TITLE: Tessellating the sphere
QUESTION [5 upvotes]: It is a famous result that the plane can be tessellated by regular triangles, squares, and hexagons.
Which regular polygons can tessellate the sphere?

REPLY [2 votes]: If the following restrictions are satisfied:

All polygons are regular
Only one type of polygon is permitted and all instances are congruent
The entire surface of the sphere is tessellated exactly once

then I believe the only tessellations that are admitted are precisely the Platonic solids (projected onto their circumscribing sphere) and no others.
If only condition (2) is relaxed to allow distinct but finitely many polygons, then this allows Archimedean solids and the regular prisms and antiprisms, but it may also allow other non-polyhedral solutions--I'm not sure.
If only condition (1) is relaxed, then Archimedean duals will work as well as the prism and antiprism duals.  Again, there may be additional solutions.
If conditions (2) and (3) are relaxed, then this can admit the projections of non-convex uniform polyhedra.

Indeed, there is a case that I had not previously considered, that satisfies all three conditions stated above.  Consider the projection of a regular octahedron onto its circumscribing sphere.  This projection divides the sphere along three mutually intersecting great circles.  For any such great circle, rotate the hemispheres on either side by $\pi/4$ relative to each other.  The result is a (distinct) tessellation of the sphere by equilateral triangles, but it does not correspond to the projection of a polyhedron.