arXiv subjects

Egon Schulte

Publications and source records attributed to Egon Schulte.

At least 19 recordsLinked to original sources

arXiv2026

String C-groups of 2-power order project onto a common string C-group

String C-groups are precisely the automorphism groups of abstract regular polytopes. A certain regular d-polytope C_d with an automorphism group of order 2^{2d-1}, discovered by Conder and shown to have the smallest number of flags among all regular d-polytopes of high ranks, also has the important extremal property to be the unique minimal d-polytope, with respect to combinatorial covering, among all finite regular d-polytopes with 2-power automorphism groups. In other words, the automorphism group of C_d is a quotient group of every finite string C-group of rank d and 2-power order; and every finite regular d-polytope with an automorphism groups of 2-power order covers C_d. The existence of a unique minimal element among string C-groups of 2-power order and given rank is remarkable in itself.

math.GR↗

arXiv2026

Two-Orbit Polytopes

Abstract polytopes are combinatorial structures with distinctive geometric, algebraic, or topological characteristics, that generalize (the face lattice of) traditional polyhedra, polytopes or tessellations. Most research has focused on abstract polytopes with the highest possible symmetry, in particular those that are regular or chiral. In this paper we study two-orbit polytopes, that is, abstract polytopes whose automorphism groups have exactly two orbits on flags. Such polytopes of rank $n$ fall into $2^n-1$ classes, determined by their local flag configuration. We develop a general structural theory of two-orbit polytopes of arbitrary rank. In particular, we determine their face- and section-transitivity properties and describe the structure of their automorphism groups via distinguished generating sets and face stabilizer subgroups. These results yield a characterization of the partial order { on the polytope} in terms of the automorphism group. Two-orbit polytopes in different classes behave quite differently. Our approach extends the group-theoretic framework for regular and chiral polytopes and provides a systematic foundation for the study of polytopes with two flag orbits.

math.CO↗

arXiv2026

Chiral Polyhedra from AGL(1,q)

We present a construction of chiral and regular polyhedra from subgroups of the general affine group AGL(1,q) for odd prime powers q. In particular, we show that the full group AGL(1,q) occurs as the automorphism group of a chiral polyhedron of type {q-1, q-1} when q=1 mod 4, or types {q-1,(q-1)/2} or {(q-1)/2, q-1} when q=3 mod 4, and we compute the genus in each case. We also establish that subgroups of AGL(1,q) cannot serve as full automorphism groups of regular polytopes of rank 3 or higher, nor of chiral polytopes of rank 4 or higher, demonstrating that our construction captures all polytopes that can arise from this class of affine groups.

math.CO↗

arXiv2026

Skeletal Snub Polyhedra in Ordinary Space, I

Skeletal polyhedra are discrete connected structures consisting of finite (planar or skew) or infinite (linear, planar, or spatial) polygons as faces, with two faces on each edge and a circular vertex figure at each vertex. The present paper describes the blueprint for the snub construction and shows that it can be applied to both regular and chiral skeletal polyhedra in ordinary space. The resulting skeletal snub polyhedra are vertex-transitive and highly locally symmetric. Their properties - from a combinatorial, topological, and geometric perspective - are described and illustrated on some particularly interesting examples. We examine when the construction yields uniform skeletal polyhedra and discuss the completeness of our list of generated structures.

math.CO↗

arXiv2025

Symmetric polytopes whose automorphism groups are 2-groups

The present work investigates regular, semiregular, and chiral polytopes of any rank $d\geq 3$, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with automorphism groups of 2-power order: for such polytopes with toroidal sections of rank 3, the various sections of rank 3 can be entirely prescribed (possibly with one exception in the semiregular case). It is also shown that having a 2-group as automorphism group is hereditary under taking universal extensions: the universal extension of a given regular, chiral, or alternating semiregular polytope with a finite or infinite 2-group as automorphism group, is a polytope of one rank higher with an infinite 2-group an automorphism group.

math.GR↗

arXiv2023

Bounds for the Regularity Radius of Delone Sets

Delone sets are discrete point sets $X$ in $\mathbb{R}^d$ characterized by parameters $(r,R)$, where (usually) $2r$ is the smallest inter-point distance of $X$, and $R$ is the radius of a largest ``empty ball" that can be inserted into the interstices of $X$. The regularity radius $\hatρ_d$ is defined as the smallest positive number $ρ$ such that each Delone set with congruent clusters of radius $ρ$ is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our ``Weak Conjecture" states that $\hatρ_{d}={{\rm O}(d^2\log d)}R$ as $d\rightarrow\infty$, independent of~$r$. This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius $2r$ and those with full-dimensional sets of $d$-reachable points. We also offer support for the plausibility of a ``Strong Conjecture", stating that $\hatρ_{d}={{\rm O}(d\log d)}R$ as $d\rightarrow\infty$, independent of $r$.

math.MG↗

arXiv2019

On the regularity radius of Delone sets in $\mathbb{R}^3$

We complete the proof of the upper bound $\hatρ_3\leq 10R$ for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all $10R$-clusters of a Delone set $X$ with parameters $(r,R)$ are equivalent, then $X$ is a regular system.

