Platonic Solids

A Platonic solid is a polyhedron where (1) each face is the same regular polygon, and (2) each vertex joins the same number of faces.

The Platonic solids are highly symmetrical, and, in three dimensions, only five such solids can exist: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

This was first proven in Euclid’s Elements around 300 B.C., and has since been more rigorously proven using the Euler characteristic. The proofs are relatively easy to follow, and if you’re interested you can check them out both here and here.

Mathematica code:

pSolids={"Tetrahedron","Cube","Octahedron","Dodecahedron","Icosahedron"}
Manipulate[Graphics3D[
 {Opacity[0.8],Rotate[PolyhedronData[pSolids[[n]],"Faces"],th,{0,0,1}],
  Opacity[0],Circumsphere[PolyhedronData[pSolids[[n]],
    "VertexCoordinates"][[1;;4]]]},
 Boxed->False,SphericalRegion->True],{n,1,5,1},{th,0,2\[Pi]}]