Tessellations of 3-sphere

Question

How many are there regular geodesic tessellations of the 3-sphere? What kind polyhedrons are used in those?

Answer 1

The regular tessellations of an n-sphere are equivalent to the regular polytopes in n+1 dimensions. There is a precise one-to-one correspondence which may be obtained via central projection of the polytope onto a concentric sphere. Indeed, such spherical tessellations are often called spherical polytopes.

Some tessellations wrap around, or cover, the sphere multiple times. These are equivalent to star polytopes, and regular examples of these exist. So you need to be clear whether you are interested in these or wish to stick with single coverings, which are equivalent to convex polytopes.

In the case of the 3-sphere, there are six regular convex 4-polytopes (also called polycells or polychora) and ten star ones.

By definition, all the faces of a regular n-polytope are regular (n−1) polytopes. The various regular polychora have faces which are regular polyhedra. All are represented in one regular polychoron or another.

The link to the relevant Wikipedia article has already been posted in a comment but it bears repeating here.