Dihedral angles and orthogonal polyhedra

Consider an orthogonal polyhedron, i.e., a polyhedron where (at least after a suitable rotation) all faces are perpendicular to a coordinate axis, and hence all edges are parallel to a coordinate axis. Clearly, any facial angle (i.e., the angle of a face at an incident vertex) is a multiple of π/2. Also, any dihedral angle (i.e., the angle between two planes that support to faces with a common edge) is a multiple of π/2. In this note we explore the converse: if the facial and/or dihedral angles are all multiples of π/2, is the polyhedron necessarily orthogonal? The case of facial angles was answered previously in two papers at CCCG 2002 [DO02, BCD02]: If a polyhedron with connected graph has genus at most 2, then facial angles that are multiples of π/2 imply that the polyhedron is orthogonal, while for genus 6 or higher there exist examples of non-orthogonal polyhedra with connected graph where all facial angles are π/2. In this note we show that if both the facial and dihedral angles are multiples of π/2 then the polyhedron is orthogonal (presuming connectivity), and we give examples to show that the condition for dihedral angles alone does not suffice.