordering by S , H has type (l0, l1, l2) = (3, 2, 3), (3, 3, 3), (4, 2, 3) and ( 4 , 4 , 3 ) parenthesis 3 comma 2 comma 3 closing parenthesis comma open parenthesis1 3 commaP −1 3 comma 3 closing parenthesis comma open parenthesis 4 when G = G1,..., G4, and so H has Euler characteristic χ = 2 | G | ( li − 1) = 2, 0, −1, −2 commarespectively 2 comma 3 .closing parenthesis and open parenthesis 4 comma 4 comma 3 closing parenthesis \noindentwhen overbaran G isomorphism = overbar G sub . 1\ commah f i l l periodS imilarly period period $G comma{ 2 overbar} ,G G sub{ 4 comma3 }$ and\ h fso i l H l hasand Euler $ characteristicG { 4 }$ chi correspond = 1 divided to S − orbits of lengths The only subgroup of index 2 in S is A . In the cases G and G each r i s odd , so H is by$ \ 2sigma bar G bar= open 1 parenthesis , 6 to$ the and power 3 of4 sum l4 sub i to the power of1 minus 12 minus 1 closingi parenthesis = 2 comma 0 comma minus 1 comma minus 2 \noindentrespectivelyrespectively period . \quad Thusthereare ,uptoisomorphism $, 3 + 1 + 6 + 3 = 13$ r e gThe u l a only r $ subgroup S { 4 of} index − 2$ in hypermapsS sub 4 is A sub H ; 4 period .. In the cases overbar G sub 1 and overbar G sub 2 each r sub i i s odd comma so H is this agrees with P . Hall ’ s result ( [ 13 ] , see also \S 8 ) that the number $ d {\Delta } (S { 4 } ) $ of normal subgroups H $ \ vartriangleleft \Delta $ with $ \Delta / $ H $ \sim = S { 4 }$ i s equal to 13 .

\noindent 4 . 2 . \quad Description of the hypermaps

\noindent By using Figure 4 . 1 to find the orders $ l { i }$ of the permutations $ r { j } r { k } ( \{ $ i , j , k $ \} = \{ 0 , 1 , 2 \} ) , $ we s ee that , up to a reordering by S , Hhas type $ ( l { 0 } , l { 1 } , l { 2 } ) = ( 3 , 2 , 3 ) , ( 3 , 3 , 3 ) , ( 4 , 2 , 3 )$and(4,4,3)

\noindent when $\ overline {\}{ G } = \ overline {\}{ G } { 1 , } ..., \ overline {\}{ G } { 4 , }$ and so H has Euler characteristic $ \ chi = \ f r a c { 1 }{ 2 }\mid $ G $ \mid ( ˆ{\sum } l ˆ{ − 1 } { i } − 1 ) = 2 , 0 , − 1 , − 2 $

\noindent respectively .

\ hspace ∗{\ f i l l }The only subgroup of index 2 in $ S { 4 }$ i s $ A { 4 } . $ \quad In the cases $\ overline {\}{ G } { 1 }$ and $\ overline {\}{ G } { 2 }$ each $ r { i }$ i sodd , soHis A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 9 \ hspace4 period∗{\ 3f period i l l }A .. .Construction J . Breda of d the ’ hypermaps Azevedo ; G . A . Jones : \quad Platonic Hypermaps \quad 9 hline \noindent 4 . 3 . \quad ConstructionA . J . Breda d of ’ Azevedo the hypermaps ; G . A . Jones : Platonic Hypermaps 9 The4 . first 3 . row in Construction Table 3 comma corresponding of the hypermaps to G sub 1 comma gives an S hyphen orbit of three orientable hypermaps H of genus 0 period .. The chosen representative S sub 1 .. i s the t etrahedron T comma .. which is a map of type \ [ open\ r u l brace e {3em 3}{ comma0.4 pt 3}\ closing] brace .. and thus a hypermap .. of type .. open parenthesis 3 comma 2 comma 3 closing parenthesis semicolon .. Figure .. 3 period 2 .. shows T drawn as a map .. and as aThe hypermap first row period in .... Table Now 3 T , correspondingis self hyphen dual to ....G1, opengives parenthesis an S - orbit that of i s three comma orientable .... T thicksim hypermaps = T to the H power of open parenthesis 2 \noindent The first row in Table 3 , corresponding to $ G { 1 } , $ g i v e s an S − orbit of three orientable hypermaps closingof parenthesis genus 0 . closing The parenthesis chosen representative comma .... so thisS1 orbiti s .... the of t S etrahedron has length sigma T , = 3 which is a map of type Hrather o{ f3 genus , than 3 } .. 0 6and . semicolon\quad thusaThe ..hypermap the chosen .. other two representative of .. typeassociates ( .. 3 of , 2 T $ , are S 3 ) ..{ ; T1 to Figure} the$ power\quad 3of .i open 2 s the parenthesis shows t etrahedron T 0drawn 1 closing T parenthesis , \quad ..which and .. T is to a map of type the\{ poweras3 a ,map of 3 open\}\ parenthesisquad and asand 1 thus 2 closing a hypermap parenthesis\ commaquad ..o f spherical type \ hypermapsquad ( 3 .. , of 2 , 3 ) ; \quad Figure \quad 3 . 2 \quad shows T drawn as a map \quad and as typesa hypermap .. open parenthesis . Now 2T comma is self 3 - comma dual (3 thatclosing i sparenthesis , T ∼= ..T and(2)), ..so open this parenthesis orbit of 3 S comma has length 3 commaσ = 2 3closing parenthesis .. obtained from\noindentrather T by transposing thana hypermap 6 hyperedges ; . the\ h withf i l other l hyperverticesNow two T i s associates s e l f − dual of\ h T f iare l l ( thatT (01) iand s , \ hT f(12) i l l, Tspherical $ \sim = T ˆ{ ( 2 ) } )andhypermaps , .. $ hyperfaces\ h f i l l .. ofso respectively types this orbit (period 2 ,\ 3h .. f, Fori 3 l l ) ..of example and S has comma ( length 3 , 3.. ,Figure 2 $ )\sigma .. obtained 4 period= 2 from.. shows 3 $ T .. by two transposing .. drawings .. of T to the power of open parenthesishyperedges 1 2 closing with parenthesis hypervertices comma .. with \noindenthyperverticesand hyperfacesr a t comma h e r than hyperedges respectively\quad and6 hyperfaces; .\quad Forthe black example\ commaquad ,other grey Figure and two white\quad 4 semicolon . 2a s shows s o it c alsoi a t e shows s two\quad the drawings Walsho f T map are \quad $ T ˆ{ ( 0Wof 1 openT (12) ) parenthesis}, $ with\quad T hypervertices toand the\ powerquad of , hyperedges$ open T ˆ parenthesis{ ( and 1 1 hyperfaces 2 2 closing ) } parenthesis black, $ ,\ greyquad closing andspherical parenthesis white ; itcomma hypermaps also withshows black\quad ando white f vertices corre- spondingtypesthe Walsh\ toquad the hypervertices map( 2 , 3 , and 3 ) hyperedges\quad and \quad ( 3 , 3 , 2 ) \quad obtained from T by transposing hyperedges with hypervertices ofW T to(T the(12) power), with of black open parenthesis and white 1 vertices 2 closing correspondingparenthesis period to the hypervertices and hyperedges \noindentFigureof T (12) 4 periodand. \ 2quad periodh .. y pThe e r f hypermapa c e s \quad T torespectively the power of open . parenthesis\quad For 1 2\ closingquad example parenthesis , and\quad it s WalshFigure map\quad 4 . 2 \quad shows \quad two \quad drawings \quad o f $ Thline ˆ{ ( 1 2 )Figure} , 4 $ . 2\quad . Thewith hypermap T (12) and it s Walsh map hyperverticesThe second S hyphen , hyperedges orbit comma and corresponding hyperfaces to G subblack 2 comma , grey consists and of white a single ; S it hyphen also invariant shows orientable the Walsh hypermap map H .. = S sub 2 .. of type .. open parenthesis 3 comma 3 comma 3 closing parenthesis .. and genus .. 1 semicolon .. it .. has four hypervertices comma\noindent .. fourW hyperedges $ ( T .. ˆand{ ( four 1 2 ) } ) , $ with black and white vertices corresponding to the hypervertices and hyperedges The second S - orbit , corresponding to G , consists of a single S - invariant orientable hypermap H hyperfaces period .. We .. can .. construct .. S sub2 2 .. from .. T using the .. method .. outlined .. in .. S 2 .. open parenthesis see .. also .. = S of type ( 3 , 3 , 3 ) and genus 1 ; it has four hypervertices , four hyperedges open\noindent square2 bracketo f $ 4 T closing ˆ{ ( square 1 bracket 2 closing) } . parenthesis $ period and four hyperfaces . We can construct S from T using the method outlined Being a non hyphen bipartite regular map of type open brace 32 comma 3 closing brace comma T has a bipartite double cover T to the power of \ centerlinein §2{ Figure ( see 4 also . 2 . [\quad 4 ] ) .The Being hypermap a non $ - bipartiteT ˆ{ ( regular 1 2 map ) } of$ type and{ it3 , s 3 Walsh} ,T map } circumflex-zero = T times B to the power0ˆ of zero-circumflex0ˆ has a bipartite double cover T = T ×B which i s the Walsh map W (S2) of some regular which i s the Walsh map W−1 openT −circumflex parenthesis−zero S sub 2 closing parenthesis .. of some regular hypermap S sub 2 = W to the power of minus 1 open parenthesis\ [ \hypermapr u l e { to3em the}{S power0.42 pt= of}\W T-circumflex-zero]( closing) of parenthesis type ..( 3of , type 3 , ..3 open) , with parenthesis Aut S 32 comma=∼ Aut 3 comma T =∼ 3S closing4. parenthesis comma withThe Aut underlying S sub 2 = surfacesub thicksim of S Aut2 i s T a = torus sub thicksim ( Figure S sub 4 . 4 3 period ) ; as .. a The double underlying covering surface of theof S spheresub 2 i s , a it torus i open parenthesis Figure 4 s branched at the four face - centres of T . The construction of S2 i s described in detail in [ 4 period 3 closing parenthesis semicolon as a double0 \noindentcovering, §1 ] , of whereThe the sphere second it i s comma denoted S − itorbit i bys branchedT ,; correspondingone at can the alsofour face obtain to hyphen $S2 G centresby{ putting2 of} T period, b $= ..2 consists, Thec = construction 0 in ofTheorem a single of S 1 sub 2S − invariant orientable hypermap H i\ s2quad described of [ 7$ ] . in= detail S in{ open2 } square$ \quad bracketo f 4 typecomma\ Squad 1 closing( 3 square , 3 bracket , 3 ) comma\quad whereand genusit i s denoted\quad by1 T to ; the\quad poweri t of\ primequad semicolonhas four hypervertices , \quad four hyperedges \quad and fo ur onehyperfaces can also obtain . \ Squad sub 2We by putting\quadFigurecan 4 .\ 3quad .c The o n s thypermap r u c t \quadS2 on$ Sa torus{ 2 }$ \quad from \quad T using the \quad method \quad o u t l i n e d \quad in \quad \S 2 \quad ( see \quad a l s o \quad [ 4 ] ) . Beingb = 2 acomma non − c =bipartite 0 in Theorem regular 1 2 of open map square of type bracket\{ 7 closing3 , 3 square\} , bracket T has period a bipartite double cover $ T ˆ{\hat{0}} = $Figure T 4 $ period\times 3 periodB .. ˆ{\ Thehat hypermap{0}}$ S sub 2 on a torus which i s the Walsh mapW $ ( S { 2 } ) $ \quad of some regular hypermap $ S { 2 } = W ˆ{ − 1 } hlineThe third S - orbit , corresponding to G , consists of six non - orientable hypermaps , including ( ˆ{ T−circumflex −zero } ) $ \quad o f3 type \quad ( 3 , 3 , 3 ) , Thethe thirdprojective S hyphen orbit octahedron comma correspondingS3 = toP G O sub , 3 commawhich weconsists will of six non now hyphen construct orientable . hypermaps The comma including withtheoctahedron projective Aut $ S.. octahedron O{ 2 is} a= S sub{\ regular 3sim = .. P}$ O spherical comma Aut T .. which $ hypermap = we{\ ..sim will ..} of now typeS ..{ construct4 (} 4 , 2. period , $ 3 )\quad .. with TheThe .. octahedron Aut underlying .. O .. surface is of $ S { 2 }$ i s a torus ( Figure 4 . 3 ) ; as a double aO .. regular=∼ ..S spherical4 × ..C2 hypermap; these .. of two type .. openfactors parenthesis represent 4 comma the rotation 2 comma group3 closing and parenthesis the subgroup .. with .. Aut O .. = sub thicksim S subcovering 4generated times C of sub by the 2 semicolonthe sphere antipodal .. ,these it automorphism .. i two s branched .. factors . at By the identifying four face antipodal− centres points of of T O . one\quad obtainsThe a construction of $ S { 2 }$ i s described in detail in [ 4 , \S 1 ] , where it i s denoted by $Tˆ{\prime } ; $ one can also obtain representregular the hypermap rotation group P O and= O the/C subgroup2 of type generated by the antipodal automorphism period $ SBy{ identifying2 }$ by antipodal putting points of O one obtains a regular hypermap P O = O slash C sub 2 of type b $= 2 ,$ c $= 0$ inTheorem12of[7].

\ centerline { Figure 4 . 3 . \quad The hypermap $ S { 2 }$ on a torus }

\ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent The t h i r d S − orbit , corresponding to $ G { 3 } , $ consists of six non − orientable hypermaps , including the projective \quad octahedron $ S { 3 } = $ \quad PO, \quad which we \quad w i l l \quad now \quad c o n s t r u c t . \quad The \quad octahedron \quad O \quad i s a \quad r e g u l a r \quad s p h e r i c a l \quad hypermap \quad o f type \quad ( 4 , 2 , 3 ) \quad with \quad Aut O \quad $ = {\sim } S { 4 }\times C { 2 } ; $ \quad these \quad two \quad f a c t o r s represent the rotation group and the subgroup generated by the antipodal automorphism . By identifying antipodal points of O one obtains a regular hypermap PO $ = $ O $ / C { 2 }$ o f type 10 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentopen parenthesis10 \quad 4 commaA . 2 J comma . Breda 3 closing d ’ parenthesis Azevedo on ; G the . projective A . Jones plane : comma Platonic with Aut Hypermaps P O = sub thicksim S sub 4 semicolon this is the map10 open A .brace J . Breda 3 comma d’ 4 Azevedo closing brace ; G slash . A 2. Jones= open : brace Platonic 3 comma Hypermaps 4 closing brace 3 of open square bracket 1 0 comma \noindentS 8 period( 6closing 4 , 2 square , 3 ) bracket on the comma projective shown in Figure plane 4 period , with 4 where AutPO opposite $= boundary{\sim points} areS identified{ 4 } period; $ this is the map ( 4 , 2 , 3 ) on the projective plane , with Aut P O = S ; this is the map {3, 4}/2 = {3, 4}3 of [ 1 $ \{Figure3 4 period , 4 4 period\} .. The/ hypermaps 2 = P\{ O and3 P C , 4∼ 4\} 3 $ o f [ 1 0 , \S 08 , .§8 6 . ] 6 ,] , shown shown in in Figure 4 4 . .4 where4 where opposite opposite boundary boundary points points are identified are identified . . There are three hyperverticesFigure comma six 4 . hyperedges 4 . The comma hypermaps and four P hyperfaces O and P period C .... Among the associates of PThere O i s it are s dual three open hypervertices parenthesis P O, six closing hyperedges parenthesis , and to the four power hyperfaces of open parenthesis . Among 2 closing the associates parenthesis of = P open parenthesis O to the power\ centerline of open parenthesis{ Figure 4 2 closing . 4 . parenthesis\quad The closing hypermaps parenthesis P O= P and C comma P C } the projective cube comma also shown in Figure 4 period 4 period P O i s it s dual ( P O )(2) = P (O(2)) = P C , the projective cube , also shown in Figure 4 . 4 .. This \noindent. ThisThere i s the are regular three map hypervertices{4, 3}/2 = {4 ,, 3 six}3 of hyperedges [ 10 , §8 . 6 , ] and , sometimes four hyperfaces called the .Purse\ h f i of l l Among the associates of i sFortunatus the regular map [ 9 open ] ; brace it is 4 an comma interesting 3 closing experiment brace slash 2 to = open take brace three 4 square comma 3 handkerchiefs closing brace 3 and of open try square bracket 10 comma S 8 periodsewing 6 closing them square together bracket tocomma form sometimes P C ! called the Purse of Fortunatus \noindentopen squarePOi bracket s 9 it closing sdual(PO square bracket $)ˆ semicolon{ ( .. it 2 is an ) interesting} = $ experiment P $ ( to O take ˆ{ three( square 2 ) handkerchiefs} ) =$ and PCtry sewing , the them projective cube , also shown in Figure 4 . 4 . \quad This itogether s the to regular form P C map ! $ \{ 4 , 3 \} / 2 = \{ 4 , 3 \} 3 $ o f [ 10 , \S 8 . 6 ] , sometimes called the Purse of Fortunatus [hline 9 ] ; \quad it is an interesting experiment to take three square handkerchiefs and try sewing them together to form P C ! The lastThe S last hyphen S - orbit orbit comma , corresponding corresponding to to G4, subcontains 4 comma three contains non three - orientable non hyphen hypermaps orientable hypermaps of char - of char hyphen acteristicacteristic .. minus−2; 2 semicolonthe chosen .. representative the chosen representativeS = SW sub−1( 4P =O W0ˆ), to theof type power of ( minus 4 , 4 ,1 3 open ) , parenthesis is P O to the power of \ [ \ r u l e {3em}{0.4 pt }\ ] 4 circumflex-zero closing parenthesis comma .. of type .. open parenthesis 4 comma−1 40ˆ comma 3 closing parenthesis comma .. is formed from formed from S3 = P O in the same way as S2 = W (T ) is formed from S1 = T: S sub 3 = .. P O .. in the same way as S sub 2 = W to the power of minus 1 open parenthesis T to the power0ˆ of circumflex-zero closing parenthesis it i s the hypermap whose Walsh map W (S4) is the bipartite double covering P O of the non .. is formed from S sub 1 = T : .. it .. i s the hypermap The- l bipartitea s t S − maporbit P O, corresponding ( see Figure 4 . to 5 ) . $ G { 4 } , $ contains three non − orientable hypermaps of char − whose Walsh map W open parenthesis S sub 4 closing parenthesis is the bipartite0ˆ double covering P O to the power of circumflex-zero of the non a c t e r i s t i c \quad $ − Figure2 ; 4 $ . 5\ .quad Pthe O covered chosen by representative P O = W (S4) $ S { 4 } = W ˆ{ − 1 } ( $ P $ O ˆ{\hat{0}} hyphenThis bipartite is a double map P O covering of the proj ective plane , branched at the four face - centres of P O . )open , $ parenthesis\quad seeo f Figure type \ 4quad period( 5 4closing , 4 parenthesis , 3 ) , \ periodquad is formed from 0ˆ $FigureThus S { 4 each3 period} of= 5 the period $ four\quad .. P triangular OPO covered\quad byfaces Pin O of to the P the O same power l ift way s of to circumflex-zero asa hexagonal $ S { =2 face W} open of= P parenthesis WO ˆ,{representing − S sub1 } 4 closing( a T parenthesis ˆ{\hat{0}} ) $ \quadThishyperfaceis a formed double of coveringS4 from; each of of$ the Sthe proj{ three1 ective} vertices= plane $ commaT 1 , : 2\ ,quad branched 3 of Pi t O at\ liquad the ft s four toi aface s pair the hyphen1 hypermap+, 1 centres− etc . of P of O vertices period whose Walsh0ˆ mapW $ ( S { 4 } ) $ is the bipartite double covering P $ O ˆ{\hat{0}}$ o f the non − bipartite map P O Thusof P eachO of, thecoloured four triangular black and faces white of P O l ift and s to representing a hexagonal face a hypervertex of P O to the and power a of hyperedge circumflex-zero of commaS4 representing a ( see Figure 4 . 5 ) . 0ˆ hyperfacerespectively of S sub . 4 semicolon Each edge each a of , .the . three. , f verticesof P O 1 l comma ift s to 2 acomma pair 3a of+, Pa− Oetc li ft . s to aof pair edges 1 sub of plus P O comma. 1 sub minus etc period .. of verticesThe covering P O0ˆ → P O i s shown in Figure 4 . 5 , where pairs of boundary edges with the \ centerlineofsame P O to lab the{ elFigure power are of identified 4 circumflex-zero . 5 . ,\ asquad indicated commaPO .. covered coloured by their byblack incident P and $Oˆ white vertices{\ .. andhat ;{ representing thus0}} horizontal= $ a hypervertex W and $ ( vertical and S a{ hyperedge4 } ) .. $ of} S sub 4 respectivelyboundary period edges .. of Each P edgeO0ˆ( labelled a comma periodc , d , period e , f ) periodare identified comma f of orientably P O l ift s to, as a pair in the a sub construction plus comma a sub minus etc period .. of \noindent This is a double covering+ of+ the+ + proj ective plane , branched at the four face − centres of PO . edgesof of a P torus O to the from power a ofsquare circumflex-zero , while diagonal period .. The boundary covering edges ( labelled a+, a−, b+, b−) are identified ThusPnon O toeach - the orientably power of the of circumflex-zero ,four so the triangular underlying right arrow faces surface P O of i iss PO shown a torus l in ift Figure with s to two 4 period a cross hexagonal 5 comma - caps whereface . pairsThe of P ofresulting boundary$ O ˆ{\ edgeshat{ with0}} the, same $ lab representing el a hyperface of $ S { 4 } ; $ each of the three vertices 1 , 2 , 3 of PO li ft s to a pair $ 1 { + } , are hypermap S4 i s shown in Figure 4 . 6 , where the identification of sides of the 1 2 - gon i s 1 identified{indicated − }$ comma e byt c the . as\ indicatedquad surfaceo f -by symbol v their e r t i cincident e s vertices semicolon thus horizontal and vertical boundary edges o fof P P O $ to O the ˆ{\ powerhat{ of0}} circumflex-zero, $ \quad opencoloured parenthesis black labelled and c sub white plus comma\quad dand sub plus representing comma e sub a plus hypervertex comma f sub and plus a closing hyperedge \quad o f parenthesis$ S { 4 are}$ identified orientably comma as in the construction of a torus from respectivelya square comma . while\quad diagonalEachedgea boundary edges , . open . . parenthesis , f ofPOl labelled ift a sub s plustoa comma pair a sub $a minus{ + comma} , b sub a plus{ − comma }$ b e t sub c . minus\quad o f edges o f P closing$ O ˆ{\ parenthesishat{0}} are identified. $ \quad non hyphenThe coveringorientably comma Pso $ the O underlying ˆ{\hat{ surface0}}\ is arightarrow torus with two$ cross PO hyphen i s shown caps period in Figure .. The resulting 4 . 5 hypermap , where S pairs sub 4 i sof shown boundary edges with the same lab el are identifiedin Figure 4 period , as 6 indicated comma where by the their identification incident of sides vertices of the 1 2 ; hyphen thus gon horizontal i s indicated and by the vertical surface hyphen boundary symbol edges o f P $ O ˆ{\hat{0}} ( $ labelled $ c { + } , d { + } , e { + } , f { + } ) $ are identified orientably , as in the construction of a torus from a square , while diagonal boundary edges ( labelled $ a { + } , a { − } , b { + } , b { − } ) $ are identified non − orientably , so the underlying surface is a torus with two cross − caps . \quad The resulting hypermap $ S { 4 }$ i s shown in Figure 4 . 6 , where the identification of sides of the 1 2 − gon i s indicated by the surface − symbol A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 1 1 \ hspaceopen parenthesis∗{\ f i l l }A abcdeb . J .to Breda the power d ’ of Azevedo minus 1 afe ; Gto the. A power . Jones of minus : \quad 1 dc toPlatonic the power Hypermaps of minus 1 f\ toquad the power1 1 of minus 1 closing parenthesis : .... thus the pair ofA sides . J . lab Breda elled ad comma ’ Azevedo a are ; j G oined . A non . Jones hyphen : orientably Platonic comma Hypermaps while 1 1 \noindentb comma b$ to (the power abcdeb of minus ˆ{ − 1 indicates1 } ana f e orientable ˆ{ − j1 oin} commadc ˆ etc{ − period1 } f ˆ{ − 1 } ) : $ \ h f i l l thus the pair of sides lab elled a , a are j oined non − orientably , while (abcdeb−1afe−1dc−1f −1): thus the pair of sides lab elled a , a are j oined non - orientably , while Figure 4 period 6 period .. The hypermap S sub 4 b , b−1 indicates an orientable j oin , etc . \noindentThis completesb $ the , construction b ˆ{ − of1 the}$ 1 3 regular indicates S sub an4 hyphen orientable hypermaps j semicolon oin , etc note . that four of them are Figure 4 . 6 . The hypermap S orientable comma and three are maps open parenthesis namely T comma P O4 and P C closing parenthesis period \ centerlineThis completes{ Figure the 4 construction . 6 . \quad ofThe the hypermap 1 3 regular $S4 S− hypermaps{ 4 }$ } ; note that four of them are 5orientable period .. Regular , and hypermaps three are with maps automorphism ( namely T group , P O A andsub 5 P C ) . 5 period 1 period .. Enumeration of the hypermaps \noindent5 .This Regular completes hypermaps the construction with automorphism of the 1 group 3 regularA5 5 $ . S 1 .{ 4 } Enumeration − $ hypermaps of ; note that four of them are Wethe will hypermaps represent elements of G = A sub 5 as even permutations of open brace 1 comma period period period comma 5 closing brace comma so overbar G is now a 5 hyphen We will represent elements of G = A as even permutations of { 1 , . . . , 5 } , so G is now a 5 - \noindentvertex graphorientable period .... S , ince and Aut three A sub are5 thicksim5 maps = ( S namely sub 5 comma T , isomorphismPO and PC classes ) . of regular A sub 5 hyphen hypermaps H correspond vertex graph . S ince Aut A5 ∼= S5, isomorphism classes of regular A5− hypermaps H hlinecorrespond \noindentto unlabelled5 . permutation\quad Regular graphs G hypermaps comma as in with S 4 for automorphism S sub 4 period group $ A { 5 }$ 5The . 1 15 . involutions\quad Enumeration in A sub 5 are of all conjugate the hypermaps semicolon being double hyphen transpositions comma they are repre hyphen sented as disj oint pairs of edges in overbar G sub period .... It i s not hard to see that .... open parenthesis up to graph hyphen isomorphisms \noindent We will represent elements of G $ = A { 5 }$ as even permutations of \{ 1 , . . . , 5 \} , so andto permutations unlabelled permutation of the edge hyphen graphs colours G , closing as in § parenthesis4 for S4. .. the only connected permutation graphs overbar G for three $\ overline {\}{ G }$ i s now a 5 − doubleThe hyphen 15 transpositions involutions rin subA5 i inare A all sub conjugate 5 are those ; shown being in double Figure 5- periodtranspositions 1 period , they are repre - Figuresented 5 period as disj 1 ointperiod pairs .. Possibilities of edges for in overbarG. It i G s when not hard G thicksim to see = that A sub ( 5 up to graph - isomorphisms \noindentBeingand transitive permutationsvertex comma graph of the the .subgroup\ edgeh f i l - l .... coloursS angbracketleft i n c e ) Aut the $r only right A { connectedangbracket5 }\sim .... permutation of A= sub S 5 generated graphs{ 5 } by,G r$for sub isomorphismthree 0 comma r sub classes 1 and r sub of 2 regular has $ A { 5 } − $ hypermaps H correspond orderdouble divisible - by transpositions 5 period ri ∈ A5 are those shown in Figure 5 . 1 . In the cases overbar G subFigure 1 comma 5 period . 1 . period Possibilities period comma for overbarG when G G sub∼ 5= oneA5 can inspect overbar G to find an element of order 3 in \ [ \ r u l e {3em}{0.4 pt }\ ] angbracketleftBeing transitive r right angbracket , the subgroup comma soh angbracketleftr i of A5 generated r right angbracket by r0, r has1 and orderr2 has order divisible by 5 . divisibleIn the by cases .. 15 ..G1 and,..., mustG5 one .. therefore can inspect be .. AG subto 5 find period an .. element In the final of case order comma 3 in ..h howeverr i, so commah r ..i thehas second drawing oforder overbar divisible G sub 7 showsby that 15 there and i must s a cyclic therefore ordering of be the verticesA5. In inverted the final by each case r sub , i however comma so , .... angbracketleftthe r right angbracket thicksim\noindentsecond = D drawingto sub unlabelled 5 period permutation graphs G , as in \S 4 f o r $ S { 4 } . $ Thusof thereG7 shows are six that S hyphen there orbitsi s a cyclic of regular ordering A sub of5 hyphen the vertices hypermaps inverted H comma by each .... correspondingri, so h r i to overbar ∼= D5. G = overbar G sub 1 comma \ hspace ∗{\ f i l l }The 15 involutions in $ A { 5 }$ are all conjugate ; being double − transpositions , they are repre − periodThus period there period are comma six S overbar - orbits G of sub regular 6 periodA5− hypermaps H , corresponding to G = G1,..., G6. ByBy considering considering which which permutations permutations of edge hyphenof edge labels - labels induce induce graph graph hyphen - isomorphisms isomorphisms comma , we.. we see see that that \noindentthesethese S hyphen Ssented - orbits orbits haveas have disj lengths lengths oint 6 6 , pairscomma 3 , 3 , 3 of 3 comma , edges 3 and 3 comma 1in respectively $\ 3overline comma , 3 so{\}{ and there 1 respectivelyG are} { , up. } commato$ isomorphism\ h so f i there l l It are i comma s not up hard to isomorphism to see that \ h f i l l ( up to graph − isomorphisms comma, 19 19 regular A5− hypermaps H ( in accordance with P . Hall ’ s result [ 1 3 ] that d∆(A5) = 19, \noindentregularsee §8 A ) suband . 5 permutations hyphen hypermaps of H the open edge parenthesis− c o l in o u accordance r s ) \quad withthe P period only Hall connected quoteright permutation s result .. open graphs square bracket $\ overline 1 3 closing{\}{ G }$ squaref o r thr bracket ee that d sub Capital Delta open parenthesis A sub 5 closing parenthesis = 19 comma see S 8 closing parenthesis period double − transpositions $ r { i }\ in A { 5 }$ are those shown in Figure 5 . 1 .

\ centerline { Figure 5 . 1 . \quad Possibilities for $\ overline {\}{ G }$ when G $ \sim = A { 5 }$ }

\noindent Being transitive , the subgroup \ h f i l l $ \ langle $ r $ \rangle $ \ h f i l l o f $ A { 5 }$ generated by $ r { 0 } , r { 1 }$ and $ r { 2 }$ has order divisible by 5 .

