What a coincidence! Just a few minutes ago, I finished reading Chapter 20 ("Regular Polygons") from the book <i>Galois Theory</i>, 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.<p>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.<p>The more general result can be stated as follows: The regular n-gon can be constructed by an unmarked ruler and a compass <i>if and only if</i> n has the form<p><pre><code> n = 2^r p_1 ... p_s
</code></pre>
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.<p>Remarkably, the only Fermat primes known so far are 3, 5, 17, 257, 65537. See also <<a href="https://oeis.org/A019434" rel="nofollow">https://oeis.org/A019434</a>>.
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.<p>You then join up the bounces to their neighbours to get the desired N(prime)-sided polygon.<p>The easier Heptadecagon (17-sided) was illuminating here:<p><a href="https://en.wikipedia.org/wiki/Heptadecagon" rel="nofollow">https://en.wikipedia.org/wiki/Heptadecagon</a>
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.<p>Also see:<p><a href="https://en.wikipedia.org/wiki/Apollonian_gasket" rel="nofollow">https://en.wikipedia.org/wiki/Apollonian_gasket</a><p>Computer screens can only display the 65535-gon, not even the 65536 or 65537-gon.