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

polytopes

symmetries

useful

pretty

f-vectors, Theorems, structure steinitz

refs studies in higher dimensions [billera1981simplicial] Billera, Lee simplicial

[brinkmann2018small] small $f$-vectors of 4-polytopes

[ziegler2002fatness] fatness, complexity 4-polytopes

Mani, orthogonal groups, grove benson

Mani: In LABEL:M71 note: ref it is shown that for every abstract 3-polytope, invariant under some combinatorial symmetry, there is a realization that is symmetric under an orthogonal symmetry that in turn induces the given combinatorial symmetry.

In this paper we characterize $f$-vectors of 3-dimensional polytopes that are symmetric with respect to a finite rotation group. Let $P\subset\mathbb{R}^{3}$ be a 3-dimensional polytope. For $i\in\{1,2,3\}$, let $f_{i}(P)$ denote the number of $i$-dimensional faces of $P$. By [Euler] and [Steinitz], we know that there are certain dependencies among the numbers of $i$-dimensional faces of a 3-dimensional polytope:

For a polytope $P\subset\mathbb{R}^{3}$, it thus suffices to know two of the three numbers $f_{0}(P)$, $f_{1}(P)$ and $f_{2}(P)$; the missing one can then be computed using Theorem 1.1 Equation (1). That leads us to the following definition of an $f$-vector that differs slightly from the definition for higher dimensional polytopes, where all numbers $f_{i}(P)$ occur in the $f$-vector.

Let further $F$ be the set of all possible $f$-vectors of 3-dimensional polytopes,

It turns out that the conditions of Theorem 1.1 are sufficient for a characterization of $F$.

There are many studies of $f$-vectors in higher dimensions note: extensive list of references. But 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.

Let $G$ be a matrix group. We say that a polytope $P$ is $G$-symmetric if $G$ acts on $P$, i.e. $G\cdot P=P$. The set of all possible $f$-vectors of $G$-symmetric polytopes is denoted by

For any set $M$ of tuples we denote the symmetric set arising from $M$ as

Considering the $f$-vector of the dual polytope $f(P^{\vee})=(f_{2}(P),f_{0}(P))$ shows that $\mathit{F}$ is invariant under the $\diamond$ operation. Let $P$ be a $G$-symmetric polytope and let $b$ be the barycenter of $P$. The polytope $P^{*}=\{x\in\mathbb{R}^{3}\ :\ \left<x,y\right>\leq 1\text{ for all }y\in P-b\}$ is a $G$-symmetric polytope with $f(P^{*})=(f_{2}(P),f_{0}(P))$. That shows that $\mathit{F}(G)$ is also invariant under the $\diamond$ operation. For the remainder of this article, we will denote by $P^{\vee}$ any dual polytope of $P$ that is symmetric under $G$ (for example $P^{*}$).

Note that since $P\subset\mathbb{R}^{3}$ is full dimensional, $P$ being $G$-symmetric implies that $G$ is finite and that $\det(A)\in\{-1,1\}$ for all $A\in G$. Our goal is to characterize $\mathit{F}(G)$ for all rotation groups, i.e. finite orthogonal subgroups of $SO_{3}(\mathbb{R})$. By the following well known fact, this implies the characterization for all finite symmetry groups that are generated by elements with positive determinant.

Thus, for any finite matrix group $G$ there is an orthogonal group $G^{\prime}$ with $\mathit{F}(G)=\mathit{F}(G^{\prime})$. Luckily, the orthogonal groups are well known. In fact, there are only finitely many families. A full list can be found in [GroveBenson]. Here, we will only state the characterization of all finite rotation groups, since these are our main subject of interest.

A detailed description of each group is given in Section 4.