\noindent In the cases $\ overline {\}{ G } { 1 , } ..., \ overline {\}{ G } { 5 }$ one can inspect $\ overline {\}{ G }$ to find an element of order 3 in $ \ langle $ r $ \rangle , $ so $ \ langle $ r $ \rangle $ has order divisible by \quad 15 \quad and must \quad therefore be \quad $ A { 5 } . $ \quad In the final case , \quad however , \quad the second drawing

\noindent o f $\ overline {\}{ G } { 7 }$ shows that there i s a cyclic ordering of the vertices inverted by each $ r { i } , $ so \ h f i l l $ \ langle $ r $ \rangle \sim = D { 5 } . $

\noindent Thus there are six S − orbits of regular $ A { 5 } − $ hypermaps H , \ h f i l l corresponding to $\ overline {\}{ G } = \ overline {\}{ G } { 1 , } ..., \ overline {\}{ G } { 6 . }$

\noindent By considering which permutations of edge − labels induce graph − isomorphisms , \quad we s ee that these S − orbits have lengths 6 , 3 , 3 , 3 , 3 and 1 respectively , so there are , up to isomorphism , 19 r e g u l a r $ A { 5 } − $ hypermaps H ( in accordance with P . Hall ’ s result \quad [ 1 3 ] that $ d {\Delta } (A { 5 } ) = 19 , $ se e \S 8 ) . 1 2 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindent5 period 21 period 2 \quad .. DescriptionA . J of. theBreda hypermaps d ’ Azevedo ; G . A . Jones : Platonic Hypermaps S ince A sub 5 has no subgroup of index 2 comma all 19 hypermaps H are non hyphen orientable period .... By inspecting \noindent1 2 A5 . .J .2 Breda . \quad d ’ AzevedoDescription ; G . ofA . the Jones hypermaps : Platonic Hypermaps hline5 . hline 2 . hline Description of the hypermaps each .. G .. = G sub 1 comma period period period comma G sub 6 .. one .. can compute the type and hence the .. Euler .. characteristic .. chi S ince A has no subgroup of index 2 , all 19 hypermaps H are non - orientable . By inspecting ..\noindent and S5 i n c e $ A { 5 }$ has no subgroup of index 2 , all 19 hypermaps H are non − orientable . \ h f i l l By inspecting genus g .. of each H comma .. as in S 4 period 2 period .. This information is given in Table 4 comma .. where comma .. as in Table 3 \ [ for\ r Gu l =e { S3em sub}{ 40.4 comma pt }\ each rowr u l e describes{3em}{0.4 a representative pt }\ r u l H e { =3em A sub}{0.4 r chosen pt }\ from] the S hyphen orbit of length sigmaeach corresponding G = toG1 overbar, ..., G6 Gone sub r open can parenthesiscompute the r = type 1 comma and period hence period the period Euler comma characteristic 6 closing parenthesis period .. Notice that overbarχ Gand sub 4genus and overbar g of G each sub 5 H both , yield as in hypermaps§4 . 2 . of This type information is given in Table 4 , where , \noindent each \quad G \quad $ = G { 1 } ,...,G { 6 }$ \quad one \quad can compute the type and hence the \quad Euler \quad characteristic \quad openas in parenthesis Table 3 5 for comma G = 5 commaS4, each 3 closing row describes parenthesis a semicolon representative they are H not= isomorphicAr chosen since from r sub the 0 S r sub- orbit 1 r sub 0 r sub 2 has order 2 and 3$ respectively\ chiof length$ \ openquad parenthesisand alternatively comma genus g \quad o f each H , \quad as in \S 4 . 2 . \quad This information is given in Table 4 , \quad where , \quad as in Table 3 theσ corresponding elements r sub 1 tor subG 2r and( r r sub= 1, 2 ..., r6) sub. 0Notice of order that 5 are conjugateG4 and andG5 both not conjugate yield hypermaps respectively of closing type parenthesis ( period f o r G $ = S { 4 } , $ each row describes a representative H $ = A { r }$ chosen from the S − orbit of length H5 .. , sigma 5 , 3 N) ; sub they 0 N are sub not 1 N isomorphic sub 2 chi .. orient since periodr0r1r0 ..r2 ghas order 2 and 3 respectively ( alternatively , Tablethe elementsignored! r1r2 and r2r0 of order 5 are conjugate and not conjugate respectively ) . \noindent $ \sigma $ corresponding to $\ overline {\}{ G } { r } ($r$=1 , . . . ,6 Table 4 period .. The 19 regular hypermapsH σ H N0 withN1 AutN H2 thicksimχ orient = A . sub 5g )5 period . $ 3\quad periodNotice .. Construction that of $\ theoverline hypermaps{\}{ G } { 4 }$ and $\ overline {\}{ G } { 5 }$ both yield hypermaps of type ( 5 , 5 , 3 ) ; they are not isomorphic since $ r { 0 } r { 1 } r { 0 } r { 2 }$ has order 2 and 3 respectively ( alternatively , The first S hyphen orbit comma .. corresponding toTable overbar ignored! G sub 1 comma .. i s represented by the projective dodecahedron A sub 1 = P D = D slash C sub 2 semicolon this is the regular map open brace 5 comma 3 closing brace slash 2 on the proj ective plane formed by identifying \noindentantipodal pointsthe elements of the dodecahedron $ r { D1 comma} r and{ 2 i s} also$ the and map $ open r { brace2 } 5 commar { 0 3 closing}$ of brace order 5 of open 5 are square conjugate bracket 10 and comma not S conjugate respectively ) . Table 4 . The 19 regular hypermaps H with Aut H ∼= A5 8 period5 . 6 3 closing . square Construction bracket period of .. the This hypermaps S hyphen orbit \ centerlinealso contains{H it s\quad dual comma$ \sigma the projectiveN { icosahedron0 } N A{ sub1 } 1 toN the{ power2 }\ of openchi parenthesis$ \quad 2o closing r i e n t parenthesis . \quad =g P} I = I slash C sub 2 The first S - orbit , corresponding to G i s represented by the projective dodecahedron = open brace 3 comma 5 closing brace 5 period .. Both maps1, A = PD = D /C ; this is the regular map { 5 , 3 } / 2 on the proj ective plane formed by are1 shown in Figure 5 period2 2 comma where opposite boundary points are identified to form the proj ective identifying antipodal points of the dodecahedron D , and i s also the map { 5 , 3 } 5 of [ 10 , §8 plane period \ vspace. 6 ]∗{ .3 ex This}\ centerline S - orbit {\ framebox[.5 \ l i n e w i d t h ] {\ t e x t b f { Table ignored!}}}\ vspace ∗{3 ex} Figure 5 period 2 period .. The hypermaps P D and P I (2) Salso imilarly contains comma it the s great dual dodecahedon , the projective GD comma icosahedron an orientableA1 regular= PI map= ofI type/C2 open= {3 brace, 5}5. 5 commaBoth maps 5 closing brace and genus 4 shownare shown in Figure in 3 Figure period 54 comma . 2 , where yields opposite a non hyphen boundary orientable points regular are map identified A sub 2 of to type form open the brace proj 5 ective comma 5 closing brace and genus 5 semicolonplane this . \ centerlinei s the projective{ Table great 4 dodecahedron . \quadFigureThe P 5 19 GD. 2 regular .= GD The slash hypermaps hypermaps C sub 2 comma H P with D isomorphic and Aut P I H to $ open\sim brace= 5 comma A { 55 closing}$ brace} 3 in open square bracketS imilarly 10 comma , theS 8 period great 6 dodecahedon closing square GDbracket , an comma orientable lying regular map of type { 5 , 5 } and genus 4 \noindenthlineshown in5 Figure . 3 . \ 3quad . 4 ,Construction yields a non - orientable of the hypermaps regular map A2 of type { 5 , 5 } and genus 5 ; inthis the S i hyphen s the projective orbit of length great sigma dodecahedron = 3 correspondingP GD to G= subGD 2 period/C2, isomorphic .. This map is to shown{ 5 , in 5 Figure} 3 in 5 [ period 10 , 3 comma where \noindentboundary§8 . 6 ] edgesThe , lying fare i r s identified t S − o in r b pairs i t , as\ indicatedquad corresponding by the lab elling toof the $ vertices\ overline period{\}{ .. JustG as} { 1 , }$ \quad i s represented by the projective dodecahedron $ AP D{ and1 } P I= are $ related by duality comma .. P GD and P I are related by Wilson quoteright s Petrie operation Popen D $ square = $ bracket D $ 29 / semicolon C { 2 ..} 1 0 comma; $ this .. S 8 is period the 6 semicolonregular.. map 20 closing\{ 5 square, 3 \} bracket/ 2 on comma the which proj transposes ective planefaces and formed Petrie by identifying antipodal points of the dodecahedron D , and i s also the map \{ 5 , 3 \} 5 o f [ 10 , \S 8 . 6 ] . \quad This S − o r b i t polygonsin the of maps S - orbit while of preserving length theσ = 3 corresponding to G2. This map is shown in Figure 5 . 3 , where 1 hyphen skeleton period \noindentboundaryalso edges contains are identified it s dualin pairs , theas indicated projective by the icosahedron lab elling of the $ A vertices ˆ{ ( . 2 Just ) } as{ 1 P } = $ P I $ = $ I $ /D and C { P2 I are} related= \{ by duality3 , , 5 P GD\} and5 P I . are $ related\quad byBoth Wilson maps ’ s Petrie operation [ 29 are; shown 1 0 , in§ Figure8 . 6 ; 5 20 . 2 ] , , which where transposes opposite faces boundary and Petrie points polygons are identified of maps while to preserving form the proj ective planethe . 1 - skeleton .

\ centerline { Figure 5 . 2 . \quad The hypermaps P D and P I }

\noindent S imilarly , the great dodecahedon GD , an orientable regular map of type \{ 5 , 5 \} and genus 4 shown in Figure 3 . 4 , yields a non − orientable regular map $ A { 2 }$ o f type \{ 5 , 5 \} and genus 5 ; this i s the projective great dodecahedron PGD $ = $ GD $ / C { 2 } , $ isomorphic to \{ 5 , 5 \} 3 in [ 10 , \S 8 . 6 ] , l y i n g

\ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent in the S − orbit of length $ \sigma = 3 $ corresponding to $G { 2 } . $ \quad This map is shown in Figure 5 . 3 , where boundary edges are identified in pairs as indicated by the lab elling of the vertices . \quad Just as PD and P I are related by duality , \quad PGD and P I are related by Wilson ’ s Petrie operation [ 29 ; \quad 1 0 , \quad \S 8 . 6 ; \quad 20 ] , which transposes faces and Petrie polygons of maps while preserving the 1 − s k e l e t o n . A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 13 \ hspaceFigure∗{\ 5 periodf i l l } 3A period . J .. . The Breda hypermap d ’ Azevedo A sub 2 = ; P G GD . A . Jones : \quad Platonic Hypermaps \quad 13 The proj ective dodecahedronA P . D J is . Bredaa non hyphen d ’ Azevedo bipartite ; map G . ofA type . Jones open : brace Platonic 5 comma Hypermaps 3 closing brace comma 13 so it s bipartite double \ centerlinecover P D to{ Figure the power 5 of . circumflex-zero 3 . \quad The = Phypermap D times B to $ the A power{ 2 of} zero-circumflex= $ P GD is} the Walsh map of a hypermap A sub 3 = W to the Figure 5 . 3 . The hypermap A = P GD power of minus 1 open parenthesis P D to the power of circumflex-zero closing2 parenthesis of type open parenthesis 3 comma 3 comma 5 closing \noindentThe projThe ective proj dodecahedron ective dodecahedron P D is a non P D - is bipartite a non − mapbipartite of type { map5 , 3of} type, so it\{ s5 bipartite , 3 \} , so it s bipartite double parenthesis semicolon l ike 0ˆ 0ˆ −1 0ˆ coverPdouble D comma P $ cover AD sub ˆ{\ P 3 ishatD regular={0PD}} comma×B= $ withis P the DAut Walsh $ A\ subtimes map 3 = sub ofB athicksim hypermapˆ{\hat Aut{0 P}}A D3 $ == subW is thicksim( theP D Walsh A) of sub type map 5 period ( of 3 ,a .. It hypermap s underlying $ surface A { i3 s a} =non hyphen3 W , ˆ 5{ ) − orientable ; l ike1 } P D( $, A P3 is $ regular D ˆ{\ ,hat with{0 Aut}} A)$3 =∼ Aut oftype(3 P D =∼ A ,35. ,5)It s underlying ; l ike surface i Pdouble Ds a $ non .. , covering - Aorientable{ of3 the}$ proj double is ective regular plane covering comma , with of .. the branchedAut proj $A ective at ..{ the3 plane six} face= , hyphen{\ branchedsim centres}$ at Autof P the D P comma D six $ face = .. so -{\ .. itsim .. has} A { 5 } . $ \quadcharacteristiccentresIt s of underlying P 2 period D , 1 so minus surface it 6 = minushas i s 4 a and non genus− o 6 r periodi e n t a b.. l Thee Walsh map W open parenthesis A sub 3 closing parenthesis = P D to the 0ˆ powerdoublecharacteristic of circumflex-zero\quad covering2.1 is− shown6 = of−4 inand the Figure genus proj6 ective . The plane Walsh , \ mapquad Wbranched(A3) = P atD \isquad shownthe in s Figure i x f a c e 5 − centres of PD , \quad so \quad i t \quad has 5. period 4 , with 4 comma black with and black white and vertices white vertices representing representing the the hypervertices hypervertices and and hyperedges hyperedges of ofA subA3; 3part semicolon of part \noindentofthe the surface surfacecharacteristic symbol symbol open ( ab parenthesis . $. . 2 cb ab . . . period . 1 cdperiod .− . . ad6 period . =. cb . ) period− is shown4 period $ , and periodindicating genus cd period 6the . periodidentifications\quad periodThe adWalsh period mapW period period $ ( closing A { 3 } parenthesis)needed = $ is P shown to $ form D comma ˆ{\ thehat indicating shaded{0}} face$ the is identifications ; shown rotation in needed Figureabout the centre gives four more faces , s imilarly 5to .formed form 4 , the with , shaded black face and semicolon white .. rotation vertices about representing the centre gives the four morehypervertices faces comma sand imilarly hyperedges formed comma of $ A { 3 } ; $ part ˆ ofandand the the the surface central central face symbol face completes completes ( abthe six . the. faces . six cb of Pfaces . D . to .of the cd P powerD .0, .corresponding of . circumflex-zero ad . . . ) to commais the shown s ixcorresponding hyperfaces , indicating to of the s the ix hyperfaces identifications of needed toA form sub 3 period the shaded face ; \quad rotation about the centre gives four more faces , s imilarly formed , Figure 5 period 4 period .. The map W open parenthesis A sub 3 closing parenthesis = P D to the power of circumflex-zero \noindentRotation Pand D to the the power central of circumflex-zero face completes comma the interchangingA3 six. faces to the of power P $ of D of ˆ{\ Figurehat 5{0 period}} 4, the $ through corresponding sets of black to pi about the s sub ix hyperfaces of and it s centre white sub vertices sub semicolon to the power of induces sub this an automorphism shows sub that A sub 3 = sub thicksim of A sub 0ˆ 3\ begin order open{ a l i gparenthesis n ∗} 1 closing parenthesisFigure 5 sub . 4 comma . The so to map the W power(A of3) 2= subP theD to the power of of A S{ hyphen3 } orbit. containing A sub 3 has length sigma = 3 period .. The quotient of P D to the power of circumflex-zero by this automorphism is \end{ a l i g n ∗} the ofFigure induces 2 of Rotation ˆ 5.4thethrough πaboutanditscentrewhite ;thisanautomorphismshowsthatA = ofA3order map P D shownP D0,interchanging in Figure 5 period 2 periodsetsofblack vertices 3 ∼ (1),sothe \ centerlineS imilarly comma{ Figure .. the 5 non . 4 hyphen . \quad bipartiteThe map maps W P I $ and ( P AGD{ ..3 of} types) .. open = $ brace P $3comma D ˆ{\ 5hat closing{0}} brace$ } .. and .. open brace 5 0ˆ commaS - 5 orbit closing containing brace .. giveA rise3 has to length σ = 3. The quotient of P D by this automorphism is the \ [non Rotationmap hyphen P D orientable shown{ P in D .. Figure regular ˆ{\hat ..5 A.{0 2 sub}} . 5 .., hypermaps interchanging .. A sub 4 =} Wˆ{ too the f power Figure of minus} 15 open . parenthesis 4 { the P I-circumflex-zero} through { subs closing e t s parenthesiso f blackS imilarly .. and}\ .. ,pi A sub the 6about =non W - to bipartite{ theand power} maps ofi t minus P s I 1 and open c e P n parenthesis t GD r e { white of P types GD} to{ the{v e3 power r , t i5 c} e of s zero-circumflex}andˆ{ induces{ 5 , 5 closing}} { ; parenthesis} { t h i s ..} of typesan automorphism { shows } { that A { 3 } = {\sim }} o f { A } { −31 } order { ( 1 ) { , } so }ˆ{ 2 } { the }ˆ{ o f }\ ] opengive parenthesis rise to non 5 comma - orientable 5 comma 3 regular closing parenthesisA5 hypermaps and open parenthesisA4 = 5 commaW ( P 5 commaI − circumflex 5 closing− parenthesiszero) comma corresponding to −1 0ˆ overbarand G subA 46 and= overbarW ( GP subGD 6) periodof types .. The underlying surface of A sub 4 i s a double covering( 5 , 5 of , the3 ) proj and ective ( 5 , plane 5 , 5 comma ) , corresponding branched at the to 10G face4 and hyphenG6. centresThe of underlying P I comma so surface it has characteristic of A4 i s \noindent2a period double 1 minusS − coveringorbit 1 0 = minus ofcontaining the 8 and proj genus ective $ 10 A period plane{ 3 .. Figure},$ branched has 5 period length at 5 theshows 10$ W\ facesigma open - parenthesis centres= of3 A P sub .I 4 , $ closing so\ itquad has parenthesisThe quotient = P I-circumflex-zero of P $ D ˆ{\hat{0}}$ by this automorphism is the as a bipartitecharacteristic double covering2.1−10 = of−8 and genus 10 . Figure 5 . 5 shows W (A4) = P I −circumflex−zero as a mapPbipartite I PD : .. the shown vertex double in hyphen covering Figure labelling 5of . P and 2 I : . colouring the vertex indicate - labelling the identifications and colouring of boundary indicate edges the period identifications .. S ince of boundary edges . S ince S i m i l a r l y , \quad the non − bipartite maps P I and P GD \quad o f types \quad \{ 3 , 5 \}\quad and \quad \{ 5 , 5 \}\quad g i v e r i s e to non − o r i e n t a b l e \quad r e g u l a r \quad $ A { 5 }$ \quad hypermaps \quad $ A { 4 } = W ˆ{ − 1 } ( $ P $ I−circumflex −zero { ) }$ \quad and \quad $ A { 6 } = W ˆ{ − 1 } ( $ P $ GD ˆ{\hat{0}} ) $ \quad o f types

\noindent ( 5 , 5 , 3 ) and ( 5 , 5 , 5 ) , corresponding to $\ overline {\}{ G } { 4 }$ and $\ overline {\}{ G } { 6 . }$ \quad The underlying surface of $ A { 4 }$ i s a double covering of the proj ective plane , branched at the 10 face − centres of P I , so it has characteristic $ 2 . 1 − 1 0 = − 8 $ and genus 10 . \quad Figure 5 . 5 showsW $ ( A { 4 } ) = $ P $ I−circumflex −zero $ as a bipartite double covering of PI: \quad the vertex − labelling and colouring indicate the identifications of boundary edges . \quad S i n c e 14 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentA sub 4 =14 sub\ thicksimquad A A . sub J 4. to Breda the power d ’ of Azevedo open parenthesis ; G . A0 1 . closing Jones parenthesis : Platonic we have Hypermaps sigma = 3 for this S hyphen orbit period $ A { 4 } = {\sim } A ˆ{ ( 0 1 ) } { 4 }$ we have $ \sigma = 3$ for thisS − o r b i t . Figure 5 period 5 period .. The double covering W open parenthesis A sub 4 closing parenthesis(01) = P I-circumflex-zero right arrow P I 14 A . J . Breda d ’ Azevedo ; G . A . Jones : Platonic Hypermaps A4 =∼ A4 we have σ = 3 Infor the this case S of - orbitA sub . 6 comma .. the surface i s a double covering of P GD .. open parenthesis which has characteristic .. minus 3 closing parenthesis\noindent commaFigure 5 . 5 . \quad The double coveringW $ ( A { 4 } ) = $ P $ I−circumflex −zero \rightarrow $ Figure 5 . 5 . The double covering W (A ) = P I − circumflex − zero → P I In the case of A , PIbranched at the 6 face hyphen centres comma so it has4 characteristic 2 period open parenthesis minus 3 closing parenthesis6 minus 6 = minus 1 2 the surface i s a double covering of P GD ( which has characteristic −3), andIn genus the 14 case period of .... $A Figure{ 6 } , $ \quad the surface i s a double covering of PGD \quad ( which has characteristic \quad $ − branched3 ) at ,the $ 6 face - centres , so it has characteristic 2.(−3) − 6 = −12 and genus 14 . Figure 5 period 6 shows W open parenthesis0ˆ A sub 6 closing parenthesis = .... P GD to the power of circumflex-zero .... as a branched double covering of the5 map . 6 shows P GD in W Figure(A6) 5 period = P GD 3 periodas a branched double covering of the map P GD in Figure 5 . 3 . \noindentBeingBeing the the uniquebranched unique regular regular at A the subA 5 65 hyphen− facehypermap hypermap− centres of of type type , so(5 open, 5 it, 5) parenthesis, hasA6 is characteristic S - 5 invariant comma 5 comma, that $ i 5 2 s closing , it . s S parenthesis - ( orbit− comma3 ) A sub− 6 is6 S hyphen = −invarianthas1 commalength 2$ that andσ = i 1 genus s. comma 14 it s . S\ hyphenh f i l l orbitFigure 0ˆ has length sigma = 1 period Figure 5 . 6 . The map W (A6) = P GD \noindentFigure 5 period5.6showsW 6 period .. The $( map W A open{ parenthesis6 } )A = sub $ 6\ closingh f i l l parenthesisP $ GD =ˆ{\ P GDhat to{0 the}} power$ \ h of f circumflex-zeroi l l as a branched double covering of the map PGD in Figure 5 . 3 . hline hline \noindentLike G subBeing 4 comma the G subunique 5 corresponds regular to an $ AS hyphen{ 5 } orbit − of$ length hypermapoftype 3 containing a non hyphen $( orientable 5 , hypermap 5 , of 5 ) , A { 6 }$ i s SLike− invariantG4, G5 corresponds , that to i san , S it - orbit s S − of lengtho r b i t 3 containing a non - orientable hypermap of type open parenthesis 5 comma 5 comma 3 closing parenthesis and genus 10∼ comma namely A sub 5 period .. However comma A sub 5 similar-equal type ( 5 , 5 , 3 ) and genus 10 , namely A5. However , A5 = A4 : when G = G4 the elements A sub 4 : .. when overbar G = overbar G sub 4 the elements r sub 1 r sub 2 r1r2 and r2r0( which rotate all blades around hypervertices and hyperedges ) are conjugate in A5, \noindentand r subhas 2 r sub length 0 open parenthesis $ \sigma which= rotate 1 all . blades $ around hypervertices and hyperedges closing parenthesis are conjugate in A sub 5 comma \ centerline { Figure 5 . 6 . \quad The map W $ ( A { 6 } ) = $ P $ GD ˆ{\hat{0}}$ }

\ [ \ r u l e {3em}{0.4 pt }\ r u l e {3em}{0.4 pt }\ ]

\noindent Like $ G { 4 } ,G { 5 }$ corresponds to an S − orbit of length 3 containing a non − orientable hypermap of

\noindent type ( 5 , 5 , 3 ) and genus 10 , namely $A { 5 } . $ \quad However $ , A { 5 }\cong A { 4 } : $ \quad when $\ overline {\}{ G } = \ overline {\}{ G } { 4 }$ the elements $ r { 1 } r { 2 }$ and $ r { 2 } r { 0 } ( $ which rotate all blades around hypervertices and hyperedges ) are conjugate in $ A { 5 } , $ A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 15 \ hspacesince by∗{\ inspectionf i l l }A r . sub J 2 . r Breda sub 0 = d open ’ Azevedo parenthesis ; r G sub . 1 A r sub. Jones 2 closing : \ parenthesisquad Platonic to the power Hypermaps of r sub 0\ rquad sub 115 semicolon when overbar G = overbar G sub 5 comma onA the . other J . Breda hand comma d ’ Azevedo each of ; r G sub . A1 r . sub Jones 2 and : r sub Platonic 2 r sub Hypermaps0 15 \noindenti s conjugatesince in A subby 5 inspection to the square of $ the r other{ 2 comma} r so{ they0 } are= conjugate ( r in S{ sub1 } 5 butr not{ in2 A} sub) 5 ˆ period{ r ..{ To0 } r { 1 }} ; $ since by inspection r r = (r r )r0r1 ; when G = G on the other hand , each of r r and r r i s whensee that $\ overline .. W to the{\}{ powerG2 of0} minus=1 12\ openoverline parenthesis{\}{ PG I-circumflex-zero5,} { 5 , } sub$ closingon the parenthesis other1 hand2 .. i s isomorphic , each2 0 of to A $ sub r 4{ and1 } not Ar sub{ 52 }$ conjugate in A to the square of the other , so they are conjugate in S but not in A . To see commaand $ note r that{ 2 in} Figure5r { 5 period0 }$ 5 the rotations 5 5 that W −1( P I − circumflex − zero i s isomorphic to A and not A , note that in Figure 5 . 5 iaround s conjugate the black in and white $A vertices{ 5 }$ of P to I-circumflex-zero) the square are of both the l otherift4 ed from , so the5 they same rotations are conjugate around in $ S { 5 }$ but not in $ A the{ rotations5 } . $ around\quad theTo black and white vertices of P I − circumflex − zero are both l ift ed from thethe vertices same of rotations P I comma around so r sub 1 r sub 2 and r sub 2 r sub 0 are conjugate period .. An alternative way of distinguishing A sub 4 seefrom that A sub\quad 5 open parenthesis$ W ˆ{ − and1 of} showing( $ that P $.. WI−circumflex to the power− ofzero minus{ 1 open) }$ parenthesis\quad i P s I-circumflex-zero isomorphic subto closing $ A parenthesis{ 4 }$ andthe not vertices $ A { of5 P} I , so, $r1r note2 and thatr2r0 are in conjugate Figure 5 . . 5 An the alternative rotations way of distinguishing A4 thicksim = A sub 4 closing parenthesis .. is to note−1 that r sub 0 r sub 1 r sub 0 r sub 2 has order 2 and 3 aroundfrom theA5 black( and of and showing white that verticesW ( ofP PI − circumflex $ I−circumflex− zero)−zero∼= A $4) areis to both note l that iftr ed0r1r from0r2 the same rotations around respectivelyhas order in 2 these and 3 two respectively maps period in these two maps . We will construct A sub 5 .. from the Walsh map of it s orientable double cover A sub 5 to the power of plus period .. First we We will construct A from the Walsh map of it s orientable double cover A+. First we \noindentconstruct athe bipartite vertices orientable of mapP5 I M , sub so 5 comma$ r { then1 } we taker { A2 sub}$ 5 to and the power $ r of{ plus2 } to5 ber the{ corresponding0 }$ are hypermap conjugate . \quad An alternative way of distinguishing construct a bipartite orientable map M , then we take A+ to be the corresponding hypermap $ AW to{ the4 } power$ of minus 1 open parenthesis M sub5 5 closing parenthesis5 comma .... a regular orientable hypermap with Aut A sub 5 to the power W −1(M ), a regular orientable hypermap with Aut A+ ∼= A × C , and finally we take A offrom plus thicksim $ A5 { = A5 sub} 5( times $ and C sub of 2 comma showing and that finally\ wequad take A$ Wsub5 ˆ 5{ − 51 } 2 ( $ P $ I−circumflex5 −zero { ) }\sim to be the antipodal quotient P A+ = A+/C , a non - orientable regular A − hypermap . ( It is =to A be the{ 4 antipodal} ) $ quotient\quad P Ais sub to 5 noteto5 the power that5 of2 $ plus r ={ A0 sub} 5 tor the{ power1 } ofr plus{5 slash0 } Cr sub{ 22 comma}$ a has non orderhyphen 2orientable and 3 regular straightforward to check that W −1(M ) i s the orientable double cover of A , so the notation A+ Arespectively sub 5 hyphen hypermap in these period two .... maps open parenthesis . 5 It is 5 5 is j ustified . ) We take the 1 - skeleton of M5 to be that of I −circumflex−zero, or equivalently straightforwardof the small to stellated check that triacontahedron W to the power of shown minus 1in open Figure parenthesis 6 . 3 : M sub each 5 closing vertex parenthesis v of I corresponds i s the orientable double cover of A sub 5 comma\ hspace so∗{\ thef notation i l l }We will construct $ A { 5 }$ \quad from the Walsh map of it s orientable double cover $ A ˆ{ + } { 5 } to a pair v and v of black and white vertices of M , and each edge vw of I corresponds to . $A sub\quad 5 to theF i+ r power s t we of− plus is j ustified period closing parenthesis ..5 We take the 1 hyphen skeleton of M sub 5 to be that of I-circumflex-zero two edges v w and v w of M . This gives us a connected bipartite 5 - valent graph with 24 sub comma or equivalently+ − of the− small+ 5 \noindentverticesconstruct and 60 edges a bipartite . For each orientable face ( uvw map ) of $ I weM take{ 5 a} six, - $ sided then face we( takeu−z+v− $Aˆx+w−y{+)+, } { 5 }$ to be the corresponding hypermap stellatedas shown triacontahedron in Figure 5 shown . 7 , in where Figure x 6 , period y and 3 z : are.. each the vertex other v vertices of I corresponds of I adj to acent a pair tov sub v and plus w , to andw andv sub u minus , and .. toof black u and and v whiterespectively vertices of . M sub 5 comma and each edge vw of I corresponds to two edges \noindentv sub plus w$ Wsub ˆ minus{ − and1 } v sub(M minus w{ sub5 } plus) of M , sub $ 5\ periodh f i l l ..a This regular gives us orientable a connected hypermapbipartite 5 hyphen with Aut valent $ graph A ˆ{ with+ } 24{ 5 } Figure 5 . 7 . A face of M vertices\sim = A { 5 }\times C { 2 } , $ and finally we5 take $A { 5 }$ We now have an orientable bipartite map M with 20 hexagonal faces . Figure 5 . 7 shows and 60 edges period .. For each face open parenthesis uvw5 closing parenthesis of I we take a six hyphen sided face open parenthesis u sub minus that each face of I is covered by four faces of M , so the surface of M i s a 4 - sheeted covering z\noindent sub plus v subto minus be the x sub antipodal plus w sub minusquotient y sub P plus $ closing A5 ˆ{ parenthesis+ } { 5 } comma= as5 A shown ˆ{ + } { 5 } /C { 2 } , $ a non − orientable regular of the sphere , with branch - points of order 2 at the 1 2 white vertices v : Figure 5 . 8 $ Ain Figure{ 5 } 5 period − $ 7 hypermap comma where . \ xh comma f i l l ( y and I t z i s are the other vertices of I adj acent to v and− w comma to w and u comma shows that i f an orientation of I induces the cyclic order ρ = ( abcde ) of neighbouring vertices and to u and v respectively period of v in I , then around v the order i s (a b c d e ), giving a single unbranched sheet , whereas \noindentFigure 5 periodstraightforward 7 period .. A face+ to of check M sub 5 that− $W− − ˆ−{− − 1 } (M { 5 } ) $ i s the orientable double cover of $ A around{ 5 } v−, $the so order the is notation(a+d+b+e+c+), so three sheets are j o ined to give a branch - point of Weorder now 2 have , as an in orientable Figure 5 bipartite . 9 . map M sub 5 with 20 hexagonal faces period .. Figure 5 period 7 shows that $each A ˆface{ + of} I is{ covered5 }$ by is four j faces ustified of M sub . )5 comma\quad soWe the take surface the of M 1 − subskeleton 5 i s a 4 hyphen of $Msheeted{ covering5 }$ to be that of $I−circumflex −zero { , }$ Being bipartite , M i s the Walsh map of an orientable hypermap A+ = W −1(M ) of type orof equivalently the sphere comma of .. the with5 small branch hyphen points of order 2 at the .. 1 2 white vertices5 v sub minus5 : .. Figure 5 period 8 shows ( 5 , 5 , 3 ) and characteristic −16 : the 1 2 hypervertices , 1 2 hyperedges and 20 hyperfaces of stellatedthat i f an orientation triacontahedron of I induces shown the cyclic in order Figure rho = 6 open . 3 parenthesis : \quad each abcde closingvertex parenthesis v of I correspondsof neighbouring verticesto a pair of v $ v { + }$ A+ correspond to the black vertices , white vertices and faces of M . By the symmetry of the andin I5 comma $ v { then − around }$ \quad v sub plusof the black order and i s open white parenthesis vertices a sub of minus $M b sub5{ minus5 } c sub, $ minus and d sub each minus edge e sub vw minus of I closing corresponds parenthesis to two edges method of construction , Aut A+ contains Aut I ∼= A × C , of order 1 20 ; since A+ has 1 20 comma$ v giving{ + a} singlew unbranched{ − }$ and sheet comma$5 v { whereas − } w { + }$5 o f2 $ M { 5 } . $ \quad5 This gives us a connected bipartite 5 − valent graph with 24 vertices blades ( two for each edge of M ), A+ must be regular , with Aut A+ ∼= A × C . The antipodal andaround 60 edgesv sub minus . \quad .. theFor order each is .. open face5 parenthesis5 ( uvw ) a of sub I plus we d take sub plus a six b sub5− plussided5 e sub face2 plus c $ sub ( plus u closing{ − parenthesis} z { + comma} v ..{ so − } factor C induces an orientation - reversing fixed - point - free automorphism of A+, and the threex { sheets+ } arew2 j{ o ined − } toy give{ a+ branch} ) hyphen , $ point as of shown 5 inorder Figure 2 comma 5 . as 7 in , Figure where 5 period x , y 9 andperiod z are the other vertices of I adj acent to v and w , to w and u , andBeing to bipartite u and vcomma respectively M sub 5 i s the . Walsh map of an orientable hypermap A sub 5 to the power of plus = W to the power of minus 1 open parenthesis M sub 5 closing parenthesis .... of type \ centerlineopen parenthesis{ Figure 5 comma 5 . 5 7 comma . \quad 3 closingA f a parenthesis c e o f $ and M characteristic{ 5 }$ } minus 16 : .. the 1 2 hypervertices comma 1 2 hyperedges and 20 hyperfaces of A sub 5 to the power of plus \noindentcorrespondWe to nowthe black have vertices an orientable comma white bipartite vertices and facesmap of $ M M sub{ 55 period}$ .. with By the 20 symmetry hexagonal of the faces . \quad Figure 5 . 7 shows that eachmethod face of construction of I is covered comma Aut by A four sub 5 faces to the ofpower $M of plus{ ....5 } contains, $ Aut so I the thicksim surface = A sub of 5 $M times C{ sub5 } 2$ comma i s a of 4 order− sheeted 1 20 covering semicolon .... since A sub 5 to the power of plus has 1 20 \noindentblades openof parenthesis the sphere two for , each\quad edgewith of M branch sub 5 closing− points parenthesis of ordercomma A2 sub at 5 the to the\quad power1 of 2 plus white must vertices be regular comma $ v with{ − Aut } A: sub $ 5\quad to the powerFigure of plus5 . thicksim 8 shows = A sub 5 times C sub 2 period .... The antipodal thatfactor i .... f C an sub orientation 2 .... induces .... of an I .... induces orientation the hyphen cyclic reversing order .... fixed$ \rho hyphen= point ( hyphen $ abcde free .... ) automorphism of neighbouring .... of A vertices sub 5 to the of v powerin I of , plus then comma around .... and $ .... v the{ + }$ theorderis $( a { − } b { − } c { − } d { − } e { − } ) , $ giving a single unbranched sheet , whereas around $ v { − }$ \quad the order i s \quad $ ( a { + } d { + } b { + } e { + } c { + } ) , $ \quad so three sheets are j o ined to give a branch − point o f order 2 , as in Figure 5 . 9 .

