| Regular 257-gon | |
|---|---|
 A regular 257-gon | |
| Type | Regular polygon |
| Edges andvertices | 257 |
| Schläfli symbol | {257} |
| Coxeter–Dynkin diagrams |      |
| Symmetry group | Dihedral (D257), order 2×257 |
| Internal angle (degrees) | 178.599222° |
| Properties | Convex,cyclic,equilateral,isogonal,isotoxal |
| Dual polygon | Self |
Ingeometry, a257-gon is apolygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900°.
The area of aregular 257-gon is (witht = edge length)
A whole regular 257-gon is not visually discernible from acircle, and its perimeter differs from that of thecircumscribed circle by about 24parts per million.
The regular 257-gon (one with all sides equal and all angles equal) is of interest for being aconstructible polygon: that is, it can beconstructed using a compass and an unmarked straightedge. This is because 257 is aFermat prime, being of the form 22n + 1 (in this casen = 3). Thus, the values and are 128-degreealgebraic numbers, and like allconstructible numbers they can be written usingsquare roots and no higher-order roots.
Although it was known toGauss by 1801 that the regular 257-gon was constructible, the first explicit constructions of a regular 257-gon were given byMagnus Georg Paucker (1822)[1] andFriedrich Julius Richelot (1832).[2] Another method involves the use of 150 circles, 24 beingCarlyle circles: this method is pictured below, along with a full construction showing all steps. One of these Carlyle circles solves thequadratic equationx2 + x − 64 = 0.[3]
Theregular 257-gon hasDih257 symmetry, order 514. Since 257 is aprime number there is one subgroup with dihedral symmetry: Dih1, and 2cyclic group symmetries: Z257, and Z1.
A 257-gram is a 257-sidedstar polygon. As 257 is prime, there are 127 regular forms generated bySchläfli symbols {257/n} for allintegers 2 ≤ n ≤ 128 as.
Below is a view of {257/128}, with 257 nearly radial edges, with its star vertexinternal angles 180°/257 (~0.7°).