The Hasse diagram of the icosahedral and octahedral rotation group:

Now we can state our main theorem which allows us to classify the $f$-vectors of $G$-symmetric polytopes where $G$ is a finite rotation group:

Furthermore, for $G$ a finite rotary reflection group, the classification of the $f$-vectors of $G$-symmetric polytopes is:

If we visualize the set $F$, it looks like a cone translated by $(4,4)$ (cf. Figure LABEL: note: ref). To show that all integer points in the cone exist as $f$-vectors of 3-polytopes, Steinitz starts with pyramids over $n$-gons, whose $f$-vectors are $(n+1,n+1)$ for $n\in\mathbb{Z}_{>2}$. He then shows that by stacking a point on a simplicial facet and by cutting a simple vertex, the $f$-vector of the polytope changes by $+(1,2)$ and $+(2,1)$, respectively. The resulting polytope again has simple vertices and simplicial facets, which means that the construction can be repeated. With this method, it can be shown that for all points $f$ in the set $C=\{(f_{0},f_{2})\in\mathbb{Z}^{2}\colon 2f_{0}-f_{2}\geq 4\textnormal{ and }2f_{2}-f_{0}\geq 4\}$ we can construct a 3-polytope $P$ with $f(P)=f$. Here, the pyramids over $n$-gons in a way act as generators. In fact, the pyramids over a $3$-, a $4$- and a $5$-gon would already suffice to generate all $f$-vectors by stacking on simplicial facets and cutting simple vertices.

The sets $F(\operatorname{G})$ for finite symmetry groups $\operatorname{G}$ are subsets of $F$. Theorems 1.7 and 1.8 state that for most groups, $F(\operatorname{G})$ is coarser, since only some values modulo $n$ are allowed. For some groups of $C_{n}$ and $D_{d}$ there are also inequalities restricting some residue classes. The third alteration we observe is that one or several small points are left out as it happens for $\mathcal{D}_{2}$ and $\operatorname{G}_{1}$.

The coarser structure of $F(\operatorname{G})$ is due to the composition of orbits that the group $\operatorname{G}$ admits. This can be described in general and will be shown in Lemma 2.2. 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 describing $F(\operatorname{G})$ is the construction of $G$-symmetric polytopes with a given $f$-vector. In Section 3 we introduce so called base polytopes, symmetric polytopes that can be used to generate an infinite class of $f$-vectors, analogous to the pyramids over $n$-gons in Steinitz’s work. 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 obtain all $f$-vectors in one of these coarser integer cones we introduce three types of certificates in Section 4. In Corollary 4.5 we describe for which $f$-vectors certificates are needed to obtain all $f$-vectors conjectured to be in $F(G)$. To find polytopes for this finite number of $f$-vectors, we introduce in Section 5 useful constructions on polytopes that change the $f$-vectors, butpreserve the symmetry. We then give a list of some well known polytopes taken from the Platonic, the Archimedean and their duals, the Catalan solids, that will be used later on. In Section 6 we are finally able to connect the thoery with explicit constructions of polytopes to proof Theorems 1.7 and 1.8. Lastly, we conclude the paper with some open questions and conjectures in Section 7, putting an emphasis on the adaptability of our work to symmetry groups that contain reflections.

In this section we will deduce conditions on the sets $\mathit{F}(G)$ in dependence of the group $G$. These conditions mostly depend on the structures of orbits under the action of $G$ on $\mathbb{R}^{3}$.

A ray in $\mathbb{R}^{3}$ is a set of the form $\mathbb{R}_{+}x=\{\lambda\cdot x\ :\ \lambda>0\}$ for some $x\in\mathbb{R}^{3}\backslash\{0\}$. We also say that $\mathbb{R}_{+}x$ is the ray generated by $x$. If we consider two points in the same ray, say $v\in V\backslash\{0\}$ and $\lambda v$ with $\lambda>0$, we notice that the orbit polytopes $\operatorname{conv}\{G\cdot v\}$ and $\operatorname{conv}\{G\cdot\lambda v\}=\lambda\cdot\operatorname{conv}\{G\cdot v\}$ are the same up to dilation. It is thus useful to consider different types of orbits of rays as in the following definition:

The following lemma can be used to define a set $\mathit{F}^{\prime}$ such that $\mathit{F}(G)\subset\mathit{F}^{\prime}$. It can be seen in Section 6 that this outer approximation is not far from being exact and for certain groups these conditions are already sufficient to describe $F(G)$. For a set $M\subset\mathbb{Z}^{2}$ and an integer $n$, we define

The characterization of $f$-vectors for a given group $G$ mainly consists of two parts. First, we need to find a precise conditions on $\mathit{F}(G)$ to show that $\mathit{F}(G)\subset\mathit{F}^{\prime}$ for a given set $\mathit{F}^{\prime}$. Then we need to construct explicit $G$-symmetric polytopes for each $f\in\mathit{F}^{\prime}$ to show that $\mathit{F}^{\prime}\subset\mathit{F}(G)$. If we have a polytope with certain propterties, we can use it to construct an infinite family of $G$-symmetric polytopes. These so called base polytopes form the foundations for our constructions.