\noindent Being bipartite $ , M { 5 }$ i s the Walsh map of an orientable hypermap $ A ˆ{ + } { 5 } = W ˆ{ − 1 } (M { 5 } ) $ \ h f i l l o f type

\noindent ( 5 , 5 , 3 ) and characteristic $ − 16 : $ \quad the 1 2 hypervertices , 1 2 hyperedges and 20 hyperfaces of $ A ˆ{ + } { 5 }$ correspond to the black vertices , white vertices and faces of $ M { 5 } . $ \quad By the symmetry of the

\noindent method of construction , Aut $ A ˆ{ + } { 5 }$ \ h f i l l contains Aut I $ \sim = A { 5 }\times C { 2 } , $ of order 1 20 ; \ h f i l l s i n c e $ A ˆ{ + } { 5 }$ has 1 20

\noindent blades ( two for each edge of $M { 5 } ) , A ˆ{ + } { 5 }$ must be regular , with Aut $ A ˆ{ + } { 5 } \sim = A { 5 }\times C { 2 } . $ \ h f i l l The antipodal

\noindent f a c t o r \ h f i l l $ C { 2 }$ \ h f i l l induces \ h f i l l an \ h f i l l orientation − r e v e r s i n g \ h f i l l f i x e d − point − f r e e \ h f i l l automorphism \ h f i l l o f $ A ˆ{ + } { 5 } , $ \ h f i l l and \ h f i l l the 16 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentFigure 5 period16 \quad 8 periodA .. . Rotations J . Breda around d ’ vertices Azevedo of I and ; G M . sub A .5 Jones : Platonic Hypermaps Figure16 5 A period . J . 9 Breda period d .. ’ Branching Azevedo of ; G M .sub A 5. Jonesover I : Platonic Hypermaps \ centerlinequotient ....{ iFigure s .... a non 5 hyphen. 8 . \ orientablequad Rotations .... regular around.... A sub vertices 5 hyphen hypermap of I and .... A $ sub M 5{ =5 A} sub$ 5} to the power of plus slash C sub 2 Figure 5 . 8 . Rotations around vertices of I and M = W to the power of minus 1 open parenthesis M sub 5 slash C sub 2 closing parenthesis ....5 of type \ centerline { Figure 5 . 9 . Figure\quad 5Branching . 9 . Branching of $ M of{ M5 5}$over over I I } open parenthesis 5 comma 5 comma 3 closing parenthesis and characteristic minus+ 8 period .. Equivalently−1 comma one can construct M sub 5 quotient i s a non - orientable regular A5− hypermap A5 = A5 /C2 = W (M5/C2) of slashtype C sub 2 directly from the 1 hyphen \noindentskeleton ofquotient P I-circumflex-zero\ h f i l l byi adding s \ h f 6i l hyphen l a non s ided− o faces r i e n ast a bin le Figure\ h f 5i l period l r e g u7 l comma a r \ h andf i l l then$ defineA { A5 sub} 5− =$ W hypermap to the power\ h of f i l l ( 5 , 5 , 3 ) and characteristic −8. Equivalently , one can construct M /C directly from the 1 - minus$ A 1{ open5 } parenthesis= A ˆ M{ + sub} 5{ slash5 } C sub/C 2 closing{ 2 parenthesis} = W comma ˆ{ − 1 } (M5 2 { 5 } /C { 2 } ) $ \ h f i l l o f type skeleton of P I − circumflex − zero by adding 6 - s ided faces as in Figure 5 . 7 , and then define as shown in−1 Figure 5 period 1 0 period .... open parenthesis An alternative construction for A sub 5 to the power of plus endash and hence for A \noindentA5 = W (( 5M5 ,/C 52), 3 ) and characteristic $ − 8 . $ \quad Equivalently , one can construct $ M { 5 } sub 5 endash i s to take + /Cas shown{ 2 } in$ Figure directly 5 . 1 from 0 . the ( An 1 − alternative construction for A5 – and hence for A5 – i s to Figuretake 5 period 1 0 period .. The construction of W open parenthesis A sub 5 closing parenthesis from P I skeletonthe graph Capitalof P Gamma$ I−circumflex consisting− ofzero the 20 $ vertices by adding and 60 face 6 − hyphens ided diagonals faces of as the in dodecahedron Figure 5 D . semicolon 7 , and each then define $ A { 5 } Figure 5 . 1 0 . The construction of W (A ) from P I =of W the ˆ{ .. −1 2 faces1 } Capital(M Phi of{ D5 carries} /C a pentagram{ 2 } in Capital) , Gamma $ comma5 .. to which we attach a disc representing a the graph Γ consisting of the 20 vertices and 60 face - diagonals of the dodecahedron D ; each of hyperedge Capital Phi sub minus of A sub 5 to the power of plus open parenthesis see Figure 5 period 1 1 open parenthesis a closing parenthesis \noindentthe 1as 2 faces shownΦ inof D Figure carries 5 a . pentagram 1 0 . \ h inf i l lΓ, ( Anto which alternative we attach construction a disc representing for $ a A ˆ{ + } { 5 }$ −− and hence for closing parenthesis semicolon+ the vertices of D adj acent in D to those in Capital Phi form $ A hyperedge{ 5 }$ −−Φ− iof sA to5 take( see Figure 5 . 1 1 ( a ) ) ; the vertices of D adj acent in D to those in Φ aform pentagon in Capital Gamma comma to which we attach a disc representing a hypervertex Capital Phi sub plus of A sub 5 to the power of plus open parenthesis see Figure a pentagon in Γ, to which we attach a disc representing a hypervertex Φ of A+ ( see Figure \ centerline5 period 1 1{ openFigure parenthesis 5 . 1 b 0 closing . \quad parenthesisThe construction closing parenthesis ofW period $ .... ( This A gives+{ 5 us} a5 2) hyphen $ from face hyphenP I } coloured map endash the 5 . 1 1 ( b ) ) . This gives us a 2 - face - coloured map – the dual of M5 – from which we obtain dual of+ M sub 5 endash from which we obtain A sub 5 to the power of plus \noindentA5 the graph $ \Gamma $ consisting of the 20 vertices and 60 face − diagonals of the dodecahedron D ; each o f the \quad 1 2 f a c e s $ \Phi $ of D carries a pentagram in $ \Gamma , $ \quad to which we attach a disc representing a

\noindent hyperedge $ \Phi { − }$ o f $ A ˆ{ + } { 5 } ( $ see Figure 5 . 1 1 ( a ) ) ; the vertices of D adj acent in D to those in $ \Phi $ form

\noindent a pentagon in $ \Gamma , $ to which we attach a disc representing a hypervertex $ \Phi { + }$ o f $ A ˆ{ + } { 5 } ( $ see Figure

\noindent 5 . 1 1 ( b ) ) . \ h f i l l This gives us a 2 − f a c e − coloured map −− the dual of $M { 5 }$ −− from which we obtain $ A ˆ{ + } { 5 }$ A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 1 7 \ hspaceby expanding∗{\ f i l each l }A of . the J .20 Breda vertices d to ’ a smallAzevedo disc representing ; G . A . a Jones hyperface : comma\quad incidentPlatonic with Hypermaps \quad 1 7 three hypervertices and threeA hyperedges . J . Breda period d ’Azevedo closing parenthesis ; G . A . Jones : Platonic Hypermaps 1 7 \noindentFigure 5 periodby expanding 1 1 period Capital each ofPhi thesub minus 20 vertices and Capital to Phi a sub small plus disc representing a hyperface , incident with threeby expanding hypervertices each of and the three20 vertices hyperedges to a small . disc ) representing a hyperface , incident with three Usinghypervertices either of these and two three models hyperedges comma it i. s ) now easily seen that A sub 5 corresponds to overbar G sub 5 rather than overbar G sub 4 period .... For example comma in A sub 5 to the power of plus the permutations r sub 1 r sub 2 and r sub 2 r sub 0 of order 5 Figure 5.11. Φ and Φ around\ centerline hypervertices{ Figure and $5 . 1 1 . \Phi −{ − }$+ and $ \Phi { + }$ } Using either of these two models , it i s now easily seen that A corresponds to G rather than hyperedges correspond to rotations rho and rho to the power of 2 by angles5 2 pi slash 5 and 4 pi slash5 5 about vertices of I G For example , in A+ the permutations r r and r r of order 5 around hypervertices and \noindentopen4. parenthesisUsing see either Figures of 5 period5 these 8 and two 5 period models 1 1 , closing1 it2 i parenthesis s now2 0 easily comma seen so they that are not $ conjugate A { 5 in}$ A sub corresponds 5 times C sub to 2 semicolon $\ overline {\}{ G } { 5 }$ hyperedges correspond to rotations ρ and ρ2 by angles 2π/5 and 4π/5 about vertices of I whenr a t h e we r factor than out C sub 2 comma A sub 5 ( see Figures 5 . 8 and 5 . 1 1 ) , so they are not conjugate in A5 × C2; when we factor out C2, A5 alsoalso has has this this property property period . .. Alternatively Alternatively comma , one one can can consider consider the the action on on the the blades blades of the of the element element \noindentg = r sub 0$ r\ suboverline 1 r sub{\}{ 0 r sub 2G comma} { 4 shown . in}$ Figure\ h f 5 i l period l For 1 2example comma where , in we $ regard A ˆ{ a+ blade} { as5 an} incident$ the edge permutations hyphen face pair $ in r { 1 } r {g 2=}r$0r1r and0r2, shown $ r in{ Figure2 } r 5 .{ 10 2} ,$ where of order we regard 5 around a blade hypervertices as an incident edgeand - face pair in thethe Walsh Walsh map map period . Figure 5 period 1 2 period .. The action of g = r sub 0 r sub 1 r sub 0 r sub 2 on a blade beta Figure 5 . 1 2 . The action of g = r r r r on a blade β \noindentWe see fromhyperedges Figure 5 period correspond 5 that g has to order rotations 2 on A sub $4 comma\rho 0$ whereas1 0 and2 Figure $ \rho 5 periodˆ{ 2 10} shows$ by that angles g has order $ 2 \ pi / 5 $ We see from Figure 5 . 5 that g has order 2 on A , whereas Figure 5 . 10 shows that g has order andhline $ 4 \ pi / 5 $ about vertices of I 4 3 on A sub 5 comma confirming that A sub 5 corresponds to G sub 5 period \noindentHaving constructed( see Figures A sub 1 comma5 . 8 period and 5 period . 1 1 period ) , comma so they A sub are 6 notcomma conjugate we have now in accounted $A { for5 all}\ 1 9times regular AC sub{ 52 hyphen} ; $ when we factor out $ C { 2 } ,A { 5 }$ hypermaps3 on A5, confirming that A5 corresponds to G5. in Table projHaving ective constructed to the power ofA 41, period ..., A6, icosahedronwe have now to the accounted power of Among for all these 1 9 regular sub P I toA the5− hypermaps power of comma to the power of there = A to\noindent the power ofalso open has parenthesis this property 2 closing parenthesis . \quad fromAlternatively 1 comma to are , and one to thecan power consider of 3 maps the the action sub proj on ective the to blades the power of ofthe : to the element powerg $ of = the proj r { sub0 great} r to the{ 1 power} r of ective{ 0 } dodecahedronr { 2 } to the, $ power shown of dodecahedron in Figure P 5 P . D 1 GD 2 = ,to where the power we regard of = A sub a blade 2 period as to an incident edge − f a c e p a i r in the Walsh map . 4. Amongthese, there (2)1 , 3 : theprojective dodecahedron = A1, the powerinTableprojective of A sub 1 commaicosahedron the P I = A are and mapstheprojective great dodecahedron PP DGD = A2. the 6 period .. Regular hypermaps with automorphism group A sub 5 times C sub 2 \ centerline66 period . 1 Regularperiod{ Figure .. Enumeration hypermaps 5 . 1 2 .of\ thequad with hypermapsThe automorphism action of g group $=A r 5 { ×0C}2 r { 1 } r { 0 } r { 2 }$ on a blade $ \Inbeta6 this . 1 case$ .} the Enumeration hypermap subgroups of theare the hypermaps normal subgroups H less or equal Capital Delta with Capital Delta slash H = sub thicksim G = AIn sub this 5 times case C the sub hypermap 2 semicolon subgroups these are the are intersections the normal H subgroups = A cap B of H normal≤ ∆ with subgroups∆/ H A= comma∼ G = BA5 of× CapitalC2; Delta with Capital Delta slash\noindentthese A thicksim areWe the = see A intersections sub from 5 Figure H 5= .A 5∩ thatB of g normal has order subgroups 2 on A $ , BA of{ ∆4 with} ,∆ $/ A whereas∼= A5 Figureand 5 . 10 shows that g has order and∆/ CapitalB ∼ Delta= C slash2, so B .. the thicksim hypermaps = C sub H 2 we comma require .. so arethe hypermaps the disj oint H we products require are H the= disj ointA × productsB of H .. = .. A times B of \ [ regular\regularr u l e { hypermaps3em hypermaps}{0.4 A pt and}\ A B] and with B automorphism with automorphism groups A sub groups 5 and CA5 suband 2 periodC2. ..S S ince G G decomposes decomposes as as AA sub5 × 5C times2 in a C unique sub 2 in way a unique , way each comma H determines .. each H determines A and B A uniquely and B uniquely , so each comma H corresponds so each H corresponds to a to aunique unique pair pair A A comma , B . B Byperiod§5 ..there By S are 5 there 1 9 possible are 1 9 possible hypermaps hypermaps A , A and comma by ..§4 and of [ by 4 ]S 4 of there open are square bracket 4 closing square bracket\noindent7 possibilities.. there3 are on 7 $ for A B{ , so5 we} obtain, $ confirming19.7 = 133 hypermaps that $ A H .{ 5 }$ corresponds to $ G { 5 } . $ possibilities for B comma so we obtain 1 9 period 7 = 133 hypermaps H period \ hspace ∗{\ f i l l }Having constructed $ A { 1 } ,...,A { 6 } , $ we have now accounted for all 1 9 regular $ A { 5 } − $ hypermaps

\ [ in Table { pr oj } e c t i v e ˆ{ 4 . } icosahedron ˆ{ Among these }ˆ{ , } { PI }ˆ{ ther e } = A ˆ{ ( 2 ) }ˆ{ 1 , } { are } and ˆ{ 3 } maps{ the }ˆ{ : } { pr oj e c t i v e }ˆ{ the pr oj }ˆ{ e c t i v e } { gre at } dodecahedron ˆ{ dodecahedron } PPD { GD } = ˆ{ = } A ˆ{ A { 1 } , } { 2 . } the \ ]

\noindent 6 . \quad Regular hypermaps with automorphism group $ A { 5 }\times C { 2 }$

\noindent 6 . 1 . \quad Enumeration of the hypermaps

\noindent In this case the hypermap subgroups are the normal subgroups H $ \ leq \Delta $ with $ \Delta / $ H $ = {\sim }$ G $ = $ $ A { 5 }\times C { 2 } ; $ these are the intersections H $=$ A $ \cap $ B of normal subgroups A , B of $ \Delta $ with $ \Delta / $ A $ \sim = A { 5 }$ and $ \Delta / $ B \quad $ \sim = C { 2 } , $ \quad so the hypermaps H we require are the disj oint products H \quad $ = $ \quad A $ \times $ B o f regular hypermaps A and B with automorphism groups $ A { 5 }$ and $ C { 2 } . $ \quad S ince G decomposes as $ A { 5 }\times C { 2 }$ in a unique way , \quad each H determines A and B uniquely , so each H corresponds to a unique pair A , B . \quad By \S 5 there are 1 9 possible hypermaps A , \quad and by \S 4 o f [ 4 ] \quad the re are 7 possibilities for B , so we obtain $ 1 9 . 7 = 133 $ hypermapsH. 18 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindent6 period 218 period\quad .. DescriptionA . J . of Breda the hypermaps d ’ Azevedo ; G . A . Jones : Platonic Hypermaps The 2 hyphen blade hypermaps B are open parenthesis where open brace i comma j comma k closing brace = open brace 0 comma 1 comma 2 \noindent18 A6 . J . . 2 Breda . \quad d ’ AzevedoDescription ; G . A of . Jones the hypermaps : Platonic Hypermaps closing6 .brace 2 . closing Description parenthesis : of the hypermaps B to the power of plus comma in which R sub i comma R sub j and R sub k transpose the two blades semicolon The 2 - blade hypermaps B are ( where { i , j , k } = {0, 1, 2}): \noindentB to the powerThe of2 − i commablade in hypermaps which R sub B i fixes are the ( two where blades\{ whilei , R j sub , k j comma $ \} R sub= k transpose\{ 0 them , semicolon 1 , 2 \} ) : $ B+, in which R ,R and R transpose the two blades ; B to the power of circumflex-i comma in whichi j R sub i transposesk the two blades while R sub j comma R sub k fix them period Bi, in which R fixes the two blades while R ,R transpose them ; \ centerlineThe .. 19 regular{ $ B A ˆ sub{ + 5 hyphen} , $ hypermapsi in which A in .. $R S 5 ..{ arei all} non,R hyphenj k { orientablej }$ and and without $ R { .. boundaryk }$ transpose comma .. so the .. it two blades ; } Bˆı, in which R transposes the two blades while R ,R fix them . follows from open square bracket 4 commai S 5 closing square bracket that theirj doublek coverings H = A times B are without boundary comma The 19 regular A − hypermaps A in §5 are all non - orientable and without boundary and\ centerline that { $ B ˆ{ i5 } , $ in which $R { i }$ fixes the two blades while $ R { j } ,R { k }$ transpose, so them it follows ; } from [ 4 , §5 ] that their double coverings H = A × B are without boundary of, these and that A to the power of plus = A times B to the power of plus i s orientable comma with rotation group Aut to the power of plus A to the power of plus thicksim = A sub 5 comma while A to the power of i = A times B to the power of i of these A+ = A ×B+ i s orientable , with rotation group Aut+A+ ∼= A , while Ai = A ×Bi and \ centerlineand A to the{ power$ B ˆ of{\ circumflex-ihat{\imath = A}} times, B $ to in the which power of $R i-circumflex{ i } are$ non transposes5 hyphen orientable the two period blades .. This while gives us $ 19 R orientable{ j } Aˆı = A ×Bˆı are non - orientable . This gives us 19 orientable hypermaps H = A+, and 1 14 hypermaps,R { Hk =}$ A to f thei x power them of . plus} comma and non - orientable hypermaps H = Ai and Aˆı( i = 0, 1, 2). 1 14 non hyphen orientable hypermaps H = A to the power of i and A to the power of circumflex-i open parenthesis i = 0 comma 1 comma 2 \noindentIf AThe has type\quad ( l19 , m regular , n ) then $ so A has{ 5A+}: −it$ is hypermaps, in fact , the A orientablein \quad \ doubleS 5 \quad coverare of A a , l l non − orientable and without \quad boundary , \quad so \quad i t closing parenthesis period + + 0 0ˆ 0 followsIfso A has that typefrom A iopen s [ the 4 parenthesis , antipodal\S 5 ] l commathat quotient their m comma P A double n= closingA /C coverings2 parenthesis. S ince H thenB $=$and so hasB Ahave to $ the type\times power(1, of2$, 2) plus B, A are:and .. it without is comma boundary in fact comma , and the that 0ˆ 0 0 0 0 1 1ˆ orientableA have double type cover ( l of, mA comman ), where so m = lcm ( m , 2 ) and n = lcm ( n , 2 ) ; similarly , A and A ˆ \noindentthathave A i type s theo f antipodal these(l0, m , n $ quotient0) A, while ˆ{ + PA }2 toand= the $ powerA A2 have $ of\ plus typetimes = A(l0 to, mB the0, n ˆ{ power )+ . }$ of plus i s slash orientable C sub 2 period , with .. S ince rotation B to thegroup power of 0$ and Aut B ˆ to{ + } Athe ˆ power{ +Being}\ of circumflex-zerosim an unbranched= A have{ double type5 } open covering, parenthesis $ while of A 1 comma, $A A+ has ˆ{ 2 commacharacteristici } = 2 $closing A2 parenthesis $χ(\Atimes ) and comma henceB ˆ A{ has toi the} genus$ power of 0 and and $ A ˆ{\hat{\imath}} = $ A $ \timesˆ B ˆ{\hat{\imath}}$ are non − orientable . \quad This gives us 19 orientable hypermaps H A1 to− theχ( powerA ) . of circumflex-zero The coverings have H type= openA0 and parenthesisA0 have l comma branch m to - the points power of of orderprime n 1 to on the the power 30 of / prime m closing parenthesis comma where$ =hyperedges m A to ˆ the{ + power} of A of, prime$if m and is = odd lcm open , and parenthesis on the 30 m comma/ n hyperfaces 2 closing parenthesis i f n is odd and , n with to the similar power of conditions prime = lcm open parenthesis n comma 1 14 non − orientable hypermaps H $ = A ˆ{ i }$ and $ A ˆ{\hat{\imath}} ( $ i $ = 0 , 1 , 2 closing parenthesis semicolon1 similarly1ˆ comma A to the power of 1 and2 A to the power2ˆ of one-circumflex have 2on ) l . and $ n for A and A , and on l and m for A and A . In each of these s ix cases typewe open can parenthesis therefore l compute to the powerχ( ofH prime) comma = 2χ( mA comma) − β nwhere to the powerβ i s of the prime total closing order parenthesis of branching comma ; while A to the power of 2 and A tobeing the power non of - circumflex-two orientable , Hhave has type genus open2 parenthesis− χ( H). l to the power of prime comma m to the power of prime comma n closing parenthesis periodIf Ahas type ( l ,m , n ) then so has $Aˆ{ + } : $ \quad it is , in fact , the orientable double cover of A , so that A i s the antipodal quotient P $ A ˆ{ + } = A ˆ{ + } /C { 2 } . $ \quad S i n c e $ B ˆ{ 0 }$ Being an unbranched double covering of A comma+ 0 A0ˆ to the1 power1ˆ 2 of plus2ˆ has characteristic 2 chi open parenthesis A closing parenthesis and Ar Ar, Ar Ar, Ar Ar, Ar henceand has $ B ˆ{\hat{0}}$havetype$( 1 , 2 , 2 ) , Aˆ{ 0 }$ and $genus A ˆ{\ 1 minushat{ chi0}} open$ parenthesis havetype( A closing l parenthesis $ , mˆ period{\prime .. The} coveringsn ˆ{\ Hprime = A to} the) power ,$ of 0 and where A to the $mˆ power{\ ofprime circumflex-zero} = $ havelcm branch (m,2)and hyphen points $nˆ of order{\ 1prime on the 30} slash=$Table m lcm(n ignored! ,2) ; similarly $ , Aˆ{ 1 }$ and $ A ˆ{\hat{1}}$ havehyperedges of A if m is odd comma and on the 30 slash n hyperfaces i f n is odd comma with similar conditions typeon l .. $ and ( n for l A ˆ{\ to theprime power} of 1, .. $ and m A $to the , power n ˆ{\ of circumflex-oneprime } ) comma ,$ .. whileand on l .. $Aˆ and m{ for2 A}$ to the and power $ A of ˆ 2{\ .. andhat A{2 to}} the$ powerhavetype of two-circumflex $( l period ˆ{\prime .. In each} of, these m six ˆ{\ casesprime we } , $ n ) . can therefore compute chi open parenthesis H closing parenthesis = 2 chi open parenthesis A closing parenthesis minus beta where beta i s the totalBeing order an of unbranched branching semicolon double .. coveringbeing of A $ , A ˆ{ + }$ has characteristic $ 2 \ chi ( $ A ) and hence has genusnon hyphen $ 1 orientable− \ chi comma H( has $ genusA ) .2 minus\quad chiThe open coverings parenthesis H closing $= parenthesis Aˆ{ 0 period}$ and $ A ˆ{\hat{0}}$ have branch − points of order 1 on the 30 / m hyperedgesA sub r to the of power A if of m plus is A odd sub ,r to and the on power the of 30 comma / n to hyperfaces the power of i 0 Af nsub is r to odd the , power with of similar circumflex-zero conditions A sub r to the power ofon comma l \quad to theand power n off o r1 A $ sub A r ˆ{ to1 the}$ power\quad of one-circumflexand $ A ˆ A{\ subhat r to{1}} the power, $ of\ commaquad and to the on power l \quad of 2 Aand sub m r to f o the r power $ A ˆ of{ 2 }$ circumflex-two\quad and $ A ˆ{\hat{2}} . $ \quad In each of these s ix cases we canTable therefore ignored! compute $ \ chi ( $ H $ ) = 2 \ chi ( $ A $ ) − \beta $ where $ \beta $ i s the total order of branching ; \quad being non − orientable , H has genus $ 2 − \ chi ( $ H ) .

\ [ A ˆ{ + } { r } A ˆ{ 0 } { r ˆ{ , }} A ˆ{\hat{0}} { r } A ˆ{ 1 } { r ˆ{ , }} A ˆ{\hat{1}} { r } A ˆ{ 2 } { r ˆ{ , }} A ˆ{\hat{2}} { r }\ ]

\ vspace ∗{3 ex }\ centerline {\ framebox[.5 \ l i n e w i d t h ] {\ t e x t b f { Table ignored!}}}\ vspace ∗{3 ex} A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 19 \ hspace6 period∗{\ 3f period i l l }A .. .Construction J . Breda of d the ’ hypermaps Azevedo ; G . A . Jones : \quad Platonic Hypermaps \quad 19 Row .. 1 period .. The .. first .. row .. of Table .. 5 .. gives the .. 42 .. double .. coverings .. of A sub 1 = .. P D .. and their \noindent 6 . 3 . \quad AConstruction . J . Breda d of ’ Azevedo the hypermaps ; G . A . Jones : Platonic Hypermaps 19 associates6 . 3 . period Construction .. The first entry of represents the hypermaps the orientable double cover open parenthesis P D closing parenthesis to the power of plus = D of typeRow open parenthesis 1 . 3 commaThe 2 first comma 5row closing of parenthesis Table comma 5 gives the 42 double coverings of \noindenttogether withRow it\ squad five associates1 . \quad commaThe including\quad thef i r s icosahedron t \quad row I = D\quad to theo power f Table of open\quad parenthesis5 \quad 2 closingg i v e s parenthesis the \quad of type42 \ openquad double \quad c o v e r i n g s \quad o f A = P D and their associates . The first entry represents the orientable double cover ( parenthesis$ A 1{ 1 5} comma= $ 2\ commaquad PD 3 closing\quad parenthesisand t h period e i r PD )+ = D of type ( 3 , 2 , 5 ) , associatesThe second entry . \quad representsThe a first pair of entry hypermaps represents P D to the the power orientable of 0 .. and P double D to the cover power of ( circumflex-zero PD $ ) ˆ{ comma+ } ..=$ both Doftype(3 non hyphen ,2 ,5) , together with it s five associates , including the icosahedron I = D(2) of type ( 5 , 2 , 3 ) . orientable comma 0 0ˆ \noindentof typeThe .. secondtogether open parenthesis entry with represents 3 it comma s five a 2 pair comma associates of hypermaps 10 closing , parenthesis including P D and .. theand P D characteristic icosahedron, both non .. minusI- orientable $ = 4 comma D , ˆ of{ ..( together 2 with ) }$ their oftype(5,2,3). .. 10 .. associatestype period ( 3 .. , 2 Both , 10 P ) D to and the characteristic power of 0 .. and −4, together with their 10 associates . Both 0 0ˆ ThePP secondD toD theand power entry P of representsD circumflex-zeroarise as double a arise pair as coverings ofdouble hypermaps coverings of P D ofP with P $ D branchD with ˆ{ branch0 -} points$ hyphen\quad at points theand six at P face the $ six D - centres faceˆ{\ hyphenhat{0}} centres, comma $ \quad but theyboth non − orientable , areo f, type but they\quad are( not3 , isomorphic 2 , 10 ) \ sincequad theyand have characteristic different patterns\quad of$ cuts− between4 ,$ these\quad branchtogether - with their \quad 10 \quad associates . \quad Both P ˆ ˆ $ Dnotpoints ˆ{ isomorphic0 } .$ \ For sincequad example theyand have the different map patterns P D0, proj of cuts ecting between onto theseB0 branch, i s bipartite hyphen points , whereas period P .. ForD0 i s not Pexample. $ D Alternatively ˆ{\ thehat map{ P0}} Dto ,$ the arise power permutation as of circumflex-zero double g = coveringsr r commar , having ofproj P orders ecting D with onto 5 , branch 1B and to the 2− power inpoints P D of zero-circumflex, B at0 and theB six0ˆ, comma face − i scentres bipartite comma , but they are not isomorphic since they have different0 patterns1 2 of cuts between these branch − p o i n t s . \quad For whereashas P orders D to the 5 andpower 10 of 0in i sP notD0 periodand P .. AlternativelyD0ˆ respectively comma , so P D0equal − similar P D0ˆ. In fact , this example the map P $ D ˆ{\hat{0}} , $ proj ecting onto $Bˆ{\hat{0}} , $ i s bipartite , whereas P theshows permutation that the g = Petrie r sub 0 polygons r sub 1 r sub of the2 comma maps having P D orders0 and 5 P commaD0ˆ have 1 and lengths 2 in P D5 andcomma 10 B , to since the power g is of 0 and B to the power of circumflex-zero$ D ˆ{ 0 }$ comma i s nothas orders . \quad 5 andAlternatively 10 , thethe permutation basic “ z gig ” $= ( or r { “ zag0 } ” )r from{ 1 which} r such{ 2 paths} , are $ formed having ; orders see Figure 5 , 6 1and2 . 1 for a inPD $ , Bˆ{ 0 }$ intypical P D toPetrie the power polygon of 0 and in P P D D to . the power of circumflex-zero respectively comma so P D to the power of 0 equal-similar P D to the power ofand circumflex-zero $ B ˆ{\hat period{0}} .. In fact, $ comma has this orders shows 5 that and the 10 Petrie polygons in P $ D ˆ{ 0 }$ and PFigure $ D ˆ{\ 6 .hat 1 .{0}} A$ Petrie respectively polygon in P , D so P $Dˆ{ 0 } equal−similar $ P $Dˆ{\hat{0}} of the maps P D to the power of 0 and P D to the0 power of circumflex-zero have lengths 5 and 10 comma since g is the basic .. quotedblleft z ig . $ (\ Thisquad In also fact shows , this shows that that P D thecan Petrie be polygons formed by applying Wilson ’ s quotedblrightPetrie .. operation open parenthesis [ 29 or ].. quotedblleft to D , transposing zag quotedblright faces closing and Petrie parenthesis polygons from while retaining the ofwhich the such mapsP paths are $Dˆ formed{ 0 semicolon}$ and see P Figure $ D 6 ˆ period{\hat 1{ for0}} a typical$ have Petrie lengths polygon in5 Pand D period 10 , since g is the basic \quad ‘ ‘ z i g ’ ’ \quad ( or \quad ‘‘ zag ’’ ) from which1 - skeleton such paths : are since formed D i s a ; non see - Figure bipartite 6 . 1 map for a of typical type Petrie{ 5 , 3 polygon} , with in PD Petrie . Figurepolygons 6 period of 1 period length .. A 10Petrie , polygon such in an P D operation must produce a non - bipartite regular open parenthesis This .. also .. shows .. that .. P D to the power of 0 .. can .. be .. formed .. by .. applying .. Wilson quoteright s .. Petrie .. \ centerline(A5 × C2)−{ Figuremap of 6 type . 1{ .10\ ,quad 3 } ,A with Petrie Petrie polygon polygons in of P length D } operation5 ; .. by open inspection square bracket of Table 29 closing 5 and square by the bracket preceding .. to remarks , it must be P D0. It follows that DP commaD0 is transposing covered by faces the and map Petrie{ 10 polygons , 3 } while5 in retaining Table 8 the of .. [ 1 hyphen 0 ] , and skeleton since : ..they since both D i s have a non 1 hyphen 20 \noindentbipartite ..( map This .. of\quad type ..a openl s o \ bracequad 5shows comma 3\quad closingthat brace\ commaquad P .. with $ D .. ˆ Petrie{ 0 } ..$ polygons\quad ..can of length\quad ..be 10 comma\quad ..formed such .. an\quad .. by \quad applying \quad Wilson ’ s \quad P e t r i e \quad operation \quad [ 29 ] \quad to D ,blades transposing faces and Petrie polygons while retaining the \quad 1 − s k e l e t o n : \quad since D i s a non − operationwe have .. must P D0 ∼= {10, 3}5.) b iproduce p a r t i t ae non\quad hyphenmap bipartite\quad regularo f type open\ parenthesisquad \{ 5 A sub, 3 5\} times, \ Cquad sub 2with closing\ parenthesisquad P e t rhyphen i e \quad map ofpolygons type open\ bracequad 10o fcomma length 3 \quad 10 , \quad such \quad an \quad operation \quad must produceS a imilarly non − bipartite the third and regular fourth entries $ ( inA the{ 5 first}\ rowtimes each representC { 2 } two) S -− orbits$ map , con o -f type \{ 10 , 3 \} , with Petrie polygons of length closingtaining brace comma 1 2 non with - i Petriesomorphic polygons hypermaps of length ; they are all non - orientable , those in the third entry 5having semicolon characteristic .. by inspection of−14 Table, branched 5 and by the over preceding the remarks 10 vertices comma it must and be 6P D to face the - power centres of 0 period of .. It follows that \noindentP D to the5 power ; \quad of 0 isby covered inspection by the map of open Table brace 5 10 and comma by 3 the closing preceding brace 5 in remarksTable 8 of open , it square must bracket be P 1 $ 0 Dclosing ˆ{ 0 square} . bracket $ \quad It follows that PP $ D D , ˆ{ while0 }$ those is covered in the fourth by the entry map have\{ 10 characteristic , 3 \} 5 in−8 Tableand are 8 branched of [ 1 0 over ] , the and vertices since they both have 1 20 blades commaof andP D since . they both have 1 20 blades we have P D to the power of 0 thicksim = open brace 1 0 comma 3 closing brace 5 period closing parenthesis \noindentRowwe 2 . haveThe P $ second D ˆ{ row0 }\ of Tablesim 5= consists\{ of double1 0 coverings , 3 of\}A2 5= P . GD ) and $ their S imilarlyassociates, the thirdthe and fourthsignificant entries in the first row each represent two S hyphen orbits comma con hyphen i (2) = onlyP GD ), so there differenceareisomorphisms from thewhich first rowbeingreducethethatnumber taining 1 2s,P nonGD hyphen∼ i somorphic hypermaps semicolon .... they are all non hyphen orientable comma those in the third entry isofself −dual (that \ hspacehavingP GD∗{\ characteristicf i l l }hypermapsS imilarly .. minusin 14 the comma this third row .. branched from and 42 fourth over to 2 the 1 entries ... 10 .. vertices in the .. and first .. 6 .. row face each hyphen represent centres .. of two P D commaS − o r .. b whilei t s , con − + + thoseThe in the first fourth entry entry represents have characteristic the great minus dodecahedron 8 and are branchedA2 = over ( P the GD vertices) = ofGD P D , period described in §3 \noindentRowand .. i 2 llustrated periodtaining .. The in 1 second Figure 2 non row− 3 . ofi 4 Tablesomorphic , together 5 consistswith hypermaps of double it s two coverings ; other\ h f i l of associates l Athey sub 2 are = , .. giving allP GD non threeand− their orientable , those in the third entry i tohypermaps the power of of associates genus 4 sub. s comma P GD to the power of open parenthesis 2 closing parenthesis to the power of comma to the power of the sub\noindent = sub thicksimAshaving in only characteristic Pthe GD sub case closing of parenthesis\quadA1, $the comma− 14 second to the , power $ entry\quad of s ignificantbranched represents so there over two difference the S\quad - are orbits isomorphisms10 \quad fromv e r t the i c e sub s \ whichquad and \quad 6 \quad f a c e − c e n t r e s \quad o f P D , \quad while firstthose rowcontaining being in the reduce fourth 1 sub 2 the non entry that - sub have number characteristic P GD i s of to the $ power− 8 of $ self and sub hypermaps are branched to the powerover ofthe hyphen vertices dual to of the PD power . of open parenthesisi somorphic that in hypermaps ; these have characteristic −12 and are branched over the six \noindent Row \quad 2 . \quad The second row0ˆ of Table0 5 consists of double coverings of $ A { 2 } = $ \quad P GD and their thisface row - from centres 42 to of 2 P 1 period GD . They include A2 and A2, bipartite and non - bipartite maps of type $The{ i1 first ˆ 0{ ,a entry 5 s s} o; c represents i a t e s }ˆ the{ great, } { dodecahedrons , P A sub GD 2 ˆto{ the( power 2 of ) plus}}ˆ ={ openthe parenthesis} { = {\ P GDsim closing}} parenthesisonly { P to the GD power}ˆ{ ofs plus =i g GD n i comma f i cant described} { ), in S 3 }$ so there $ di ff erence { are isomorphisms }$ from $ the { which }$ f i r s t $ rowand i llustrated being { inreduce Figure 3 period} { the 4 comma} that together{ withnumber it s two}$ other P GD associates $ i comma s { o giving f }ˆ{ threes e l orientable f }ˆ{ − dual } { hypermaps }ˆ{ ( thathypermaps}$ in of genus 4 period thisAs .. row in .. fromthe .. case 42 to.. of 2 .. 1 A .sub 1 comma .. the .. second .. entry .. represents .. two .. S hyphen orbits .. containing .. 1 2 .. non hyphen i somorphic hypermaps semicolon .. these have .. characteristic .. minus 1 2 .. and are branched over the .. six face hyphen Thecentres first of Pentry GD period represents .. They include the great A sub 2 dodecahedron to the power of circumflex-zero $ A ˆ{ + } and{ 2 A} sub= 2 to the( $ power P GD of comma $ ) ˆ to{ the+ } power= $ of 0 GD bipartite , described in \S 3 andand non i hyphen llustrated bipartite in maps Figure of type 3 open . 4 brace , together 1 0 comma with 5 closing it braces two semicolon other associates , giving three orientable hypermaps of genus 4 .

