By definition, all of the outer vertices of the Platonic Solids lie upon a sphere.
The Platonic Spheres were shown to me in a series of lucid dreams by Archangel Metatron.
4 out of 5 of the Platonic Spheres are created by joining the outer points of the faces with a ring that touches 3 or more points on the face of a Platonic Solid.
The Icosasphere is based upon 12 rings instead of 20 (which is what it would be if it followed the rules of construction of the other four). This is how the Icosasphere was revealed to me.
From L to R:
Tetrasphere, Icosasphere,
Octasphere, Hexasphere and Dodecasphere
Also available
with nested Platonic Solids
10.47 x 1.89 x 1.89 inches