Orthogonal polyhedron

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. The angle between Jessen's icosahedron's faces is right, but the edges are not axis-parallel, which 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 in Sydler (1965) in which he showed that any polyhedron is equivalent to a cube: it can be decomposed into pieces which later can be used to construct a cube. This showed the requirements for the polyhedral equivalence conditions by 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 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]