257-gon

Regular 257-gon

The area of a regular 257-gon is (with ) :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, along with a full construction showing all steps. One of these Carlyle circles solves the quadratic equation x2 + x − 64 = 0.

257-gon-step-1.png|Step 1 257-gon-step-2.png|Step 2 257-gon-step-3.png|Step 3 257-gon-step-4.png|Step 4 257-gon-step-5.png|Step 5 257-gon-step-6.png|Step 6 257-gon-step-7.png|Step 7 257-gon-step-8.png|Step 8 257-gon-step-9.png|Step 9

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°). :[[File:star polygon 257-128.svg|400px]]

References

References

1. Magnus Georg Paucker. (1822). "Das regelmäßige Zweyhundersiebenundfunfzig-Eck im Kreise.". Jahresverhandlungen der Kurländischen Gesellschaft für Literatur und Kunst.
2. Friedrich Julius Richelot. (1832). "De resolutione algebraica aequationis x²⁵⁷ = 1, ...". Journal für die reine und angewandte Mathematik.
3. DeTemple, Duane W.. (Feb 1991). "Carlyle circles and Lemoine simplicity of polygon constructions". The American Mathematical Monthly.