257-gon

Regular 257-gon
A regular 257-gon
Type	Regular polygon
Edges and vertices	257
Schläfli symbol	{257}
Coxeter diagram
Symmetry group	Dihedral (D257), order 2×257
Internal angle (degrees)	≈178.60°
Dual polygon	self
Properties	convex, 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 91800°.

Regular 257-gon

The area of a regular 257-gon is (with t = edge length)

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

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)^[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]

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°).

 

Approximate construction of the first side of the regular 257-gon

Since the exact construction of the 257-gon is very extensive and can not be clearly displayed, hereinafter the first side is shown as an approximate construction.

$\scriptstyle\angle{}$ BME₁ = 1.400778210117941...° ; 360° ÷ 257 = 1.400778210116731...° ; $\scriptstyle\angle{}$ BME₁ - 360° ÷ 257 = 1.209...E-12°

Example to illustrate the error: At a circumscribed circle R = 1 billion km (the light needed for this distance about 55 minutes), the absolute error of the 1st side would be approximately 21 mm.

For details, see:

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, ...". Source: 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): 97–208. doi:10.2307/2323939. Archived from the original (PDF) on 2016-01-27. Retrieved 6 November 2011.

External links

• 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.
• Wikibooks 257-Eck (German) Approximate construction of the first side