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E COLE NORMALE SUPERIEURE

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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

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

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

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

k b a f g are listed there The graphs of all R ful lerenes

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

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

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

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

Prism edgetruncated on n edges separated by at least edges of only one

ngon

faced Prism vertex truncated on all vertices of only one ngon

faced Barrel truncated on all vertices of only one ngon

In fact many of the frp in the lists are some partial truncations of Prism and

Barrel For example there are exactly frp which are partial truncations of the

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

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

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

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

As a well known consequence of Eulers formula we get ip in the

P P

n n

ip ip for valent p olyhedra and valent case

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

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

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 h h

F I Nrs have three types of faces and last seven p olyhedra Nrs

have four types of faces

Among the p olyhedra of the List three are regular ones Tetrahedron

Cub e and Do decahedron ve are semiregular gonal prisms truncated

o ctahehedron and truncated Icosahedron and no one is regularfaced from the list of

in Joh But there are three which are dual to regularfaced snub disphenoid

augmented gonal prism and gyro elongated square dipyramid last three have

number and resp ectively in the list of Joh Together with three

regular ones and gonal prisms it gives the duals of all eight convex deltahedra

Nr v Nr v Nr v

p p p

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Group T Group O Group I

d h h

Nr v Nr v Nr v Nr v

p p p p

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Group D Group D Group D Group D

h d h d

Nr v Nr v

p p

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Group D Group D

h d

Nr v Nr v Nr v Nr v

p p p p

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Group T Group T Group D Group T

d d h

Nr v Nr v Nr v Nr v

p p p p

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Group D Group D Group D Group S

h h d

Nr v Nr v Nr v Nr v

p p p p

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Group O Group D Group D Group O

h h d h

Nr v

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Group O

Nr v Nr v Nr v Nr v

p p p p

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Group D Group T Group D Group C

d d h v

Nr v Nr v Nr v Nr v

p p p p

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Group T Group T Group T Group I

d h

Nr v Nr v Nr v Nr v

p p p p

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Group T Group I Group D Group I

d h h

Nr v Nr v Nr v

p p p

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Group C Group C Group D

v v h

Nr v Nr v Nr v Nr v

p p p p

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Group C Group C Group D Group D

v v d h

Nr v Nr v

p p

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Group C Group D

h h

Nr v Nr v Nr v Nr v

p p p p

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Group D Group D Group D Group D

h h h d

Nr v Nr v Nr v Nr v

p p p p

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Group C Group D Group D Group D

v d h h

Nr v Nr v Nr v Nr v

p p p p

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Group D Group D Group D Group D

Nr v Nr v

p p

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Group D Group D

h d

Nr v Nr v Nr v Nr v

p p p p

Groupsize Groupsize Groupsize Groupsize

Group C Group C Group C Group C

v v

Nr v Nr v Nr v

p p p

Groupsize Groupsize Groupsize

Group C Group C Group C

v v h

List all faceregular simple p olyhedra

with b = 7 and up to faces

Nr v Nr v Nr v

p p p

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Nr v Nr v Nr v Nr v

p p p p

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Nr v Nr v

p p

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Nr v Nr v Nr v Nr v

p p p p

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v

p p

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

p p p p

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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Nr v Nr v Nr v Nr v

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List Selected faceregular simple p olyhedra

with b 8

Nr v Nr v Nr v

p p p

Groupsize Groupsize Groupsize

Nr v Nr v Nr v

p p p

p Groupsize p

Groupsize Groupsize

Nr v Nr v Nr v

p p p

p Groupsize p

p p

Groupsize Groupsize

List all faceregular valent p olyhedra

with b = 4

For the p olyhedra in this list the graph induced by the gons is interesting in Nrs

the graphs are two C C and the truncated Octahedron resp ectively

Nr v Nr v Nr v

p p p

Groupsize p p

Group O Groupsize Groupsize

Group D Group D

d h

Nr v Nr v Nr v

p p p

Groupsize Groupsize Groupsize

Group O Group D Group D

h h h

Nr v Nr v Nr v

p p p

Groupsize Groupsize Groupsize

Group D Group D Group O

d d

List all faceregular valent p olyhedra with

b = 5 and up to faces

Among the p olyhedra of Lists and there are the Octahedron three semiregular

ones gonal antiprisms and the Cub o ctahedron and three regularfaced Nr

of List and Nrs and of List which are the elongated square dipyramid the

p entagonal gyrobicup ola and the p entagonal orthobicup ola having number

and resp ectively in the list of p olyhedra in Joh

Nr in list is the Octahedron truncated and capp ed on vertices of an induced

C Nr is the elongated antiprism Nr is the dual rhombic Icosahedron

elongated gonal antiprism

Nr v Nr v Nr v

p p p

Groupsize Groupsize p

Group D Group D Groupsize

d h

Group D

Nr v Nr v Nr v

p p p

Groupsize Groupsize Groupsize

Group D Group D Group D

d d h

List all 6R Fullerenes

The p olyhedra of this list are faceregular They are the

p olyhedra of List resp ectively

Nr v Nr v Nr v Nr v

p p p p

Groupsize Groupsize Groupsize Groupsize

Nr v Nr v Nr v Nr v

p p p p

Groupsize Groupsize Groupsize Groupsize

Nr v Nr v Nr v Nr v

p p p p

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Nr v Nr v Nr v Nr v

p p p p

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Nr v Nr v Nr v Nr v

p p p p

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Nr v Nr v Nr v Nr v

p p p p

Groupsize Groupsize Groupsize Groupsize

Nr v Nr v

p p

Groupsize Groupsize

Faceregular k valent bifaced p olyhedra

In this section we will study the case where faces of exactly two sizes a b o ccur

