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Several years ago, on a flight of fancy when I discovered that our actual universe is a metric four-sphere, I had asked about the tesselations or regular arrangements possible.  The only finite group that can act freely on $S^4$ is $\mathbf{Z}/2$ so the the regular arragements of $S^3$ doubled is the soft and useless answer but it turns out that this is a highly nontrivial solved problem surveyed in this paper:  Spherical_space_forms
A nice explicit calculation for discrete subgroups of $SO(3)$ is provided here: disc3d