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?
Following on comments below, more generally in $\mathbb{R}^n$, if there are $k$ regular polytopes, and we look at the mapping from these regular polytopes to $\mathbb{Z}_k$ given by $f|k$ ($f$ modulo $k$) where $f$ is the number of faces in the polytope, then (if comments below are correct), the mapping is a bijection iff $n=3|6$ or $n=5|6$. And so the question is, is there a deep reason for this?
