Canonical Polyhedron

A canonical polyhedron is a polyhedron for which all its polyhedron edges touch a sphere and the geometric centroid of their contact points is the center of that sphere. Thus it possesses a midsphere satisfying an additional contact-point geometric centroid condition (Springborn 2005, Theorem 2).

A dual polyhedron can be constructed from a canonical polyhedron that possesses these properties as well. Moreover, the edges of the canonical polyhedron and its dual cross at right angles.

With its midsphere normalized to the unit sphere centered at the origin, there is a unique canonical version (modulo rotations and reflections) for each combinatorial type of convex polyhedron (Schramm 1992; Ziegler 1995, pp. 117-118; Springborn 2005). Many symmetric polyhedra are canonical in their "natural" form, including the Platonic solids as well as the Archimedean solids and their duals. Equilateral antiprisms and prisms are also canonical.