A four-dimensional cube at pendant size.  Is it adorable?  Why yes, it is. Some feel it is a little large to wear as jewelry, so I've uploaded a smaller one here.

More polytopes are here.

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 sixth, the hypericosahedron, with 600 tetrahedral cells.

This was the hardest of this group to make a printable model of. For a Schlegel diagram one would need quite a large size to allow the amount of interior complexity required, and it gets difficult to build as well as expensive, so I used this face-first projection suggested by Henry Cohn. Some of the tetrahedral are collapsed and become planar, but on the plus side the complexity is on the outside where you can see it!

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 bigger model is here.

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This is the second, analogous to the octahedron, called the cross polytope.

This is close to a vertex-first projection, but rotated a little so the central vertices don't quite overlap and you can see all 16 tetrahedral cells. The cross polytope is dual to the hypercube, so its 16 cells correspond to the 4-cube's 16 corners.

A different projection is here.

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This is the second, analogous to the octahedron, called the cross polytope.

The cross polytope is dual to the hypercube, so its 16 cells correspond to the 4-cube's 16 corners. This projection is a straightforward Schlegel diagram. A different projection is here.

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This is the first, the pentachoron or hyperpyramid, in a vertex-first projection. It has 5 tetrahedral cells, and like the tetrahedron is its own dual.

In every dimension there's one polytope like this: all triangles, self-dual, analogous to the tetrahedron. As a group they're called simplexes, so this is the 4-simplex.

This isn't one of the 4D "Platonic" solids, just a polytope that a friend of mine likes. It has icosahedral and tetrahedral cells.