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Snub Disphenoid� E COLE NORMALE SUPERIEURE ________________________ Lists of faceregular p olyhedra Gunnar BRINKMANN Michel DEZA LIENS ________________________ Département de Mathématiques et Informatique CNRS URA 1327 Lists of faceregular p olyhedra Gunnar BRINKMANN Michel DEZA LIENS November Lab oratoire dInformatique de lEcole Normale Superieure rue dUlm PARIS Cedex Tel Adresse electronique dezadmiensfr Fakultat f ur Mathematik Universitat Bielefeld D Bielefeld Adresse electronique gunnarmathematikunibielefeldde Lists of Faceregular Polyhedra Gunnar Brinkmann Michel Deza Fakultat f ur Mathematik CNRS and LIENSDMI Universitat Bielefeld Ecole Normale Superieure y D Bielefeld Paris France Abstract We introduce a new notion that connects the combinatorial concept of regularity with the geometrical notion of facetransitivity This new notion implies niteness results in case of b ounded maximal face size We give lists of structures for some classes and investigate p olyhedra with constant vertex degree and faces of only two sizes Introduction A planar nite or innite graph is called facetransitive if the automorphism group acts transitively on the set of faces For nite p olyhedra see Ma as well as for innite graphs in the plane with nite faces and nite vertex degree that is tilings see BaDe it is well known that the graph can b e realized with its full com binatorial automorphism group as its group of geometrical symmetries Restricting the attention to p olyhedra with constant vertex degree up to combinatorial equiva lence only the Platonic solids have an automorphism group acting transitively on their faces In the remaining text we will restrict our attention to p olyhedra with constant vertex degree A natural generalisation of this concept let us call it weakly facetransitive is to require that only faces of the same size are equivalent under the automorphism group If we dene the th corona of a face to b e the face itself and the nth corona to b e the set of all those faces that are contained in the nth corona or share an edge with it we can further relax this concept and only require some coronas of xed size to b e isomorphic by an isomorphism mapping the central faces onto each other A p olyhedron with all ncoronas of faces of the same size isomorphic is called weakly ntransitive Obviously all p olyhedra are weakly transitive and if a p olyhedron is weakly ntransitive it is also weakly ntransitive So the rst interesting case to study is the case of weakly transitive p olyhedra Still relaxing this condition by not requiring the rst coronas to b e isomorphic but just to b e isomorphic as multisets that is every face of a given size i must have the same number of neighbours of size i for every i still gives a very restrictive condition email gunnarmathematikunibielefeldde y email dezadmiensfr and as we will see it already implies niteness in case the maximal size of a face is b ounded We call this condition strong faceregularity So the class of all face regular p olyhedra contains all weakly nfacetransitive p olyhedra for any n and therefore also the weakly facetransitive or even facetransitive ones The same concept can b e reached by strengthening the notion of a mono chro matic regular dual Let p denote the number of igons in a given p olyhedron We use i the notation p p p p p for the facevector or pvector of a p olyhedron i b b is the maximal number for which a face with size b exists A less restrictive denition of faceregularity but only for bifaced p olyhedra was considered in DGrc Namely if only p and p are nonzeros and a b a b then the number f of ifaces edgeadjacent to any given iface was required to b e indep endent of the choice of the iface for i either a or b For a k valent p olyhedron we write aR or bR if this partial or weak faceregularity holds for agonal or f f resp ectively bgonal faces All such simple p olyhedra with b as well as all valent ones with b except the cases R for k a b and aR aR for k a b f g were found in DGrc For example all resp p olyhedra bR for all ve p ossible cases k k b and f k b a f g are listed there The graphs of all R ful lerenes f ie k a b are given in list b elow In these cases resp p olyhedra are also aR ie faceregular in the sense of the present pap er f The faceregularity which we consider is a purely combinatorial prop erty of the skeleton of a p olyhedron It is dierent from the ane notion of regularfaced ie all faces b eing regular p olygons p olyhedra We use the abbreviation