The midsphere, also called the intersphere, reciprocating sphere, or inversion sphere, is a sphere with respect to which the polyhedron
vertices of a polyhedron are the inversion
poles of the planes of the faces of the dual polyhedron
(and vice versa). The radius 
of the midsphere is called the midradius.
The midsphere touches all polyhedron edges as well as the edges of the dual of that solid Note that the midsphere does not necessarily pass through the midpoints of the edges a polyhedron dual, but is rather only tangent to the edges at some point along their lengths.
A polyhedron that possesses a midsphere is said to be a canonical polyhedron. An interesting theorem states that each topological type (genus 0) of convex polyhedron possesses a canonical polyhedron (Ziegler 1995, pp. 117-118).
The figure above shows the Platonic solids and their duals, with the circumsphere of the solid, midsphere, and insphere of the dual superposed.
