Dihedral angles of convex polyhedra
Let \(P\) be a compact convex polyhedron in hyperbolic 3-space \(H^3.\) It is classical to ask: Is \(P\) determined, among convex polyhedra, by its combinatorics and its dihedral angles? The author shows that some similar or related questions have negative answers. There exist pairs of non-congruent convex space-like polyhedra in the de Sitter space \(S_1^3\) with the same edge lengths or with the same dihedral angles. The same holds for convex polyhedra in \(S^3\subset\mathbb R^4.\) The examples are built from a well-known one-parameter family of polyhedra in \(\mathbb R^3\) with the same edge lengths, through a transformation discovered by Pogorelov for \(H^3\) and \(S^3.\)
scientific article; zbMATH DE number 1436022