65537-gon

In geometry, a 65537-gon is a polygon with 65,537 (2¹⁶ + 1) sides. The sum of the interior angles of any non-self-intersecting 65537-gon is 11796300°.

Regular 65537-gon
A regular 65537-gon
Type	Regular polygon
Edges and vertices	65537
Schläfli symbol	{65537}
Coxeter diagram
Symmetry group	Dihedral (D65537), order 2×65537
Internal angle (degrees)	≈179.994 507°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

Regular 65537-gon

The area of a regular 65537-gon is (with t = edge length)

${\displaystyle A={\frac {65537}{4}}t^{2}\cot {\frac {\pi }{65537}}}$

A whole regular 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 2²ⁿ + 1 (in this case n = 4). Thus, the values ${\displaystyle \cos {\frac {\pi }{65537}}}$ and ${\displaystyle \cos {\frac {2\pi }{65537}}}$ are 32768-degree algebraic numbers, and like any constructible numbers, they can be written in terms of square roots and no higher-order roots.

Although it was known to 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.[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 x² + x − 16384 = 0 (16384 being 2¹⁴).[2]

Symmetry

The regular 65537-gon has Dih₆₅₅₃₇ symmetry, order 131074. Since 65,537 is a prime number there is one subgroup with dihedral symmetry: Dih₁, and 2 cyclic group symmetries: Z₆₅₅₃₇, and Z₁.

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 symbols {65537/n} for all integers 2 ≤ n ≤ 32768 as ${\displaystyle \left\lfloor {\frac {65537}{2}}\right\rfloor =32768}$.

See also

• Equilateral triangle
• Pentagon
• Heptadecagon (17-sides)
• 257-gon

References

1. Johann Gustav Hermes (1894). "Über die Teilung des Kreises in 65537 gleiche Teile". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German). Göttingen. 3: 170–186.
2. 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 2015-12-21. Retrieved 6 November 2011.

Bibliography

• Weisstein, Eric W. "65537-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.

External links

• 65537-gon mathematik-olympiaden.de (German), with images of the documentation HERMES; retrieved on July 9, 2018
• Wikibooks 65573-Eck (German) Approximate construction of the first side in two main steps
• 65537-gon, exact construction for the 1st side, using the Quadratrix of Hippias and GeoGebra as additional aids, with brief description (German)