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) ≈178.599°
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 45,900°.

Regular 257-gon

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

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]

257-gon, as a neusis construction for the first side, using the quadratrix of Hippias as an additional aid.
For the central angle of the sector of a circle applies
taking into account the center angle 90° of the quadrant is obtained:

For the length of the following segment is valid : [LE]
The decimal number and the fraction are constructed using the third intercept theorem. An animation with a description see here

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 .

Below is a view of {257/128}, with 257 nearly radial edges, with its star vertex internal angles 180°/257 (~0.7°).

 

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): 97–108. doi:10.2307/2323939. Archived from the original (PDF) on 2015-12-21. Retrieved 6 November 2011.
  • 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)
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