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>.
tetris11 · 15d ago
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:
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.
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
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>.
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
Also see:
https://en.wikipedia.org/wiki/Apollonian_gasket
Computer screens can only display the 65535-gon, not even the 65536 or 65537-gon.