Shapes of Centrally Symmetric Octahedra with Prescribed Cone-Deficits

TL;DR

This paper investigates the geometric structure of centrally symmetric octahedra with specific cone-deficits, revealing that their shape space forms a real hyperbolic ideal tetrahedron with angles related to the deficits.

Contribution

It extends Thurston's work by characterizing the shape space of symmetric octahedra with prescribed cone-deficits as a hyperbolic tetrahedron.

Findings

Shape space forms a real hyperbolic ideal tetrahedron.

Dihedral angles are half of the prescribed cone-deficits.

Provides geometric insight into symmetric polyhedra with cone-deficits.

Abstract

In his paper "Shapes of Polyhedra and Triangulations of the Sphere", Thurston found that the set of shapes of convex polyhedra with prescribed cone-deficits has a complex hyperbolic structure. Inspired by his work, this paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits. We show that this set forms a real hyperbolic ideal tetrahedron, whose dihedral angles are half of the prescribed cone-deficits.