# 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)

${\displaystyle A={\frac {257}{4}}t^{2}\cot {\frac {\pi }{257}}\approx 5255.751t^{2}.}$

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 ${\displaystyle \cos {\frac {\pi }{257}}}$ and ${\displaystyle \cos {\frac {2\pi }{257}}}$ 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)[1] and Friedrich Julius Richelot (1832).[2] 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.[3]

257-gon, as a neusis construction for the first side, using the quadratrix of Hippias as an additional aid.
For the central angle ${\displaystyle \Theta }$ of the sector of a circle ${\displaystyle OE_{257}E_{16}}$ applies ${\displaystyle \Theta =16\cdot {\frac {360^{\circ }}{257}}=22.41245...^{\circ },}$
taking into account the center angle 90° of the quadrant is obtained:
${\displaystyle {\frac {1}{\Theta }}\cdot 90^{\circ }={\frac {257\cdot 90^{\circ }}{16\cdot 360^{\circ }}}=4{\frac {1}{64}}=4.015625.}$
For the length of the following segment is valid :${\displaystyle {\overline {OM}}={\frac {\Theta }{90^{\circ }}}={\frac {16\cdot 360^{\circ }}{257\cdot 90^{\circ }}}={\frac {64}{257}}={\frac {1}{4.015625}}=0.249027\ldots }$ [LE]
The decimal number ${\displaystyle 4.015625}$ and the fraction ${\displaystyle {\frac {1}{4.015625}}}$ 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 ${\displaystyle \left\lfloor {\frac {257}{2}}\right\rfloor =128}$.

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

## References

1. ^ Magnus Georg Paucker (1822). "Das regelmäßige Zweyhundersiebenundfunfzig-Eck im Kreise". Jahresverhandlungen der Kurländischen Gesellschaft für Literatur und Kunst (in German). 2: 188. Retrieved 8. December 2015.
2. ^ Friedrich Julius Richelot (1832). "De resolutione algebraica aequationis x257 = 1, ..." Journal für die reine und angewandte Mathematik (in Latin). 9: 1–26, 146–161, 209–230, 337–358. Retrieved 8. December 2015.
3. ^ DeTemple, Duane W. (Feb 1991). "Carlyle circles and Lemoine simplicity of polygon constructions" (PDF). The American Mathematical Monthly. 98 (2): 97–108. doi:10.2307/2323939. Archived from the original (PDF) on 2015-12-21. Retrieved 6 November 2011.