The mapping from the regular solids to $\mathbb{Z}_5$ given by the number of faces in the solid mod 5, interestingly, is a bijection. Any geometers or algebraists know if there is a significant reason for this? I mean, the fact that it is a possibility (i.e. that there are exactly 5 regular solids) is a well known proof, but why is this particular mapping bijective? Just seems too much of a coincidence.

Its been 40 years since I understood the proof in grad school, and it's long since gone from my memory - perhaps this fact is even buried in the proof? Any grad students fresh off understanding the proof?