Regular Heptadecagon Inscribed in a Circle

For anyone who's wondering: why a 17-gon? the answer is an astonishing theorem: with the traditional Euclidean tools of ruler and compass, you can construct an n-sided polygon if and only if n is the product of a power of 2 (1,2,4,8,...) and some set of "Fermat primes": prime numbers of the form 2^2^n+1 (2,3,5,17,65537, and no one knows whether there are any more but there probably aren't). You aren't allowed to use any of the Fermat primes more than once.

So you can construct a regular heptadecagon but not a regular nonagon or a regular heptagon.

Why? Here's a super-handwavy sketch of the ideas involved. Points in the plane are complex numbers. Constructing a regular n-gon is like constructing z = exp(2 pi i / n). That satisfies the equation z^n=1. We only need to consider prime-power values of n, because if m,n have no common factor then you can do mn if and only if you can do m and n. (Proof left as an exercise for the reader.) For such n, it's not hard to figure out the minimal polynomial in z whose value is 0; when n is prime, e.g., it's z^(n-1) + z^(n-2) + ... + z + 1.

Now, imagine doing any ruler-and-compass construction you like. Start with two points, which we'll call 0 and 1. Then every individual construction you can do involves solving either a linear or a quadratic equation. Conversely, you can solve any quadratic equation by doing ruler-and-compass constructions. (Details again left as an exercise; or look it up.)

Now think about those "minimal polynomials" for all the numbers you construct along the way. The degree of such a polynomial (i.e., the highest exponent it contains) is an important quantity. It turns out (super-handwavy, again) that when you solve a quadratic equation with coefficients whose minimal polynomials have degree d, you get something whose minimal polynomial has degree d or 2d. So at any stage in your ruler-and-compass construction, all the degrees are powers of 2. In particular, you can never construct anything whose minimal polynomial has degree that isn't a power of 2. And that is enough to tell you that when n is a power of a prime number, you can't construct a regular n-gon unless n is a power of 2 or a Fermat prime.

The other direction is conceptually easier but fiddlier, and I shan't try to explain it here.