I should've written "regular tetrahedra". Jim On Monday, October 16, 2017, Warren D Smith <warren.wds@gmail.com> wrote:

On 10/16/17, James Propp <jamespropp@gmail.com <javascript:;>> wrote:

...

More specifically, it doesn't bode well for a nice answer for the tetrahedron. Tetrahedra don't tile space, so Warren's type of argument does not seem to be available.

Actually, you are wrong: Regular tets do not tile, but there exist nonregular tets that do tile (and in a lovely-face-to-face manner). The nicest of them is the "BCC tetrahedron" which arises in the Delaunay triangulation of the BCC point-lattice.