Everyone that has a polyhedron web site needs to have a Platonic Solids page. If you are visiting my pages, you have probably seen dozens of these. So I will try to do something a little different.
This page was inspired by a discussion with some friends. Someone asked what was a good way to arrange the Platonic polyhedra inside of each other in an aesthetically pleasing fashion. Obviously, when putting something like this together, you need to focus on the Platonic Relationships - symmetries that the polyhedra have in common, and different ways that they can be inscribed in each other.
One recommended method was to inscribe them in a way that the edges (and sometimes entire faces) of each polyhedron lay in the faces of another polyhedron. This was accomplished by having an icosahedron inside an octahedron inside a tetrahedron (all with common faces). The tetrahedron could be inscribed in a cube where each edge of the tetrahedron was a diagonal of the square faces of the cube. And each of the edges of the cube were diagonals of the pentagonal faces of the dodecahedron. John Conway has named this arrangement the "Cosmogram" in homage to Johannes Keppler.
Although I think this arrangement is quite amazing, and the size of the innermost polyhedron is pretty large compared to the outermost, I also think that it looks a bit crowded. I prefer to have them spaced out a bit more evenly, and I came up with an arrangement that has each polyhedron touching its neighbors vertex-to-face-center. The vertices of the tetrahedron in the center of my figure touches four of the eight face centers of the octahedron. The octahedron's six vertices touch all of the face centers of the cube (because of their dual relationship). The eight vertices of the cube touch eight of the twenty face centers of the icosahedron. And the twelve vertices of the icosahedron touch all of the face centers of the dodecahedron (again because of their dual relationship).
Apparently other people like this arrangement as well. If you ever want to see a beautiful paper model of this, check out Ulrich Mikloweit's Cosmogram Page
Back to the main Polyhedron Page.
Link to this page as http://Polyhedra.Doskey.com/PlatonicRelationships.html