257-gon


In geometry, a 257-gon 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
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 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. 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 and Friedrich Julius Richelot. 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.
for the first side, using the quadratrix of Hippias as an additional aid.
For the central angle of the sector of a circle applies
taking into account the center angle 90° of the quadrant is obtained:
For the length of the following segment is valid :
The decimal number and the fraction are :de:Konstruktion mit Zirkel und Lineal#Algebraische Operationen|constructed using the third intercept theorem. An animation with a description

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 for all integers 2 ≤ n ≤ 128 as.
Below is a view of, with 257 nearly radial edges, with its star vertex internal angles 180°/257.