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 edit

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 edit

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.

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°).

• 17-gon

• Weisstein, Eric W. "257-gon". MathWorld.
• Robert Dixon Mathographics. New York: Dover, p. 53, 1991.
• Benjamin Bold, Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982. ISBN 978-0486242972
• H. S. M. Coxeter Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Chapter 2, Regular polygons
• Leonard Eugene Dickson Constructions with Ruler and Compasses; Regular Polygons. Ch. 8 in Monographs on Topics of Modern Mathematics *Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352–386, 1955.
• 257-gon, exact construction the 1st side using the quadratrix according of Hippias as an additional aid (German)