The most familiar polychoron, to those who have heard of any of them, is the hypercube, or tesseract. It is analogous to the cube, but in four dimensions. All polychora are four-dimensional. With numbers of spatial dimensions above four, only the term “polytope” is used. Polyhedra are 3-polytopes, and polychora are 4-polytopes.
This is a three-dimensional projection of a tesseract, as it rotates in hyperspace, casting a “shadow” into our space:
In three dimensions, a cube is not the simplest polyhedron. A tetrahedron (a regular triangle-based pyramid) is simpler.
The simplest polychoron is composed of five tetrahedral cells, and is analogous to the tetrahedron, but in hyperspace. Here is a rotating “hypertetrahedron.”
There are even more names for these two polychora, based on the number of cells (cubes or tetrahedra). The tesseract/hypercube is composed of eight cubes, so it is called the 8-cell, as well as the octachoron. The preferred names for the hypertetrahedron are the 5-cell and the pentachoron, as it is composed of five (tetrahedral) cells.
Just as there are other Platonic solids not mentioned here, there are other regular polychora as well. The others will be subjects of upcoming posts, and one has already appeared here once (the 120-cell, or hyperdodecahedron), just three posts back.
Software note: these .gifs were made using Stella 4d, which may be purchased, and/or tried for free (on a trial basis), at http://www.software3d.com/Stella.php.