\ hspace ∗{\ f i l l }As \quad in \quad the \quad case \quad o f \quad $ A { 1 } , $ \quad the \quad second \quad entry \quad r e p r e s e n t s \quad two \quad S − o r b i t s \quad c o n t a i n i n g \quad 1 2 \quad non −

\noindent i somorphic hypermaps ; \quad these have \quad characteristic \quad $ − 1 2 $ \quad and are branched over the \quad s i x f a c e − centres of PGD . \quad They include $ A ˆ{\hat{0}} { 2 }$ and $ A ˆ{ 0 } { 2 ˆ{ , }}$ bipartite and non − bipartite maps of type \{ 1 0 , 5 \} ; 20 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentA sub 2 to20 the\ powerquad ofA comma . J . to Breda the power d ’ of Azevedo 0 which is ; obtained G . A by . applying Jones : Wilson Platonic quoteright Hypermaps s Petrie operation to I comma is i somorphic to the map \noindent20 A .$ J A . Breda ˆ{ 0 } d{ ’ Azevedo2 ˆ{ , ;}} G$ . A which . Jones is : obtained Platonic Hypermaps by applying Wilson ’ s Petrie operation to I , is i somorphic to the map open0 brace 10 comma 5 closing brace 3 in Table 8 of open square bracket 10 closing square bracket period A , which is obtained by applying Wilson ’ s Petrie operation to I , is i somorphic to the map { \{S ince102 ,P 5GD\} comma3 in B Table to the power 8 of of [ 1 10 .. and ] . B to the power of circumflex-one .. are all self hyphen dual comma .. so are P GD to the power 10 , 5 } 3 in Table 8 of [ 10 ] . of 1 .. and P GD to the power of one-circumflex semicolon .. thus the third entry 1 1ˆ 1 1ˆ \ hspacerepresents∗{\S twof i ince l l ....}SincePGD P S hyphenGD , B orbitsand $of length ,B Bˆ 3are comma{ all1 } self$ .... - giving\ dualquad 6 ,and .... sonon $are hyphen B P ˆ{\GD i somorphichat{and1}} P non$ GD\ hyphenquad; thus orientableare the a l l hypermaps s e l f − dual of , \quad so are P $ GDcharacteristicthird ˆ{ 1 entry}$ minus\quad 18 commaand P branched $ GD ˆ at{\ thehat 6 vertices{1}} and; $ 6 face\quad hyphenthus centres the of third P GD period entry .. The 1 2 hypermaps representedrepresents by two the fourth S - orbits entry of are length not new 3 : , .. giving P GD is 6 self non hyphen - i somorphic dual comma non while - orientable B to the power hypermaps of 2 and B of to the power of circumflex-two are\noindent thecharacteristic duals represents−18, branched two \ h f at i l l theS − 6 verticesorbits and of 6length face - centres 3 , \ h of f i lP l GDg i v .i n g The 6 \ 1h f2 i hypermapsl l non − i somorphic non − orientable hypermaps of 2 2ˆ ofrepresented B to the power by of the 0 and fourth B to the entry power are of notcircumflex-zero new : comma P GD so is these self -hypermaps dual , while are i somorphicB and B to thoseare the given by the second entry period ˆ ..\noindent Thusduals ofcharacteristicB0 and B0, so these $ − hypermaps18 , $ are branched i somorphic at the to those 6 vertices given by and the 6 second face − entrycentres . of PGD . \quad The 1 2 hypermaps representedtheThus second the row second by yields the 3row plus fourth yields 2 period entry3 + 6 2plus.6 are + 2 2 period.3 not = 21 new 3hypermaps = 2 : 1\ hypermapsquad . P GD period i s s e l f − dual , while $Bˆ{ 2 }$ and $ B ˆ{\hat{2}}$ areRowsRows the 3 dualsendash 3 – 6 6 period . .. SS imilarly imilarly comma , the the remaining remaining rows rows of Tableof Table 5 give 5 double give double coverings coverings of the hypermaps of the o f $ B ˆ{ 0 }$ and−1 $ B0ˆ {\hat{0−}}1 , $ so these hypermaps+ are i somorphic−1 to those0ˆ given by the second entry . \quad Thus Ahypermaps sub 3 = W toA the3 = powerW ( ofP minusD ), 1A open4 = W parenthesis( P zero P− Dcircumflex to the power− I of),A circumflex-zero5 = P A5 and closingA6 = parenthesisW ( P GD comma) A sub 4 = W to the power ofthesecondrowyields minusconstructed 1 open parenthesis in §5 . 3 P . zero-circumflex-I $3 + 2 sub .closing 6 parenthesis + 2 comma . 3 A sub = 5 = 2 P A sub1$ 5 to hypermaps. the power of plus and A sub 6 = W to the power of minus 1 open parenthesis P GD to the power of circumflex-zero closing parenthesis(1) constructed in S 5 period 3 period In the cases A3, A4 and A5 we have A − r =∼ Ar , so the first entry \noindentIn the .. casesRows A3 sub−− 3 comma6 . \quad A subS 4 .. imilarly and .. A sub , the 5 .. we remaining .. have .. A-r rows = sub of thicksim Table A5 sub give r to double the power coverings of open parenthesis of the 1 hypermaps closing $of A each{ 3 } of= rows W ˆ{ − 3 ,1 } 4 and( $ P 5 $ D represents ˆ{\hat{0}} a),A single S{ -4 orbit} = of W length ˆ{ − 1 } ( $ P $ zero−circumflex −I { ) parenthesis comma .. so .. the .. first .. entry .. of each .. of rows .. 3+ comma .. 4−1 0ˆ −1 zero−circumflex−I , } 3A containing{ 5 } = $ P the $ A orientable ˆ{ + } { 5 } hypermap$ and $ AAr{ 6 =} W=(D W),W ˆ{ −( 1 } ( $) Pand $ GD ˆ{\hat{0}} ) $ and− ..1 5 .. represents .. a .. single .. S hyphen orbit .. of length .. 3 .. containing .. the .. orientable .. hypermap .. A sub r to the power of plus W (M5). The second entry represents two S - orbits of length 6 , =constructed in \S 5 . 3 . W to the power of minus 1 open parenthesis D to the power of circumflex-zero closing parenthesis comma W to the power of minus 1 open \noindent In the \quadnon− c a s e s $ Ahypermaps{ 3 } ,A branched{ 4 over}$ \thequadhyperedgesand \andquadhyperfaces$ A of { 5 }$r \quad we \quad have \quad parenthesis to the power of zero-circumflex-I0 closing parenthesisˆ .. and .. W to the power of minus 1 open parenthesis M sub 5 closing parenthesis consistingSinceB1 =of( B )orientable 1ˆ = (B0)(01), thethirdentryof these3rows representsAthe. period$ A−r .. The = second{\sim entry} representsA ˆ{ ( two 1 S1 hyphen() ) and} B{ orbitsr } of length, $ 6\quad commaso \quad the \quad f i r s t \quad entry \quad o f each \quad o f rows \quad 3 , \quad 4 and \quad 5 \quad r e p r e s e n t s \quad a \quad s i n g l e \quad S − o r b i t \quad o f length \quad 3 \quad c o n t a i n i n g \quad the \quad o r i e n t a b l e \quad hypermap \quad consisting S ince B to the power of 1 = of sub open parenthesis sub B to the powerˆ of 0 closingˆ parenthesis to the power of non hyphen orientable $ Asame ˆ{ + hypermaps} { r } = as $ the second entry ; since B2 = (B2)(1) and B2 = (B2)(1), the fourth entry 1 openrepresents parenthesis two closing S -parenthesis orbits of three and B non to the - orientable power of circumflex-one hypermaps to , the branched power of over hypermaps the hypervertices sub = open parenthesis B to the power of zero-circumflex$ W ˆ{ − closing1 } parenthesis( D ˆ{\ openhat parenthesis{0}} ) 0 1 closing , W parenthesis ˆ{ − 1 sub} comma( ˆ{ tozero the− powercircumflex of branched−I sub} the) $ third\quad to theand power\quad of over and hyperfaces of A . sub$ W entry ˆ{ − to the1 power} (M of ther sub{ of5 to} the) power . $ of hyperedges\quad The sub second these 3 to entry the power represents of and sub two rows S to− theorbits power of of hyperfaces length sub 6 , represents −1 0ˆ to the powerNow ofA of6 A= the toW the( powerP GD of) r periodi s S - invariant , so the first entry in the final row represents \ [ consisting { S i n c e B ˆ{ 1 } = } +o f {−1( }ˆ0ˆ{ non − } { B ˆ{ 0 } ) } o r i e n t a b l e { 1 { ( } ) samea single hypermaps S - invariant as the second orientable entry semicolon hypermap sinceA6 B= toW the( powerGD ). ofThe 2 = open second parenthesis entry represents B to the power two ofS 2 closing parenthesis to the powerand- orbitsof B open ˆ{\ parenthesishat of length{1}}} 1ˆ closing{ 3hypermaps : parenthesis these} areand{ = nonB to (- the orientable power B ˆ{\ ofhat circumflex-two hypermaps{0}} ) =} open of( characteristic parenthesis 0 1 B ) to{ the−,36 power,}ˆ{ ofbranched two-circumflex} { closingthe t h i r d }ˆ{ over } { entry }ˆ{ the } { o f }ˆ{ hyperedges } { these 3 }ˆ{ and } { rows }ˆ{ h y p e r f a c e s } { r e p r e s e n t s }ˆ{ o f } parenthesisbranched to the over power the of open6 hyperedges parenthesis and 1 closing 6 hyperfaces parenthesis of commaA6. theBy fourth the S entry - invariance of A6, these 6 A represents{hypermapsthe }ˆ two{ r S reappear hyphen} . \ orbits] in the of thirdthree non and hyphen fourth orientable entries , hypermaps so this row comma yields branched only 1 over+ 2.3 the = 7 hyperverticeshypermaps and. hyperfaces of A sub r period + + Now AThis sub 6gives = W a to total the power of 133 of hypermaps minus 1 open , parenthesis of which 1 P 9 GD ( namely to the powerA1 ,of ..., circumflex-zeroA6 and their closing associates parenthesis ) .. i s S hyphen invariant comma\noindentare .. so orientable thesame first hypermaps .. entry . in Among the as final the these row second represents 133 entry hypermaps a ; since there $ B are ˆ{ 2 } 2 1= maps ( B, ˆ{ of2 which} ) 3 ˆ{ ( 1 ) }$ and $ Bsingle( ˆ namely{\ Shat hyphen{ D2}} , invariant I and= GD orientable ( ) B ˆ are{\ hypermap orientablehat{2}} A sub . 6) to ˆ These{ the( power maps 1 of plus) , which} =, W $ are to the the all power obtained fourth of minus entry by applying 1 open parenthesis GD to the power of circumflex-zerorepresentsthe seven closing twodouble S parenthesis− orbits period of three .. The secondnon − entryorientable represents hypermaps two , branched over the hypervertices andScoverings hyphen hyperfaces orbits in [ .. 4 of ] length to $ P A D .. ,3{ P :r .. I} these and. P .. $ areGD non , are hyphen l ist orientable ed and described .. hypermaps in Table .. of characteristic 6 . .. minus 36 comma .. branched over the 6 hyperedges and 6 hyperfaces of A sub 6 period .. By the S hyphen invariance of A sub 6 comma these 6 hypermaps Now $ A { 6 } = W ˆ{+ − 0ˆ 1 }0 (1ˆ $1 P $2ˆ GD2 ˆ{\+hat0ˆ{0}}0 )1ˆ $ 1\quad i s S − i n v a r i a n t , \quad so the f i r s t \quad entry in the final row represents a reappear in the third and fourthA1 entriesA1 A comma1 A1 soA1 thisA row1 A yields1 A only2 A 12 plusA2 2 periodA2 A 32 = 7 hypermaps period s iThis n g l e gives S − a totalinvariant of 133 hypermaps orientable comma hypermap of which 1 $ 9 A open ˆ{ parenthesis+ } { 6 } namely= AW sub ˆ{ 1 − to the1 } power( of GD plus ˆ comma{\hat period{0}} period) period . $ comma\quad AThe sub second 6 to the power entry of represents plus and their two associates closing parenthesis S are− o .. r orientable b i t s \quad periodo f .. length Among these\quad .. 1333 :hypermaps\quadTablethese ignored! there ..\ arequad .. 2are 1 .. maps non − commao r i e .. n t of a b which l e \quad 3 .. openhypermaps parenthesis\ namelyquad of D comma characteristic I \quad $ −and GD36 closing , $ parenthesis\quad branched .. are orientable period .. These maps comma which are all obtained by applying the seven double overcoverings the in 6 open hyperedges square bracket and 4 6 closing hyperfaces square bracket of $ to A P D{ comma6 } P. I $ and\quad P GDBy comma the are S l− istinvariance ed and described of in $ Table A { 6 period6 } , $ theseA sub 6 1 hypermaps to the power of plus A sub 1 to the power of circumflex-zero A sub 1 to the power of 0 A sub 1 to the power of one-circumflex A sub 1 to thereappear power of 1 in A sub the 1 to third the power and of fourth circumflex-two entries A sub , so1 to this the power row of yields 2 A sub 2only to the $ power 1 of + plus 2 A sub . 2 to 3 the power = of7 zero-circumflex $ hypermaps . A sub 2 to the power of 0 A sub 2 to the power of circumflex-one A sub 2 to the power of 1 \ hspaceTable ignored!∗{\ f i l l } This gives a total of 133 hypermaps , of which 1 9 ( namely $ A ˆ{ + } { 1 } ,... , A ˆ{ + } { 6 }$ and their associates )

\noindent are \quad orientable . \quad Among these \quad 133 hypermaps there \quad are \quad 2 1 \quad maps , \quad o f which 3 \quad ( namely D , I and GD ) \quad are orientable . \quad These maps , which are all obtained by applying the seven double

\noindent coverings in [ 4 ] to PD , P I and PGD , are l ist ed and described in Table 6 .

\ [ A ˆ{ + } { 1 } A ˆ{\hat{0}} { 1 } A ˆ{ 0 } { 1 } A ˆ{\hat{1}} { 1 } A ˆ{ 1 } { 1 } A ˆ{\hat{2}} { 1 } A ˆ{ 2 } { 1 } A ˆ{ + } { 2 } A ˆ{\hat{0}} { 2 } A ˆ{ 0 } { 2 } A ˆ{\hat{1}} { 2 } A ˆ{ 1 } { 2 }\ ]

\ vspace ∗{3 ex }\ centerline {\ framebox[.5 \ l i n e w i d t h ] {\ t e x t b f { Table ignored!}}}\ vspace ∗{3 ex} A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 2 1 \ hspace6 period∗{\ 4f period i l l }A .. .Two J t . riacontahedra Breda d ’ Azevedo ; G . A . Jones : \quad Platonic Hypermaps \quad 2 1 In the next section we will describe how the Walsh maps of suitable associates of A sub 1 to the power of plus comma period period period comma \noindent 6 . 4 . \quadATwo . J .t Breda riacontahedra d ’ Azevedo ; G . A . Jones : Platonic Hypermaps 2 1 A sub6 6 . to 4 the . power Two of plus t riacontahedra may be obtained from j ust two polyhedra comma namely the rhombic triacontahedron RT open square+ bracket+ 8 comma S 2 period 7 closing \noindentIn the nextIn the section next we section will describe we will how describe the Walsh how maps the of Walsh suitable maps associates of suitable of A1 , ..., associatesA6 may of $ A ˆ{ + } { 1 } squarebe bracket obtained from j ust two polyhedra , namely the rhombic triacontahedron RT [ 8 , §2 . 7 ] ,and . the . small . ste l , lated A triacontahedron ˆ{ + } { 6 ST}$ open square bracket 8 comma S 6 period 4 closing square bracket comma so that in a sense comma mayand be the obtainedsmall ste from l lated j usttriacontahedron two polyhedraST [ , 8 namely , §6 . 4 ] the , so rhombic that in a triacontahedronsense , RT and ST RT generate [ 8 , \S 2 . 7 ] RT andall ST 19generate of the orientable hypermaps in the previous section . For us , allthe .. 19 .. most of the .. importantorientable .. hypermaps properties .. of in ..these the .. triacontahedra previous .. section are period that .. they For .. are us comma bipartite .. the maps .. most , .. important \noindentpropertiesand of these the triacontahedra small ste are l lated that they triacontahedron are bipartite maps commaST [ 8 with , \ 1S 26 plus . 20 4 and ] , 1 so 2 plus that 1 2 in a sense , RTand ST generate with 12 + 20 and 12 + 12 vertices respectively , and that they have automorphism groups a lvertices l \quad respectively19 \quad commao f the .. and\quad that ..o they r i e n have t a b ..l e automorphism\quad hypermaps groups containing\quad in A\ subquad 5 timesthe C\ subquad 2 periodprevious .. This\quad s e c t i o n . \quad For \quad us , \quad the \quad most \quad important propertiescontaining ofA5 these× C triacontahedra2. This section are is devoted that they to their are construction bipartite maps and description , with $ . 1 2 + 20 $ and $ 1 section is devotedTo construct to their construction the rhombic and triacontahedron description period RT we take a dual pair D and I ( as convex 2To + construct 1 2 the $ rhombic triacontahedron RT we take a dual pair D and I open parenthesis as convex polyhedra ) , with their relative s iz es and positions in R3 chosen so that corresponding edges verticespolyhedra respectivelyclosing parenthesis , comma\quad withand their that relative\quad s izthey es and have positions\quad in Rautomorphism to the power of 3 groups chosen so containing that corresponding $ A edges{ 5 }\times of D and I bisect each other at right - angles . Then RT is the convex hull of D ∪ I in R3, C of{ D2 and} I. bisect $ \ eachquad otherThis at right hyphen angles period .. Then RT is the convex hull of D cup I in R to the power of 3 comma sectionhaving is 30 devoted rhombic to their faces construction ( with these andpairs description of edges as . diagonals ) and 60 edges having, both 30 .. sets rhombic permuted .. faces .. transitively open parenthesis by with these pairs .. of edges as .. diagonals closing parenthesis .. and 60 .. edges comma .. both sets \ hspace ∗{\ f i l l }To constructAut the RT rhombic= Aut triacontahedron I = Aut D ∼= A RT5 × weC2. take a dual pair D and I ( as convex permutedThis group transitively has two by orbits on the vertices of RT , namely the 1 2 5 - valent vertices of I and the 20 Aut RT = Aut I = Aut D thicksim = A sub 5 times C sub 2 period \noindent3 - valentpolyhedra vertices of ) D , , with coloured their black relative and white s iz in Figure es and 6 positions . 2 . As ain map $ R, RT ˆ{ i3 s spherical}$ chosen so that corresponding edges Thisand group bipartite has two , but orbits not on regular the vertices . of RT comma namely the 1 2 5 hyphen valent vertices of I and the 20 3 hyphen valent verticesFigure of D comma 6 . 2 coloured . The black rhombic and white triacontahedron in Figure 6 period RT 2 period .. As a map comma RT i s spherical \noindentand bipartiteof comma D and but I bisectnot regular each period other at right − a n g l e s . \quad Then RT is the convex hull of D $ \cup $ We construct the small stellated triacontahedron ST from concentric copies I and I = I inFigure $6 R period ˆ{ 3 2} period, $ .. The rhombic triacontahedron RT − + lambda − I (λ > 1) of I , the vertices v of I and v = λv of I coloured white and black respec - havingWe construct 30 \−quad the smallrhombic stellated\quad triacontahedronf a c e− s \quad ST− from( with concentric+ these− copies pairs+ I sub\ minusquad ..o and f edges I sub plus as =\quad d i a g o n a l s ) \quad and 60 \quad edges , \quad both s e t s t ively . Each edge vw of I separates two triangular faces uvw and xvw , and we may choose λ permutedlambda-I sub transitively minus open parenthesis by lambda greater 1 closing parenthesis of I comma the vertices v sub minus of I sub minus and v sub plus = so that the corresponding edge v w of I and the l ine - segment u x meet , bisecting each lambda v sub minus of I sub plus coloured− white− and black− respec hyphen + + other at right - angles ; then u , v , x and w are coplanar , and are the vertices of a rhombus \ centerlinet ively period{AutRT .. Each edge $=$ vw of IAutI separates+ − $=$ two+ triangular AutD− faces $ \ uvwsim and= xvw comma A { 5 and}\ wetimes may chooseC lambda{ 2 } . $ } u v x w , one of the 30 such rhombic faces of ST ( one for each edge of I ). This is a so+ that− the+ − corresponding edge v sub minus w sub minus .. of I sub minus .. and the l ine hyphen− segment u sub plus x sub plus meet comma \noindentnon - convexThis polyhedrongroup has two, with orbits the 24 on vertices the vertices of I+ and ofI− RT( ,the namely latter arethe hidden 1 2 5 inside− valent ST ; vertices of I and the bisectingin Figure each 6 . 3 , several faces are made transparent to reveal three of them ) . 20other 3 − atvalent right hyphen vertices angles semicolon of D , thencoloured u sub plus black comma and v subwhite minus in comma Figure x sub 6 plus . 2 and . \ wquad sub minusAs a are map coplanar , RT comma i s spherical and are the There are 60 edges , a pair v w and v w corresponding to each edge vw of I ; in particular verticesand bipartite of a rhombus , but not regular+ − . − + ST i s bipartite , it s 1 - skeleton being isomorphic to that of I − circumflex − zero As a map , u sub plus v sub minus x sub plus w sub minus comma .. one of the 30 .. such rhombic .. faces of ST. open parenthesis one for each edge of I sub \ centerlineST has type{ Figure{ 4 , 5 6} .and 2 . characteristic\quad The rhombic24 − 60 + triacontahedron 30 = −6; being orientable RT } , it has genus 4 . By minusradial closing proj parenthesis ection period it is a .. 3 This - sheeted is a covering of the sphere , with branch - points of order 1 at the non hyphen convex polyhedron comma with the 24 vertices of I sub plus and I sub minus open parenthesis the latter are hidden inside ST \noindent1 2 verticesWe constructv− of I− the( where small the stellated vertex - figure triacontahedron i s a pentagram ST{ from5 / 2 concentric} , resembling copies that for $ I { − }$ \quad and semicolona vertex of GD ) . $ Iin Figure{ + } 6 period= $ 3 comma several faces are made transparent to reveal three of them closing parenthesis period $There lambda are− 60I edges{ − comma } ( a pair\lambda v sub plus> w sub1 minus )$ and v of sub I minus , the w sub vertices plus corresponding $v { − to }each$ edge o f vw $ Iof I{ semicolon − }$ in and particular $ v { + } = ST\lambda i s bipartitev comma{ − it }$ s 1 hyphen o f $ skeleton I { + being}$ isomorphic coloured to white that of and I-circumflex-zero black respec sub− period .. As a map comma ST has type open bracet i v 4 e comma l y . \ 5quad closingEach brace edge vw of I separates two triangular faces uvw and xvw , and we may choose $ \lambda $ soand that characteristic the corresponding 24 minus 60 plus edge 30 = minus $ v 6{ semicolon − } w being{ − orientable }$ \quad commao f it $ has I genus{ − 4 } period$ \quad .. By radialand the proj ectionl i n e − segment $ uit is{ a+ 3 hyphen} x sheeted{ + }$ covering meet of , the bisecting sphere comma each with branch hyphen points of order 1 at the 1 2 vertices v sub minus otherof I sub at minus right open− parenthesisangles ; where then the $ vertex u { hyphen+ } figure, v i s a{ pentagram − } , open x brace{ + 5}$ slash and 2 closing $ w brace{ − comma }$ are resembling coplanar that for , aand are the vertices of a rhombus vertex$ u of GD{ + closing} v parenthesis{ − } x period{ + } w { − } , $ \quad one o f the 30 \quad such rhombic \quad faces of ST ( one for each edge of $ I { − } ) . $ \quad This i s a

\noindent non − convex polyhedron , with the 24 vertices of $ I { + }$ and $ I { − } ( $ the latter are hidden inside ST ;

\noindent in Figure 6 . 3 , several faces are made transparent to reveal three of them ) .

\noindent There are 60 edges , a pair $ v { + } w { − }$ and $ v { − } w { + }$ corresponding to each edge vw of I ; in particular

\noindent ST i s bipartite , it s 1 − skeleton being isomorphic to that of $ I−circumflex −zero { . }$ \quad As a map , ST has type \{ 4 , 5 \} and characteristic $ 24 − 60 + 30 = − 6 ; $ being orientable , it has genus 4 . \quad By radial proj ection i t i s a 3 − sheeted covering of the sphere , with branch − points of order 1 at the 1 2 vertices $ v { − }$ o f $ I { − } ( $ where the vertex − figure i s a pentagram \{ 5 / 2 \} , resembling that for a vertex of GD ) . 22 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentFigure 6 period22 \quad 3 periodA .. . The J . small Breda stellated d ’ triacontahedronAzevedo ; G ST. A comma . Jones with :one Platonic Hypermaps face shaded and three inner vertices revealed \ centerline22 A .{ JFigure . Breda 6 d . ’ 3Azevedo . \quad ; GThe . A small . Jones stellated : Platonic triacontahedron Hypermaps ST , with one } The Euclidean isometryFigure group 6 . 3 of . ST i The s equal small to stellated triacontahedron ST , with one Autface I sub shaded minus and = Aut three I sub inner plus verticesthicksim = revealed A sub 5 times The EuclideanC sub 2 comma isometry group of ST i s equal to \noindentacting transitivelyface shaded on it s white and vertices three comma inner black vertices vertices revealed comma edges and faces period .. However comma as a map TheST Euclideanhas the larger isometry automorphism group group of ST i s equal to AutI = AutI ∼= A × C , Aut ST thicksim = S sub 5 times C sub 2 comma− + 5 2 \ [and Autacting i s therefore I transitively{ − regular } = on : .. it Aut it s is white comma I vertices{ in+ fact}\ comma , blacksim the vertices= map Aopen , edges{ brace5 }\ and 4 commatimes faces 5 . closingC However{ brace2 } 6 , comma as, \ a] map the dual of the regular map open braceST 5 comma has the 4 closing larger brace automorphism 6 group in .. open square bracket 10 comma .. Table .. 8 closing square bracket .. and .. open square bracket 1 2 comma .. Tabelle .. I I closing square \noindent acting transitively on it s white vertices , black vertices , edges and faces . \quad However , as a map bracket period .. To .. see .. this comma .. one .. checksAutST .. ∼that= S ..5 × a ..C2 cyclic, .. permutation STopen has parenthesis the larger u sub automorphism plus v sub minus group x sub plus w sub minus closing parenthesis of vertices around one face extends to an automorphism of and i s therefore regular : it is , in fact , the map { 4 , 5 } 6 , the dual of the regular map { 5 order 4 interchang hyphen , 4 } 6 in [ 10 , Table 8 ] and [ 1 2 , Tabelle I I ] . To see this , one \ [ing Aut black ST and white\sim vertices= semicolon S { 5 ..}\ thustimes Aut to theC power{ 2 of} plus, ST\ ] has order 1 20 comma .. contains the Euclidean rotation checks that a cyclic permutation (u+v−x+w−) of vertices around one face extends to an groupautomorphism .. isomorphic of to order .. A sub 4 interchang5 comma .. and - has .. an .. element .. of order 4 comma .. so .. it .. must .. be .. isomorphic to .. S sub 5 comma with the antipodal symmetry generating the factor C sub 2 period open parenthesis Note that the .. quotedblleft obvious quotedblright involution ing black and white vertices ; thus Aut+ ST has order 1 20 , contains the Euclidean rotation comma\noindent and i s therefore regular : \quad it is , in fact , the map \{ 4 , 5 \} 6 , the dual of the regular map \{ 5 , 4 \} 6 group isomorphic to A , and has an element of order 4 , so it must be intransposing\quad [ pairs 10 , v sub\quad plusTable comma5 \ vquad sub minus8 ] comma\quad iand s not\ anquad automorphism[ 1 2 , period\quad closingTabelle parenthesis\quad II]. \quad To \quad see \quad t h i s , \quad one \quad checks \quad that \quad a \quad c y c l i c \quad permutation isomorphic to S , with the antipodal symmetry generating the factor C . ( Note that the “ $6 period ( u 5 period{ + } .. Alternativev 5 { − } constructionsx { + } w { − } ) $ of vertices around2 one face extends to an automorphism of order 4 interchang − obvious ” involution , transposing pairs v , v , i s not an automorphism . ) We will now use the triacontahedra RT and ST to construct+ − Walsh maps for representatives \noindent6 . 5 .ing Alternative black and white constructions vertices ; \quad thus $ Aut ˆ{ + }$ ST has order 1 20 , \quad contains the Euclidean rotation ofWe each will of the now 6 S use hyphen the triacontahedraorbits of regular orientable RT and .. ST open to parenthesisconstruct WalshA sub 5 maps times C for sub representatives 2 closing parenthesis of hyphen hypermaps .. open parenthesisgroup \quad and henceisomorphic comma indirectly to \quad comma$ A { 5 } , $ \quad and has \quad an \quad element \quad o f order 4 , \quad so \quad i t \quad must \quad be \quad isomorphic to \quad $ S each{ 5 of} the, 6 $ S - orbits of regular orientable (A5 × C2)− hypermaps ( and hence , indirectly , toto obtain obtain all 152all 152 of the of hypermaps the hypermaps in Tables in 4 Tables and 5 closing 4 and parenthesis 5 ) . period withFirstly the ST antipodal comma being symmetry bipartite comma generating i s the Walsh the map factor W open $ parenthesis C { 2 } H closing. ( parenthesis $ Note for that a hypermap the \quad H semicolon‘‘ obvious by inspection ’’ involution , transposingFirstly ST pairs , being $ vbipartite{ + } , i, s the v Walsh{ − } map, W $ ( i H s ) not for a an hypermap automorphism H ; by . inspection ) of of ST , H is orientable , of genus 4 and type ( 5 , 5 , 2 ) , with 1 2 hypervertices , 1 2 hyperedges ST comma H is orientable comma of genus 4 and type open parenthesis 5 comma 5 comma 2 closing parenthesis comma with 1 2 hypervertices \noindentand 306 hyperfaces . 5 . \quad correspondingAlternative to the constructions black vertices , white vertices and faces of ST . The i commasometry 1 2 hyperedges group andof ST 30 acts as a group of automorphisms of H , permuting the blades transitively hyperfaces corresponding to the black vertices comma white vertices and faces of ST period .. The i sometry , so H i s regular with Aut H ∼= A × C . We have classified all such hypermaps H , and by \noindentgroup of STWe acts will as a now group use of automorphisms the triacontahedra5 of H2 comma RT permuting and ST the to blades construct transitively Walsh comma maps so H for representatives ofinspection each of the 6 S − orbits of regular orientable \quad $ ( A { 5 }\times C { 2 } ) − $ hypermaps \quad ( and hence , indirectly , i s regular with Aut H thicksim = A sub 5 times C sub 2 period .. We have+ (12) classified all(12) such hypermaps H comma and by inspection toof obtain Table5 all in § 1526 . 2 of we the see hypermaps that H must in be Tables the associate 4 and 5 )(A .2 ) = GD of the orientable of Table 5 in S 6+ period 2 we see that H must be the associate .... open parenthesis A sub 2 to the power of plus closing parenthesis to the power hypermap A2 = GD in the second row . of open parenthesisIf we 1 retain 2 closing parenthesis the 1 -= skeleton GD to the powerof ST of open , parenthesis but replace 1 2 closing it sparenthesis rhombic .... of thefaces orientable Firstlyhypermap ST A , sub being 2 to the bipartite power of plus , i = s GD the in the Walsh second mapW row period ( H ) for a hypermap H ; by inspection of with 20 hexagons (u v w u v w ), one covering each face ( uvw ) of I as in Figure 6 . 4 , STIf .. , weH ..is retain orientable .. the .. 1 ,hyphen of+ genus− skeleton+ − 4+ .. and of− .. type ST comma ( 5 .., but5 , .. 2 replace ) ,with .. it s .. 1 rhombic 2 hypervertices .. faces .. with , .. 1 20 2 .. hyperedges hexagons and 30 we obtain I − circumflex − zero hyperfacesopen parenthesis corresponding u sub plus v sub to minus the. w black sub plus vertices u sub minus , whitev sub plus vertices w sub minus and closing faces parenthesis of ST . comma\quad oneThe covering i sometry each face open parenthesisgroup of uvw ST closing acts parenthesis as a group of I of as in automorphisms Figure 6 period 4 of comma H , we permuting obtain I-circumflex-zero the blades sub transitively period , so H i s regular with Aut H $ \sim = A { 5 }\times C { 2 } . $ \quad We have classified all such hypermaps H , and by inspection

\noindent o f Table 5 in \S 6 . 2 we see that H must be the associate \ h f i l l $ ( A ˆ{ + } { 2 } ) ˆ{ ( 1 2 ) } = GD ˆ{ ( 1 2 ) }$ \ h f i l l of the orientable

\noindent hypermap $ A ˆ{ + } { 2 } = $ GD in the second row .

