I saw a resolution that showed the general case starting at $ n = 8 $. It has been found that the polygon species should be prime numbers relative to $ 100 $ and less than $ 50 $. Why should it be less than $ 50 $? And how would I make a formal generalization with demonstration for a $ n $ sided polygon?
Point: The definition of regular polygon really only takes into account equal sides and equal internal angles, does not consider whether or not there is any concavity.
If you take steps of $57$, that's equivalent to taking steps of $-43$ and gives the same shape as taking steps of $43$.
The coprimality is easy: show that if you take steps of $s$ then the number of vertices is $\frac n{\gcd(n, s)}$
The choice of only one step from $\{s, n-s\}$ is easy: generalise my example above.
The slightly tricky part is showing that these conditions are sufficient. Perhaps the best approach is to calculate the angle as a function of $s$ and $n$ and use known properties of trig functions to show that there are no coincidences.