Compact hyperbolic tetrahedra with non-obtuse dihedral angles

TL;DR

This paper investigates the space of dihedral angles of compact hyperbolic tetrahedra with non-obtuse angles, demonstrating that unlike polyhedra with five or more faces, this space is non-convex, highlighting unique geometric properties.

Contribution

It explains why the space of non-obtuse dihedral angles for hyperbolic tetrahedra is non-convex, contrasting with higher-face polyhedra, and provides a simple proof using the method of continuity.

Findings

The space of non-obtuse dihedral angles for hyperbolic tetrahedra is non-convex.

Convexity holds for polyhedra with five or more faces, but not for tetrahedra.

The proof illustrates the method of continuity in geometric classification.

Abstract

Given a combinatorial description $C$ of a polyhedron having $E$ edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize $C$ is generally not a convex subset of $\mathbb{R}^E$ \cite{DIAZ}. If $C$ has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angles $A_C$ obtained by restricting to {\em non-obtuse} angles is a convex polytope. In this paper we explain why Andreev did not consider tetrahedra, the only polyhedra having fewer than five faces, by demonstrating that the space of dihedral angles of compact hyperbolic tetrahedra, after restricting to non-obtuse angles, is non-convex. Our proof provides a simple example of the ``method of continuity'', the technique used in classification theorems on polyhedra by Alexandrow \cite{ALEX}, Andreev \cite{AND}, and Rivin-Hodgson \cite{RH}.