[Paper Review] Coxeter Decompositions of Bounded Hyperbolic Pyramids and Triangular Prisms

Anna Felikson|ArXiv.org|Dec 15, 2002

Geometric and Algebraic Topology2 references3 citations

TL;DR

This paper classifies Coxeter decompositions of bounded convex hyperbolic pyramids and triangular prisms in 3-dimensional hyperbolic space. Using geometric and combinatorial methods, it proves that fundamental polyhedra in such decompositions must be tetrahedra or triangular prisms, and fully enumerates all possible decomposition types via dihedral angle configurations and mirror arrangements.

ABSTRACT

Coxeter decompositions of hyperbolic simplices where studied in math.MG/0212010 and math.MG/0210067. In this paper we use the methods of these works to classify Coxeter decompositions of bounded convex pyramids and triangular prisms in the hyperbolic space H^3.

Motivation & Objective

• To classify all possible Coxeter decompositions of bounded convex hyperbolic pyramids in H³.
• To classify all Coxeter decompositions of bounded triangular prisms in H³.
• To determine the structure of fundamental polyhedra in such decompositions, proving they must be tetrahedra or triangular prisms.
• To characterize the role of mirrors and fundamental edges/vertices in the decomposition process.
• To enumerate all valid dihedral angle configurations for these decompositions using geometric and combinatorial constraints.

Proposed method

• Applies methods from prior works on Coxeter decompositions of hyperbolic simplices to analyze bounded pyramids and prisms.
• Uses the definition of a quasi-Coxeter polyhedron as a polyhedron admitting a Coxeter decomposition into congruent Coxeter polyhedra.
• Employs geometric arguments based on face intersections and mirror arrangements to prove that non-tetrahedral fundamental polyhedra cannot exist in pyramids.
• Introduces the concept of a minimal triangular prism to reduce the classification problem to fundamental cases.
• Analyzes pentagonal mirrors—mirrors intersecting prisms in a specific 5-sided configuration—as key structural elements in prism decompositions.
• Uses combinatorial enumeration of dihedral angles (expressed as rational multiples of π) to classify all possible decomposition types.

Experimental results

Research questions

• RQ1 What are the possible Coxeter decompositions of bounded convex hyperbolic pyramids in H³?
• RQ2 What are the possible Coxeter decompositions of bounded triangular prisms in H³?
• RQ3 What types of fundamental polyhedra can appear in such decompositions—specifically, can they be non-tetrahedral or non-prismatic?
• RQ4 Under what conditions do mirrors in a prism decomposition intersect the prism in a pentagonal cross-section?
• RQ5 How do the dihedral angles of the original polyhedron constrain the possible decomposition types?

Key findings

• Any bounded convex hyperbolic pyramid admits a Coxeter decomposition only if its fundamental polyhedra are tetrahedra.
• For bounded triangular prisms, the fundamental polyhedra in a Coxeter decomposition are either tetrahedra or triangular prisms.
• In minimal triangular prisms, all mirrors are pentagonal and all dihedral angles of the prism are fundamental.
• The paper provides a complete enumeration of all possible Coxeter decompositions, listing 12 distinct types with specific dihedral angle configurations.
• Each decomposition type is characterized by a triple of rational multiples of π for dihedral angles, with explicit counts of tetrahedral and pyramidal tiles.
• The classification includes 12 distinct decomposition types, with the number of fundamental tiles and tile types explicitly tabulated (e.g., types 1–12 with 3–23 tiles, 0–3 tetrahedra, etc.).