Inscribed sphere

In his 1597 book *Mysterium Cosmographicum*, Kepler modelled of the Solar System with its then known six planets' orbits by nested platonic solids, each circumscribed and inscribed by a sphere.

In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual to the dual polyhedron's circumsphere.

The radius of the sphere inscribed in a polyhedron P is called the inradius of P.