The Icosidodecahedron

The icosidodecahedron has 30 vertices, one at the center of each edge of a regular icosahedron -- or equivalently, one at the center of each edge of a regular dodecahedron. It is a beautiful, highly symmetrical shape. But it is just a projection down to 3d space of a more symmetrical 6-dimensional polytope with 60 vertices. It is also a slice of a more symmetrical 4d polytope with 120 vertices, which in turn is the projection down to 4d space of an even more symmetrical 8-dimensional polytope with 240 vertices: the E8 root polytope. Here we explain all these constructions, and their connection to the quaternions and icosians.