To maintain symmetry, a general approach to construct a $G$-symmetric polytope is to take a given symmetric polytope $P$ and a family of some vectors $v_{1},\dots,v_{k}$ and consider the convex hull $\operatorname{conv}(P\cup G\cdot\{v_{1},\dots,v_{k}\})$. In order to keep track of how the number of vertices and faces change due to the construction with respect to the faces and vertices of $P$, the following definition is useful: We say $X$ sees $Y$ with respect to $P$, if any of the line segments $\operatorname{conv}(x,y)$ with $x\in X$ and $y\in Y$ does not intersect the interior of $P$.

The next lemma is a technical result ensuring that many operations known for general polytopes can also be deployed for symmetric polytopes (by adding whole orbits instead of points) without getting unexpected edges and facets.

The most important technique to generate new $G$-symmetric polytopes is by stacking vertices on facets. This is a generalization of the constructions of Steinitz [??].

For any operation $\delta$ we can define the dual operation which sends $P$ to $(\delta(P^{\vee}))^{\vee})$. For example the following:

Our next aim is to establish a class of polytopes, such that the operations $\operatorname{CS}$ and $\operatorname{CC}$ can be applied successively to generate infinite families of $G$-symmetric polytopes. Recall that the (edge)-degree of a vertex $v$ is the number of edges that contain $v$. A vertex of degree $3$ is called simple. The dual concept is the degree of a facet $F$ which is the number of edges that are contained in $F$. A facet of degree $3$, namely a triangular facet, is also called a simplicial facet.

The following Corollary shows that the existence of a base polytope guarantees the existence of a whole integer cone of $f$-vectors. Using the notation $\operatorname{Cf}=(2,1)\mathbb{N}+(1,2)\mathbb{N}$ we have:

This is the main tool for the construction of polytopes with a given $f$-vector.

Let $G$ be a rotation group. Since it is often impossible to construct $G$-symmetric base polytopes with small entries in their $f$-vectors, we introduce the concept of certificates. The central idea is that giving a certificate for an $f$-vector is sufficient to construct an integer cone of $f$-vectors, similar to Corollary 3.5. To this extend we introduce the notion of left type and right type polytopes which can be interpreted as ’half base’:

Note that a left type polytope $P$ can be used to construct a base polytope $Q$ with $f(Q)=f(P)+(n,2n)$ by a single $\operatorname{CS}_{3,n}$ operation. On the other hand, a right type polytope $P$ can be used to construct a base polytope $Q$ with $f(Q)=f(P)+(2n,n)$ by a single $\operatorname{CC}_{3,n}$ operation. Now we are able to define certificates, which are sufficient to construct certain integer cones:

While there are in theory infinite possibilities for certificates, these four are the only relevant in practice. The benefit of certificates is the following theorem:

Note that Lemma 2.2 conditions $\mathit{F}(G)$ to be contained in certain subsets given by congruence relations. But the certificates construct integer cones of the form $f+n\operatorname{Cf}$. The following lemma relates these two approaches:

With Theorem 4.3 and Lemma 4.4 as well as the fact $\mathit{F}(G)^{\diamond}=\mathit{F}(G)$ it easily follows:

Consequently, to show that $\mathit{F}(G)$ contains a set $\mathit{F}^{\prime}$ given by Lemma 2.2 it suffices to state a table of certificates.

In this section, we will describe some general operations which can be applied to $G$-symmetric polytopes. We will empathize the implications for the $f$-vector and the types of the polytope. First we discuss how small narrowed prisms can be stacked on facets of a polytope.