65537-gon

What a coincidence! Just a few minutes ago, I finished reading Chapter 20 ("Regular Polygons") from the book Galois Theory, 5th ed. by Ian Stewart. It presents a rigorous proof of why the regular 65537-gon can be constructed with an unmarked ruler and compass. In fact, the book proves a more general result.

Firstly, the regular 65537-gon can be constructed using only an unmarked ruler and compass because 65537 is a Fermat prime, i.e., a prime of the form, 2^(2^r) + 1, where r is a non-negative integer. Indeed 65536 = 2^(2^4) + 1.

The more general result can be stated as follows: The regular n-gon can be constructed by an unmarked ruler and a compass if and only if n has the form

  n = 2^r p_1 ... p_s

where the integers r, s >= 0 and p_1, ..., p_s are distinct Fermat primes. As a result, if n is a Fermat prime, the regular n-gon is constructible. This also explains why, for example, the regular 7-gon, 9-gon, etc. are not constructible by unmarked ruler and compass.

Remarkably, the only Fermat primes known so far are 3, 5, 17, 257, 65537. See also <https://oeis.org/A019434>.

So the general idea is to find with compass and pencil the exact needed separation angle that can be used to bounce around the interior circumference of a circle with perfect overlap.

You then join up the bounces to their neighbours to get the desired N(prime)-sided polygon.

The easier Heptadecagon (17-sided) was illuminating here:

https://en.wikipedia.org/wiki/Heptadecagon

Do circles exist, or are they just spirals? In space, they’re all spiral galaxies - the only sphere is the observable universe based on the speed of light in every direction.

Also see:

https://en.wikipedia.org/wiki/Apollonian_gasket

Computer screens can only display the 65535-gon, not even the 65536 or 65537-gon.