Sestrigintagons

A sestrigintagon is a polygon with 63 sides and 63 vertices. The term is derived from Latin, with "sestri" meaning six and "ginta" meaning ten, combined with "gon" meaning angle or corner. As with any polygon, a sestrigintagon can be either convex or concave. In a convex sestrigintagon, all interior angles are less than 180 degrees, and all vertices point outwards. In a concave sestrigintagon, at least one interior angle is greater than 180 degrees, meaning at least one vertex points inwards.

The sum of the interior angles of any sestrigintagon can be calculated using the formula (n-2) * 180 degrees, where n is the number of sides. For a sestrigintagon, this sum is (63-2) * 180 = 61 * 180 = 10980 degrees. If the sestrigintagon is regular, meaning all its sides and angles are equal, then each interior angle measures 10980 / 63 degrees, which is approximately 174.29 degrees.

A regular sestrigintagon has 63 equal sides and 63 equal interior angles. Like all regular polygons with an odd number of sides, it cannot be constructed using only a compass and straightedge. This is a consequence of the Gauss-Wantzel theorem, which states that a regular n-gon can be constructed if and only if the odd prime factors of n are distinct Fermat primes. Since 63 has prime factors 3, 3, and 7, and 3 is not a Fermat prime, a regular sestrigintagon is not constructible.