Dihedral Angles Do Not Determine Face Angles of Nonconvex Polyhedra: An Explicit Pair of Triangulated Bipyramids

At the 2015 Oberwolfach workshop on discrete differential geometry N. Amenta asked whether Stoker's conjecture extends to nonconvex polytopes: for a polytope with given combinatorics, do the dihedral angles, all different from \(0\) and \(\pi\), determine the face angles? One may assume that the polytope is triangulated and homeomorphic to a sphere (Oberwolfach Report 13/2015, Problem 4). We show by one explicit example that the answer is no. Two embedded nonconvex polyhedra \(P\) and \(P'\) realize the same labelled triangulation of the sphere, the boundary of the triangular bipyramid. No two adjacent faces are coplanar, all nine interior dihedral angles of \(P'\) are equal to those of \(P\) (one edge is reflex in both, with angle about \(273.82^\circ\)), and twelve of the eighteen face angles differ. The coordinates lie in a real quadratic field, and the proof is an exact finite certificate, checked by a short program in rational arithmetic, together with elementary lemmas. The pair also refutes, for this mesh, the conjecture of Amenta and Rojas (2018, restated in the journal version of 2020) that a vector of dihedral angles is consistent with at most one set of inner face angles of an embedded triangle mesh. For convex polyhedra the answer is yes, by a theorem in a preprint of Wang and Xie, and by that theorem no pair of the kind found here can have a convex member. We claim one explicit pair only: no family of pairs and nothing on continuous deformations with constant dihedral angles; the offset question of Rote, merged with Amenta's question in the problem record, is not addressed. The papers of Stoker and of Karcher (1968), the closest prior sources, and the later papers of Milka (1970) and Pogorelov (2002) on Stoker's problem could not be read. No printed example was found in the sources searched; a search that finds nothing is not a proof of novelty. This is an unrefereed note.