I f \quad we \quad r e t a i n \quad the \quad 1 − s k e l e t o n \quad o f \quad ST , \quad but \quad r e p l a c e \quad i t s \quad rhombic \quad f a c e s \quad with \quad 20 \quad hexagons $ ( u { + } v { − } w { + } u { − } v { + } w { − } ) , $ one covering each face ( uvw ) of I as in Figure 6 . 4 , we obtain $ I−circumflex −zero { . }$ A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 23 \ hspaceAs we∗{\ sawf in i lS l 6}A period . J 3 . comma Breda .. thisd ’ i Azevedos the Walsh ; map G . W A open . Jones parenthesis : \quad A subPlatonic 4 to the power Hypermaps of plus closing\quad parenthesis23 .. for the regular orientable hypermap \ hspace ∗{\ f i l l }As we sawA in . J\ .S Breda6 . 3 d ,’ Azevedo\quad this ; G . i A s . the Jones WalshmapW : Platonic $ Hypermaps ( Aˆ{ + 23} { 4 } ) $ \quad for the regular orientable hypermap A sub 4 to the power of plus of type open parenthesis 5 comma 5 comma+ 3 closing parenthesis comma a double covering of the sphere with branch As we saw in §6 . 3 , this i s the Walsh map W (A4 ) for the regular orientable hypermap hyphen+ points of order 1 at the 20 \noindentA4 of type$ A ( 5 ˆ{ , 5+ ,} 3 ){ ,4 a double}$ of covering type ( of5 the, 5 sphere , 3 ) with , a doublebranch - covering points of orderof the 1 at sphere the 20 with branch − points of order 1 at the 20 faceface hyphen - centres centres of of I . I period f aFigure c e − 6centres period 4 period of I .. . A face of I-circumflex-zero covering a face of I Figure 6 . 4 . A face of I − circumflex − zero covering a face of I As explained in S 5 period 3 comma the map M sub 5 = W open parenthesis A sub 5 to the power of plus closing parenthesis .. can be constructed As explained in §5 . 3 , the map M = W (A+) can be constructed from the 1 - skeleton of from\ centerline the 1 hyphen{ Figure skeleton 6 of . I-circumflex-zero 4 . \quad A f5 a c e o f $5 I−circumflex −zero $ covering a face of I } I − circumflex − zero ( or equivalently of ST ) by attaching 20 hexagonal faces as in Figure 5 . 7 ; open parenthesis or equivalently of ST closing parenthesis by attaching 20 hexagonal faces as in Figure 5 period 7 semicolon in this case we have \noindentin this caseAs explained we have a 4 in - sheeted\S 5 . covering 3 , themap of the sphere $M { ,5 with} = branch $ W - $ points ( A of ˆ order{ + } 2{ at5 the} ) $ \quad can be constructed from the 1 − s k e l e t o n o f a1 4 2hyphen white sheeted vertices covering . S imilarly of the sphere , if we comma replace .. with the branch faces hyphenof ST with points 1 of 2 order decagons 2 at the , each .. 1 2 covering white vertices 10 period $ IS− imilarlycircumflex comma−zero if we replace $ the faces of ST with 1 2 decagons comma each covering 10 faces of I as in ( orfaces equivalently of I as in of ST ) by attaching 20 hexagonal faces as in Figure 5 . 7 ; in this case we have Figure 6 period 5 comma .. we obtain W+ open parenthesis A sub 6 to the power of plus closing parenthesis .. as a 6 hyphen sheeted covering of a 4Figure− sheeted 6 . 5 , covering we obtain of W the(A sphere6 ) as a, 6\quad - sheetedwith covering branch of− thepoints sphere of with order branch 2 at - thepoints\quad 1 2 white vertices . the sphereof order with 3 branch at the hyphen 1 2 white points vertices of . Sorder imilarly 3 at the , 1 if 2 white we replace vertices period the faces of ST with 1 2 decagons , each covering 10 faces of I as in Figure 6 . 5 . A face of W (A+) covering 1 0 faces of I Figure 6 period 5 period .. A face of W open parenthesis A sub6 6 to the power of plus closing parenthesis covering 1 0 faces of I \noindentRepresentativesFigure 6 of . the 5 remaining, \quad we S obtainW- orbits ( and $ ( of at Aˆ least{ + one} { of6 those} ) considered $ \quad aboveas a 6) can− sheeted covering of the sphere with branch − p o i n t s o f Representativessimilarly be obtained of the remaining from theS hyphen rhombic orbits triacontahedron open parenthesis and RT of . at Forleast example one of those , RT considered i s the above Walsh closing parenthesis can ordersimilarly 3at be obtained the 1 2 from white the rhombic vertices triacontahedron . RT period .. For example comma RT i s the Walsh map W (I(12)) of the hypermap I(12) of type ( 5 , 3 , 2 ) , an associate of I and of D = A+. If we map W open parenthesis I to the power of open parenthesis 1 2 closing parenthesis closing parenthesis of1 the hypermap I to the power of open \ centerlineretain the{ Figure 1 - skeleton 6 . of5 .RT\quad and replaceAface it ofW s faces $( with 20 Aˆ hexagons{ + } { ,6 corresponding} ) $ covering to triples 1 0 faces of I } parenthesis 1 2 closing parenthesis of type open parenthesis 5 comma 3 comma 2 closing parenthesis+ comma(2) an associate of I and of D = A sub 1 to of faces RT around a common white vertex as in Figure 6 . 6 , we obtain W ((A3 ) ), where the power+ (2) of plus period .. If we + \noindent(A3 ) , ofRepresentatives type ( 5 , 3 , 3 of ) , the is remaining the dual of S −A3 ;orbitsthis i ( s and a 2 - of sheeted at least covering one of of the those sphere considered above ) can retainwith the branch 1 hyphen - points skeleton of of order RT and 1 at replace the 1 it 2 s blackfaces with vertices 20 hexagons of RT comma . corresponding to triples similarlyof faces RT bearound obtained a common from white the vertex rhombic as in Figure triacontahedron 6 period 6 comma RT we . obtain\quad WFor open exampleparenthesis , open RT parenthesis i s the Walsh A sub 3 to the power Figure 6 . 6 . A face of W ((A+)(2)) covering 3 faces of RT of plus closing parenthesis to the power of open parenthesis 2 closing3 parenthesis closing parenthesis comma where \noindentIf , insteadmap , W we $ replace ( I the ˆ{ faces( 1 of RT 2 with ) } 1 2) decagons $ of the , corresponding hypermap $ to I quintuples ˆ{ ( 1 of faces 2 ) }$ of type ( 5 , 3 , 2 ) , an associate of I and ofD open parenthesis A sub 3 to the power of plus closing parenthesis to the power of open parenthesis+ (12) 2 closing parenthesis comma of type .. open $ =of ART ˆ{ around+ } { a1 common} . $ black\quad vertexI f we as in Figure 6 . 7 , we obtain W ((A4 ) ) as a double parenthesiscovering 5 comma of the 3 sphere comma with 3 closing branch parenthesis - points comma of order .. is the 1 dual at of the A sub 20 white 3 to the vertices power of of plus RT semicolon ; here .. this i s a 2 hyphen sheeted coveringr e t a i n of the the sphere 1 − skeleton with of RT and replace it s faces with 20 hexagons , corresponding to triples (A+)(12) has type ( 5 , 3 , 5 ) . branch4 hyphen points of order 1 at the 1 2 black vertices of RT period \noindentFigure 6 periodof faces 6 period RT .. around A face of a W common open parenthesis white vertex open parenthesis as in FigureA sub 3 to 6 the . 6 power , we of plusobtainW closing parenthesis $ ( ( to theA ˆ power{ + } of{ open3 } parenthesis) ˆ{ ( 2 2 closing ) } parenthesis) , $closing where parenthesis covering 3 faces of RT If comma instead comma we replace the faces of RT with 1 2 decagons comma corresponding to quintuples of faces \noindentof RT around$ a( common A ˆ{ black+ } vertex{ 3 } as in)Figure ˆ{ ( 6 period 2 ) 7} comma, $ we o obtain f type W\ openquad parenthesis( 5 , 3 open , 3 parenthesis ) , \quad A subis 4the to the dual power of of plus$ A closing ˆ{ + parenthesis} { 3 } to; the $ power\quad oft open h i s parenthesis i s a 2 − 1 2sheeted closing parenthesis covering closing of the parenthesis sphere as witha double branchcovering− ofpoints the sphere of with order branch 1 athyphen the points 1 2 ofblack order vertices.. 1 .. at the of 20 RT white . vertices of RT semicolon .. here open parenthesis A sub 4 to the power of plus closing parenthesis to the power of open parenthesis 1 2 closing parenthesis has type open parenthesis 5\ centerline comma 3 comma{ Figure 5 closing 6 parenthesis. 6 . \quad periodAfaceofW $( ( Aˆ{ + } { 3 } ) ˆ{ ( 2 ) } ) $ covering 3 faces of RT } \noindent If , instead , we replace the faces of RT with 1 2 decagons , corresponding to quintuples of faces

\noindent of RT around a common black vertex as in Figure 6 . 7 , we obtainW $ ( ( A ˆ{ + } { 4 } ) ˆ{ ( 1 2 ) } ) $ as a double covering of the sphere with branch − points of order \quad 1 \quad at the 20 white vertices of RT ; \quad here

\noindent $ ( A ˆ{ + } { 4 } ) ˆ{ ( 1 2 ) }$ hastype(5 ,3 ,5) . 24 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentFigure 6 period24 \quad 7 periodA .. . A J face . Breda of W open d ’ parenthesis Azevedo open ; G parenthesis . A . Jones A sub : 4 Platonic to the power Hypermaps of plus closing parenthesis to the power of open parenthesis24 A 1 2 . closing J . Breda parenthesis d ’ Azevedo closing ;parenthesis G . A . Jones covering : Platonic 5 faces of Hypermaps RT \ centerlineThis S hyphen{ Figure orbit comma 6 . 7 arising . \quad from bothAfaceofW ST and RT comma $( thus ( provides Aˆ{ + a l} ink{ between4 } ) these ˆ{ ( two triacon 1 2 hyphen ) } ) $ covering 5 faces of RT } Figure 6 . 7 . A face of W ((A+)(12)) covering 5 faces of RT tahedra period 4 \noindentThis S -This orbit S ,− arisingorbit from , arising both ST from and RT both , thus ST and provides RT , a thus l ink provides between these a l ink two betweentriacon - these two triacon − Wetahedra have now . accounted for all six S hyphen orbits of regular orientable .. open parenthesis A sub 5 times C sub 2 closing parenthesis hyphen hypermapstahedra period . We have now accounted for all six S - orbits of regular orientable (A5 ×C2)− hypermaps . By Bytaking .. taking antipodal .. antipodal .. quotients comma , and .. and applying .. applying .. the the .. above above .. face face hyphen - transformations transformations .. to to .. the .. non hyphen Weorientable have now maps accounted P ST = ST for slash all C sub six 2 .. S and− orbits P RT = RTof slash regular C sub orientable 2 comma we l\ ikewisequad obtain$ ( the A Walsh{ 5 maps}\times C { 2 } the non - orientable maps P ST = ST /C and P RT = RT /C , we l ikewise obtain the ) for− representatives$ hypermaps of the . S hyphen orbits of regular A sub2 5 hyphen hypermaps in S 52 period .. S ince the remaining open parenthesis non hyphen By Walsh\quad mapstaking for\quad representativesantipodal of\ thequad S -q orbitsu o t i e n of t s regular , \quadA5and− hypermaps\quad applying in §5 .\quad S incethe the\quad above \quad f a c e − transformations \quad to \quad the \quad non − orientableremaining closing ( non parenthesis - regular open parenthesis A sub 5 times C sub 2 closing parenthesis hyphen hypermaps can all be obtained from the regularorientable A sub 5 hyphenmapsPST hypermaps $=$ ST $ / C { 2 }$ \quad andPRT $=$ RT $/ C { 2 } , $ we l ikewise obtain the Walsh maps orientable ) regular (A × C )− hypermaps can all be obtained from the regular A − hypermaps forby taking representatives double coverings of comma5 the2 S we− haveorbits shown of how regular all the 19 plus $ A 133{ =5 152} regular − $ hypermaps hypermaps5 in in \S 5 . \quad S ince the remaining ( non − by taking double coverings , we have shown how all the 19 + 133 = 152 regular hypermaps in §§5 S– S 6 5 canendash be 6 derived can be derived from ST from and ST RTand RT. period \noindent7 period ..orientable Regular hypermaps ) regular with automorphism $ ( A group{ 5 ..}\ S subtimes 4 timesC C sub{ 22 } ) − $ hypermaps can all be obtained from the regular $ A 7{ .5 } Regular − $ hypermaps hypermaps with automorphism group S4 × C2 77 period . 1 .1 period Enumeration .. Enumeration of the the hypermaps hypermaps byThe taking arguments double for G coverings = S sub 4 times , we C sub have 2 are shown s imilar how to all those the in S S $ 6 19 period + 1 endash 133 6 period = 152 3 for $ A sub regular 5 times hypermapsC sub 2 comma in but \S The\S 5 arguments−− 6 can for be G derived= S4 × C from2 are ST s imilar and RT to those . in §§6 . 1 – 6 . 3 for A5 × C2, but with two withextra two complications : extra complications : 1 ) Unlike A ,S has a subgroup of index 2 , namely A , so there exist orientable , as well as \noindent1 closing parenthesis7 . \quad5 ..4 UnlikeRegular A sub hypermaps 5 comma S subwith 4 has automorphism a subgroup4 of index group 2 comma\quad namely$ S A{ sub4 4}\ commatimes so thereC exist{ orientable2 }$ comma non - orientable regular S − hypermaps S . A further consequence is that each S covers one of as well as 4 the seven 2 - blade hypermaps B ( namely S /A ), so we can form only s ix , rather than seven \noindentnon hyphen7 orientable . 1 . \quad regularEnumeration S sub 4 hyphen of hypermaps the hypermaps S4 period .. A further consequence is that each S covers one of disj oint products H = S × B from S . the seven 2 hyphen blade hypermaps B open parenthesis namely S slash A sub 4 closing parenthesis comma so we can form only s ix comma 2 ) A second complication arises from the fact that the direct decomposition of G as S ×C is not rather\noindent than sevenThe arguments for G $ = S { 4 }\times C { 2 }$ are s imilar4 2 to those in \S \S 6 . 1 −− 6 . 3 f o r unique : although the factor C , being the centre of G , i s unique , there are two subgroups $ Adisj{ oint5 products}\times H = SC times{ B2 from} S, period $2 but with two S0,S00 = S , of index 2 in G , with G = S0 × C = S00 × C . ( One can take G to be the i sometry extra2 closing complications parenthesis∼ 4 A second : complication arises from2 the fact that2 the direct decomposition of G as S sub 4 times C sub 2 is group of a cube , with S0 = S × {1} it s rotation group and S00 = (A × {1}) ∪ ((S \ A ) × {−1}) the not unique : .. although the factor C4 sub 2 comma being the centre of G comma4 i s unique comma4 4 there are two subgroups \noindentsubgroup1 leaving) \quad invariantUnlike the $ A two{ inscribed5 } ,S t etrahedra{ 4 }$ , shown has a in subgroup Figure 7 . of 1 . index ) 2 , namely $ A { 4 } , $ S to the powerFigure of 7 prime . 1 . comma Two S to t etrahedra the power of inscrib prime prime ed in = a sub cube thicksim , forming S sub a 4 stella comma o of ctangula index 2 in G comma with G = S to the power ofso prime there times exist C sub orientable 2 = S to the power , as of well prime as prime times C sub 2 period open parenthesis One can take G to be the i sometry nongroup− oforientable a cube comma regular with S to $ the S power{ 4 of} prime − $ = S hypermaps sub 4 times open S . brace\quad 1 closingA further brace it consequence s rotation group is and that S to theeach power S covers of prime one of primethe = seven open parenthesis 2 − blade A hypermapssub 4 times open B ( brace namely 1 closing S $ brace / closing A { parenthesis4 } ) cup , $ open so parenthesis we can form open parenthesis only s ix S sub , rather 4 backslash than A seven subdisj 4 closing oint parenthesis products times H open$=$ brace S minus $ \times 1 closing$ brace B from closing S parenthesis . the subgroup leaving invariant the two inscribed t etrahedra comma shown in Figure 7 period 1 period closing parenthesis \noindentFigure 7 period2 ) A 1 period second .. Two complication t etrahedra inscrib arises ed in from a cube the comma fact forming that a the stella direct o ctangula decomposition of G as $ S { 4 } \times C { 2 }$ i s not unique : \quad although the factor $ C { 2 } , $ being the centre of G , i s unique , there are two subgroups $ S ˆ{\prime } , S ˆ{\prime \prime } = {\sim } S { 4 } ,$ of index2inG, withG $= Sˆ{\prime } \times C { 2 } = S ˆ{\prime \prime }\times C { 2 } . ( $ One can take G to be the i sometry

\noindent group of a cube , with $ S ˆ{\prime } = S { 4 }\times \{ 1 \} $ it s rotation group and $ S ˆ{\prime \prime } = ( A { 4 }\times \{ 1 \} ) \cup ((S { 4 }\setminus A { 4 } ) \times \{ − 1 \} ) $ the subgroup leaving invariant the two inscribed t etrahedra , shown in Figure 7 . 1 . )

\ centerline { Figure 7 . 1 . \quad Two t etrahedra inscrib ed in a cube , forming a stella o ctangula } A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 25 A .It J follows . Breda that for d each ’ Azevedo regular S ; sub G 4 . hyphen A . Jones hypermap : \ Squad commaPlatonic the six 2 hyphenHypermaps blade\ hypermapsquad 25 B disj oint from S It follows that for each regular $ S { 4 } − $ hypermap S , the six 2 − blade hypermaps B disj oint from S are groupedA . J . into Breda pairs d B ’ to Azevedo the power ; ofG prime . A . comma Jones B : to the Platonic power of Hypermaps prime prime satisfying 25 It S follows times B that to the for power of prime = sub thicksim S times B to the power of prime prime period .... S ince there are 1 3 such hypermaps each regular S − hypermap S , the six 2 - blade hypermaps B disj oint from S \noindentS with to theare power grouped4 of open into parenthesis pairs see $ S Aut B ˆ to{\ theprime power} of 4, period B sub ˆ{\ Hprime to the power\prime of 1 closing}$ parenthesis satisfying comma S sub$ \ =times sub thicksimB ˆ{\prime } are grouped into pairs B0, B00 satisfying S ×B0 = S ×B00. S ince there are 1 3 such hypermaps =to the{\ powersim of}$ we S S sub $ \ 4times to the powerB ˆ of{\ thereforeprime times\prime C∼ sub} 2 period. $ The\ h tof i l the l S power ince of there obtain hline are from 1 3 i such s determined hypermaps by the rule that comma i f S denotes(see to 11) 3, period pairing 1 divided by 2 to the power of period 6 = 39 non hyphen isomorphic regular hypermaps to the power of H S §4. we S∴ × C . Theobtain isdeterminedbytherulethat,ifSdenotes H = S×B \ [S { with with}ˆ{Aut(H =∼ see4 }\2 S { Aut }ˆ{ 4˚ 1 ..6 }ˆ{ 1 ) , } { H } the{ unique= {\sim }}ˆ{ we } S ˆ{\ t h e r e f o r e } { 4 } sub the to the power of = sub unique to the power of S times13.pai B2 =39non−isomorphicregularhypermaps \times C { 2 } . The ˆ{ obtain }\ r u l e {3em}{0.4 pt } ˆ{ i s determined by the rule that , 2 hyphen blade hypermap S slash A sub 4 covered by S comma0 then S times00 B to the power of prime0 thicksim00 = S times B to the power of prime 2 - blade hypermap S /A4 covered by S , then S ×B ∼= S ×B i f and only if S × B =∼ S × B ; all primei f i f and S only denotes if overbar} S{ times1 B 3 to the . power{\ mathring of prime{ =pai sub}}\ thicksimf r aoverbar c { 1 }{ S times2 }ˆ B{ to. the 6 power} = of prime 39 prime non semicolon− isomorphic regularsuch isomorphisms hypermaps between}ˆ{ H } products{ the }ˆ of{ = pairs} { ofunique 2 - blade}ˆ hypermaps{ S \times can beB read}\ ] off from Figure all8 such of [ isomorphisms4 ] . between products of pairs of 2 hyphen blade hypermaps can be read off from Figure7 . 2 8 . of open Description square bracket of 4 closing the hypermaps square bracket period 7 period 2 period .. Description of the hypermaps −1 T −circumflex−zero The hypermaps S = S1 and S2 in §4 ( namely T and W ( )) are both orientable , \noindentThe hypermaps2 − blade S = S sub hypermap 1 and S Ssub $ 2 in / S 4 A open{ parenthesis4 }$ covered namely by T and S , W then to the S power $ \times of minus 1B open ˆ{\ parenthesisprime }\ to thesim power= of $ S $that\times i s , B ˆ{\prime \prime }$ i f andonly if $\ overline {\}{ S }\times B ˆ{\prime } = {\sim }\ overline {\}{ S } T-circumflex-zero closing parenthesis closing+ parenthesis are both orientable comma that i+ s comma + \timesr0, r1, r2B∈ ˆ{\S4 \primeA4, so S\prime= B ; }thus;there $ is no double covering H = S = S ×B , and s ince r sub+ 0 commai + r subˆı 1 comma r sub 2 in S sub 4 backslash A sub 4 comma so overbar S = B to the power of plus semicolon thus there is no allB such× B = isomorphisms∼ B × B for each between i the products pairing of ofthe pairs other six of double 2 − blade coverings hypermaps of S is can given be by read off from double covering H = S to the power of plusi = S timesi B to theˆı powerˆı of plus comma and s ince Figure 8 of [ 4 ] . S = S ×B =∼ S ×B = S ( i = 0, 1, 2). B to the power of plus times B to the power of i = sub thicksim B to the power of plus times B to the power2 of circumflex-i for each i the pairing In the cases S = S3 and S4, we have r0, r1 ∈ S4 \ A4 and r2 ∈ A4, so S = B ; hence of theS2 other= S six×B double2 does coverings not exist of S , andis given by by [ 4 , Fig . 8 ] the pairing is \noindentS to the power7 . of2 i . =\ Squad timesDescription B to the power of of i the = sub hypermaps thicksim S times B to the power of circumflex-i = S to the power of i-circumflex open parenthesis i = 0 comma 1 comma 2 closing0 parenthesis1 period0ˆ 1ˆ + 2ˆ S ∼= S , S =∼ S , S =∼ S . \noindentIn the ....The cases hypermaps .... S .... = S S sub $ 3= .... and S ....{ S1 sub}$ 4 comma and $ .... S we{ ....2 have}$ .... in r sub\S 04 comma ( namely r sub T 1 in and S sub $Wˆ 4 backslash{ − A1 sub} 4( .... ˆ{ andT−circumflex −zero } ) ) $ are both orientable , that i s , .... rS sub ince 2 inS A1, sub ..., S 44 commaare without .... so .... boundary overbar S , = so B are to the all power 39 of of the 2 semicolon hypermaps .... hence H = S × B . S ince S1 Sand to theS2 power, lying of in 2= S -S orbits times B of to lengths the power 3 and of 2 1 does , are not both exist orientable comma and , by so open are the square(1 +bracket 3).3 = 4 12 commanon - Fig period .. 8 closing square + bracket\noindenti somorphic the pairing$ r hypermaps is { 0 } , H arising r { 1 from} , these r orbits{ 2 }\ ; thein rotationS { group4 }\AutsetminusH , correspondingA { 4 } , $ so $\ overline {\}{ S } = B ˆ{ + } ; $ thus there is no double covering H $= S ˆ{ + } = $ S $ \times B ˆ{ + } , $ and s i n c e Sto to the the power subgroup of 0 thicksim of G = =S4 S× toC2 thegenerated power of 1by commar0r1, r S1r to2 and the powerr2r0, ofiscircumflex-zero isomorphic to =A sub4 × thicksimC2 in each S to the power of one-circumflex $ B ˆ{ + }\times B ˆ{ i } = {\sim } B ˆ{ + }\times B ˆ{\hat{\imath}}$ for each i the pairing of the other six double coverings of S is given by commacase S to . the The power remaining of plus = sub 9 hypermaps thicksim S to S the ( in power S - orbits of circumflex-two of lengths period 6 and 3 containing S3 and + SS ince4) are S sub non 1 comma - orientable period ;period each period gives risecomma to S a sub single 4 are orientable without boundary covering comma H = soS are, with all 39 rotation of the hypermaps H = S times B period .. \ centerline { +$ S ˆ{ i } = $ S $ \times B ˆ{ i } = {\sim }$ S $ \times B ˆ{\hat{\imath}} = S ˆ{\hat{\imath}} S incegroup S sub 1Aut H ∼= S4, while the other two non - isomorphic coverings H of S are non - orientable ($i$=0 , 1 , 2 ) .$ } 0 00 and( since S sub 2 comma one lying can inS check hyphen that orbits eachof lengths of 3 the and three 1 comma subgroupsare both orientableS ,S commaor A so4 × areC the2 openof parenthesis 1 plus 3 closing parenthesisindex period 2 3 in = 1 2 G non contains hyphen at least one of the three generators ri of G ) . \noindenti somorphicIn hypermaps the \ h f H i l arising l c a s efrom s \ theseh f i l l orbitsS \ semicolonh f i l l $ the = rotation S { group3 }$ Aut\ toh f the i l l powerand of\ h plus f i ll H comma$ S corresponding{ 4 } , $ \ h f i l l we \ h f i l l have \ h f i l l $ r { 0 } , r { 1 }\ in S {+ 4 }\0 0ˆ setminus1 1ˆ 2 A2ˆ { 4 }$ \ h f i l l and \ h f i l l $ r { 2 }\ in A { 4 } to the subgroup of G = S sub 4 times C subS 2r generatedSr , Sr bySr , rS subr S 0r r subSr 1 comma r sub 1 r sub 2 and r sub 2 r sub 0 comma is isomorphic to A sub, $ 4 times\ h f i C l l subso 2\ inh f i l l $\ overline {\}{ S } = B ˆ{ 2 } ; $ \ h f i l l hence each case period .. The remaining 9 hypermaps S open parenthesis in S hyphen orbits of lengths 6 and 3 containing S sub 3 and Table ignored! \noindentS sub 4 closing$ S parenthesis ˆ{ 2 } are= $non S hyphen $ \times orientableB semicolon ˆ{ 2 } each$ doesgives rise not to exista single orientable, and by covering [ 4 , H Fig = S . to\ thequad power8 ]of theplus comma pairing is with rotation \ [group S ˆ{ Aut0 }\ to thesim power= of plus S ˆ H{ thicksim1 } , = S S sub ˆ{\ 4hat comma{0}} while= the{\ othersim two} nonS hyphen ˆ{\hat isomorphic{1}} coverings, S ˆ{ H+ of} S are= non{\ hyphensim } orientableS ˆ{\hat{2}} . \ ] open parenthesis since .. one .. can .. check that .. each .. of the three .. subgroups S to the power of prime comma S to the power of prime prime .. or A sub 4 times C sub 2 .. of index .. 2 .. in .. G \noindentcontains atS least i n c onee of$ theS three{ 1 generators} ,...,S r sub i of G closing parenthesis{ 4 }$ period are without boundary , so are all 39 of the hypermaps H $ =S sub$ rS to $the\ powertimes of$ plus B S . sub\quad r to theS power i n c e of $ 0 commaS { 1 S sub}$ r to the power of circumflex-zero S sub r to the power of 1 comma S sub r to theand power $ of S one-circumflex{ 2 } , $ S sub l y i r n to g the in power S − oforbits 2 S sub of r to lengths the power 3 of and circumflex-two 1 , are both orientable , so are the $ ( 1 +Table 3 ignored! ) . 3 = 1 2$non − \noindent i somorphic hypermaps H arising from these orbits ; the rotation group $ Aut ˆ{ + }$ H , corresponding to the subgroup of G $= S { 4 }\times C { 2 }$ generated by $ r { 0 } r { 1 } , r { 1 } r { 2 }$ and $ r { 2 } r { 0 } , $ is isomorphic to $A { 4 }\times C { 2 }$ in each case . \quad The remaining 9 hypermaps S ( in S − orbits of lengths 6 and 3 containing $ S { 3 }$ and

\noindent $ S { 4 } ) $ are non − orientable ; each gives rise to a single orientable covering H $ = S ˆ{ + } , $ with rotation

\noindent group $ Aut ˆ{ + }$ H $ \sim = S { 4 } , $ while the other two non − isomorphic coverings H of S are non − o r i e n t a b l e ( s i n c e \quad one \quad can \quad check that \quad each \quad o f the th ree \quad subgroups $ S ˆ{\prime } , S ˆ{\prime \prime }$ \quad or $ A { 4 }\times C { 2 }$ \quad o f index \quad 2 \quad in \quad G contains at least one of the three generators $ r { i }$ o f G ) .

\ [ S ˆ{ + } { r } S ˆ{ 0 } { r } , S ˆ{\hat{0}} { r } S ˆ{ 1 } { r } , S ˆ{\hat{1}} { r } S ˆ{ 2 } { r } S ˆ{\hat{2}} { r }\ ]

\ vspace ∗{3 ex }\ centerline {\ framebox[.5 \ l i n e w i d t h ] {\ t e x t b f { Table ignored!}}}\ vspace ∗{3 ex} 26 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentAs in .. S 626 period\quad 2 ..A one . .. J can . Breda .. compute d ’ the Azevedo .. type .. ; and G .. . characteristicA . Jones .. : of Platonic each H .. from Hypermaps that .. of S period .. This information26 A . i J s . given Breda in Table d ’ Azevedo .. 7 comma ; G .. . each A . entryJones describing : Platonic a representative Hypermaps H .. = S sub r times B of an \noindentS hyphen orbitAs in of length\quad sigma\S 6 period . 2 ..\quad For spaceone saving\quad commacan some\quad columnscompute were thegrouped\quad togethertype semicolon\quad forand example\quad commacharacteristic the \quad o f each H \quad from that \quad o f S . \quad This informationAs in §6 i . 2s given one in can Table compute\quad 7 the , \quad typeeach and entry characteristic describing a of representative each H from H \quad $ = S { r } columnthat headed of S .by .. This quotedblleft information S sub r ito s the given power in of Table 0 comma 7 S , sub each r to the entry power describing of circumflex-zero a representative quotedblright .. represents two columns S\times sub r to$ the B power o f an of 0 and S sub r to the power of zero-circumflex comma .. so that the second entry of S −H orbit= ofSr length× B of an $ \ Ssigma - orbit of. $ length\quadσ. ForFor space space saving saving , , some some columns columns were were grouped grouped together ; for example , the eachtogether row refers ; for to example S sub r to , the the power of 0 and the third entry refers to S sub r to the power of circumflex-zero period 7 period 3 period .. Construction of the hypermaps S0,S0ˆ S0 S0ˆ, \noindentRowcolumn .. 1 periodcolumn headed .. The byheaded first “ row byr of\rquad Table” 7 represents‘ gives ‘ $ the S doubleˆ{ two0 columns} coverings{ r } Hr, ofand the S t ˆ etrahedron{\r hatso that{0 S}} sub the{ 1 second=r T}$ period entry ’ ’ \quad represents two columns $ S ˆ{ 0 } { r }$ and0 $ S ˆ{\hat{0}} { r } , $ 0ˆ \quad so that the second entry of Thoseof each three row which refers are maps to Sr haveand been the determined third entry by refers Vince to .. openSr . square bracket 27 comma .. S 7 comma .. Fig period .. 5 closing square bracketeach7 . semicolon row 3 . refers .. Construction they to are $ his S ˆ{ of0 the} { hypermapsr }$ and the third entry refers to $ S ˆ{\hat{0}} { r } . $ GRow sub 1 to the 1 . power ofThe comma first to row the of power Table of prime 7 gives G subthe 2 double to the power coverings of prime H of and the G tsub etrahedron 3 to the powerS1 = of prime of types open brace 3 comma\noindentT. 6 closing Those7 . brace three 3 . comma\ whichquad open areConstruction mapsbrace 6 have comma been of 3 the closing determined hypermaps brace comma by Vince open brace [ 27 6 , comma§7 ,6 closing Fig. brace 5 and ] ; genera 1 comma 1 comma 3 openthey parenthesis are his not 0 comma 0 comma 3 comma an obvious misprint closing parenthesis period \noindent0 0 Row \0 quad 1 . \quad The first row of Table 7 gives the double coverings H of the t etrahedron $ S { 1 } BeingG1, G orientable2 and G3 commaof types T yields{ 3 , 6no} double, { 6 , covering 3 } , { H6 =, 6 T} toand the genera power of 1 plus , 1 , comma 3 ( not so 0 the , 0 first , 3 entry , an obvious in this row is = $blankmisprint T period . ) . + ThoseThe hypermapBeing three orientable which S sub 1 are to , T the maps yields power have no of 0double =been T to covering determined the power H of= 0 byT .. in Vince, so the the second\ firstquad entry entry[ 27 is in the , this torus\quad row map\ isS .. blank7 open , \ bracequad . 3Fig plus .3 comma\quad 35 closing ] ; \quad they are h i s 0 0 brace = open brace 6 comma 3 closing braceThe 2 hypermap comma 0 = S1 = T in the second entry is the torus map \noindentopen{3 + brace 3, 3} 6$ comma= G ˆ{{\6, 33prime} closing2, 0 = brace} { 1 4 .... ˆ{ in, the}} notationG ˆ{\ ofprime open square} { bracket2 }$ 10 and comma $ G .... ˆ{\ Chprime period ....} 8{ closing3 }$ square o f types bracket\{ semicolon3 , 6 \} , \{ 6 , 3 \} , \{ 6 , 6 \} and genera 1 , 1 , 3 ( not 0 , 0 , 3 , an obvious misprint ) . .... it{ ....6 ,is 3 a} double4 in covering the notation of T branched of [ 10 over , Ch it s . four 8 ] ; it is a double covering of T branched over it s Beingfacefour hyphen orientable centres comma, T yields .. as shown no double in Figure covering 7 period 2 ..H open $ = parenthesis T ˆ{ + see} also, .. $ open so square the bracket first 27 entry comma in .. thisFig period row .. is 5 open parenthesisblankface . - d centres closing ,parenthesis as shown closing in squareFigure bracket 7 . 2 closing ( see parenthesis also [ comma 27 , with Fig the . edge 5 ( hyphen d ) ] ) identifications , with the givenedge by - theidentifications vertex hyphen given labelling by period the vertex - labelling . \ hspace ∗{\ f i l l }The hypermap $ S ˆ{ 0 } {0 1 }0 = T ˆ{ 0 }$ \quad in the second entry is the torus map \quad Figure 7 period 2 period S sub 1 toFigure the power7.2 of. 0S1 == TT to theas a power covering of 0 as of a T covering of T $ \{ 3 + 3 , 3+ 0\} =0 \{ 6 , 3 \} 2 , 00 = $ TheThe rotation rotation group group Aut toAut theT powerof ofT plusi s TA to4 × theC2 power. This of map 0 .... , of denoted T to the power by G of2 in 0 .... [ i27 s A ] , sub has 4 times five C sub 2 period .... This map 0 commaother .... denoted associates by G , sub including 2 to the it power s dual of prime , which .... in is .... Vince open ’ square s map bracketG1 of 27 type closing{ 3 square , 6 } bracketand Coxeter comma .... has five \noindent \{ 6 , 3 \} 4 \ h f i l l in the notation of [ 10 , \ h f i l l Ch0 . \ h f i l l 8 ] ; 0ˆ\ h f i0ˆ l l i t \ h f i l l is a double covering of T branched over it s four otherand associates Moser ’ comma s {3, 3 including + 3} = {3 it, 6 s}2 dual, 0 = comma{3, 6}4. whichS ince is Vince T i quoteright s orientable s map, T G subis isomorphic 1 to the power to ofS prime1 = T of type open brace 3 comma 6 closingin brace the third and Coxeter entry , and and hence to the map W constructed and il lustrated in [ 4 , §1 ] . \noindent f a c e − c e n t r e s , \quad as shown in Figure 71 . 2 \quad1ˆ ( see a l s o \quad [ 27 , \quad Fig . \quad 5 ( d ) ] ) , with the edge − identifications Moser quoteright s open brace 3 commaFigure 3 plus 7 3 . closing 3 . brace The =map openT brace=∼ 3T comma 6 closing brace 2 comma 0 = open brace 3 comma 6 closing given by the vertex − l a b e l l i n g . 1 1ˆ braceThe 4 period fourth .. S and ince fifth T i s entriesorientable represent comma T the to the same power self of - 0 dual is isomorphic orientable to S map sub 1 toT the power=∼ ofT circumflex-zeroof = T to the power of zero-circumflextype { 6 , 6 } and genus 3 , branched over the vertices and face - centres of T . This i s Vince \ centerlinein the third{ entryFigure0 comma $7 and hence . to 2 the map . W Sˆ constructed{ 0 } { and1 } il lustrated= Tˆ in{ open0 }$ square as bracket a covering 4 comma of S 1 T closing} square bracket period ’ s map G3 Figure 7 period 3 period .. The map T to the power of 1 = sub thicksim T to the power of circumflex-one \noindentThe fourthThe and rotation fifth entries group represent $ the Aut same ˆ{ + self} hyphenT ˆ{ dual0 } orientable$ \ h f i l map l o f .. T$ Tto ˆthe{ 0 power}$ of\ h 1 f = i l subl i thicksim s $ A T{ to4 the}\ powertimes of Ccircumflex-one{ 2 } . .. $ of type\ h f i l l This map , \ h f i l l denoted by $ G ˆ{\prime } { 2 }$ \ h f i l l in \ h f i l l [ 27 ] , \ h f i l l has f i v e open brace 6 comma 6 closing brace and genus 3 comma branched over the vertices and face hyphen centres of T period .. This i s Vince quoteright s\noindent map G sub 3other to the powerassociates of prime , including it s dual , which is Vince ’ s map $ G ˆ{\prime } { 1 }$ o f type \{ 3 , 6 \} and Coxeter and Moser ’ s $ \{ 3 , 3 + 3 \} = \{ 3 , 6 \} 2 , 0 = \{ 3 , 6 \} 4 . $ \quad S inceT i s orientable $ , Tˆ{ 0 }$ is isomorphic to $ S ˆ{\hat{0}} { 1 } = T ˆ{\hat{0}}$

\noindent in the third entry , and hence to the mapW constructed and il lustrated in [ 4 , \S 1 ] .

