Mathematician Sean Eberhard has a nice post about constructible regular polygons, giving a proof of a characterization of the n-sided polygons (aka n-gons) which are constructible only with a ruler and a compass. Those which are so constructible correspond to n being decomposable into a power of 2 and a product of primes of a certain form:

Theorem The regular n-gon is constructible by ruler and compass if and only if n has the form p_1 * . . . . * p_l * 2^k, where p_1, . . . , p_l are distinct primes of the form 2^{2^m} + 1.

That physical geometric actions should map to – and from – certain prime numbers is a good example of some of the deep interactions that exist between different parts of mathematics, interactions that often take us by surprise and usually compel our wonder.
One question that immediately occurs to me is whether there are other instruments besides ruler and compass which, jointly with those two instruments, would enable n-gon construction for other values of n.   Indeed, is there a collection of instruments (presumably some of them “non-constructible” or infinite in themselves) which would eventually garner all n, or at least other interesting subsets of the natural numbers?