Regular 257-gon
A regular 257-gon
Type	Regular polygon
Edges and vertices	257
Schläfli symbol	{257}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D257), order 2×257
Internal angle (degrees)	≈178.599°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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 (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]

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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°).