\ centerline { Figure 7 . 3 . \quad The map $ T ˆ{ 1 } = {\sim } T ˆ{\hat{1}}$ }

\noindent The fourth and fifth entries represent the same self − dual orientable map \quad $ T ˆ{ 1 } = {\sim } T ˆ{\hat{1}}$ \quad o f type \{ 6 , 6 \} and genus 3 , branched over the vertices and face − centres of T . \quad This i s Vince ’ s map $ G ˆ{\prime } { 3 }$ A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 27 \ hspaceopen square∗{\ f i bracket l l }A . 27 J comma . Breda .. S d 7 comma’ Azevedo Fig period ; G . .. A 5 open . Jones parenthesis : \quad e closingPlatonic parenthesis Hypermaps closing square\quad bracket27 comma and Sherk quoteright s open brace 2 times 3A comma . J . Breda 2 times d 3 closing ’ Azevedo brace ; open G . squareA . Jones bracket : 23 Platonic comma Table Hypermaps V comma Fig 27 period 14 closing square bracket period\noindent .. It i s[ il 27 lustrated , \quad in Figure\S 7 , Fig . \quad 5(e) ] ,andSherk ’ s $ \{ 2 \cdot 3 , 2 \cdot [ 27 , §7 , Fig . 5 ( e ) ] , and Sherk ’ s {2 · 3, 2 · 3} [23, Table V , Fig . 14 ] . It i s 3 7 period\} 3[ comma 23 where ,$ boundary TableV,Fig.14] edges with the same vertex . \quad hyphenIt lab i s els il and lustrated the same colours in Figure are identified comma 7 .il 3 lustrated , where in boundary Figure 7 edges . 3 , where with boundary the same edges vertex with− thelab same els and vertex the - lab same els colours and the aresame identified , socolours each edge are of identified T i s covered , by two edges of T to the power of 1 period .. The rotation group i s A sub 4 times C sub 2 comma and there are two other associates period so each edge of T i s covered by two edges of T 1. The rotation group i s A × C , and there are \noindentS ince T isso self each hyphen edge dual of comma T i the s covered hypermap by T to two the edgespower of of 2 thicksim $ T ˆ{ =1 T} to the4. $power2 \quad of circumflex-twoThe rotation in the group s ixth andi s seventh $ A { 4 } \timestwo otherC { associates2 } , $ . and t here entries i s j ust the ˆ S ince T is self - dual , the hypermap T 2 ∼= T 2 in the s ixth and seventh entries i s j ust the aredual two of T other to the power associates of 0 = sub . thicksim T to the power of circumflex-zero in the second and third entries period .... Thus we find no further T 0 = T 0ˆ S hyphendual orbits of of double∼ in the second and third entries . Thus we find no further S - orbits of double 6 + 3 = 9 \ hspacecoveringscoverings∗{\ off i T of l l comma} TS , ince so so this this T row isrow selfyields − a a totaldual total of of, 6 plus the 3 hypermap = 9 hypermapshypermaps $ T period ˆ .{ 2 }\sim = T ˆ{\hat{2}}$ in the s ixth and seventh entries i s j ust the S = RowRow .... 2 2 period . The .... The second .... second row row of .... Table of Table 7 gives .... 7 thegives coverings the .... coverings H of H the .... torus of the hypermaptorus hypermap2 .... S sub 2 = −1 0ˆ \noindentWW to the(T power)dualof type of o f minus ( $ 3 T , 1 3 ˆopen{ , 30 ) parenthesis} shown= {\ in Tsim to Figure the} powerT 4 ˆ .{\ of 3 circumflex-zero.hat As{0 with}}$S1 inclosing= theT parenthesis , second the orientability and .... of third type .... of entries open parenthesis . \ h f i l 3 l commaThus we3 find no further S − orbits of double commaS2 3 closing parenthesis .... shown .... in .... Figure .... 4 period 3 period .... As with .... S sub 1 = .... T comma .... the .... orientability .... of \noindent coverings of T , so this+ row yields ai totalˆı of $ 6 + 3 = 9 $ hypermaps . S subimplies 2 that there i s no entry for S2 , and that S2 =∼ S2 for i = 0, 1 and 2 . The orientable 0 0ˆ implieshypermap that thereS2 i s∼ no= entryS2 foroftype S sub 2 to ( the 3 , power 6 , 6 of ) plus has comma genus .. and 5 , that being S sub branched 2 to the power over of the i = subfour thicksim S sub 2 to the power \noindent Row \ h f i l l 2 . \ h f i l l The \ h f i l l second row \ h f i l l o f Table \ h f i l l 7 g i v e s the \ h f i l l c o v e r i n g s H \ h f i l l of the torus hypermap \ h f i l l of circumflex-ihyperedges .. for and i = four 0 comma hyperfaces 1 and 2 period of S ..2. TheIt orientable has two other associates , and s ince $ S { 2 } = $ hypermapS2 i s S sub S - 2 invariant to the power of these 0 thicksim three = hypermaps S sub 2 to the reappear power of circumflex-zero in the remaining .. oftype entries .. open in this parenthesis row ; 3 comma 6 comma 6 closing parenthesisthey all .. has have genus rotation 5 comma group .. being branched over the four hyperedges \noindentand four hyperfaces$ W ˆ{ .. − of S1 sub} 2 period( T .. ˆ It{\ .. hashat ..{0 two}} .. other) $ ..\ associatesh f i l l o comma f type ..\ andh f i.. l ls ince( 3 S sub, 3 2 , .. 3 i s ) ..\ Sh hyphen f i l l shown invariant\ h .. f these i l l in \ h f i l l Figure \ h f i l l 4 . 3 . \ h f i l l As with \ h f i l l $ Sthree{ hypermaps1 } = $ reappear\ h f i l inl T, the remaining\ h f i l l entriesthe \ inh f this i l l roworientability semicolon they all\ h have f i l l rotationo f $ group S { 2 }$ A sub 4 times C sub 2 period A4 × C2. \noindentFigure 7 periodimplies 4 period that .. W there open parenthesis i s no entry S sub for2 to the $ power S ˆ{ of+ 2} closing{ 2 } parenthesis, $ \quad = openand brace that 6 comma $ S 6closing ˆ{ i } brace{ 2 2} comma= {\ 0 sim } Figure 7 . 4 . W (S2) = {6, 6}2, 0 as a double covering of the dual maps T 0 = {6, 3}2, 0 and asS aˆ{\ doublehat covering{\imath of}} the { 2 }$2 \quad fori $= 0 , 1$ and2. \quad The orientable T 2 = {3, 6}2, 0. hypermapdual maps T $ to S the ˆ{ power0 } of{ 02 =}\ opensim brace 6= comma S ˆ 3{\ closinghat{ brace0}} 2{ comma2 }$ 0 and\quad T to theo f power type of\quad 2 = open( 3 brace , 6 3 ,comma 6 ) \ 6quad closinghas brace genus 5 , \quad being branched over the four hyperedges 2and comma four 0 period hyperfaces \quad o f $ S { 2 } . $ \quad I t \quad has \quad two \quad other \quad associates , \quad and \quad s i n c e $ S { 2 }$ \quad i s \quad S − i n v a r i a n t \quad these three hypermaps reappear in the remaining entries in this row ; they all have rotation group

\ begin { a l i g n ∗} A { 4 }\times C { 2 } . \end{ a l i g n ∗}

\ begin { c e n t e r } Figure 7 . 4 . \quad W $ ( S ˆ{ 2 } { 2 } ) = \{ 6 , 6 \} 2 , 0 $ as a double covering of the dual maps $ T ˆ{ 0 } = \{ 6 , 3 \} 2 , 0 $ and $ T ˆ{ 2 } = \{ 3 , 6 \} 2 , 0 . $ \end{ c e n t e r } 28 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentAmong these28 comma\quad ..A the . J .. dual. Breda .. hypermap d ’ Azevedo .. S sub 2; to G the . A power . Jones of 2 = : open Platonic parenthesis Hypermaps S sub 2 to the power of 0 closing parenthesis to the28 power A of . Jopen . Breda parenthesis d ’ Azevedo 2 closing ; parenthesis G . A . Jones comma : Platonic .. of type ..Hypermaps open parenthesis 6 comma 6 comma 3 closing parenthesis comma .. has Walsh\noindent map ..Among W open these parenthesis , \quad S sub 2the to the\quad powerdual of 2 closing\quad parenthesishypermap \quad $ S ˆ{ 2 } { 2 } = ( S ˆ{ 0 } { 2 } Among these , the dual hypermap S2 = (S0)(2), of type ( 6 , 6 , 3 ) , has ) ˆi{ somorphic( 2 to ) Sherk} quoteright, $ \quad s regularo f type map open\quad brace(2 6 6 comma , 6 , 6 32 closing ) , \ bracequad 2has comma Walsh 0 of genus map \ 5quad open squareW $ bracket ( S 24 ˆ{ semicolon2 } { 2 1 2} Walsh map W (S2) i somorphic to Sherk ’ s regular map { 6 , 6 } 2 , 0 of genus 5 [ 24 ; 1 2 , comma) $ Tabelle I I I closing square2 bracket period This i s il lustrated i somorphicTabelle I I Ito ] . Sherk This i ’ sil s lustrated regular map in Figure\{ 6 7 , . 6 4\} as2 a double , 0 of covering genus 5 of [ the 24 pair ; 1 of 2 dual , Tabelle maps I I I ] . This i s il lustrated in Figure 7 period0ˆ 4 as a double covering of the pair of dual2ˆ maps T-zero = T to the power of zero-circumflex = open brace 6 comma 3 closing braceinT Figure 2− commazero 7 0= .. .T and 4= as T-two{6 a, 3 double} =2, 0 and coveringT − two of= theT pair= of{3, 6 dual}2, 0; mapsthere $ are T− eightzero hexagonal = T ˆ{\ faceshat{0}} = \{ 6 ,T, 3 to the six\} power around2 of circumflex-two the , vertex 0 $ \quad =1 open+, andand brace $two 3 T comma− formedtwo 6 closing = from $ brace the 2 comma small triangles 0 semicolon ; .. there only are aeight few hexagonal faces comma .. six $edge T ˆ{\ - identificationshat{2}} = are\{ 3 shown , , 6 s ince\} the2 rest , 0 can ; be $ deduced\quad there by symmetry are eight . hexagonal S faces , \quad six around the vertex \quad around the vertex .. 1 sub plus2 comma .. and two formed 2 $ 1 ince{ +T}− zero, $ and\quadT andare two double formed coverings of T , W (S2 ) i s a 4 - sheeted from the .. small triangles0 2 semicolon .. only .. a few .. edge hyphen identifications .. are .. shown comma .. s ince the .. rest .. can be fromdeducedcovering the by\quad Tsymmetry×Bsmall× Bperiodof triangles T .. , S with ince T-zero two ; \quad branch .. andonly T- points to\ thequad power of ordera few of 2 1 ..\ overquad are double eachedge vertex coverings− identifications and of T face comma - centre .. W\quad open parenthesisare \quad S subshown 2 to , \quad s i n c e the \quad r e s t \quad can be thededuced powerof T of. by 2 closing symmetry parenthesis . \quad .. i sS a 4 i hyphenn c e $ sheeted T−zero $ \quad and $ T ˆ{ 2 }$ \quad are double coverings of T , \quad W $ (coveringRow S ˆ 3 T{ . times2 } { BThe to2 the} third power) $ row of\quad gives 0 times thei B s todouble a the 4 − power coveringssheeted of 2 of ofT comma the proj with ective two branch o ctahedron hyphen pointsS3 = PO, of order 1 over each vertex and face hyphena noncentre - orientable map of type { 3 , 4 } and characteristic 1 ( Figure 4 . 4 ) . The first entry \noindentrepresentscovering T $ \times B ˆ{ 0 }\times B ˆ{ 2 }$ of T , with two branch − points of order 1 over each vertex and face − c e n t r e of T period + o fRowit T s .3 orientable period .. The double third row cover gives , the the double o ctahedron coverings O of= the (projPO ective) o ctahedron= {3, 4}, Sone sub of3 = an P SO -comma orbit a of six non hyphen orientable map of type open brace 3 comma 4 closing brace and(2) characteristic 1 open parenthesis Figure 4 period 4 closing parenthesis period\noindentspherical .. TheRow first hypermaps entry 3 . represents\quad includingThe third it s row dual gives , the cube the double C = O coverings= {4, 3}; both of the of these proj maps ective are o i ctahedron $ S { 3 } = $itllustrated s P orientable O , a in double Figure cover 3 . comma 1 . .... the o ctahedron O .... = open parenthesis P O closing parenthesis to the power of plus = open brace 3 0 0ˆ commanon − 4The closingorientable second brace and comma map third of .... entries type one of\{ an in S3 this hyphen , 4row\} orbit areand ....a pair characteristic of six of non - orientable 1 ( Figure maps P 4O . 4and ) P . \Oquadof The first entry represents sphericaltype { hypermaps6 , 4 } and including characteristic it s dual comma−2, shown the cube in Figure C = O to 7 . the 5 power( where of open the edge parenthesis - identifications 2 closing parenthesis can = open brace 4 comma 3 closing\noindentbe brace deduced semicolonit s , by orientable bothsymmetry of these double about maps are the cover vertex , \1h+ f, ifrom l l the those o ctahedron indicated ) O ; all\ h twelvef i l l $ hypermaps = ( $ inP O $ ) ˆ{ + } = \{ 3i llustratedtheir , 4two in S\} Figure - orbits, 3 $ period are\ h branchedf 1 i periodl l one over o f an the S four− o face r b i t -\ centresh f i l l ofo f P s O i x . 0 0ˆ The second and third entriesFigure in this 7 row . 5 are . a pair P O of nonand hyphen P O orientableas coverings maps of P P O O to the power of 0 and P O to the power of circumflex-zero ˆ ˆ \noindentofTo type see open thatspherical brace P 6O comma0 =∼ hypermapsP 4O closing0, note brace including that and P characteristicO it0 i s s bipartite dual minus , the2 , comma whereas cube shown C P $inO = Figure0 is not O 7 ˆ period{ ( s( ince 5 open 2 it s parenthesis ) 1} - = where\{ the4 edge , hyphen 3 identifications\} skeleton; $ both can contains of these circuits maps of odd are length ) . Alternatively , one can demonstrate this non - i llustrated in Figure 3 . 1 . 0 beisomorphism deduced comma by by noting symmetry that about the the element vertex g1 sub= r plus0r1r2 comma, which from has those orders indicated 3 , closing 1 and parenthesis 2 in P O semicolon, B all twelve hypermaps inand theirB two0ˆ, has S hyphen orders orbits 3 and are 6 branched in P O0 overand the P fourO0ˆ, faceso the hyphen Petrie centres polygons of P O ofperiod these maps have lengths TheFigure3 second and 7 6period andrespectively 5 third period .. entries ( P see O to Figure the in power this 7 . 6 of row ) 0 . and are P O a to pair the power of non of circumflex-zero− orientable as coverings maps P of $ P O ˆ{ 0 }$ and P $ O ˆ{\hat{0}}$ o fTo type see that\{ P6 O , to 4 the\} powerand of characteristic 0 similar-equal P O to$ − the power2 of , zero-circumflex $ shown in comma Figure note 7 that . 5 P ( O where to the power the edge of 0 i s− bipartiteidentifications comma can whereasbe deduced P O to the , by power symmetry of circumflex-zero about the is not vertex open parenthesis $ 1 { s+ ince} it, s1 $ hyphen from skeleton those indicated ) ; all twelve hypermaps incontains their circuits two S of− oddorbits length closingare branched parenthesis over period the .. Alternatively four face comma− centres .. one can of demonstratePO . this non hyphen isomorphism by noting that the element g = r sub 0 r sub 1 r sub 2 comma .. which has orders 3 comma 1 and 2 in P O comma B to the power of 0 and B to the\ centerline power of circumflex-zero{ Figure 7 comma. 5 . has\quad P $ O ˆ{ 0 }$ and P $ O ˆ{\hat{0}}$ as coverings of PO } orders 3 and 6 in P O to the power of 0 and P O to the power of circumflex-zero comma so the Petrie polygons of these maps have lengths 3 and 6\noindent To see that P $Oˆ{ 0 }\cong $ P $ O ˆ{\hat{0}} , $ note thatP $Oˆ{ 0 }$ i s bipartite , whereas P $ Orespectively ˆ{\hat{ open0}} parenthesis$ is not see ( Figure s ince 7 period it s1 6 closing− s k parenthesis e l e t o n period contains circuits of odd length ) . \quad Alternatively , \quad one can demonstrate this non − isomorphism by noting that the element g $= r { 0 } r { 1 } r { 2 } , $ \quad which has orders 3 , 1 and 2 in PO $ , B ˆ{ 0 }$ and $ B ˆ{\hat{0}} , $ has orders 3 and 6 in P $Oˆ{ 0 }$ and P $ O ˆ{\hat{0}} , $ so the Petrie polygons of these maps have lengths 3 and 6 respectively ( see Figure 7 . 6 ) . A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 29 \ hspaceFigure∗{\ 7 periodf i l l } 6A period . J .. . Petrie Breda polygons d ’ Azevedo in P O comma ; G . P A O to. Jonesthe power : of\quad circumflex-zeroPlatonic and Hypermaps O \quad 29 It follows that P O to the powerA . of J 0 . is Breda the map d’ open Azevedo brace 6 ; comma G . A 4. Jonesclosing :brace Platonic 3 obtained Hypermaps by applying Wilson 29 quoteright s Petrie operation open\ centerline square bracket{ Figure 29 closing 7 . square 6 . \quad bracketPetrie comma polygons in PO , P $ O ˆ{\hat{0}}$ and O } O0ˆ transposing faces and PetrieFigure polygons 7 . 6comma . Petrie to O = polygons open brace in 3 comma P O , 4P closingand brace O 6 open parenthesis see open square bracket 1 0 comma O0 3 S\noindent 8 periodIt follows 6 andIt that Table follows P 8 closing thatis the square P map $O bracket{ 6 ˆ{ , comma 40 }}$ andobtained i s compare the mapby applying\{ 6 , Wilson 4 \} 3 ’ obtained s Petrie operation by applying [ 29 Wilson ’ s Petrie operation [ 29 ] , with], the i somorphism P D to the power of 0 thicksim = open brace 1 0 comma 3 closing brace 5 discussed in S 6 period 3 closing parenthesis = {3, 4}6( period\noindenttransposingtransposing faces and faces Petrie and polygons Petrie , to polygons O , to Osee $[ 1 = 0 , \{§8 .3 6 and , Table 4 8\} ] , and6 ( $ s ee [ 1 0 , \S 8 . 6 and Table 8 ] , and compare Thesecompare three regular open parenthesis S sub 4 times C sub 2 closing parenthesis hyphen hypermaps O comma P O to the power of 0 and P O to D0 ∼= {10, 3}5 the\noindent powerwith of the circumflex-zerowith i somorphism the i somorphismare P all quotients P of $ the Ddiscussed ori ˆ{ hyphen0 }\ insim§6 . 3= ) . \{ 1 0 , 3 \} 5 $ discussed in \S 6 . 3 ) . 0 0ˆ entableThese map O threeto the powerregular of 0( =S4 sub× C thicksim2)− hypermaps O to the power O , P ofO circumflex-zeroand P O are of type all open quotients brace 6 of comma the ori 4 closing - brace and genus 3 shown \ hspace ∗{\ f i l l }These0 three0ˆ regular $ ( S { 4 }\times C { 2 } ) − $ hypermapsO , P $Oˆ{ 0 }$ in Figureentable 7 period map 7 periodO =∼ O of type { 6 , 4 } and genus 3 shown in Figure 7 . 7 . andFigure P $ 7 O period ˆ{\ 7hat period{0}}Figure .. The$ map are 7 . O 7 all to. the quotients The power map of 0 ofO thicksim0 ∼ the= O ori=0ˆ as O− toa covering the power of of Ocircumflex-zero as a covering of O ThisThis map map .... open( Sherk parenthesis ’ s {2 · Sherk3, 4} in quoteright Table V s of open [ 23 brace ] , the2 times dual 3 comma of his Figure4 closing 18 brace ) i .... s bipartite in Table V and of open square bracket 23 closing square\noindentregular bracket ,entable comma with automorphism .... map the dual $ O of ˆ his{ group0 Figure} = ....{\ 18 closingsim } parenthesisO ˆ{\hat ....{ i0 s}} bipartite$ o f and type \{ 6 , 4 \} and genus 3 shown in Figure 7 . 7 . 0ˆ 0ˆ regular comma with automorphismAut groupO ∼= Aut O × Aut B ∼= S4 × C2 × C2. \ centerlineAutThere O to are the{ Figure threepower ofcentral 7 circumflex-zero . 7 involutions . \quad thicksimThe in this map = Aut group $ O O times ; ˆ in{ tAut0 erms}\ B to ofsim the the power lab= ofelling O zero-circumflex ˆ{\ inhat Figure{0}} thicksim 7$ . 7 theyas = a S subcovering 4 times Cof sub O 2} times C sub 2are period given by \noindentThere are threeThis central map \ involutionsh f i l l ( Sherk in this group ’ s semicolon $ \{ in2 t erms\cdot of the lab3 elling , in 4 Figure\} 7$ period\ h f7 i theyl l in TableV of [ 23 ] , \ h f i l l the dual of his Figure \ h f i l l 18 ) \ h f i l l i s bipartite and are given by a : n± 7→ (7 − n)±, \noindent regular , with automorphism group Line 1 a : n sub plusminux mapsto-arrowright openb parenthesis: n± 7→ n 7∓ minus, n closing parenthesis plusminux comma Line 2 b : n sub plusminux arrowright-mapsto n sub minusplus comma Line 3 c : n sub plusminux mapsto-arrowright open parenthesis 7 minus n closing parenthesis sub c : n 7→ (7 − n) . minusplus\ centerline period{Aut $ O ˆ{\hat{0}}\sim =± $ Aut O∓ $ \times $ Aut $ B ˆ{\hat{0}}\sim = S { 4 } \timesThusThus a open aC ( which{ parenthesis2 }\ reversestimes which orientation reversesC { orientation2 and} preserves. $ and} preserves vertex vertex - colours hyphen ) colours is induced closing by parenthesis the antipodal is induced by the antipodal automorphismautomorphism n mapsto-arrowright n 7→ 7− n of O7 minus , while n of b O ( comma which while preserves b open orientation parenthesis which and preserves transposes orientation vertex and - transposes vertex hyphen \noindent There are three central involutions in this group ; in t erms of the lab elling in Figure 7 . 7 they colourscolours closing ) parenthesisis the covering .. is the ..- transformation covering hyphen transformation of O0ˆ = .. ofO O to×B the0ˆ power→ O of circumflex-zero induced by = the .. O .. times B to the power of zero-circumflex right arrow .. O .. induced by the non hyphen trivial \noindentnon - trivialare given by

\ [ \ begin { a l i g n e d } a : n {\pm }\mapsto ( 7 − n ) \pm , \\ b : n {\pm }\mapsto n {\mp } , \\ c : n {\pm }\mapsto ( 7 − n ) {\mp } . \end{ a l i g n e d }\ ]

\noindent Thus a ( which reverses orientation and preserves vertex − colours ) is induced by the antipodal

\noindent automorphism n $ \mapsto 7 − $ n of O , while b ( which preserves orientation and transposes vertex − c o l o u r s ) \quad i s the \quad covering − transformation \quad o f $ O ˆ{\hat{0}} = $ \quad O \quad $ \times B ˆ{\hat{0}}\rightarrow $ \quad O \quad induced by the non − t r i v i a l 30 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentautomorphism30 \ ofquad B toA the . power J . ofBreda circumflex-zero d ’ Azevedo comma ; and G . finally A . c Jones = ab = : ba Platonic period .... HypermapsBy inspection of Figures 7 period 7 and 7 period 5 it i30 s easily A . J . Breda d ’ Azevedo ; G . A . Jones : Platonic Hypermaps \noindentseen that automorphism of $ B ˆ{\hat{0}} ,$ and finally c $=$ ab $=$ ba . \ h f i l l By inspection of Figures 7 . 7 and 7 . 5 it i s easily B0ˆ, = = Oautomorphism to the power of circumflex-zero of and finally slash cangbracketleftab ba a . right By angbracket inspection = sub of thicksim Figures P 7 O . to 7 and the power 7 . 5 of it zero-circumflex i s comma O to the power\noindenteasily of circumflex-zeroseen that slash angbracketleft b right angbracket thicksim = O comma O to the power of zero-circumflex slash angbracketleft c right angbracketseen that thicksim = P O to the power of 0 comma O to the power of circumflex-zero slash angbracketleft a comma b right angbracket = sub O0ˆ/h i = O0ˆ, O0ˆ/h i ∼= , O0ˆ/h i ∼= O0, O0ˆ/h i = thicksim\noindent Pa O period$∼ OP ˆ{\hat{0}}b / O\ langle $c a $P\rangle =a ,{\ b sim∼ P}$ O .P The $ O ˆ double{\hat{0}} , O ˆ{\hat{0}} / Thecoverings\ langle .. double$ .. betweencoverings b $ \rangle .. between these ..\sim these maps ..= maps $ ( induced O.. open $ , parenthesis O by ˆ{\ inducedinclusionshat{0}} .. by ../ between inclusions\ langle .. between subgroups$ c .. $ subgroups\rangle .. of \sim = $ Pangbracketleft $of O ˆ{ 0 } a comma, O b ˆ right{\hat angbracket{0}} thicksim/ \ langle = C sub$ 2 times a , C b sub $ 2\ closingrangle parenthesis= {\ aresim shown}$ in P Figure O . 7 period 8 period h i ∼= C × C ) TheLinea\quad 1, bO thicksimdouble2 = O\quad2 toare thec shown power o v e r i of n in g circumflex-zero s Figure\quad 7between . 8 Line . 2\ d117-d117-d117-d117-d117-d117-d117quad these \quad maps \quad d15-d15-line( induced d73-d73-d73-d73-d73-d73-d73\quad by \quad i n c l u s i o n s \quad between \quad subgroups \quad o f P O to the power of 0 .. O .. P O to the power of circumflex-zero \noindent $ \ langle $ a , b $ \rangle \sim = C { 2 }\times C { 2 } ) $ are shown in Figure 7 . 8 . d74-d74-d74-d74-d74-d74-d74 d15-d15 d116-d116-d116-d116-d116-d116 O ∼= O0ˆ PO \ [ \Figurebegind117 7{− periodad l117 i g n− e 8 dd period}117O− d ..117\ Somesim− d coverings117=− d117 O of ˆ−{\ Pd O117hatd{150}}\\− d15 − line d73 − d73 − d73 − d73 − d73 − d73 − d73 d117As explained−d117−d117 in S 7− periodd117− 2d117 we have−d117 P− Od117 to the power d15− ofd15 1 =− subl i n e thicksim d73− Pd73 O to−d73 the− powerd73− ofd73 0− andd73 P− Od73 to the\end power{ a l ofi g ncircumflex-one e d }\ ] thicksim 0 0ˆ = P O to the power of zero-circumflex comma so theP associatesO OP of P O to the power of 0 and of P O to the power of circumflex-zero also correspond to the fourth and fifth entries of this row comma respectively period .. S ince P O covers \ centerlineB to the powerd74{P− d of $74 2 O− thered ˆ74{ − is0 d no}74$ entry− d\74quad for− d P74O O−\ toquadd74 thedP power15 − $d15 Oof 2 ˆd{\ comma116hat− d{ and1160}}− s inced$116} P− Od116 to− thed116 power− d of116 circumflex-two thicksim = P O to the power of plus thicksim = O the hypermaps correspondingPO to \ [the d74 final−d74 entry−d74 are−d74 the−d74 same−d74 as those−d74Figure for the d15 7 first.− 8d15 . comma Some d116 .. so− coverings thisd116 row−d116 contributes of− Pd116 O−d116 6 plus− 6d116 plus 6\ ] = 18 hypermaps period 1 0 1ˆ 0ˆ 0 As explained in §7 . 2 we have P O =∼ P O and P O ∼= P O , so the associates of P O and Rowof 4P periodO0ˆ also .. The correspond final row of to Table the 7fourth gives the and double fifth coverings entries of of this the non row hyphen , respectively orientable . hypermap S ince P O \ centerlineS sub 4 = W{PO to the} power of minus 1 open parenthesis P O to the power of circumflex-zero closing parenthesis .. of type .. open parenthesis 4 covers B2 there is no entry for P O2, and s ince P O2ˆ ∼= P O+ ∼= O the hypermaps corresponding comma 4 comma 3 closing parenthesis .. and .. characteristic .. minus 2 .. shown .. in .. Figure .. 4 period 6 .. open parenthesis see .. Figures \ centerlineto the final{ Figure entry 7 are . the 8 . same\quad asSome those coveringsfor the first of , P so O this} row contributes 6 + 6 + 6 = 18 4hypermaps period 5 .. and . .. 7 period 5 .. for .. W open parenthesis S sub 4 closing parenthesis = .. P O to the power of circumflex-zero closing parenthesis period .. The .. first .. entry .. is .. the .. orientable .. double .. covering S sub 4 to the power of plus = \noindentRow 4As . explainedThe final in row\S of7 Table . 2wehaveP 7 gives the double $Oˆ coverings{ 1 } = of the{\sim non -} orientable$ P $ O hypermap ˆ{ 0 }$ and P $ O ˆ{\hat{1}} W to the power−1 of minus0ˆ 1 open parenthesis O to the power of circumflex-zero closing parenthesis = W to the power of minus 1 open parenthesis O\sim toS the4 power== $W of P 0( closing$P OO ˆ{\) parenthesishatof type{0}} comma ( 4, ,.. $ 4 of , so type 3 ) the .. openand associates parenthesis characteristic of 4 comma P $Oˆ 4− comma2 { shown0 3}$ closing and in parenthesis o f Figure .. and genus 3 semicolon .. it P $ O ˆ{\hat{0}}$ also correspond to the fourth and fifth entries0ˆ of this row , respectively . \quad S ince PO covers s Walsh4 . map 6 O ( tosee the power Figures of zero-circumflex 4 . 5 and = O 7 to . 5the power for of W 0 i s(S the4) double = P O ). The first entry $coveringis B ˆ{ the open2 }$ brace orientable there 2 times is 3 no double comma entry 4 closing coveringfor P brace $OˆS of+ O{= shown2W}−1 in(O,$ Figure0ˆ) = ands 7W period−1(O inceP0) 7, periodof type $Oˆ .. This{\ (S 4 hyphenhat , 4{ ,2}}\ 3 orbit ) opensim parenthesis= $ P of length $ O ˆ{ + } \sim = $ O the hypermaps corresponding to4 sigmaand = 3 genus since the 3 ; element it s Walsh map O0ˆ = O0 i s the double covering {2 · 3, 4} of O shown in Figure the final entry are the same as those for the first , \quad so this row contributes $6 + 6 + 6 b7 in . .... 7 . Aut This O to theS - powerorbit (of of circumflex-zero length σ = 3 ....since induces the S element sub 4 to the power of plus thicksim = open parenthesis S sub 4 to the power of = 18 $ plus closing parenthesis0ˆ to the power+ of open parenthesis+ (01) 0 1 closing parenthesis2ˆ closing∼ + parenthesis .... also corresponds to S sub 4 to the power of b ∈ Aut O induces S4 ∼= (S4 ) ) also corresponds to S4 ( = S4 ). The second and third two-circumflex open parenthesis to the power of thicksim = S sub 4 to the power of plus closing parenthesis period ....0 The second and third \noindententries yieldhypermaps two dist . inct S - orbits of length 3 , each containing a non - orientable hypermap S4 entriesor S0ˆ yieldof type two dist ( 4 inct , 4 , S 6 hyphen ) and orbits characteristic of length 3 comma−8, branched each containing over the a nonfour hyphen face - orientable centres of hypermapS . As S sub 4 to the power of 0 or S sub4 4 to the power of circumflex-zero of type open parenthesis 4 comma 4 comma 6 closing parenthesis4 and characteristic minus 8 comma \noindentin the thirdRow 4 row . \ ,quad theseThe two Sfinal - orbits row reappear of Table in 7the gives fourth the and double fifth entries coverings , while of there the i snon no − orientable hypermap branchedentry over for theS2 four. This face hyphen row therefore centres of yields S sub 43 period + 3 + ..3 = As 9 inhypermaps . $the S third{ 4 row} comma4= W these ˆ{ two− S1 hyphen} ( orbits $ P reappear $ O ˆ in{\ thehat fourth{0}} and fifth) $ entries\quad commao f type while there\quad i s( no 4 , 4 , 3 ) \quad and \quad characteristic \quad $ − This2 gives$ \quad a totalshown of 39\ hypermapsquad in \quad , of whichFigure 2 1\quad are orientable4 . 6 \quad . Among( see these\quad areFigures 9 maps M , entry for S sub 4 to the power of 2 period .. This row therefore yields 3 plus 3 plus 3 = 9 hypermaps period + 4 .l ist 5 \ edquad in Tableand \ 8quad ; five7 . of 5 these\quad f are o r orientable\quad W $, ( and S for{ these4 } the) rotation = $ \quad groupPAut $ OM ˆ{\hat{0}} ) . $ Thisis indicated gives a total . of 39 As hypermaps usual , χ commaand g of denote which 2 the 1 are characteristic orientable period and .... genus Among of these M . are 9 maps M comma \quadl ist edThe in Table\quad 8 semicolonf i r s t \quad .. fiveentry of these\ ..quad are orientablei s \quad commathe ..\quad and foro these r i e n thet a b rotation l e \quad groupdouble Aut to\ thequad powercovering of plus M .. $ is S ˆ{ + } { 4 } = $indicated period .. As usual comma chi and g denote the characteristic and genus of M period $ W ˆ{ − 1 } ( O ˆ{\hat{0}} ) = W ˆ{ − 1 } ( O ˆ{ 0 } ) , $ \quad o f type \quad ( 4 , 4 , 3 ) \quad and genus 3 ; \quad it s Walsh map $ O ˆ{\hat{0}} = O ˆ{ 0 }$ i s the double covering $ \{ 2 \cdot 3 , 4 \} $ of Oshown in Figure 7 . 7 . \quad This S − orbit ( of length $ \sigma = 3 $ since the element

\noindent b $ \ in $ \ h f i l l Aut $ O ˆ{\hat{0}}$ \ h f i l l induces $ S ˆ{ + } { 4 }\sim = ( S ˆ{ + } { 4 } ) ˆ{ ( 0 1 ) } ) $ \ h f i l l also corresponds to $ S ˆ{\hat{2}} { 4 } ( ˆ{\sim } = S ˆ{ + } { 4 } ) . $ \ h f i l l The second and third

\noindent entries yield two dist inct S − orbits of length 3 , each containing a non − orientable hypermap $ S ˆ{ 0 } { 4 }$ or $ S ˆ{\hat{0}} { 4 }$ of type ( 4 , 4 , 6 ) and characteristic $ − 8 , $ branched over the four face − c e n t r e s o f $ S { 4 } . $ \quad As in the third row , these two S − orbits reappear in the fourth and fifth entries , while there i s no entry for $ S ˆ{ 2 } { 4 } . $ \quad This row therefore yields $3 + 3 + 3 = 9$ hypermaps .