math.MG↗

arXiv2019

Prescribing Symmetries and Automorphisms for Polytopes

We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When $Γ$ is a subgroup of the combinatorial automorphism group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with combinatorial automorphism group exactly $Γ$. When $Γ$ is a subgroup of the geometric symmetry group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with both geometric symmetry group and combinatorial automorphism group exactly $Γ$. These symmetry-breaking results then are applied to show that for every abelian group $Γ$ of even order and every involution $σ$ of $Γ$, there is a centrally symmetric convex polytope with geometric symmetry group $Γ$ such that $σ$ corresponds to the central symmetry.

math.CO↗

arXiv2018

On the Origin of Crystallinity: a Lower Bound for the Regularity Radius of Delone Sets

The local theory of regular or multi-regular systems aims at finding sufficient local conditions for a Delone set $X$ to be a regular or multi-regular system. One of the main goals is to estimate the regularity radius $\hatρ_d$ for Delone sets $X$ in terms of the radius $R$ of the largest "empty ball" for $X$. The present paper establishes the lower bound $\hat{ρ_d}\geq 2dR$ for all $d$, which is linear in $d$. The best previously known lower bound had been $\hatρ_d\geq 4R$ for $d\geq 2$. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent $(2dR-\varepsilon)$-clusters, which are not regular systems.

math.MG↗

arXiv2017

Regular Incidence Complexes, Polytopes, and C-Groups

Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-groups. Further, extensions of regular incidence complexes are studied, and certain incidence complexes particularly close to abstract polytopes, called abstract polytope complexes, are investigated.

math.CO↗

arXiv2016

Wythoffian Skeletal Polyhedra in Ordinary Space, I

Skeletal polyhedra are discrete structures made up of finite, flat or skew, or infinite, helical or zigzag, polygons as faces, with two faces on each edge and a circular vertex-figure at each vertex. When a variant of Wythoff's construction is applied to the forty-eight regular skeletal polyhedra (Grunbaum-Dress polyhedra) in ordinary space, new highly symmetric skeletal polyhedra arise as "truncations" of the original polyhedra. These Wythoffians are vertex-transitive and often feature vertex configurations with an attractive mix of different face shapes. The present paper describes the blueprint for the construction and treats the Wythoffians for distinguished classes of regular polyhedra. The Wythoffians for the remaining classes of regular polyhedra will be discussed in Part II, by the second author. We also examine when the construction produces uniform skeletal polyhedra.

math.MG↗

arXiv2016

Skeletal Geometric Complexes and Their Symmetries

Skeletal polyhedra and polygonal complexes are finite or infinite periodic structures in 3-space with interesting geometric, combinatorial, and algebraic properties. These structures can be viewed as finite or infinite periodic graphs (nets) equipped with a polyhedral superstructure imposed by the faces, allowed to be skew, zigzag, or helical. The article describes skeletal structures with maximal symmetry.

math.MG↗

arXiv2015

Polytopes with Preassigned Automorphism Groups

We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be realized as a convex polytope.

math.CO↗

arXiv2015

Groups of Ree type in characteristic 3 acting on polytopes

Every Ree group $R(q)$, with $q\neq 3$ an odd power of 3, is the automorphism group of an abstract regular polytope, and any such polytope is necessarily a regular polyhedron (a map on a surface). However, an almost simple group $G$ with $R(q) < G \leq \mathsf{Aut}(R(q))$ is not a C-group and therefore not the automorphism group of an abstract regular polytope of any rank.

math.GR↗

arXiv2014

Cube-like Incidence Complexes and Their Groups

The article studies power complexes and generalized power complexes, and investigates the algebraic structure of their automorphism groups. The combinatorial incidence structures involved are cube-like, in the sense that they have many structural properties in common with higher-dimensional cubes and cubical tessellations on manifolds. Power complexes have repeatedly appeared in applications.

math.CO↗

arXiv2014

Colorful Associahedra and Cyclohedra

Every n-edge colored n-regular graph G naturally gives rise to a simple abstract n-polytope, the colorful polytope of G, whose 1-skeleton is isomorphic to G. The paper describes colorful polytope versions of the associahedron and cyclohedron. Like their classical counterparts, the colorful associahedron and cyclohedron encode triangulations and flips, but now with the added feature that the diagonals of the triangulations are colored and adjacency of triangulations requires color preserving flips. The colorful associahedron and cyclohedron are derived as colorful polytopes from the edge colored graph whose vertices represent these triangulations and whose colors on edges represent the colors of flipped diagonals.

math.CO↗

arXiv2014

Regular Polygonal Complexes of Higher Ranks in E^3

The paper establishes that the rank of a regular polygonal complex in 3-space E^3 cannot exceed 4, and that the only regular polygonal complexes of rank 4 in 3-space are the eight regular 4-apeirotopes.

math.MG↗

arXiv2014

Polyhedra, Complexes, Nets and Symmetry

Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zig-zag, or helical. A polyhedron or complex is "regular" if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra. Their edge graphs are nets well-known to crystallographers, and we identify them explicitly. There also are 6 infinite families of "chiral" apeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits.

math.MG↗