Bifaced p olyhedra and similar concepts are well studied eg in Mal GM

GZ Go GoZa JT JT JJ Ow Ow JO

Clearly in this case the fgraph from the pro of of Theorem is the K so we

f ab

get the equation p p

b a

f ba

By using k v e ap bp and the Euler formula v e p p we get

a b a b

f (ab)

k aak bk bk

f (ba)

We will use the following notation for op erations on p olyhedra

tetrakis the tetrakis of a p olyhedron is obtained by putting a pyramid on gonal

faces

triakon the triakon of a p olyhedron is obtained by partitioning each triangle

into a ring of gons by putting a vertex in the middle and connecting it to

the midp oint of every edge in the b oundary

triakon the triakon of a p olyhedron is obtained by partitioning each hexagon

into a ring of p entagons by putting a vertex in the middle and connecting it

to the midp oint of every second edge in the b oundary

Theorem For k there is only one innite series of faceregular a bpolyhedra

that is the Antiprisms AP r ism for any b

Apart from this al l faceregular k valent a bpolyhedra have k a and

They are

b polyhedra given as Nrs in List

b the Icosidodecahedron and Nr in List the tetrakis of the Octahedron

truncated on al l but two opposite vertices

b the tetrakis of the ful ly truncated Octahedron

Theorem will follow from the following Lemmata

Remark

Theorem shows that the largest valent faceregular a bp olyhedra have

vertices ie faces and a They have p b f g

and are Nr in List the Icosido decahedron the tetrakis of the fully truncated

Octahedron resp ectively The largest valent faceregular bp olyhedron also has

faces It is the fully truncated Do decahedron with p

The largest valent faceregular b and the largest known valent b

p olyhedron b oth have vertices They are the triakon of the truncated Do

decahedron so p p and the triakon of the truncated Icosahedron

so p p given b elow

Lemma The only possibilities for k a b are b b b

and b

f ab

In all other cases the denominator in will b e nonp ositive even for

f ba b

the smallest p ossible value

Lemma The case k a b is not possible so there is no faceregular

valent polyhedron

Pro of

The denominator in is p ositive only for f a b f b a

In the rst case any gon is surrounded by gons which implies neighbouring

gons in the next layer a contradiction In the case any pair of adjacent

gons is surrounded by gons so the next layer contains a gon with gonal

neighbours again a contradiction

In the following lemma we will exclude some of the theoretically p ossible param

eters for k

Lemma Al l cases for k a not being contained in the fol lowing list are

impossible

a b the polyhedra Nrs in List

b b the Icosidodecahedron with f f v

c b the innite class of antiprisms AP r ism

d b f b f b b p p b v b

e f b f b b p p b v b

Pro of For b the denominator in is p ositive only if

f b f b f b b b bg or

b f g and f b f b b or

b and f b f b f g

The sub case b in case is not p ossible b ecause otherwise p

p The sub cases b b b of the case are resp ectively the

cases c d and e of Lemma

Cases and are not p ossible b ecause we get gons on consecutive edges

of each bgon so the gonal neighbour of the gon in the middle will b e adjacent

to gons a contradiction

The sub case of is not p ossible since the gonal neighbours of adjacent

gons containing a vertex of the intersection would share an edge

In the sub case of all triangles neighbouring a p entagon in a row

would imply a triangle neighbouring other ones so assume we have a gon and

neighbouring gons not all in a row But then one of the gonal neighbours has

all gonal neighbours in a row again a contradiction

The remaining sub case of is case b of Lemma

Lemma The cases d and e of Lemma are realized only by polyhedron Nr in

List and the tetrakis of suitably truncated Octahedra given in Theorem

Pro of

In b oth cases all gons are organized in cycles surrounded by bgons since

otherwise we will get AP r ism In case e of Lemma the number of edges b etween

p bf b

So the only p ossibility is b and Nr of List two bgons is

is unique realization it is the dual of the Octahedron truncated on opp osite

vertices In case d of Lemma the number of b bedges is This implies

b f g and we get the tetrakis of suitably truncated Octahedra the rst one

b eing Nr of List the elongated Nr of the List

Theorem Al l faceregular cubic bpolyhedra have b

They are special truncations of the Tetrahedron the Cube and the Dodecahe

dron

the and truncated Tetrahedron

b truncated Cubes one for b f g and two for b

b truncated Dodecahedra one for b and two for b

Pro of

Due to the connectedness of p olyhedra we get f b for all cubic b

p olyhedra So each triangle is isolated and p v p Together

f b

with equality and p we get f b b and f b min So

Actually this is a result by Malkevitch Mal for general that is not only

faceregular cubic bp olyhedra

The remaining p ossibilities for b are b f b v f

and The rst cases are realized by trun

cations of the Do decahedron giving only p olyhedron in the rst two cases and

nonisomorphic p olyhedra in the others The last cases are realized by trunca

tions of the Cub e giving a unique p olyhedron in every case For b all wanted

p olyhedra are Nrs and of List

Nrs of List are the truncated Cub e and two truncated Do decahedra

