Midsphere Tetrahedra and the Canonical Tetrahedron

A midsphere polyhedron is one in which all edges are tangent to a unit sphere. A canonical polyhedron is a midsphere polyhedron such that the centroid of the tangent points is at the center of the unit sphere.

There are many midsphere tetrahedra, but the canonical tetrahedron (a regular tetrahedron) is unique.