From Notices of the AMS
The Icosidodecahedron
by John C. Baez
The icosidodecahedron can be built by truncating either a regular icosahedron or a regular dodecahedron. It has 30 vertices, one at the center of each edge of the icosahedron---or equivalently, one at the center of each edge of a dodecahedron. It is a beautiful, highly symmetrical shape.
But the icosidodecahedron is just a shadow of a more symmetrical shape with twice as many vertices, which lives in a space with twice as many dimensions! Namely, it is a projection down to 3-dimensional space of a 6-dimensional polytope with 60 vertices. Even better, it is also a slice of a still more symmetrical 4-dimensional polytope with 120 vertices, which in turn is the projection down to 4-dimensional space of a even more symmetrical 8-dimensional polytope with 240 vertices. Note how the numbers keep doubling: 30, 60, 120, and 240.
To understand all this, start with the group of rotational symmetries of the icosahedron. This is a 60-element subgroup of the rotation group SO(3).
