65537gon

The 65537gon is a regular polygon with 65,537 sides. It is famously known as the Fermat polygon because its number of sides, 65,537, is a Fermat number. Specifically, 65,537 is the fifth Fermat number, denoted as F_4 = 2^(2^4) + 1.

The constructibility of a regular n-gon using only a compass and straightedge is determined by the prime factorization of n. A regular n-gon can be constructed if and only if n is the product of a power of 2 and distinct Fermat primes. The known Fermat primes are 3, 5, 17, 257, and 65537.

In 1801, Carl Friedrich Gauss proved that a regular n-gon is constructible if n is a product of distinct Fermat primes and a power of 2. Since 65537 is a prime number and a Fermat prime, the regular 65537gon is constructible. This was a significant theoretical breakthrough, although the actual geometric construction is extremely complex and impractical to perform by hand.

The construction of the 65537gon was first achieved by the German mathematician Johann Gustav Hermes in 1894. His work was a testament to the possibility, though the sheer number of steps and intricate geometry involved make it an abstract rather than a practical geometric feat. The existence of constructible polygons with such a large number of sides highlights the power of number theory in solving geometric problems.