65537gon
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Regular 65537gon  

A regular 65537gon  
Type  Regular polygon 
Edges and vertices  65537 
Schläfli symbol  {65537} 
Coxeter diagram  
Symmetry group  Dihedral (D_{65537}), order 2×65537 
Internal angle (degrees)  ≈179.994 507° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a 65537gon is a polygon with 65,537 (2^{16} + 1) sides. The sum of the interior angles of any nonselfintersecting 65537gon is 11796300°.
Regular 65537gon
The area of a regular 65537gon is (with t = edge length)
A whole regular 65537gon is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 15 parts per billion.
Construction
The regular 65537gon (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 65,537 is a Fermat prime, being of the form 2^{2n} + 1 (in this case n = 4). Thus, the values and are 32768degree algebraic numbers, and like any constructible numbers, they can be written in terms of square roots and no higherorder roots.
Although it was known to Gauss by 1801 that the regular 65537gon was constructible, the first explicit construction of a regular 65537gon was given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200page manuscript.^{[1]} Another method involves the use of at most 1332 Carlyle circles, and the first stages of this method are pictured below. This method faces practical problems, as one of these Carlyle circles solves the quadratic equation x^{2} + x − 16384 = 0 (16384 being 2^{14}).^{[2]}
Symmetry
The regular 65537gon has Dih_{65537} symmetry, order 131074. Since 65,537 is a prime number there is one subgroup with dihedral symmetry: Dih_{1}, and 2 cyclic group symmetries: Z_{65537}, and Z_{1}.
65537gram
A 65537gram is a 65,537sided star polygon. As 65,537 is prime, there are 32,767 regular forms generated by Schläfli symbols {65537/n} for all integers 2 ≤ n ≤ 32768 as .
See also
 Equilateral triangle
 Pentagon
 Heptadecagon (17sides)
 257gon
References
 ^ Johann Gustav Hermes (1894). "Über die Teilung des Kreises in 65537 gleiche Teile". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, MathematischPhysikalische Klasse (in German). Göttingen. 3: 170–186.
 ^ 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 20151221. Retrieved 6 November 2011.
Bibliography
 Weisstein, Eric W. "65537gon". 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 9780486242972
 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.
External links
 65537gon mathematikolympiaden.de (German), with images of the documentation HERMES; retrieved on July 9, 2018
 Wikibooks 65573Eck (German) Approximate construction of the first side in two main steps
 65537gon, exact construction for the 1st side, using the Quadratrix of Hippias and GeoGebra as additional aids, with brief description (German)