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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 [[compass and straightedge construction|constructed using a compass and an unmarked straightedge]]. This is because 257 is a [[Fermat prime]], being of the form 2<sup>2<sup>''n''</sup></sup> + 1 (in this case ''n'' = 3). Thus, the values <math>\cos \frac{\pi}{257}</math> and <math>\cos \frac{2\pi}{257}</math> are 128-degree [[algebraic number]]s, and like all [[constructible number]]s they can be written using [[square root]]s and no higher-order roots. |
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 [[compass and straightedge construction|constructed using a compass and an unmarked straightedge]]. This is because 257 is a [[Fermat prime]], being of the form 2<sup>2<sup>''n''</sup></sup> + 1 (in this case ''n'' = 3). Thus, the values <math>\cos \frac{\pi}{257}</math> and <math>\cos \frac{2\pi}{257}</math> are 128-degree [[algebraic number]]s, and like all [[constructible number]]s they can be written using [[square root]]s and no higher-order roots. |
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Although it was known to [[Carl Friedrich Gauss|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)<ref>{{cite journal |author=Magnus Georg Paucker |title=Das regelmäßige Zweyhundersiebenundfunfzig-Eck im Kreise. |language=German |journal=Jahresverhandlungen der Kurländischen Gesellschaft für Literatur und Kunst |volume=2 |year=1822 | pages=188|url=https://books.google.de/books?id=aUJRAAAAcAAJ&pg=PA188&dq=%22Das+regelm%C3%A4%C3%9Fige+Zweyhundertsiebenundfunfzig-Eck%22+188&hl=de&sa=X&ved=0ahUKEwjrqsHx58vJAhVHtBoKHTU4D80Q6AEIIjAB#v=onepage&q=%22Das%20regelm%C3%A4%C3%9Fige%20Zweyhundertsiebenundfunfzig-Eck%22%20188&f=false}} Retrieved 8. December 2015. </ref> and [[Friedrich Julius Richelot]] (1832).<ref>{{cite journal |author=Friedrich Julius Richelot |title=De resolutione algebraica aequationis x<sup>257</sup> = 1, ...|language=Latin |journal=Source: Journal für die reine und angewandte Mathematik |volume=9 |year=1832 | pages=1–26, 146–161, 209–230, 337–358 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0009&DMDID=DMDLOG_0004}} Retrieved 8. December 2015.</ref> Another method involves the use of 150 circles, 24 being [[Carlyle circle]]s: this method is pictured below. One of these Carlyle circles solves the [[quadratic equation]] ''x''<sup>2</sup> + ''x'' − 64 = 0.<ref name=DeTemple>{{cite journal|last=DeTemple|first=Duane W.|title=Carlyle circles and Lemoine simplicity of polygon constructions|journal=The American Mathematical Monthly|date=Feb 1991|volume=98|issue=2|pages= 97–208|url= |
Although it was known to [[Carl Friedrich Gauss|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)<ref>{{cite journal |author=Magnus Georg Paucker |title=Das regelmäßige Zweyhundersiebenundfunfzig-Eck im Kreise. |language=German |journal=Jahresverhandlungen der Kurländischen Gesellschaft für Literatur und Kunst |volume=2 |year=1822 | pages=188|url=https://books.google.de/books?id=aUJRAAAAcAAJ&pg=PA188&dq=%22Das+regelm%C3%A4%C3%9Fige+Zweyhundertsiebenundfunfzig-Eck%22+188&hl=de&sa=X&ved=0ahUKEwjrqsHx58vJAhVHtBoKHTU4D80Q6AEIIjAB#v=onepage&q=%22Das%20regelm%C3%A4%C3%9Fige%20Zweyhundertsiebenundfunfzig-Eck%22%20188&f=false}} Retrieved 8. December 2015. </ref> and [[Friedrich Julius Richelot]] (1832).<ref>{{cite journal |author=Friedrich Julius Richelot |title=De resolutione algebraica aequationis x<sup>257</sup> = 1, ...|language=Latin |journal=Source: Journal für die reine und angewandte Mathematik |volume=9 |year=1832 | pages=1–26, 146–161, 209–230, 337–358 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0009&DMDID=DMDLOG_0004}} Retrieved 8. December 2015.</ref> Another method involves the use of 150 circles, 24 being [[Carlyle circle]]s: this method is pictured below. One of these Carlyle circles solves the [[quadratic equation]] ''x''<sup>2</sup> + ''x'' − 64 = 0.<ref name=DeTemple>{{cite journal|last=DeTemple|first=Duane W.|title=Carlyle circles and Lemoine simplicity of polygon constructions|journal=The American Mathematical Monthly|date=Feb 1991|volume=98|issue=2|pages= 97–208|url=http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf#9|archiveurl=http://web.archive.org/web/20151221113614/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf|accessdate=6 November 2011|doi=10.2307/2323939|archive-date=2016-01-27|deadurl=yes}}</ref> |
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[[File:Regular 257-gon Using Carlyle Circle.gif]] |
[[File:Regular 257-gon Using Carlyle Circle.gif]] |
Revision as of 23:49, 26 January 2016
Regular 257-gon | |
---|---|
Type | Regular polygon |
Edges and vertices | 257 |
Schläfli symbol | {257} |
Coxeter–Dynkin diagrams | |
Symmetry group | Dihedral (D257), order 2×257 |
Internal angle (degrees) | ≈178.599° |
Properties | Convex, cyclic, equilateral, isogonal, isotoxal |
Dual polygon | Self |
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 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]
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°).
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.
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.
References
- ^ 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.
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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.