Inscribed sphere

For such cases, the notion of an insphere does not seem to have been properly defined and various interpretations of an insphere are to be found: Often these spheres coincide, leading to confusion as to exactly what properties define the insphere for polyhedra where they do not coincide.

For example, the regular small stellated dodecahedron has a sphere tangent to all faces, while a larger sphere can still be fitted inside the polyhedron.

Important authorities such as Coxeter or Cundy & Rollett are clear enough that the face-tangent sphere is the insphere.

Again, such authorities agree that the Archimedean polyhedra (having regular faces and equivalent vertices) have no inspheres while the Archimedean dual or Catalan polyhedra do have inspheres.

But many authors fail to respect such distinctions and assume other definitions for the 'inspheres' of their polyhedra.

Tetrahedron with insphere in red (also midsphere in green, circumsphere in blue)
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.