-vectors of 3-polytopes symmetric under rotations and rotary reflections
Introduction
There are many studies of -vectors in higher dimensions, for example see [5], [6], [7], [11], [12], [18], [20] or [21]. The set of -vectors of four dimensional polytopes has been studied in [3], [8], [9], [10], [13], [14] and [24]. Some insights about -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 -vectors of symmetric polytopes are. This question will be partially answered in this paper.
In particular, given a finite 3 × 3 matrix group , we ask to determine the set of vectors such that there is a polytope symmetric under (i.e., for all ) with vertices and 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 be the set of -vectors of three dimensional polytopes (omitting the number of edges). For we use the notation as well as to denote componentwise congruence. The -vectors of symmetric polytopes under rotation groups can be classified as follows: For rotary reflection groups, the -vectors can be classified as:
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 is due to the composition of orbits that the group 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 -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 is the construction of -symmetric polytopes with a given -vector. In Section 4 we introduce so called base polytopes, symmetric polytopes that can be used to generate an infinite class of -vectors. Since the operations on base polytopes produce -vectors in the same congruence class, we divide the set of possible -vectors in into several coarser integer cones. To certify the existence of all -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 -vectors certificates are needed to obtain all -vectors conjectured to be in . To give these certificates we need to find symmetric polytopes with ‘small’ -vector. To this end, in Section 6, we introduce useful constructions on polytopes that change the -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 . A face of a polytope is the intersection of with a hyperplane that contains no points of the relative interior of . The polytope itself and the empty set are often considered as non-proper faces of as well, but are irrelevant for the study of -vectors, since there is always exactly one of each. The dimension of a face is the dimension of its
Conditions on
For the rest of the paper, let be a finite orthogonal subgroup of of size . In this section we deduce conditions on the sets in terms of the group . These conditions mostly depend on the structures of orbits under the action of on .
We start with some notation. A ray in is a set of the form for some . We say that is the ray generated by . Furthermore, we say that a convex set is on a ray if intersects the relative interior of . On the other
Base polytopes
The characterization of -vectors for a given group mainly consists of two parts. First, we need to find conditions on to show that for a given set as the one in Lemma 3.2 with a few adjustments. Then we need to construct explicit -symmetric polytopes for each to show that . To do so, we use polytopes with certain properties to construct infinite families of -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 -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 -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 -vectors of the form for some vector and . The
Constructions of symmetric polytopes
It is easy to construct polytopes symmetric under by taking the convex hull of several orbits. However, in most cases it is hard to control the -vector of the resulting polytope. In this section, we describe some special operations that can be applied to -symmetric polytopes to obtain other -symmetric polytopes. These operations will be used in Section 7 to construct symmetric polytopes needed for certificates. We will emphasize the implications for the -vector and the types of the
Characterization of -vectors
In this section, we go through all finite orthogonal rotation and rotary reflection groups, as described in Theorem 2.4, and characterize their -vectors using the tools developed in the previous sections.
We start with the group , the cyclic group of order , which is generated by a rotation with rotation-angle around a given axis. Thus, has two non-regular orbits of size , namely the two rays of the rotation axis. These are flip-orbits if and only if .
Theorem 7.1 For we have
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 -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 -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 we define its linear symmetry group by . Furthermore, we denote Then . We conjecture that every ‘large enough’ -vector in is also contained in , while there are finitely many -vectors in . 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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