 # 257-gon

Regular 257-gon A regular 257-gon
TypeRegular polygon
Edges and vertices257
Schläfli symbol{257}
Coxeter diagram     Symmetry groupDihedral (D257), order 2×257
Internal angle (degrees)≈178.599°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

In geometry, a 257-gon (diacosipentacontaheptagon, diacosipentecontaheptagon) 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 t = edge length)

$A={\frac {257}{4}}t^{2}\cot {\frac {\pi }{257}}\approx 5255.751t^{2}.$ 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 $\cos {\frac {\pi }{257}}$ and $\cos {\frac {2\pi }{257}}$ 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 solves the quadratic equation x2 + x − 64 = 0. 257-gon, as a neusis construction for the first side, using the quadratrix of Hippias as an additional aid.
For the central angle $\Theta$ of the sector of a circle $OE_{257}E_{16}$ applies $\Theta =16\cdot {\frac {360^{\circ }}{257}}=22.41245...^{\circ },$ taking into account the center angle 90° of the quadrant is obtained:
${\frac {1}{\Theta }}\cdot 90^{\circ }={\frac {257\cdot 90^{\circ }}{16\cdot 360^{\circ }}}=4{\frac {1}{64}}=4.015625.$ For the length of the following segment is valid :${\overline {OM}}={\frac {\Theta }{90^{\circ }}}={\frac {16\cdot 360^{\circ }}{257\cdot 90^{\circ }}}={\frac {64}{257}}={\frac {1}{4.015625}}=0.249027\ldots$ [LE]
The decimal number $4.015625$ and the fraction ${\frac {1}{4.015625}}$ are constructed using the third intercept theorem. An animation with a description see here

### Symmetry

The regular 257-gon has Dih257 symmetry, order 514. Since 257 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z257, and Z1.

## 257-gram

A 257-gram is a 257-sided star polygon. As 257 is prime, there are 127 regular forms generated by Schläfli symbols {257/n} for all integers 2 ≤ n ≤ 128 as $\left\lfloor {\frac {257}{2}}\right\rfloor =128$ .

Below is a view of {257/128}, with 257 nearly radial edges, with its star vertex internal angles 180°/257 (~0.7°). 