f-vectors of 3-polytopes symmetric under rotations and rotary reflections

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Abstract

The f-vector of a polytope consists of the numbers of its i-dimensional faces. An open field of study is the characterization of all possible f-vectors. It has been solved in three dimensions by Steinitz in the early 19th century. We state a related question, i.e., to characterize f-vectors of three dimensional polytopes respecting a symmetry, given by a finite group of matrices. We give a full answer for all three dimensional polytopes that are symmetric with respect to a finite rotation or rotary reflection group. We solve these cases constructively by developing tools that generalize Steinitz’s approach.

Introduction

There are many studies of f-vectors in higher dimensions, for example see [5], [6], [7], [11], [12], [18], [20] or [21]. The set of f-vectors of four dimensional polytopes has been studied in [3], [8], [9], [10], [13], [14] and [24]. Some insights about f-vectors of centrally symmetric polytopes are given in [2], [4], [15] and [22]. It is still an open question, even in three dimensions, what the f-vectors of symmetric polytopes are. This question will be partially answered in this paper.

In particular, given a finite 3 × 3 matrix group G, we ask to determine the set F(G) of vectors (f0,f2)N×N such that there is a polytope P symmetric under G (i.e., AP=P for all AG) with f0 vertices and f2 facets (we omit the number of edges by the Euler-equation). In this paper we give an answer for all groups that do not contain a reflection summarized in the following theorem. For a detailed explanation of the mentioned groups see Theorem 2.4.

Theorem 1.1 Main Theorem

Let F be the set of f-vectors of three dimensional polytopes (omitting the number of edges). For MN×N we use the notation MM{(y,x):(x,y)M} as well as to denote componentwise congruence. The f-vectors of symmetric polytopes under rotation groups can be classified as follows: F(Cn)={fF:f(1,1)(modn)}{f=(f0,f2)F:f(0,2)(modn);2f0f22n2} for n>2,F(C2)=F,F(Dd)={fF:f(0,2),(d,2)(mod2d)}{f=(f0,f2)F:f(0,d+2),(d,d+2)(mod2d);2f0f23d2} for d>2,F(D2)={fF:f(0,0),(0,2),(2,2)(mod4)}{(6,6)},F(T)={fF:f(0,2),(0,8),(4,4),(4,10),(6,8)(mod12)},F(O)={fF:f(0,2),(0,14),(6,8),(6,20),(8,18),(12,14)(mod24)},F(I)={fF:f(0,2),(0,32),(12,20),(12,50),(20,42),(30,32)(mod60)}. For rotary reflection groups, the f-vectors can be classified as: F(Gd)={fF:f(0,2)(mod2d)} for d>2,F(G2)={fF:f(0,0),(0,2)(mod4)}.F(G1)={fF:f(0,0)(mod2)}{(4,4),(6,6)}

In this paper we generalize the elementary approach of Steinitz (see [23]). We start by introducing some fundamental terms and concepts relevant to this work in Section 2. The coarser structure of F(G) is due to the composition of orbits that the group G admits. This can be described in general and will be shown in Section 3, especially in Lemma 3.2. The extra restrictions arise from certain structures of facets and vertices, e.g., a facet on a 6-fold rotation axis must have at least 6 vertices, which forces the polytope to be ‘further away’ from being simplicial.

The main difficulty in characterizing F(G) is the construction of G-symmetric polytopes with a given f-vector. In Section 4 we introduce so called base polytopes, symmetric polytopes that can be used to generate an infinite class of f-vectors. Since the operations on base polytopes produce f-vectors in the same congruence class, we divide the set of possible f-vectors in F(G) into several coarser integer cones. To certify the existence of all f-vectors in one of these coarser integer cones we introduce four types of certificates in Section 5. In Corollary 5.6 we describe for which f-vectors certificates are needed to obtain all f-vectors conjectured to be in F(G). To give these certificates we need to find symmetric polytopes with ‘small’ f-vector. To this end, in Section 6, we introduce useful constructions on polytopes that change the f-vectors, but preserve the symmetry. As starting points, we then give a list of some well-known polytopes taken from the Platonic, the Archimedean and their duals, the Catalan solids. In Section 7 we are finally able to connect the theory with explicit constructions of polytopes to prove Theorem 1.1. Lastly, we conclude the paper with some remarks on groups that contain reflections in Section 8 as well as open questions and conjectures in Section 9.

