There are six regular convex polytopes
in 4D, which are analogous to
the five Platonic solids in 3D. This is the fifth, the
hyperdodecahedron, a remarkably beautiful object brought to my
attention by George Hart.
Here it's shown in a Schlegel diagram so you can see all 120
dodecahedral cells, though most are transformed by perspective: in
this projection, the only regular dodecahedra are the biggest one on
the outside and the tiniest one at the center.
A smaller model is here