View source for 65537-gon

{{Short description|Shape with 65537 sides}}
{{Regular polygon db|Regular polygon stat table|p65537}}

In [[geometry]], a '''65537-gon'''<!--please don't add a written-out name in Greek prefixes, since there is actually no generally agreed system in English--> is a [[polygon]] with [[65,537]] (2<sup>16</sup> + 1) sides. The sum of the interior angles of any non–[[list of self-intersecting polygons|self-intersecting]] {{nobreak|65537-gon}} is 11796300°.

==Regular 65537-gon==
The area of a ''regular {{nobreak|65537-gon}}'' is (with {{nowrap|''t'' {{=}} edge length}})
:<math>A = \frac{65537}{4} t^2 \cot \frac{\pi}{65537}</math>

A whole [[regular polygon|regular]] {{nobreak|65537-gon}} 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 65537-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 65,537 is a [[Fermat prime]], being of the form 2<sup>2<sup>''n''</sup></sup>&nbsp;+&nbsp;1 (in this case ''n''&nbsp;=&nbsp;4). 
Thus, the values <math>\cos \frac{\pi}{65537}</math> and <math>\cos \frac{2\pi}{65537}</math> are 32768-[[Degree of a polynomial|degree]] [[algebraic number]]s, and like any [[constructible number]]s, they can be written in terms of [[square root]]s and no higher-order roots.

Although it was known to [[Carl Friedrich Gauss]] by 1801 that the regular 65537-gon was constructible, the first explicit construction of a regular 65537-gon was given by [[Johann Gustav Hermes]] (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript.<ref>{{cite journal | author=Johann Gustav Hermes |title=Über die Teilung des Kreises in 65537 gleiche Teile |language=de |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse | location=Göttingen | year=1894 |volume=3 |pages=170–186 |url=http://www.digizeitschriften.de/resolveppn/GDZPPN002496585}}</ref> Another method involves the use of at most 1332 [[Carlyle circle]]s, 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''<sup>2</sup>&nbsp;+&nbsp;''x''&nbsp;&minus;&nbsp;16384&nbsp;=&nbsp;0 (16384 being 2<sup>14</sup>).<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|archive-url=https://web.archive.org/web/20151221113614/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf#3 |access-date=6 November 2011|doi=10.2307/2323939|jstor=2323939|archive-date=2015-12-21 |url-status=dead}}</ref>

[[File:Regular 65537-gon First Carlyle Circle.gif|center]]

==Symmetry==
The ''regular 65537-gon'' has [[dihedral symmetry|D<sub>65537</sub> symmetry]], order 131074. Since 65537 is a [[prime number]] there is one subgroup with dihedral symmetry: D<sub>1</sub>, and 2 [[cyclic group]] symmetries: Z<sub>65537</sub>, and Z<sub>1</sub>.

==65537-gram==
A 65537-gram is a 65,537-sided [[star polygon]]. As 65,537 is prime, there are 32,767 regular forms generated by [[Schläfli symbol]]s {65537/''n''} for all [[integer]]s 2&nbsp;≤&nbsp;''n''&nbsp;≤&nbsp;32768 as <math>\left\lfloor \frac{65537}{2} \right\rfloor = 32768</math>.

==See also==

* [[Circle]]
* [[Equilateral triangle]]
* [[Pentagon]]
* [[Heptadecagon]] (17-sides)
* [[257-gon]]

==References==
{{reflist}}

==Bibliography==
*{{MathWorld |urlname=65537-gon |title=65537-gon}}
*[[Robert Dixon (mathematician)|Robert Dixon]] ''Mathographics''. New York: Dover, p.&nbsp;53, 1991.
*Benjamin Bold, ''Famous Problems of Geometry and How to Solve Them'' New York: Dover, p.&nbsp;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.&nbsp;352–386, 1955.

==External links==
*[https://www.mathematik-olympiaden.de/moev/moev_material/Konstruktion17/index.html 65537-gon] {{Webarchive|url=https://web.archive.org/web/20230105101922/https://www.mathematik-olympiaden.de/moev/moev_material/Konstruktion17/index.html |date=2023-01-05 }} mathematik-olympiaden.de (German), with images of the documentation HERMES; retrieved on July 9, 2018
*[https://de.wikibooks.org/wiki/Mathematik:_Schulmathematik:_Planimetrie:_Polygonkonstruktionen:_65537-Eck Wikibooks 65537-Eck (German) Approximate construction of the first side in two main steps]
*[[:File:01-65.537-Eck-Quadratrix.svg|65537-gon, exact construction for the 1st side]], using the [[Quadratrix of Hippias]] and [[GeoGebra]] as additional aids, with brief description (German)
{{Polygons}}

[[Category:Constructible polygons]]
[[Category:Polygons by the number of sides]]
[[Category:Euclidean plane geometry]]
[[Category:Carl Friedrich Gauss]]