I'm looking for "nice" tessellations of $S^3$ into as many pieces as possible. Another way to think about this problem is looking for "nice" 4-polytopes with as many faces as possible, since we can project those faces onto $S^3$ to get a tessellation.
One clear candidate is the 600-cell, since it is the regular 4-polytope with the greatest number of cells. I am curious whether we can find something with more cells that is about as nice. Tetrahedral cells is a big plus.
Here's one thing I thought of trying: take each tetrahedral cell of the 600-cell, then chop off all four corners cutting through the midpoints of the edges. This gives four tetrahedra plus an octahedron. We can then cut the octahedron in half twice to get four more tetrahedra. It turns out that all eight of the tetrahedra we obtain this way have the same volume, although they are different shapes. A major disadvantage of this construction is that there is asymmetry depending on which ways we cut the octahedron in half, but the advantage is that we tessellate $S^3$ into 600 * 8 = 4800 tetrahedral cells of equal volume.
What are some other polytopes I should check out? Also, anyone who is interested in this subject should check out this project that helps visualize 4-polytopes by projecting them onto $S^3$, as I'm talking about here.