Another projection of the 4-dimensional hypercube, this one close to vertex-centered. I love the shape of its hull: almost a rhombic dodecahedron, but skewed just enough to keep the central vertices from meeting.
The more usual projection is here, other polytopes are here.

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.

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This is the third, the hypercube or tesseract, in the classic projection into 3-space, showing its 8 cubic faces in a nice straightforward visualization.
A different projection is here.