There are six regular, convex four-dimensional polytopes. Five of them correspond on a 1:1 basis with the Platonic solids (as the tesseract corresponds to the cube), leaving one four-dimensional polytope without a three-dimensional analogue among the Platonics. That polychoron is the icositetrachoron, also named the 24-cell and made of 24 octahedral cells. It also happens to be self-dual.
If, in hyperspace, the corners are cut off just right, new cells are created with 24 cubic cells created at the corners, with 24 truncated octahedral cells remaining from the original polychoron. This is the truncated icositetrachoron:
4-dimensional polytopes have 3-dimensional nets. These nets are shown below — first for the icositetrachoron, and then for the truncated icositetrachoron.
In three dimensions, there are five regular, convex polyhedra. Similarly, in five dimensions, there are five regular, convex polytopes. There are also five of them in six, seven, eight dimensions . . . and so on, for as long as care to venture into higher-dimensional realms. However, in hyperspace — that is, four dimensions — there are, strangely, six.
The five Platonic solids have analogs among these six convex polychora, and then there’s one left over — the oddball among the regular, convex polychora. It’s the figure you see above, rotating in hyperspace: the 24-cell, also known as the icositetrachoron. Its twenty-four cells are octahedra.
Like the simplest regular convex polychoron, the 5-cell (analogous to the tetrahedron), the 24-cell is self-dual. No matter how many dimensions you are dealing with, it is always possible to make a compound of any polytope and its dual. Here, then, is the compound of two 24-cells (which may be enlarged by clicking on it):
Both of these moving pictures were generated using software called Stella 4d: Polyhedron Navigator. You can buy it, or try a free trial version, right here: www.software3d.com/Stella.php.