frp for faceregular polyhedron An frp in one of the lists b elow is describ ed by i where j is the number of the List and i is its number in j List j We also use the notation i for i We call two frp frisomers if they have the same parameters as frp ie v the pvector and the numbers f a b ie the number of bfaces edgeadjacent to each aface for any a b coincide All frisomers in List are bifaced They are v v v and faced v All frisomers in List are for v for v for v for v for v for v for v for v All frisomers in Lists and are with v Considering the p olyhedra of Lists and with resp ect to collapsing of all triangular faces to p oints ie the inverse to vertextruncation we see that in List any such collapsing gives a member of List But in List there are p olyhedra such that this collapsing do es not give an frp The smallest one is Examples of sequences of frp such that each of them comes from the previous one by edge truncation are and Some innite families of valent frp Bifaced Prism and Barrel ie two ngons separated by two layers of gons n n faced Prism Barrel truncated on all n vertices of b oth ngons n n Prism edgetruncated on n disjoint edges of only one ngon n Prism edgetruncated on n edges separated by at least edges of only one n ngon faced Prism vertex truncated on all vertices of only one ngon n faced Barrel truncated on all vertices of only one ngon n In fact many of the frp in the lists are some partial truncations of Prism and n Barrel For example there are exactly frp which are partial truncations of the n Cub e There are resp p ossibilities for truncations on resp vertices Remarks i Among the chiral p olyhedra in the lists are for example Nrs in List Nr in List and esp ecially Nrs in List and in List with symmetry T O I and O resp ectively ii None of the p olyhedra in any of our Lists has a trivial symmetry group The Finiteness of Classes with Bounded Face Size Theorem For every n N there is only a nite number of faceregular polyhedra with constant vertex degree and face sizes not exceeding n Pro of We will assume that the p olyhedra in question all contain an ngon The total number can b e obtained by summing up over all m n Remind that for i j N the number f i j denotes the number of neighbouring jgons of an igon So f i j p f j ip is the number of edges b etween igons and i j f ij jgons and we can express p as p p in case igonal and jgonal faces share j j i f ji at least one edge Lo ok at the fgraph G with vertex set V fijp g and edge set E i ffi j gjf i j g This graph is connected since the dual of the underlying p oly hedron is connected We can express every other value p by a formula of the kind i f i i f i i f i i 1 2 1 b k p g ip n n f ii f i i f i i 1 1 2 k b if i i i n is a eg shortest path from i to n in G k Since for xed n all the f i j as well as the length of the path are b ounded and since the number of graphs on n vertices is also nite we have only a nite number of p ossible sets of equations p g ip i n i n P n As a well known consequence of Eulers formula we get ip in the i i P P n n ip ip for valent p olyhedra and valent case i i i i for valent p olyhedra Substituting p by g ip in this formula every set of equations gives exactly one i n solution for p and therefore also for each p So for every set of equations there is a n i well determined number of faces and therefore there is a maximum number of faces that is p ossible 2 Corollary If in the cubic case the number of nonhexagons is bounded or in the quartic case the number of nonsquares is bounded then there is only a nite number of faceregular polyhedra Pro of The fact that the number of faces smaller than resp is b ounded gives an upp er b ound on the maximum face size implying the result by the previous theorem 2 Statistics In this section we will give some statistics ab out the number of faceregular p olyhedra compared to the number of all p olyhedra for some classes vertices p olyhedra faceregular p olyhedra p olyhedra faceregular p olyhedra vertices Table Cubic p olyhedra Table Cubic p olyhedra without triangles p olyhedra faceregular p olyhedra vertices Table Cubic p olyhedra without faces larger than a hexagon For all vertex numbers not mentioned no faceregular p olyhedra exist List all faceregular simple p olyhedra with b 6 Among the p olyhedra of the List the rst three are regular then there are bifaced ones six with b four for n nine for n n n and fullerenes which are F D F T n d d F D F C F T F T F T F I F T F I F D h v d h d
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