How many species of regular 100-sided polygons are there?
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.
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$\begingroup$Why should it be less than $50$?
If you take steps of $57$, that's equivalent to taking steps of $-43$ and gives the same shape as taking steps of $43$.
And how would I make a formal generalization with demonstration for a $n$ sided polygon?
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.
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