Section snippets

Preliminaries

We start by introducing some fundamental terms and concepts relevant to this work. A polytope is the convex hull of finitely many points in Rn. A face of a polytope P is the intersection of P with a hyperplane that contains no points of the relative interior of P. The polytope P itself and the empty set are often considered as non-proper faces of P as well, but are irrelevant for the study of f-vectors, since there is always exactly one of each. The dimension of a face is the dimension of its

Conditions on F(G)

For the rest of the paper, let G be a finite orthogonal subgroup of GL3(R) of size n. In this section we deduce conditions on the sets F(G) in terms of the group G. These conditions mostly depend on the structures of orbits under the action of G on R3.

We start with some notation. A ray in R3 is a set of the form R+x={λx:λ>0}for some xR3{0}. We say that R+x is the ray generated by x. Furthermore, we say that a convex set C is on a ray r if r intersects the relative interior of C. On the other

Base polytopes

The characterization of f-vectors for a given group G mainly consists of two parts. First, we need to find conditions on F(G) to show that F(G)F for a given set F as the one in Lemma 3.2 with a few adjustments. Then we need to construct explicit G-symmetric polytopes for each fF to show that FF(G). To do so, we use polytopes with certain properties to construct infinite families of G-symmetric polytopes. These so-called base polytopes, introduced in this section, form the foundations of

Certificates

Since it is often impossible to construct symmetric base polytopes with small f-vector entries, we show that it is possible to replace one base polytope by several polytopes with certain weaker properties and still get an integer cone of f-vectors as in Corollary 4.5. We thus introduce the concept of certificates, a collection of one or more polytopes with certain properties that ensure (certify) the existence of an integer cone of f-vectors of the form f+nCf for some vector fN2 and n=|G|. The

Constructions of symmetric polytopes

It is easy to construct polytopes symmetric under G by taking the convex hull of several orbits. However, in most cases it is hard to control the f-vector of the resulting polytope. In this section, we describe some special operations that can be applied to G-symmetric polytopes to obtain other G-symmetric polytopes. These operations will be used in Section 7 to construct symmetric polytopes needed for certificates. We will emphasize the implications for the f-vector and the types of the

Characterization of f-vectors

In this section, we go through all finite orthogonal rotation and rotary reflection groups, as described in Theorem 2.4, and characterize their f-vectors using the tools developed in the previous sections.

We start with the group Cn, the cyclic group of order n, which is generated by a rotation with rotation-angle 2πn around a given axis. Thus, Cn has two non-regular orbits of size 1, namely the two rays of the rotation axis. These are flip-orbits if and only if n=2.

Theorem 7.1

For n>2 we have F(Cn)={fF:f

Thoughts on symmetry groups containing reflections

Up to this point only reflection free symmetries have been discussed. To prove the main theorem, Theorem 1.1, we mostly followed the characterization 2.4 from Grove and Benson [17, Theorem 2.5.2]. In order to characterize the f-vectors of the remaining symmetry groups, it is important to know about the reflections in the group and their arrangement. Therefore, we propose the following characterization of symmetry groups containing reflections instead:

Denote a rotation around the z-axis by an

Open questions

In this section, we state several related problems and open research questions.

First, we consider another related problem. For any three dimensional polytope P we define its linear symmetry group by Symm(P)={AR3×3:AP=P}. Furthermore, we denote F(G)¯={fF:there is a polytope P with f(P)=f,Symm(P)=G}.Then F(G)¯F(G). We conjecture that every ‘large enough’ f-vector in F(G) is also contained in F(G)¯, while there are finitely many f-vectors in F(G)F(G)¯. This conjecture seems to be related to 

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Robert Schüler has been supported by DFG, Germany grant SCHU 1503/7. We like to thank Frieder Ladisch and Michael Joswig for much helpful advice. Furthermore, we like to thank the anonymous referees for their careful reading and many valuable comments.

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