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257-gon - Wikipedia

  • ️Mon Dec 21 2015

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In geometry, a 257-gon is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900°.

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.

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, along with a full construction showing all steps. One of these Carlyle circles solves the quadratic equation x2 + x − 64 = 0.[3]

  • Step 1

    Step 1

  • Step 2

    Step 2

  • Step 3

    Step 3

  • Step 4

    Step 4

  • Step 5

    Step 5

  • Step 6

    Step 6

  • Step 7

    Step 7

  • Step 8

    Step 8

  • Step 9

    Step 9

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°).

  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, ..." 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. JSTOR 2323939. Archived from the original (PDF) on 2015-12-21. Retrieved 6 November 2011.