65537-gon

The area of a regular is (with ) :A = \frac{65537}{4} t^2 \cot \frac{\pi}{65537} The perimeter of a regular differs from that of the circumscribed circle by about 15 parts per billion. ==Construction==

The regular 65537-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 65,537 is a Fermat prime, being of the form 22n + 1 (in this case n = 4). Thus, the values \cos \frac{\pi}{65537} and \cos \frac{2\pi}{65537} are 32768-degree algebraic numbers, and like any constructible numbers, they can be written in terms of square roots and no higher-order roots. Although it was known to Carl Friedrich Gauss by 1801 that the regular 65537-gon was constructible, the first explicit construction of a regular 65537-gon was given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript. Another method involves the use of at most 1332 Carlyle circles, and the first stages of this method are pictured below. This method faces practical problems, as one of these Carlyle circles solves the quadratic equation x2 + x − 16384 = 0 (16384 being 214). ==Symmetry==

The regular 65537-gon has D65537 symmetry, order 131074. Since 65537 is a prime number, D65537 has 2 proper cyclic subgroups: Z65537, and Z2, which correspond to the subgroup generated by a rotation and a reflection respectively. ==65537-gram==

A 65537-gram is a 65,537-sided star polygon. As 65,537 is prime, there are 32,767 regular forms generated by Schläfli symbols \{{65537}/{n}\} for all integers 2 ≤ n ≤ 32768 as \left\lfloor \frac{65537}{2} \right\rfloor = 32768. ==See also==