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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 RE. If C has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angles AC obtained by restricting to 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...
In 1970, E.M.Andreev published a classification of all three-dimensional compact hyperbolic polyhedra (other than tetrahedra) having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, , Andreev’s Theorem provides five classes of linear inequalities, depending on , for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting...
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