Marquette University

Section 4.5 of the book discusses the platonic solids.

There are just five of them, the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.
To precisely define a platonic solid, it is not sufficient to say that all its surfaces must be identical regular polygons (can you give an interesting example of a solid that satisfies this weaker definition, but that is not a platonic solid?).
Besides all surfaces being identical regular polygons, one may add that the vertices must lie on a sphere. For a silly reason this still does not suffice. We must also add that the volume is positive, that is, a platonic solid is not flat.
Naturally one must verify that the five objects mentioned above really satisfy the required properties. But how do we show that there are no others? One of the tools which helps is the Euler invariant: If a solid (without holes) has V many vertices, E many edges, and S many surfaces, then 2 + E = V + S
The five platonic solids were considered so special, that for a long time researchers, often influenced by Pythagorean philosophy, wondered whether there were connections between them and some deeper properties of the universe, and beyond. Originally Kepler (some time after 1600) was among them. Experimental observations forced him to abolish his `platonic' model of the solar system, to be replaced by a model following the so-called Kepler laws. After about 2 millennia, the Pythagorean philosophy lost its support in the intellectual community.

Last updated: November 2004
Comments & suggestions: wimr@mscs.mu.edu