For b there remain p olyhedra one and one p olyhedron

Remark

If we do not require connectedness in Theorem more graphs exist eg one

b b

for every b divisible by with p p v

Theorem There is only one innite family of cubic bpolyhedra that is P r ism

for any b

The nite families are

Two vertex polyhedra coming as the truncation of the dual Rombicuboc

tahedron or dual Mil lers solid its twist on al l valent vertices

polyhedra coming from those of Theorem by the triakon decoration they

have b v f

g There are exactly nonisomorphic polyhedra

for the rd th and th case and one for the others

polyhedra besides the prism for b and for b not covered by previous

cases Nrs and of List

vertex polyhedra Nrs of List coming by suitable doubling of al l

isolated gons in Nrs of List

the vertex polyhedron coming by suitable doubling of al l isolated

gons of the truncation on al l valent vertices of the dual of the unique

vertex valent polyhedron given below the gons of which are organized

into isolated ones and isolated pairs

Pro of

Possible values for f b are and The case f b is p ossible only if

the gons form a ring giving a prism or isolated rings of gons which is exactly

case of Theorem

If f b all gons are isolated So p v The equality and p

So b and the only p ossibility for b is imply b f b min

f b p p v It is exactly case of Theorem

In the remaining case f b the gons are organized into isolated adjacent

pairs So v and using and p we get f b min

which implies b Moreover the only p ossible values for b f b are

and The last sub case is not p ossible b ecause it gives

p The sub case gives v A computer search gave that it

is exactly case of the theorem The remaining sub cases leave p ossibilities

b f b p p v f g The rst

of them can easily b e removed by a geometric argument For the last one there are

vertices contained only in gons It is easy to show that only one out of two

p ossible ways in which the pairs of gons can neighbour a gon containing such a

vertex can o ccur Using this geometric arguments give a contradiction when trying

to construct the p olyhedron The middle one is realized by the p olyhedron of case

of the theorem The uniqueness follows from its construction

Clearly all remaining wanted p olyhedra have b and so they are covered by

List it gives the last entry of case in Theorem

Remark

The largest faceregular cubic and p olyhedra are also just like in

case of Theorem the dual tetrakis of the Catalan ie dual Archimedean

p olyhedra snub cub e and snub do decahedron

Theorem There is only one innite family of faceregular cubic bpolyhedra

with b

B ar r el bgons separated by rows of b gons for any b

The nite families are

polyhedra Nrs in List

a unique vertex polyhedron organized into concentric and

ring of gons separated by and ring of gons as given below

p ossibly some vertex polyhedra with al l gons organized into iso

lated rings ie f and f ie p

a unique vertex polyhedron Nr in List a unique vertex

polyhedron given below and unique vertex polyhedron also given

below al l with f b

p ossibly other bpolyhedra with f b having b f b p p v

Pro of

The case f b gives exactly B ar r el If f b then all gons are

isolated so p v So and p imply b f b giving b

If f b then all gons are organized into isolated pairs so v Again

we get b f b and b

The case f b has gons organized in disjoint rings Let t denote the

number of rings among them so p t v The same count as ab ove gives

b f b min So b is the only p ossibility for b In

the case b we have either f and t or f and t The

rst sub case gives p p given as case in the theorem The second

sub case is f ie gons also should b e organized in isolated rings We

get p p v So and gons should b e organized in concentric

alternating rings and only vertices b elong to faces of the same type It is easy

to show that case of the theorem is the unique p ossibility for such a p olyhedron

All p ossibilities with b are covered by List it is case of the theorem The

only remaining case is f b b Using we get b f b b

f b bf b

Clearly p p and v p p So all

b b

f b f bb f bb

p ossibilities with p ositive integer v are given by f b b i b i for

i In sub case f b b the bgons are all isolated It is easy to see that

b should b e even and that for b f g the only p ossibilities are the and

vertex p olyhedra in case of the theorem In sub case f b b an attempt

to construct the structures easily gives the imp ossibility for b f g and unicity

the vertex p olyhedron of case of the theorem for b The imp ossibility of

cases b f b v were checked with the help of a computer

The remaining cases are exactly those given in case of the theorem

Remark

The p olyhedron Nr of List and the nd and rd p olyhedron in case of

Theorem come by the triakon decoration of the fully truncated Tetrahedron

Cub e and Do decahedron resp ectively

Acknowledgement We want to thank Prof A Rassat for his assistence in

determining the exact groups for the structures given in the lists

Furthermore we want to thank the Technische Fakultat der Universitat Bielefeld

and esp ecially the group of Prof I Wachsmuth for the p ossibility to use their

computers for our extensive computations

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