Hyperbolic Coxeter Pyramids

John Mcleod

doi:10.4236/apm.2013.31010

Abstract

Hyperbolic Coxeter polytopes are defined precisely by combinatorial type. Polytopes in hyperbolic n-space with n + p faces that have the combinatorial type of a pyramid over a product of simplices were classified by Tumarkin for small p. In this article we generalise Tumarkin’s methods and find the remaining hyperbolic Coxeter pyramids.

Highlights

The polytopes in n dimensions that have n 1 faces are referred to collectively as the simplices of the geometric space they inhabit
In this article we consider only Coxeter polytopes, which are precisely those that are the fundamental domains of reflection groups
The Coxeter simplices are well known in Euclidean, Spherical and Hyperbolic space

Summary

Introduction

The polytopes in n dimensions that have n 1 faces are referred to collectively as the simplices of the geometric space they inhabit. The Coxeter simplices are well known in Euclidean, Spherical and Hyperbolic space These lists illustrate an important distinction that separates Hyperbolic space from the first two spaces in this list, namely that there is an upper bound on the dimension above which there are no simplices. [9]) which has quantitatively different characteristics when it describes a pyramid, as compared to other configurations This distinction led to pyramids being classified using separate methods originally due to Vinberg. In this article we generalise Tumarkin’s classification of Hyperbolic Coxeter pyramids in terms of the Coxeter diagram, and find all remaining examples of such polytopes using simple combinatorial arguments.

Coxeter Diagrams

Pyramids