Compact Coxeter Research Articles

Infinitely many commensurability classes of compact Coxeter polyhedra in [formula omitted] and [formula omitted]

We prove that certain families of compact Coxeter polyhedra in 4- and 5-dimensional hyperbolic space constructed by Makarov give rise to infinitely many commensurability classes of reflection groups in these dimensions.

FROM GEOMETRY TO ARITHMETIC OF COMPACT HYPERBOLIC COXETER POLYTOPES

We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.

The optimal hyperball packings related to the smallest compact arithmetic 5-orbifolds

The smallest three hyperbolic compact arithmetic 5-orbifolds can be derived from two compact Coxeter polytopes that are combinatorially simplicial prisms (i.e. complete orthoschemes of degree d = 1) in five dimensional hyperbolic space H5 (see [2] and [4]). The corresponding hyperbolic tilings are generated by reflections through their delimiting hyperplanes. These involve our studies for the relating densest hyperball (hypersphere) packings with congruent hyperballs. The analogous problem was discussed in [13] and [14] in hyperbolic spaces Hn (n = 3, 4). In this paper we extend this procedure to determine the optimal hyperball packings of the above 5-dimensional prism tilings. We compute their metric data and the densities of their optimal hyperball packings, moreover, we formulate a conjecture for the candidates for the densest hyperball packings in 5-dimensional hyperbolic space H5.

Projective deformations of weakly orderable hyperbolic Coxeter orbifolds

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study the deformation space of real projective structures on a compact Coxeter $n$-orbifold $Q$ admitting a hyperbolic structure. Let $e_+(Q)$ be the number of ridges of order $\geq 3$. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension $e_+(Q) - n$ if $n=3$ and $Q$ is weakly orderable, i.e., the faces of $Q$ can be ordered so that each face contains at most $3$ edges of order $2$ in faces of higher indices, or $Q$ is based on a truncation polytope.

The Complete Set of Orderable and Deformable Compact Coxeter Hyperbolic Tetrahedrons

We classified the combinatorial structures of 3-dimensional bounded ODCH (orderable and deformable compact hyperbolic) Coxeter polyhedra up to symmetry. Using Plantri graphs and graph theory, we proved that the number of such combinatorial polyhedra is five. One of such combinatorial polyhedra is tetrahedron. Using graph theory and combinatorics, we find that there are six orderable and deformable compact hyperbolic tetrahedra up to symmetry.

Application of Plantri Graph:All Combinatorial Structure of Orderable And Deformable Compact Coxeter Hyperbolic Polyhedra

By Andreev's theorem and Choi's theorem, we proved that the degree of each vertex is three and the number of vertices of orderable compact Coxeter polyhedral is at most 10. Therefore a combinatorial polyhedron is a 3-connected planner graph. From the Plantri program, we found that the number of 3-connected planner graphs with at most 10 vertices of degree 3 is 9. We find that only five planner graphs among these 9 graphs satisfy the properties of orderable compact Coxeter polyhedra. Then we verify the polyhedra which are associated with these 5 planner graphs are orderable. Therefore the number of combinatorial polyhedra of orderable and deformable compact hyperbolic Coxeter polyhedra is five up to symmetry.

On hyperbolic Coxeter n-polytopes with n + 2 facets

Abstract A convex polytope admits a Coxeter decomposition if it is tiled by finitely many Coxeter polytopes such that any two tiles having a common facet are symmetric with respect to this facet. In this paper, we classify all Coxeter decompositions of compact hyperbolic Coxeter n-polytopes with n + 2 facets. Furthermore, going out from Schläfli‘s reduction formula for simplices we construct in a purely combinatorial way a volume formula for arbitrary polytopes and compute the volumes of all compact Coxeter polytopes in ℍ4 which are products of simplices.