There are six regular convex polytopes in 4D, which are analogous to
the five Platonic solids in 3D. This one is the odd polytope out, the one without a 3D counterpart.
It has 24 octahedral cells, all shown in this Schlegel diagram. Like the pentachoron
it's self-dual -- the only self-dual solid in any dimension > 2 that is not a simplex. And if that wasn't enough, it's also the only regular convex polytope in any dimension > 2 that tiles its space and is not a hypercube.