\noindent This gives a total of 39 hypermaps , of which 2 1 are orientable . \ h f i l l Among these are 9 maps M ,

\noindent l ist ed in Table 8 ; \quad five of these \quad are orientable , \quad and for these the rotation group $ Aut ˆ{ + }$ M \quad i s i n d i c a t e d . \quad As usual $ , \ chi $ and g denote the characteristic and genus of M . A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 3 1 \ hspaceS sub 1∗{\ to thef i l lpower}A . of J 0 S. sub Breda 1 to dthe ’ power Azevedo of 1 S ;sub G 3 . to A the . power Jones of :plus\quad S subPlatonic 3 to the power Hypermaps of 0 S sub 3\quad to the3 power 1 of circumflex-zero Table ignored! \ [ S ˆ{ 0 } { 1 } S ˆ{ A1 .} J{ . Breda1 } S d ’ ˆ Azevedo{ + } { ;3 G} . AS . ˆ Jones{ 0 } :{ 3 Platonic} S ˆ{\ Hypermapshat{0}} { 33 1 }\ ]

0 1 + 0 0ˆ S1 S1 S3 S3 S3

\ vspace ∗{3 ex }\ centerline {\ framebox[.5 \ l iTable n e w i d ignored! t h ] {\ t e x t b f { Table ignored!}}}\ vspace ∗{3 ex} 32 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentNow we can32 identify\quad EpiA open . J parenthesis . Breda d Capital ’ Azevedo Gamma ; comma G . A G . closing Jones parenthesis : Platonic with the Hypermaps set B sub Capital Gamma open parenthesis G closing32 parenthesis A . J . of Breda all Capital d ’ Azevedo Gamma hyphen ; G . A bases . Jones of G : : ....Platonic these are Hypermaps the k hyphen tuples \noindentx = open parenthesisNow we can x sub identify 1 comma Epiperiod $ period ( period\Gamma comma, x $ sub G) k closing with parenthesis the set in $B G such{\ thatGamma } ( $ G ) o f a l l $ \Gamma − $Now bases we can o f G identify : \ h f iEpi l l these(Γ, G ) are with the the k set− tB uΓ p( lG e s ) of all Γ− bases of G : these are the k - opentuples parenthesis i closing parenthesis x sub 1 comma period period period comma x sub k generate G comma and open parenthesis i i closing parenthesis R sub i open parenthesis x sub j closing parenthesis = 1 for all i in I period x = (x , ..., x ) in G such that \noindentopen parenthesisx1 $k = Given ( epsilon x { comma1 } we, define . x sub . j =. X sub , j x epsilon{ k comma} ) and$ ingiven Gsuch x comma that we define epsilon by X sub j mapsto- ( i ) x , ..., x generate G , and arrowright x sub j period closing parenthesis .. Under1 thisk identification comma ( i i ) R (x ) = 1 for all i ∈ I . ( Given ε, we define x = X ε, and given x , we define ε by \ centerlineAut G acts{ naturallyi(j i $ on ) Capital x { Gamma1 } hyphen, . bases . comma . so , d subj x Capital{ jk }$ Gamma generate open parenthesis G , and G} closing parenthesis .... i s j ust the X 7→ x .) Under this identification , numberj of orbitsj of Aut G on B sub Capital Gamma open parenthesis G closing parenthesis period \noindentAut G acts( i naturally i $ ) on R Γ{− ibases} ( , so xdΓ({Gj )} i s) j ust = the 1$number foralli of orbits of $ Aut\ in G$ on IB . Γ( G By). open parenthesis i closing parenthesis comma only the identity automorphism can fix any Capital Gamma hyphen basis comma so all orbits have( Given length bar $ Aut\ varepsilon G bar comma , $ we define $x { j } = X { j }\ varepsilon , $ and given x , we define By ( i ) , only the identity automorphism can fix any Γ− basis , so all orbits have length | Aut $ \andvarepsilon hence $ by $ X { j }\mapsto x { j } . ) $ \quad Under this identification , G |, Equation:and hence open parenthesis 8 period 1 closing parenthesis .. d sub Capital Gamma open parenthesis G closing parenthesis = bar B sub Capital Gamma\noindent openAut parenthesis G acts G closing naturally parenthesis on bar $ \Gamma divided by− bar$ Aut bases G bar = , Epi so open $d parenthesis{\Gamma Capital} ( Gamma $ G comma ) \ h f Gi l lclosingi s parenthesis j ust the number of orbits of Aut G on bar$ B divided{\Gamma by bar} Aut( G $ bar G period ) . For many groups G comma bar Aut G bar i s known| B open(G) | parenthesisEpi(Γ,G or) | is easily determined closing parenthesis comma and the main difficulty \noindent By ( i ) , only the identityd (G) = automorphismΓ = can fix. any $ \Gamma − $ basis(8.1) , so all orbits have length is Γ | AutG | | AutG | $ \tomid count$ epimorphisms Aut G $ \ ormid Capital, Gamma $ hyphen bases period .. Hall quoteright s method exploits the fact that it is generally easier to countFor homomorphisms many groups Capital G , | Aut Gamma G | righti s known arrow G ( comma or is easily or equivalently determined k hyphen ) , andtuples the x satisfying main difficulty open parenthesis i i closing parenthesis but\noindent notisnecessarily to countand epimorphismshence or Γ− bases . Hall ’ s method exploits the fact that it is generally openeasier parenthesis to count i closing homomorphisms parenthesis periodΓ → G , or equivalently k - tuples x satisfying ( i i ) but not \ beginLetnecessarily Hom{ a l i open g n ∗} parenthesis Capital Gamma comma G closing parenthesis denote the set of all homomorphisms delta : Capital Gamma right arrow Gd comma{\Gamma and define} ( G ) = \ f r a c {\mid B {\Gamma } (G) \mid }{\mid Aut G \mid } = \ f rphi a c { openEpi parenthesis ( \Gamma G closing parenthesis,G) = phi\mid Capital}{\ Gammamid openAut parenthesis G \mid G closing} . parenthesis\ tag ∗{$ ( = bar 8 Epi . open 1 parenthesis ) $} Capital Gamma\end{ a commal i g n ∗} G closing parenthesis bar comma (i). sigma open parenthesis G closing parenthesis = sigma sub Capital Gamma open parenthesis G closing parenthesis = bar Hom open parenthesis CapitalFor many Gamma groupsLet comma Hom G G $ closing(Γ ,, G\ ) parenthesismid denote$ the Aut bar set Gperiod of $ all\mid homomorphisms$ i s knownδ (:Γ or→ G is , easily and define determined ) , and the main difficulty is toEach count delta epimorphisms in Hom open parenthesis or $ \ CapitalGammaφ( G Gamma) =−φΓ($ commaG bases) =| GEpi closing . \quad(Γ, parenthesisG Hall) |, ’ maps s method Capital exploitsGamma onto the a unique fact subgroup that it H less is or generally equal easier to count homomorphisms $ \Gamma \rightarrow $ G , or equivalently k − tuples x satisfying ( i i ) but not necessarily G comma and conversely each epimor hyphenσ( G ) = σΓ( G ) =| Hom (Γ, G ) | . phismEach epsilonδ ∈ Hom : Capital(Γ, G Gamma ) maps rightΓ arrowonto Ha uniquecan be regarded subgroup as a H homomorphism≤ G , and conversely Capital Gamma each right epimor arrow - G comma so \ beginEquation:phism{ a l i gε open n:Γ∗}→ parenthesisH can be 8 regarded period 2 closing as a homomorphism parenthesis .. sigmaΓ open→ G parenthesis , so G closing parenthesis = sum phi open parenthesis H closing parenthesis( i period ) . \end{ a l i g n ∗} H less or equal G X We need to invert this equation comma expressingσ(G phi) = open parenthesisφ(H). G closing parenthesis = bar Epi open parenthesis(8.2) Capital Gamma comma G\ centerline closing parenthesis{ Let Hombar in t$ erms ( of\Gamma sigma open, parenthesis $ G ) denote G closing the parenthesis set of period all .... homomorphisms Hall does $ \ delta : \Gamma \rightarrow $ H ≤ G G ,this and by an define analogue} of the M o-dieresis bius Inversion Formula open square bracket 14 closing square bracket comma which replaces an equation We need to invert this equation , expressing φ( G ) =| Epi (Γ, G ) | in t erms of σ( G ) . Hall fdoes open parenthesis n closing parenthesis = sum g open parenthesis d closing parenthesis \ centerlined bar n { $ \phi ( $ G $ ) = \phi \Gamma ( $ G $ ) = \mid $ Epi $ ( \Gamma , $ G this by an analogue of the M o¨ bius Inversion Formula [ 14 ] , which replaces an equation $ )with \mid , $ } f ( n ) = P g ( d ) g open parenthesis n closing parenthesis = sum d bar n mu open parenthesis n divided by d closing parenthesis f open parenthesis d closing \ centerline { $ \sigma ( $ G $ ) = \dsigma| n {\Gamma } ( $ G $ ) = \mid $ Hom $ ( \Gamma parenthesiswith comma , $where G the $ )M dieresis-o\mid bius. function$ } mu : N right arrow Z is defined by g ( n ) = P µ( n ) f ( d ) , where the M o¨ bius function µ : N → Z is defined by sum d bar n mud|n opend parenthesis d closing parenthesis = Case 1 1 if n = 1 comma Case 2 0 if n greater 1 period \noindentLet S be theEach set of $ all\ subgroupsdelta H\ in less$ or equalHom G $ comma ( ( \ andGamma define the, $ Mdieresis-o G ) maps bius $ function\Gamma mu$ = mu onto G : a S right unique arrow subgroup Z H $ \ leq $ X 1 ifn = 1, G ,recursively and conversely by each epimor − µ(d) = 0 ifn > 1. Equation: open parenthesis 8 period 3 closingd| parenthesisn .. sum H greater equal K mu open parenthesis H closing parenthesis = braceleftbigg 1 0\noindent if to the powerphism of if K $ to\ thevarepsilon power of K =: less G\Gamma sub period\ torightarrow the power of$ G comma H can be regarded as a homomorphism $ \Gamma ≤ o¨ µ = µ G → \rightarrowThenLet open S parenthesis be$ the G ,set so 8 of period all subgroups 2 closing parenthesis H G ,and and open define parenthesis the M 8 periodbius 3 function closing parenthesis imply:S that Z recursively by \ begin { a l i g n ∗} \sigma ( G ) = \sum \phi (H). \ tag ∗{$ ( 8 . 2 ) $} X if K G, \end{ a l i g n ∗} µ(H) = {10 if K =

\noindent this by an analogue of the M $ \ddot{o} $ bius Inversion Formula [ 14 ] , which replaces an equation

\ centerline { f ( n $ ) = \sum $ g ( d ) }

\ centerline {d $ \mid $ n }

\noindent with

\noindent g ( n $ ) = \sum { d \mid n }\mu ( \ f r a c { n }{ d } ) $ f ( d ) , where the M $ \ddot{o} $ bius function $ \mu : $ N $ \rightarrow $ Z is defined by

\ [ \sum { d \mid n }\mu ( d ) = \ l e f t \{\ begin { a l i g n e d } & 1 i f n = 1 , \\ & 0 i f n > 1 . \end{ a l i g n e d }\ right . \ ]

Let S be the set of all subgroups H $ \ leq $ G , and define theM $ \ddot{o} $ bius function $ \mu = \mu $ G : S $ \rightarrow $ Z recursively by

\ begin { a l i g n ∗} \sum { H \geq K }\mu ( H ) = \{ 1{ 0 } i f ˆ{ i f } K ˆ{ K } ={ < } G ˆ{ G, } { . }\ tag ∗{$ ( 8 . 3 ) $} \end{ a l i g n ∗}

\noindent Then ( 8 . 2 ) and ( 8 . 3 ) imply that A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 33 A .sum J mu . Breda open parenthesis d ’ Azevedo H closing ; G parenthesis . A . Jones sigma : open\quad parenthesisPlatonic H closing Hypermaps parenthesis\quad = sum33 mu open parenthesis H closing parenthesis open$ parenthesis\sum \mu sum phi( $open H parenthesis $ ) \ Ksigma closing parenthesis( $ H closing $ ) parenthesis = \sum \mu ( $ H $ ) ( \sum \phi ( $ A . J . Breda d ’ Azevedo ; G . A . Jones : Platonic Hypermaps 33 P µ( H )σ( H K))H less or equal G .. H less or equal G .. K less or equal H ) = P µ( H )(P φ( K)) = sum open parenthesis phi open parenthesis K closing parenthesis sum mu open parenthesis H closing parenthesis closing parenthesis H ≤ GH ≤ GK ≤ H \ centerlineK less or equal{H G $ ..\ Hleq greater$ Gequal\quad K H $ \ leq $ G \quad K $ \ leq $ H } Equation: open parenthesis 8 period 4 closing parenthesisX ..X = phi open parenthesis G closing parenthesis period = (φ(K) µ(H)) \ [Substituting = \sum in open( parenthesis\phi (K) 8 period 1 closing\sum parenthesis\mu comma(H)) we obtain \ ] d sub Capital Gamma open parenthesis G closingK parenthesis≤ GH ≥ =K 1 divided by bar Aut G bar sum H less or equal G mu G open parenthesis H closing parenthesis sigma sub Capital Gamma open parenthesis H closing parenthesis period .. open parenthesis 8 period 5 closing parenthesis \ centerlineIf enough is{K known $ \ aboutleq $ .... G the\quad subgroupH .... $ \ lattgeq ice$ S K comma} .... the values of mu G open parenthesis H closing parenthesis .... can be calculated period = φ(G). (8.4) \ beginThis{ cana l i be g n a∗} t edious process comma though two straightforward observations ease the task : = openSubstituting\phi parenthesis(G). in a closing ( 8 . 1 parenthesis ) , we\ tag obtain∗{ ..$ if H( sub 8 1 comma . H4 sub ) 2 $ in} S are equivalent under Aut G comma then sigma sub G open parenthesis \end{ a l i g n ∗} 1 P H sub 1 closing parenthesis = sigma sub G open parenthesisdΓ( HG sub) = 2| closingAutG| parenthesisH≤G µ G (H semicolon)σΓ( H ) . ( 8 . 5 ) openIf enough parenthesis is known b closing about parenthesis the subgroup sigma sub G latt open ice parenthesis S , the valuesH closing of parenthesisµ G ( H )= 0can unless beH calculated i s an intersection . of m a-x imal subgroups of\noindent GThis open square canSubstituting be bracket a t edious 1 3 comma inprocess ( Theorem 8 ,. though 1 ) 2 period , twowe 3obtain straightforward closing square bracket observations period ease the task : Hall computes mu( a G ) open if parenthesisH1,H2 ∈ S H are closing equivalent parenthesis under .. for Aut several G , .. then classesσG of(H groups1) = σ ..G( GH2 comma); .. including the finite rotation groups \ hspaceand( b the∗{\) simpleσf i( lH l groups} )$ = d 0 PSLunless{\ subGamma H 2 open i s} an parenthesis( intersection $ G p $ closing ) of m = parenthesisa\−fx r aimal c { comma1 subgroups}{\ pmid prime of semicolonAut G [ 1 3 G .. , Theorem Downs\mid ..}\ open 2 sum square{ bracketH \ 1leq 1 closingG } \mu $ G (G H $ ) \sigma {\Gamma } ( $ H ) . \quad ( 8 . 5 ) square. 3bracket ] . Hall .. has computes extended thisµ G to( PSL H )sub 2 for open several parenthesis classes q closing of groupsparenthesis G.. for , all including the finite primerotation hyphen groups powers q period \noindent If enough is known about \ h f i l l the subgroup \ h f i l l l a t t i c e S , \ h f i l l the values of $ \mu $ G ( H ) \ h f i l l can be calculated . Asand a s the imple simple example groups commaPSL take2( Gp = ) A , p sub prime 4 period ; .. Downs The subgroup [ 1 1 latt ] ice has S i sextended shown in Figurethis to 8 periodPSL2( 2q : ) .. apart fromfor A all sub prime 4 and - the powers trivial q subgroup . 1 comma S contains a normal V sub 4 open parenthesis to the power of thicksim = C sub 2 times C sub 2 \noindent This can be a t edious process , though two straightforward observations ease the task : closing parenthesisAs a comma s imple four example non hyphen , take normal G = A4. The subgroup latt ice S i s shown in Figure 8 . 2 : subgroupsapart isomorphic to C sub 3 comma and three non hyphen normal subgroups i somorphic to C sub 2 period \ centerlineFigurefrom 8A period{and( a 2 the period ) \ trivialquad .. The subgroupi f subgroups $ H 1{ of ,1 S A} contains sub,H 4 a normal{ 2 }\V in (∼$= C S× areC ) equivalent, four non - normal under Aut G , then $ \sigma { G } (H { 14 } ) = \sigma { G } (H { 2 } )4 ; $ } 2 2 Applyingsubgroups open isomorphic parenthesis 8 to periodC3, and 3 closing three parenthesis non - normal .... we subgroups find that mu i open somorphic parenthesis to C A2. sub 4 closing parenthesis = 1 comma mu open parenthesis V sub 4 closing parenthesisFigure = mu open 8 . 2 parenthesis . The C subgroups sub 3 closing of parenthesisA = minus 1 comma mu open parenthesis C sub 2 closing \noindent ( b $ ) \sigma { G } ($ H $) = 0$ unlessHi4 san intersection ofm $a−x $ imal subgroups of G [ 1 3 , Theorem 2 . 3 ] . parenthesisApplying = 0 and( 8 .mu 3 open) we parenthesis find that 1 closingµ(A4) parenthesis = 1, µ(V =4) 4 comma = µ(C3 ....) so = −1, µ(C2) = 0 and µ(1) = 4, Hallopenso computes parenthesis 8 $ period\mu 4$ closing G ( parenthesis H ) \quad takesf o r the s e form v e r a l \quad classes of groups \quad G, \quad including the finite rotation groups Equation:( 8 . 4 ) open takes parenthesis the form 8 period 6 closing parenthesis .. phi open parenthesis A sub 4 closing parenthesis = sigma open parenthesis A sub 4\noindent closing parenthesisand the minus simple sigma groups open parenthesis $ PSL V{ sub2 4} closing($ parenthesis p) ,pprime minus 4 sigma ; \quad openDowns parenthesis\quad C sub[ 3 1 closing 1 ] \ parenthesisquad has plus extended 4 this to sigma$ PSL open{ parenthesis2 } ( $ 1 closing q ) \quad parenthesisf o r period a l l prime − powers q . S imilarly comma Hall obtains φ(A4) = σ(A4) − σ(V4) − 4σ(C3) + 4σ(1). (8.6) phi open parenthesis S sub 4 closing parenthesis = sigma open parenthesis S sub 4 closing parenthesis minus sigma open parenthesis A sub 4 closing\ hspaceS imilarly parenthesis∗{\ f i l , l Hall} minusAs obtains a 3 s sigma imple open example parenthesis , takeD sub G 4 closing $= parenthesis A { 4 minus} .4 $ sigma\quad openThe parenthesis subgroup S sub latt 3 closing ice parenthesis S i s shown plus 3in Figure 8 . 2 : \quad apart sigma open parenthesis V sub 4 closing parenthesis plus 4 sigma open parenthesis C sub 3 closing parenthesis plus 1 2 sigma open parenthesis C sub 2\noindent closing parenthesisfromφ minus $(S4 A) = 1{σ 2(4S sigma4)}−$σ open( andA4) parenthesis− the3σ(D trivial4) − 14 closingσ(S3 subgroup) + parenthesis 3σ(V4) + 1 4σ ,(C S3) contains + 12σ(C2) − a12 normalσ(1) $ V { 4 } ( ˆ{\sim } = C { 2 }\times C { 2 } ) , $ f ou r non − normal andand phi open parenthesis A sub 5 closing parenthesis = sigma open parenthesis A sub 5 closing parenthesis minus 5 sigma open parenthesis A sub 4 \noindent subgroups isomorphic to $ C { 3 } , $ and three non − normal subgroups i somorphic to $ C { 2 } closing parenthesis minusφ(A 65 sigma) = σ(A open5) − parenthesis5σ(A4) − 6σ D(D sub5) − 5 closing10σ(S3) parenthesis + 20σ(C3)minus + 60σ( 10C2) sigma− 60σ open(1), parenthesis S sub 3 closing parenthesis plus 20 sigma. $ open parenthesis C sub 3 closing parenthesis plus 60 sigma open parenthesis C sub 2 closing parenthesis minus 60 sigma open parenthesis 1 closing parenthesis comma \ centerline { Figure 8 . 2 . \quad The subgroups of $ A { 4 }$ }

\noindent Applying ( 8 . 3 ) \ h f i l l we find that $ \mu (A { 4 } ) = 1 , \mu (V { 4 } ) = \mu (C { 3 } ) = − 1 , \mu (C { 2 } ) = 0 $ and $ \mu ( 1 ) = 4 , $ \ h f i l l so

\noindent ( 8 . 4 ) takes the form

\ begin { a l i g n ∗} \phi (A { 4 } ) = \sigma (A { 4 } ) − \sigma (V { 4 } ) − 4 \sigma ( C { 3 } ) + 4 \sigma ( 1 ) . \ tag ∗{$ ( 8 . 6 ) $} \end{ a l i g n ∗}

\noindent S imilarly , Hall obtains

\ [ \phi (S { 4 } ) = \sigma (S { 4 } ) − \sigma (A { 4 } ) − 3 \sigma ( D { 4 } ) − 4 \sigma (S { 3 } ) + 3 \sigma (V { 4 } ) + 4 \sigma (C { 3 } ) + 1 2 \sigma (C { 2 } ) − 1 2 \sigma ( 1 ) \ ]

