257-gon
In Geometry, 257-gon, also known broadly as the Dihectapentacontakaiheptagon, is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900°. Regular 257-gon The area of a regular 257-gon is (with ) A whole regular 257-gon is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 24 parts per million. Construction The regular 257-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 257 is a Fermat prime, being of the form 22n + 1 (in this case n = 3). Thus, the values and are 128-degree algebraic numbers, and like all constructible numbers they can be written using square roots and no higher-order roots. Although it was known to Gauss by 1801 that the regular 257-gon was constructible, the first explicit constructions of a regular 257-gon were given by Magnus Georg Paucker (1822) and Friedrich Julius Richelot (1832). Another method involves the use of 150 circles, 24 being Carlyle circles: this method is pictured below. One of these Carlyle circles