An introduction to Coxeter polyhedra

TL;DR

This article offers an accessible, two-part introduction to Coxeter polyhedra across spherical, Euclidean, and hyperbolic geometries. It first develops non-obtuse polyhedra via Gram matrices, Andreev’s and Vinberg’s realization criteria, and then introduces Coxeter polyhedra and their diagrams, culminating in a systematic Wythoff construction for regular, semiregular, and uniform polyhedra and tessellations. The work surveys complete classifications in low dimensions (notably 2–3D) and highlights higher-dimensional challenges, including Gosset polytopes and right-angled hyperbolic polyhedra, while emphasizing the interplay between geometric, combinatorial, and group-theoretic structures. Overall, it ties Gram-matrix data, reflection groups, and Coxeter diagrams to concrete tessellations and symmetry classifications with broad implications for discrete groups, lattices, and manifolds of constant curvature.

Abstract

This paper is an introduction to Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries. It consists of essentially two parts that could be read independently. In the first we introduce non-obtuse polyhedra in the spherical, Euclidean, and hyperbolic spaces, and prove various fundamental theorems originated from Andreev, Coxeter, and Vinberg. In the second we introduce Coxeter polyhedra and use them to describe regular, semiregular, and uniform polyhedra and tessellations, mostly via the Wythoff construction.

Figures (32)

• Figure 1: The link of a vertex $v$ and of an edge $e$ of a three-dimensional polyhedron is a spherical triangle and a spherical arc respectively (both drawn in red).
• Figure 2: The hyperplanes $\partial H_i$ and $\partial H_j$ can be incident, parallel, or ultraparallel in ${\mathbb{H}}^n$.
• Figure 3: The only nonobtuse polyhedra in ${\mathbb{R}}^3$.
• Figure 4: A facet $F_h$ and two incident facets $F_i,F_j$ of $P$ such that $\partial H_i \cap \partial H_j \cap ( {\mathbb{H}}^n \setminus H_k) \neq \emptyset$. This is the key configuration that we want to rule out while proving Theorem \ref{['nonobtuse:teo']}. The dihedral angles of $P$ are non-obtuse, and this leads to a contradiction.
• Figure 5: A real vertex $v$ of a non-obtuse polyhedron $P\subset {\mathbb{H}}^3$. Here $\alpha_1,\alpha_2,\alpha_3 \leq \pi/2$ are the dihedral angles and $\theta_1,\theta_2,\theta_3 \leq \pi/2$ are the angles of the faces adjacent to $v$, with $\theta_i$ opposite to $\alpha_i$ (left). A tetrahedron with dihedral angles $\alpha_1,\ldots, \alpha_6$ (center) and a triangular prism with dihedral angles $\alpha_1,\ldots,\alpha_9$ (right).

Theorems & Definitions (56)

• Theorem 8
• proof
• Proposition 9
• proof
• Theorem 10
• proof
• Proposition 11
• proof
• Corollary 12
• Corollary 13