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°.
Contents
Regular 257-gon[edit]
The area of a regular 257-gon is (with t = edge length)
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 
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 (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[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.
257-gram[edit]
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 
.
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[edit]
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.
AME1 = 1.40077828746899...° ; 360° ÷ 257 = 1.40077821011673...° ; AME1 - 360° ÷ 257 = 7.73...E-8°
Example to illustrate the error: At a circumscribed circle r = 1000 km (air-line distance London - Munich ≈ 918 km), the absolute error of the 1st side would be approximately 1.35 mm.
For details, see: 
References[edit]
- ^ 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.
- ^ 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.
- ^ 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[edit]
- 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.
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