An orthogonal polyhedron is a polyhedron in which all edges are parallel to the axes of a Cartesian coordinate system, [1] resulting in the orthogonal faces and implying the dihedral angle between faces are right angles.
Though the angles between Jessen's icosahedron's faces are right angles, the edges are not axis-parallel, thus Jessen's icosahedron is not an orthogonal polyhedron. [2]
Polycubes are a special case of orthogonal polyhedra that can be decomposed into identical cubes and are three-dimensional analogs of planar polyominoes. [3] Orthogonal polyhedra can be either convex (such as rectangular cuboids) or non-convex. [2] [4]
Orthogonal polyhedra were used by Sydler (1965), who showed that any polyhedron is equivalent to a cube: it can be decomposed into pieces that later can be used to construct a cube. This showed the requirements for the polyhedral equivalence conditions in terms of the Dehn invariant. [5] [2] Orthogonal polyhedra may also be used in computational geometry, where their constrained structure has enabled advances in problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net. [6]
The simple orthogonal polyhedra, as defined by Eppstein & Mumford (2014), are the three-dimensional polyhedra such that three mutually perpendicular edges meet at each vertex and that have the topology of a sphere. [4] By using Steinitz's theorem, there are three different classes: the arbitrary orthogonal polyhedron, the skeleton of its polyhedron drawn with a hidden vertex by the isometric projection, and the polyhedron wherein each axis-parallel line through a vertex contains other vertices. All of these are polyhedral graphs that are cubic and bipartite. [7]