\noindent and

\ [ \phi (A { 5 } ) = \sigma (A { 5 } ) − 5 \sigma (A { 4 } ) − 6 \sigma (D { 5 } ) − 10 \sigma (S { 3 } ) + 20 \sigma (C { 3 } ) + 60 \sigma (C { 2 } ) − 60 \sigma ( 1 ) , \ ] 34 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentwhere D sub34 n\ denotesquad A a dihedral. J . Breda group of d order ’ Azevedo 2 n period ; G . A . Jones : Platonic Hypermaps Once34 .. Acomputed . J . Breda comma d .. ’ Azevedo the .. values ; G .. . of A mu . Jones G open : parenthesis Platonic Hypermaps H closing parenthesis .. can be .. used .. in .. open parenthesis 8 period 4 closing\noindent parenthesiswhere .. and $ D .. open{ n parenthesis}$ denotes 8 period a 5 dihedral closing parenthesis group of .. for order .. any 2 finitely n . hyphen where D denotes a dihedral group of order 2 n . generated groupn Capital Gamma period .. The values of sigma sub Capital Gamma open parenthesis H closing parenthesis are most easily Once computed , the values of µ G ( H ) can be used in ( 8 . 4 ) and calculatedOnce \quad whencomputed the defining , relations\quad the \quad values \quad o f $ \mu $ G ( H ) \quad can be \quad used \quad in \quad ( 8 . 4 ) \quad and \quad ( 8 . 5 ) \quad f o r \quad any finitely − ( 8 . 5 ) for any finitely - generated group Γ. The values of σ ( H ) are most easily generatedR sub i open group parenthesis $ \Gamma X sub j closing. $ parenthesis\quad The = 1 ofvalues Capital of Gamma $ \sigma take a simple{\ΓGamma form : ..} thus( if $ Capital H )Gamma are most i s a easily free group calculated F sub when the defining relations calculated when the defining relations R (X ) = 1 of Γ take a simple form : thus if Γ i s a free k of$ rank R { k wei have} (X sigma sub{ Capitalj } ) Gamma = open 1 $i parenthesisj o f $ \ HGamma closing$ parenthesis take a = simple form : \quad thus i f $ \Gamma $ i s a free group group F of rank k we have σ ( H ) = $ Fbar{ Hk bar}$ tok the of power rank of k k we comma haveΓ while $ \ isigma f Capital{\ GammaGamma =} Capital( $ Delta H $ thicksim ) = = $ C sub 2 * C sub 2 * C sub 2 we have sigma sub | H |k, while i f Γ = ∆ ∼= C ∗ C ∗ C we have σ ( H ) = n ( H )3, where n ( H ) i s the number of Capital Gamma open parenthesis H2 closing2 parenthesis2 = nΓ sub 2 open2 parenthesis H closing2 parenthesis to the power of 3 comma where n sub 2 so lut ions of h2 = 1 in H . open\noindent parenthesis$ H\mid closing$ parenthesis H $ \mid i s theˆ{ numberk } , of $ while i f $ \Gamma = \Delta \sim = C { 2 } ∗ C { 2 } For example , i f G = A ( so that | Aut G |=| S |= 24) then ( 8 . 5 ) and ( 8 . 6 ) give ∗ soC lut{ ions2 of}$ h to we the have power of $ 2\ =sigma4 1 in H{\ periodGamma } ( $4 H $ ) = n { 2 } ( $ H $ ) ˆ{ 3 } , $ where $ n { 2 } ( $ H) i s the number of For example comma i f G = A sub 4 open parenthesis1 so that .. bar Aut G bar = bar S sub 4 bar = 24 closing parenthesis then open parenthesis so lut ions of $h ˆ{ 2 } =d ( 1A $) = in(4 H3 − . 43 − 4.13 + 4.13) = 0 8 period 5 closing parenthesis and open parenthesis∆ 4 824 period 6 closing parenthesis give d sub Capital Delta open parenthesis A sub 4 closing parenthesis = 1 divided by 24 open parenthesis 4 to the power of 3 minus 4 to the power of A 3\ centerline minus( which 4 period i{ sFor 1 obvious to examplethe power , s ince of , 3 i plus fG4 cannot 4 period $= be 1 toA generated the{ power4 } by of( 3 involutions $ closing so thatparenthesis )\ ,quad and = 0 $ \mid $ Aut G $ \mid = \mid S { 4 } \mid = 24 )$ then(8.5)and(8.6)give } open parenthesis which i s obvious comma s ince1 A sub 4 cannot be generated by involutions closing parenthesis comma and d (A ) = (122 − 42 − 4.32 + 4.12) = 4. d sub F sub 2 open parenthesis A subF 42 closing4 parenthesis24 = 1 divided by 24 open parenthesis 1 2 to the power of 2 minus 4 to the power of 2 minus\ [ d 4{\ periodDelta 3 to the} power(A of 2{ plus4 } 4 period) =1 to\ thef r a power c { 1 of}{ 2 closing24 } parenthesis( 4 ˆ{ =3 4 period} − 4 ˆ{ 3 } − 4 . 1 ˆ{ 3 } + 4These . 1 functions ˆ{ 3 } )d and =d 0 \are] Hall ’ s c and d ; thus his results also give d (S ) = 13 and These functions d sub Capital∆ DeltaF2 and d sub F sub2,2, 2 are Hall2 quoteright s c sub 2 comma 2 comma∆ 4 2 and d sub 2 semicolon thus his results also gived∆(A d5) sub = 19 Capital, confirming Delta open our parenthesis direct enumerations S sub 4 closing of parenthesis regular S =4− 13and and A5− hypermaps in §4 and §5 . S imilarly d (S ) = 9 and d (A ) = 19, results which we will need in §9 . d sub Capital Delta openF2 4 parenthesis AF sub2 5 closing parenthesis = 19 comma confirming our direct enumerations of regular S sub 4 hyphen and A\noindent sub 5 hyphenOne( can hypermapswhich refine i Hall s in obvious S ’ 4 s and method , s to ince count $A not only{ 4 the}$ total cannot number be generated of regular byG - involutions hypermaps ) , and Sfor 5 period a given .. S group imilarly G d , sub but F also sub 2the open number parenthesis of such S sub hypermaps 4 closing parenthesis of any given = 9 and type d .sub F For sub i 2 f open we parenthesis A sub 5 closing \ [ dreplace{ F ∆{ with2 }} the(A extended{ 4 triangle} ) group = \ f rΓ a = c { ∆(1l ,}{ m24 , n} ) , obtained( 1 2by ˆ adding{ 2 } the − relations4 ˆ{ 2 } − 4 . 3 ˆ{ 2 } parenthesis =l 19 commam results whichn we will need in S 9 period +( 4R1R2) .= ( 1R2 ˆR{0)2 =} (R)0R1) == 1 4to ∆ . , we\ ] find that dΓ( G ) now gives the number of regular G - One can refine Hall quoteright0 s0 method0 to count0 not0 only the0 total number of regular G hyphen hypermaps forhypermaps a given group of G type comma(l but, m also, n ) thewhere numberl of, m suchand hypermapsn divide of l any , m given and type n ; period by doing .. For i this f we for all l replace, m and Capital n which Delta witharise the as extended orders of triangle elements group of Capital G , we Gamma obtain = Capital the required Delta open result parenthesis . ( Of l comma course m comma n closing parenthesis comma\noindent, for obtained comparitivelyThese by adding functions the “ relations straightforward $ d {\Delta ”}$ groups and G $ ,d direct{ F methods{ 2 }} are$ more are Hall efficient ’s , as $c in { 2 , 2 , 2 }$ andopen§§ $4 parenthesis –d 7{ . )2 } R sub; $ 1 R thus sub 2 his closing results parenthesis also to givethe power $ dof l{\ = openDelta parenthesis} (S R sub{ 24 R sub} 0) closing = parenthesis 13 $ and to the power of m =$9 open d .{\ parenthesisDelta Rotary} R Platonic sub(A 0 R sub{ hypermaps 15 closing} ) parenthesis = 19 to the , $ power confirming of n = 1 to Capital our direct Delta comma enumerations we find that of d sub regular Capital $Gamma S { open4 } −parenthesis$Let and H G be $ closing A any{ parenthesis orientable5 } − $ now hypermap hypermaps gives the withoutnumber in \S of boundary4 regular and , so it s hypermap subgroup H is con - \S tained5 . \quad in theS even imilarly subgroup $ d∆+{ofF∆,{and2 }} it s rotation(S { group4 } ( more) = precisely 9 $ , and it s orientation $ d { F -{ 2 }} (A { 5 } G hyphen hypermaps of type .. open parenthesis+ l to the power of prime comma m to the power of prime comma n to the power of prime closing parenthesis)preserving = 19 .. where automorphism , $ l to resultsthe power group whichweof prime) comma Aut will mH need to is the i somorphic in power\S of9 prime . to N and∆ + n ( toH the ) / power H . Weof prime call divide H rotary l comma m and n semicolon .. by i f doing this+ for all \ hspaceAut ∗{\H actsf i l l } transitivelyOne can refine on the Hall set of ’ “ s brins method ” of to H (count the cycles not only of r2 theon the total set Ω numberof blades of ) regular ; G − hypermaps l comma m and n which arise as orders+ of elements+ of G comma+ we obtain the required result period .. open parenthesis Of course comma forequivalently comparitively , ..H quotedblleft is normalin straightforward∆ , so Aut quotedblrightH ∼= ∆ / ..H groups . ( G Our comma t erminology direct methods follows are more Vince efficient and comma as in S S 4 endash 7 period\noindentWilson closing ,for parenthesis rather a given than Coxetergroup G and , but Moser also who the use number the t erms of such “ hypermaps reflexible ” of and any given “ regular type . \quad For i f we r e p” l a c e where $ \Delta we have$ used with the “ regular extended ” triangle and “ group rotary ” $ .\Gamma ) If H= is regular\Delta then( $ it i l s , m , n ) , obtained by adding the relations 9 period .. Rotary Platonic hypermaps + Letrotary H be any ( orientable since normality hypermap of without H in boundary∆ implies comma normality so it s hypermap in ∆ ), subgroupbut not H is con conversely hyphen . If \noindenttainedH is inrotary the$ even ( but subgroup R not{ regular1 Capital} R it Delta{ is2 tocalled} the) power ˆ{ chirall of} plus=; of Capital( such R hypermaps Delta{ 2 comma} R o and ccur{ 0 it in} s rotation mirror) ˆ{ m group - image} open= parenthesis ( R { more0 } preciselyR { 1 } ) ˆ{pairsn } ,= with 1 $mutually to $ reversed\Delta orienta, $ - we t ions find , that corresponding $d {\Gamma to pairs} of( normal $ G ) subgroups now gives the number of regular comma it s orientation+ hyphen G preserving−Hhypermaps of ∆ automorphismwhich of are type conjugate group\quad closing in$∆ parenthesis(. For l ˆ{\ example Autprime to the ,} the power, torus of m plus ˆmap{\ Hprime{ is4 i somorphic , 4}} b, , c toof n N [ ˆ sub 1{\ 0 Capitalprime , §8 Delta .} ) plus $ open\quad parenthesiswhere H $ l ˆ{\prime } closing,3 m ] parenthesis ˆ{\ isprime regular slash}$ if H bc periodand ( b $− ....c n We) ˆ{\ call =prime 0 H, but .... rotary is}$ chiral idivide f ( with l ,mandn mirror - image; \quad{4,by4}c, doingb ) if bc this ( b for− all lAut ,c m) to6= and the 0( see power n which Figure of plus arise 9 H . 1acts ) as . transitively orders ofon the elements set of quotedblleft of G , we brins obtain quotedblright the required .... of H open result parenthesis . \quad the cycles( Of of course r sub 2 on , the setfor Capital comparitively Omega of blades\quad closing‘‘ parenthesis straightforward semicolon ’’ \quad groups G , direct methods are more efficient , as in \S \S 4 −− 7 . ) equivalently comma H is normal in Capital Delta to the power of plus comma so Aut to the power of plus H thicksim = Capital Delta to the power\noindent of plus9 slash . \ Hquad periodRotary .. open parenthesisPlatonic Our hypermaps t erminology follows Vince and Wilson comma rather than Coxeter and Moser who use the t erms .. quotedblleft reflexible quotedblright .. and .. quotedblleft regular quotedblright\noindent Let .. where H be any orientable hypermap without boundary , so it s hypermap subgroup H is con − tainedwe have in used the .. quotedblleft even subgroup regular quotedblright $ \Delta ˆ ..{ and+ } ..$ quotedblleft o f $ \Delta rotary quotedblright, $ and period it s closing rotation parenthesis group .. If( H more is regular precisely then it , it s orientation − i s rotary .. open parenthesis since normality of \noindentH in Capitalpreserving Delta .. implies automorphism normality in Capital group Delta $ ) to the Aut power ˆ{ of+ plus}$ closing H is parenthesis i somorphic comma to .. but $N not{\ .. converselyDelta } period+ .. ( If $ H isH ) / H . \ h f i l l We c a l l H \ h f i l l rotary i f rotary but not regular it .. is \noindentcalled .. chiral$ Aut semicolon ˆ{ + ..} such$ H hypermaps acts transitively o ccur in mirror on hyphen the image set of pairs ‘‘ comma brins .. with’’ \ mutuallyh f i l l of reversed H( the orienta cycles hyphen of $ r { 2 }$ ont the ions commas e t $ ..\ correspondingOmega $ of to bladespairs of normal ) ; subgroups H of Capital Delta to the power of plus which are conjugate in Capital Delta period .. For \noindentexample commaequivalently the torus map , H open is brace normal 4 comma in $4 closing\Delta braceˆ{ b+ comma} , c $ of open sosquare $ Aut bracket ˆ{ + 1}$ 0 comma H $ ..\sim S 8 period= 3 closing\Delta squareˆ{ + } bracket/ $ H .. . is\ regularquad if( bc Our open t parenthesis erminology b minus follows c closing Vince parenthesis and = 0 comma but is chiral .. open parenthesis with Wilsonmirror hyphen , rather image than open Coxeterbrace 4 comma and 4 Moser closing who brace use sub c the comma t erms b closing\quad parenthesis‘‘ reflexible if bc open parenthesis ’’ \quad band minus\quad c closing‘‘ parenthesis regular ’’ \quad where negationslash-equalwe have used \ 0quad open parenthesis‘‘ regular see ’’Figure\quad 9 periodand 1\ closingquad parenthesis‘‘ rotary period ’’ . ) \quad If H is regular then it i s rotary \quad ( since normality of H in $ \Delta $ \quad implies normality in $ \Delta ˆ{ + } ) , $ \quad but not \quad conversely . \quad If H is rotary but not regular it \quad i s c a l l e d \quad c h i r a l ; \quad such hypermaps o ccur in mirror − image pairs , \quad with mutually reversed orienta − t i o n s , \quad corresponding to pairs of normal subgroups H of $ \Delta ˆ{ + }$ which are conjugate in $ \Delta . $ \quad For example , the torus map \{ 4 , 4 \} b , c o f [ 1 0 , \quad \S 8 . 3 ] \quad is regular if bc ( b $ − $ c $) = 0 ,$ butischiral \quad ( with mirror − image $ \{ 4 , 4 \} { c , }$ b ) i f bc ( b $ − $ c $ ) \not= 0 ($ seeFigure9.1). A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 35 \ hspaceFigure∗{\ 9 periodf i l l } 1A period . J .. . The Breda chiral d maps ’ Azevedo open brace ; G 4 comma . A . 4 Jones closing brace: \quad 2 commaPlatonic 1 and open Hypermaps brace 4 comma\quad 4 closing35 brace 1 comma 2 Now suppose that H i s any rotaryA . J hypermap . Breda d with ’ Azevedo rotation group; G . AAut . Jones to the power : Platonic of plus H Hypermaps thicksim = Aut to 35 the power of plus P for some \ centerlinePlatonic so{ lidFigure P period 9 .. . Then 1 . there\quad areThe three chiral possibilities maps : \{ 4 , 4 \} 2 , 1 and \{ 4 , 4 \} 1 , 2 } Figure 9 . 1 . The chiral maps { 4 , 4 } 2 , 1 and { 4 , 4 } 1 , 2 i closing parenthesis .. H is regular comma with Aut H thicksim = Aut P semicolon + + \noindentNow supposeNow suppose that H i that s any H rotary i s any hypermap rotary with hypermap rotation with group rotationAut H group∼= Aut $P Aut for ˆ some{ + }$ H $ \sim = Aut ˆ{ + }$ i iPlatonic closing parenthesis so lid P. .. H Then is regular there comma are withthree Aut possibilities H similar-equal : Aut P semicolon P fii o ir closing some parenthesis .. H is not regular comma that i s comma H i s chiral period i ) H is regular , with Aut H ∼= Aut P ; We have already determined the hypermaps H satisfying i closing parenthesis comma and our aim is now to show that \noindent Platonic so lidi P i ). \quad H is regularThen there , with are Aut three H =∼ Aut possibilities P ; : cases i i closing parenthesis andii i ii ) i closing H is parenthesis not regular do , not that o ccur i s , period H i s chiral . Theorem period .... If H .... is .... a rotary hypermap comma .... with Aut to the power of plus H .... thicksim = Aut to the power of plus P for \ centerlineWe have{ alreadyi ) \quad determinedH is regular the hypermaps , with AutH H satisfying $ \sim i ) , and= $ our Aut aim P is; } now to show that s omecases Platonic i i ) s and o lid ii P i comma ) do not o ccur . then H is regular with Aut H thicksim = Aut P period + + \ centerlineTheorem{ i . i )If \quadH isH a is rotary regular hypermap , with , AutH with Aut $ \congH ∼$= Aut AutP Pfor ; } s ome Platonic s o Tolid proveP , this comma we note that for any group G comma the rotary hypermaps H with Aut to the power of plus H = sub thicksim G are in bij ective correspondence with the normal subgroups H of Capital Delta to the power of plus with Capital Delta to the power of plus slash H then H is regular with Aut H ∼= Aut P . thicksim\ centerline = G semicolon{ i i i ) since\quad CapitalH is Delta not to regular the power of, plusthat i s ,H i s chiral . } To prove this , we note that for any group G , the rotary hypermaps H with Aut+ H = G are in i s a free group of rank 2 open parenthesis with basis R sub 0 R sub 1 comma R sub 1 R sub 2 closing∼ parenthesis they correspond to the orbits bij ective correspondence with the normal subgroups H of ∆+ with ∆+/ H ∼= G ; since ∆+ i s a of\noindent Aut G on We have already determined the hypermaps H satisfying i ) , and our aim is now to show that casesfree groupi i ) of and rank ii 2 i ( with ) do basis not oR0 ccurR1,R1R . 2) they correspond to the orbits of Aut G on generating generatingpairs for pairs G . for G period First let P be the t etrahedron T comma .. so that .. G = Aut to the power of plus T can be identified with A sub 4 comma .. and First let P be the t etrahedron T , so that G = Aut+ T can be identified with A , and \noindentAut G withTheorem S sub 4 acting . \ h fby i l conjugationl I f H \ h on f i lG l periodi s \ h .. f By i l l inspectiona rotary comma hypermap .. or by , Hall\ h quoterightf i l l with s4 method $ Aut .. ˆ open{ + parenthesis}$ H \ h asf i l in l S 8$ \sim Aut G with S acting by conjugation on G . By inspection , or by Hall ’ s method ( as in =closing Aut parenthesis ˆ{ + }$4 P for s ome Platonic s o lid P , §8 ) one easily finds that d ( G ) = 4, so that Aut G has four orbits one .. easily .. finds .. that .. d sub F sub 2 openF2 parenthesis G closing parenthesis = 4 comma .. so .. that .. Aut G .. has .. four .. orbits .. on \noindenton generatingthen H is pairs regular for with G : Aut the H two $ \sim generators= $ Aut can haveP . orders 2 and 3 , or .. generating3 and .. pairs 2 , .. for or 3 and 3 , and in this last case they may or may not be conjugate in GG : .. . the Thus two .. there generators are , can up have to orientation .. orders .. 2 - .. preserving and 3 comma .. or .. 3 .. and .. 2 comma .. or 3 .. and .. 3 comma .. and in this last \noindentcase they mayTo prove or may not this be conjugate, we note in G that period for .. Thus any there group are G comma , the up rotary to orientation hypermaps hyphen H preserving with $ Aut ˆ{ + }$ H $ = {\sim }$ i somorphism , four rotary hypermaps H with Aut+ H ∼= A . We saw in §4 that among the 13 G arei somorphism comma four rotary hypermaps H with Aut to the power of4 plus H thicksim = A sub 4 period .. We saw in S 4 that among the 13 regular S − hypermaps , four are orientable , with rotation group A . Clearly these are rotary , inregular bijS ective sub4 4 hyphen correspondence hypermaps comma with four the are normal orientable subgroups comma with H rotation of4 $ group\Delta A subˆ{ 4+ period}$ with.. Clearly $ these\Delta are rotaryˆ{ + comma} / $ H $so\ theysim must= $ be G the ; sfour i n c e rotary $ \Delta hypermapsˆ{ + H}$ we have j ust enumerated . ( In fact , the four so they must be the four rotary hypermaps H we have j ust enumerated(12) period(1) .. open parenthesis In− fact1 0ˆ comma the four i sorbits a free on generating group of rank pairs 2correspond ( with basis to thehypermaps $R { 0 } T R , {T 1, }T and,RS2 {= 1 W} R(T {) 2 } ) $ they correspond to the orbits of Aut G on orbits on generating pairs correspond to the hypermaps T to the power of open parenthesis+ 1 2 closing+ parenthesis comma T to the power of open parenthesisgeneratingrespectively 1 closing pairs . )parenthesis for Thus G . commaeach of T the and four S sub rotary 2 = W hypermaps to the power H of with minusAut 1 openH parenthesis∼= Aut T T is to regular the power of circumflex-zero closing parenthesis, with Aut H =∼ S4 ∼= Aut T . FirstrespectivelyA let s imilar P period be the method closing t etrahedron parenthesis can be used .. T Thus in , the\ eachquad remaining of theso four that cases rotary\quad . hypermapsG When $ = H P with= AutO Aut (ˆ{ orto+ the equivalently}$ power T ofcan plus C be H ) thicksim identified = Aut with to the power $ A { 4 } , $ , \quad and of plus T is regular comma+ Autwithwe G Autput with H G = $ sub= S Aut thicksim{ O4 }∼$ S= subS acting4, 4so thicksim that by Aut = conjugation Aut G T∼ period= S4, onacting G . on\quad it selfBy by inspection conjugation ,.\ Eitherquad or by by Hall ’ s method \quad ( as in \S 8 ) oneAinspection s\ imilarquad methode a or s i l by y can Hall\quad be used ’ sf method i in n d the s \ remainingquad we findthat cases that\quad periodS4 has ..$ When9 d conjugacy{ PF = O{ open2 classes}} parenthesis( of $ generating G or $equivalently ) pairs = , C 4 closing , $ parenthesis\quad so comma\quad that \quad Aut G \quad has \quad fo ur \quad o r b i t s \quad on \quad g e n e r a t i n g \quad p a i r s \quad f o r G:so\ therequad arethe 9 two rotary\quad hypermapsgenerators H with canAut have+ H \quad∼= Auto+ r dO e r . s \quad Now2 in \§quad7 we foundand 3 39 , regular\quad or \quad 3 \quad and \quad 2 , \quad or 3 \quad and \quad 3 , \quad and in this last we put .... G = Aut to the power of plus O .... thicksim = S sub 4 comma .... so that Aut G+ thicksim = S sub 4 comma .... acting on it self by case(S4 they× C2)− mayhypermaps or may not H , be2 1 conjugate of them orientable in G . ;\quad amongThus these there , 9 have areAut , upH to∼= orientationS4, namely − p r e s e r v i n g conjugation period .... Either by+ + −1 0 inspectionthe s ix or associates by Hall quoteright of S3 = sO method and the we find three that associates S sub 4 has of 9 conjugacyS4 = W classes(O ). ofThese generating must pairs be comma the 9 \noindentsorotary there are hypermapsi 9 somorphism rotary hypermaps previously , four H with enumerated rotary Aut to hypermaps the , so power again of H plus the with H result thicksim $ Auti s verified= ˆ Aut{ + to} .$ the H power $ of\sim plus O= period A ....{ Now4 } in S. 7 $ we\ foundquad 39We saw in \S 4 that among the 13 regularr e g u l aLikewise r $ S ,{ when4 } P −=$D or hypermaps I we put G , four∼= A5 are, so Autorientable G ∼= S5 ,, again with acting rotation by conjugation group $ A . { 4 } . $ \quad Clearly these are rotary , soBy they[13] must, d ( beG ) the = 19 fourso there rotary are 19 hypermaps rotary hypermaps H we have H jwith ustAut enumerated+ H ∼= G . , and\quad these( In must fact , the four open parenthesisF2 S sub 4 times C sub 2 closing parenthesis hyphen hypermaps H comma .... 2 1 of them orientable semicolon .... among these comma 9 have Aut to the power of plus H thicksim = S sub 4 comma namely \noindentthe s ix associatesorbits of on S sub generating 3 to the power pairs of plus correspond = O and the tothree the associates hypermaps of S sub $ 4 toT the ˆ{ power( 1 of plus 2 = W ) to} the, power T ˆ of{ minus( 1 open ) } parenthesis, $ T and O to $ the S power{ 2 of} 0 closing= W parenthesis ˆ{ − 1 period} ( .. These T ˆ{\ musthat be{ the0}} ) $ 9 rotary hypermaps previously enumerated comma so again the result i s verified period \noindentLikewise commarespectively when P = D . or ) I\ wequad putThus G thicksim each = of A sub the 5 comma four rotaryso Aut G hypermaps thicksim = S H sub with 5 comma $ Aut again ˆ acting{ + } by$ conjugation H $ \sim period= AutBy ˆ{ open+ } square$ T bracket is regular 13 closing , square bracket comma d sub F sub 2 open parenthesis G closing parenthesis = 1 9 so there are 19 rotary hypermapswith Aut H H with $ Aut = to{\ thesim power} ofS plus{ H4 thicksim}\sim = G= comma $ Aut and theseT . must \ hspace ∗{\ f i l l }A s imilar method can be used in the remaining cases . \quad WhenP $=$ O ( or equivalently C ) ,

\noindent we put \ h f i l l G $ = Aut ˆ{ + }$ O \ h f i l l $ \sim = S { 4 } , $ \ h f i l l so that AutG $ \sim = S { 4 } , $ \ h f i l l acting on it self by conjugation . \ h f i l l Either by

\noindent inspection or by Hall ’ s method we find that $ S { 4 }$ has 9 conjugacy classes of generating pairs ,

\noindent so there are 9 rotary hypermaps H with $ Aut ˆ{ + }$ H $ \sim = Aut ˆ{ + }$ O . \ h f i l l Now in \S 7 we found 39 regular

\noindent $ ( S { 4 }\times C { 2 } ) − $ hypermaps H , \ h f i l l 2 1 of them orientable ; \ h f i l l among these , 9 have $ Aut ˆ{ + }$ H $ \sim = S { 4 } , $ namely

\noindent the s ix associates of $ S ˆ{ + } { 3 } =$ Oand the three associates of $ S ˆ{ + } { 4 } = W ˆ{ − 1 } ( O ˆ{ 0 } ) . $ \quad These must be the 9 rotary hypermaps previously enumerated , so again the result i s verified .

\ hspace ∗{\ f i l l } Likewise , whenP $=$ Dor I we putG $ \sim = A { 5 } , $ so Aut G $ \sim = S { 5 } , $ again acting by conjugation .

\noindent By $ [ 13 ] , d { F { 2 }} ( $ G $ ) = 1 9 $ so there are 19 rotary hypermapsHwith $ Aut ˆ{ + }$ H $ \sim = $ G , and these must 36 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : Platonic Hypermaps \noindentbe the ....36 19 regular\quad orientableA . J . .... Breda open dparenthesis ’ Azevedo A sub ; 5G times . A C . sub Jones 2 closing : Platonic parenthesis Hypermaps hyphen hypermaps l ist ed in Table 5 .... open parenthesis36 A S 6 . periodJ . Breda 2 closing d ’ Azevedo parenthesis ; Gperiod . A .... . Jones This completes : Platonic Hypermaps \noindentthe proof periodbe the \ h f i l l 19 regular orientable \ h f i l l $ ( A { 5 }\times C { 2 } ) − $ hypermaps l ist ed in Table 5 \ h f i l l ( \S 6 . 2 ) . \ h f i l l This completes be the 19 regular orientable (A × C )− hypermaps l ist ed in Table 5 ( §6 . 2 ) . This References 5 2 \noindentcompletesthe proof . openthe square proof bracket. 1 closing square bracket .. Brahana comma H period R period : .. Regular maps and their groups period Amer period J period Math period 49 open parenthesis 1927 closing parenthesis comma 268 endash 284 period \noindentReferencesReferences open[ 1 ] square Brahana bracket ,2 H closing . R . square : Regular bracket .. maps Brahana and comma their groups H period. R Amer period . semicolon J . Math .. . Coble49 ( comma 1927 )A , period B period : .. Maps of twelve268 countries – 284 with . [ 2five ] sides Brahana with a group , H . .. R of . ; Coble , A . B . : Maps of twelve countries with five \noindentorder 120 containing[ 1 ] \quad an i cosahedralBrahana subgroup , H . R period . : Amer\quad periodRegular J period maps Math and period their 48 open groups parenthesis . Amer 1 926 . Jclosing . Math parenthesis . 49 (comma 1927 1 ) , 268 −− 284 . [ 2sides ] \quad with aBrahana group , of H . R . ; \quad Coble ,A. B . : \quad Maps of twelve countries with five sides with a group \quad o f endashorder 20 period 120 containing an i cosahedral subgroup . Amer . J . Math . 48 ( 1 926 ) , 1 – 20 . [ 3 ] open square bracket 3 closing square bracket .. Breda d quoteright Azevedo comma A period J period : .. Ph period D period thesis comma \noindentBreda dorder ’ Azevedo 120 , containing A . J . : Ph an . i D cosahedral . thesis , University subgroup of Southampton . Amer . J . , 1991Math . . [ 48 4 ] ( 1 Breda 926 ) , 1 −− 20 . Universityd ’ Azevedo of Southampton , A . J comma. ; Jones 1991 , G period . A . : Double coverings and reflexible abelian hypermaps . [open 3 ] square\quad bracketBreda 4 closing d ’ Azevedo square bracket , A ... BredaJ . : d\ quoterightquad Ph Azevedo . D . comma thesis A period , University J period semicolon of Southampton Jones comma , G 1991 period . A period [ 4 ] \quad Breda d ’ Azevedo ,A . J .Preprint ; Jones . ,G . 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J .{\ Combimath . }9 {( 1988, }$ )\ ,quad 337 –A.: 35 1 .\quad [ 8 Maps , \quad hypermaps \quad and their automorphisms : \quad a survey I , \quad II, \quad III. comma] .. Coxeter hypermaps , H .. and . S their. M . automorphisms : Regular : Polytopes .. a survey I. comma Dover .. , II New comma York .. III , 3 period rd ed . 1 973 . [ 9 Expositiones Math period .. 10 open parenthesis 1 992 closing parenthesis comma 403 endash 427 comma 429 endash 447 comma 449 endash 467 \noindent] CoxeterExpositiones , H . S . M Math . : . Geometry\quad 10. ( Wiley 1 992 , New) , 403 York−− / London427 , 429 1961−− .447 [ 10 ], 449 Coxeter−− 467 , . periodH. S, M.; Moser, W. O. I.: Generators and Relations for Discrete [open 7 ] square\quad bracketCorn 7 , closing D . square; S ingerman bracket .. Corn , D comma . : Regular D period hypermaps semicolon S ingerman . Europ comma . J . D Comb period . : Regular 9 ( 1988 hypermaps ) , 337 period−− Europ35 1 . [ 8Groups ] \quad. Coxeter ,H . 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H , . ; G Wright . A ,. : E .M.Symmetry : \ h f i. l l Introduction In : Handbook t o of the Theory of Numbers . Oxford University openApplicable square bracket Mathematics 14 closing square. Vol bracket . .... 5 , Hardy Combina comma G - period H period semicolon Wright comma E period M period : .... Introduction t\ centerline o the Theorytorics{ of andPress Numbers Geometry , Oxfordperiod ( Oxford ed 1979 . W University . .} Ledermann ) , Wiley , Chichester / New York / London Press1 985 comma , Oxford 1979 period \noindentopen square[ bracket 15 ] \ 15h f closing i l l Izquierdo square bracket , M .... .3 Izquierdo ; 29\ h – f 422 i l lcomma .S ingerman M period ,semicolon D . : \ ....h f S i l ingerman l Hypermaps comma on D period surfaces : .... withHypermaps boundary on . Europ . J . \ h f i l l Comb . surfaces[ 18 with ] boundary Jones , G period . A Europ. : periodRee groups J period and .... Riemann Comb period surfaces . J . Algebra 165 ( 1 ) ( 1994 \noindent1) 5 , .. 4 open 1 –1 parenthesis62 5 .\quad 2( closing 2 ) \ parenthesisquad ( 1994 .. open ) parenthesis , 1 59 −− 19941 72 closing . parenthesis comma 1 59 endash 1 72 period [open 16 ] square\quad bracketJames 16 closing , L .D. square : bracket\quad .. JamesOperations comma L on period hypermaps D period , : ..\quad Operationsand on outer hypermaps automo comma $ r ..−p and $ outer hisms automo . Europ r-p . J . Comb . 9 hisms period Europ period J period Comb period 9 \noindentopen parenthesis( 1 988 1 988 ) closing , 551 parenthesis−− 560 . comma 551 endash 560 period [open 1 7 square ] \quad bracketJones 1 7 closing , \quad squareG. bracket\quad ..A.: Jones comma\quad ..Symmetry G period .. A . period\quad :In .. Symmetry : \quad periodHandbook .. In : of .. Handbook Applicable of Applicable Mathematics . \quad Vol . \quad 5 , \quad Combina − Mathematics period .. Vol period .. 5 comma .. Combina hyphen \ hspacetorics and∗{\ Geometryf i l l } torics open andparenthesis Geometry ed period ( ed W . period W . ..\quad LedermannLedermann closing parenthesis ) , Wiley comma , \quad WileyChichester comma .. Chichester / New slash York New / London York 1 985 , slash London 1 985 comma \ centerline3 29 endash{ 4223 29 period−− 422 . } open square bracket 18 closing square bracket .. Jones comma G period A period : .. Ree groups and Riemann surfaces period J period Algebra 165\noindent open parenthesis[ 18 ] 1\ closingquad parenthesisJones ,G.A. .. open parenthesis : \quad 1994Ree closing groups parenthesis and Riemann comma 4 surfaces 1 endash 62 . period J . Algebra 165 ( 1 ) \quad ( 1994 ) , 4 1 −− 62 . A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps .. 37 \ hspaceopen square∗{\ f i bracket l l }A . 19 J closing . Breda square d bracket ’ Azevedo .... Jones ; G comma . A .... . Jones G period : ....\quad A periodPlatonic semicolon Hypermaps .... S i lver comma\quad ....37 S period .... A period : .... Suzuki groups .... and surfaces period .... J period .... London Math period .... Soc period .... open parenthesis 2 closing parenthesis .... 48 \noindent [ 19 ] \ h f i l l AJones . J . Breda , \ h f i d l l ’ AzevedoG. \ h f ;i l G l .A.; A . Jones\ h f i : l l S Platonic i l v e r Hypermaps , \ h f i l l S. 37\ h f i l l A.: \ h f i l l Suzuki groups \ h f i l l and surfaces . \ h f i l l J. \ h f i l l London Math . \ h f i l l Soc . \ h f i l l ( 2 ) \ h f i l l 48 open[ 19 parenthesis ] Jones , 1 G 993 . closing A . ; Sparenthesis i lver , S comma . A . 1 : 1 7Suzuki endash groups 1 25 period and surfaces . J . London Math . openSoc square . bracket 20 closing square bracket .. Jones comma ( 2 ) G period A period semicolon Thornton comma48 J period S period : .. Operations on\noindent maps comma( 1 .. and993 outer ) , automorphisms 1 1 7 −− 1 25 period . J period Com hyphen [ 20( 1 ] 993\quad ) , 1Jones 1 7 – 1 ,G 25 . . A [ 20 . ] ; Thornton Jones , G , . J A . . ;S Thornton . : \quad , JOperations . S . : Operations on maps on , \ mapsquad ,and outer automorphisms . J . Com − binatorialand outer Theory automorphisms comma Ser period. J B . Com comma - 35 open parenthesis 1 983 closing parenthesis comma 93 endash 103 period open square bracket 2 1 closing square bracket .. Mach grave-dotlessi sub comma A period : .. On the complexity of a hypermap period Discr \noindentbinatorialbinatorial Theory , Ser Theory . B , 35 , Ser( 1 983. B ) , , 3593 – ( 103 1 983 . [ 2 ) 1 , ] 93 Mach−− 103`ı, A.: . On the complexity periodof Math a hypermap period 42 open. Discr parenthesis . Math 1 982 . 42 closing( 1 parenthesis 982 ) , 22 comma 1 – 2 22 26 1 endash . [ 22 2 26] period Robertson , S . A . : [open 2 1 square ] \quad bracketMach 22 closing $ \grave square{\ bracketimath} .. Robertson{ , }$ comma A . : S\ periodquad AOn period the : complexity .. Polytopes and of Symmetry a hypermap period . London Discr Math . Math period . 42 ( 1 982 ) , 22 1 −− 2 26 . [ 22Polytopes ] \quad andRobertson Symmetry , S. London . A . : Math\quad . SocPolytopes . Lecture and Note Symmetry Series 90 . London, Math . Soc . Lecture Note Series 90 , Soc periodCambridge Lecture University Note Series Press 90 comma , Cambridge 1984 . [ 23 ] Sherk , F . A . : The regular Cambridgemaps University on a Press surface comma Cambridge of genus 1984 period three . Canadian J . Math . 1 1 \noindentopen squareCambridge bracket 23 closing University square bracket Press .. , Sherk Cambridge comma .. 1984 F period . .. A period : .. The .. regular maps .. on .. a .. surface .. of genus .. [ 23( 1 ] 959\quad ) , 452Sherk – 480 , .\quad [ 24 ]F. Sherk\quad , FA.: . A . :\quadA familyThe \quad of regularregular maps maps of\quad type on{ 6\ ,quad 6 } a \quad s u r f a c e \quad o f genus \quad thre e . \quad Canadian \quad J. \quad Math . \quad 1 1 three. period Canadian .. Canadian Math .. J . period Bull .. Math . period5 ( .. 1962 1 1 ) , open parenthesis 1 959 closing parenthesis comma 4521 3 endash – 20 . 480 period \noindentopen square( bracket 1 959 24 ) closing , 452 square−− 480 bracket . .. Sherk comma F period A period : .. A family of regular maps .. of type open brace 6 comma 6 [ 25 ] S imon , J . : Du Commentaire de Proclus sur le Tim e´ e . Paris 1839 . [ 26 closing[ 24 brace ] \quad periodSherk .. Canadian ,F .A.Math period : \quad .. BullA period family .. 5 of .. open regular parenthesis maps 1962\quad closingo f parenthesis type \{ comma6 , 6 \} . \quad Canadian Math . \quad Bull . \quad 5 \quad ( 1962 ) , ] Threlfall , W . : Gruppenbilder . Abh . s a¨ chs . Akad . Wiss . Math . - phys . Kl . 1 3 endash 20 period \ centerline, 41 ( 1932{1 3 )−− , 120 – 59 . .} [ 27 ] Vince , A . : Regular combinatorial maps .J. openCombinatorial square bracket 25 Theory closing ,square Ser bracket . B .. , S imon34 comma( 1983 J period ) , : .. Du Commentaire de Proclus sur le .. Tim acute-e e period Paris 1839 period 256 – 277 . \noindentopen square[ bracket 25 ] \ 26quad closingS imon square , bracket J . : ..\ Threlfallquad Du comma Commentaire W period : de .. Gruppenbilder Proclus sur period le \quad Abh periodTim s $ dieresis-a\acute chs{e} period$ e Akad . Paris 1839 . [ 28 ] Walsh , T . R . S . : Hypermaps versus bipar−t ite maps . J . Combinatorial Theory , period[ 26 Wiss ] \quad periodThrelfall Math period hyphen ,W . phys : \quad periodGruppenbilder Kl period comma .41 Abh open . parenthesis s $ \ddot 1932{a closing} $ parenthesis chs . Akad comma . Wiss 1 endash . Math 59 period . − phys . Kl . , 41 ( 1932 ) , 1 −− 59 . [ 27Ser ] .\ Bquad , 18 Vince , \quad A.: \quad Regular \quad combinatorial maps . \quad J. \quad Combinatorial \quad Theory , \quad Ser . \quad B, \quad 34 \quad ( 1983 ) , open( 1 square 975 ) bracket, 155 – 27 163 closing . [ square 29 ] bracket Wilson .. Vince , S . comma E . : .. AOperators period : .. over Regular regular .. combinatorial maps . Pacificmaps period J . .. J period .. Combinatorial .. TheoryMath comma . 81 .. Ser( period 2 ) .. ( B 1 comma979 ) , .. 559 34 .. – open 568 parenthesis. [ 30 ] 1983 Wilson closing , S parenthesis . E . : commaCantankerous maps \ centerline256 endash 277{256 period−− 277 . } and rotary embeddings of K . J . Combinatorial The - open square bracket 28 closing squaren bracket .... Walsh comma T period R period S period : .... Hypermaps versus bipa to the power of r-t ite \noindent [ 28 ] \ h f i l l Walsh ,T.R.ory , Ser S . B. :,\ h ( f 1989 i l l Hypermaps ) . versus $ bipa ˆ{ r−t }$ ite maps . J . Combinatorial Theory , Ser . B , 18 mapsReceived period J period August Combinatorial 30 , 1999 Theory comma Ser period B comma 18 open parenthesis 1 975 closing parenthesis comma 155 endash 163 period \noindentopen square( bracket 1 975 29 ) closing , 155 square−− 163 bracket . .. Wilson comma S period E period : .. Operators over regular maps period Pacific J period Math period[ 29 81 ] ..\quad open parenthesisWilson , 2 Sclosing . E parenthesis . : \quad .. openOperators parenthesis over 1 979 regular closing parenthesis maps . Pacific comma 559 J endash . Math 568 . period 81 \quad ( 2 ) \quad ( 1 979 ) , 559 −− 568 . [open 30 ] square\quad bracketWilson 30 closing , S . square E . bracket : \quad .. WilsonCantankerous comma S period maps andE period rotary : .. Cantankerous embeddings maps of and $ K rotary{ n embeddings} . $of J K . sub Combinatorial n The − period J period Combinatorial The hyphen \ centerlineory comma{ Serory period , Ser B comma . 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