65537-gon

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)	≈179.99°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a 65537-gon is a polygon with 65537 sides. The sum of the interior angles of any non-self-intersecting 65537-gon is 23592600°.

Regular 65537-gon

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

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

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 65537 is a Fermat prime, being of the form 2²ⁿ + 1 (in this case n = 4). Thus, the values $\cos \frac{\pi}{65537}$ and $\cos \frac{2\pi}{65537}$ are of a 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. 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).

Symmetry

The regular 65537-gon has Dih₆₅₅₃₇ symmetry, order 131074. Since 65537 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 65537-sided star polygon. As 65537 is prime, there are 32767 regular forms generated by Schläfli symbols {65537/n} for all integers 2 ≤ n ≤ 32768 as $\left\lfloor \frac{65537}{2} \right\rfloor = 32768$.

Approximate construction of the first side of the regular 65537-gon

The exact construction of the 65537-gon is not feasible in reality. To illustrate, for example, a 65537-gon with side length of 1 mm, the radius of its circumference must be about 10 m.

Hereinafter the first side is shown as an approximate construction.

The construction is accomplished out in two steps. First, the 1024th corner (E₁₀₂₄) will be constructed. Subsequently, the first side ($\overline{AE_1}$) will be destined by magnifications of the relevant area and with ten times halving of the central angle.

Constructed side of 65537-gon (see magnification in animation) $\overline{AE_1} = a = 9.58723363103371...E-5 \; [unit\;of\;length]$

Side of the 65537-gon $a_{SHOULD} = 2\cdot\sin\left(\frac{180^\circ}{65537} \right) = 9.587233631037820...E-5\; [unit\;of\;length]$

Absolute error of the constructed side, $F_a = a - a_{SOLL} = -4.107...E-17 \; [unit\;of\;length]$

Example to illustrate the error: At a circumscribed circle R = 1 billion km (the light needed for this distance about 55 minutes), the absolute error of the 1st side would be approximately -0.04 mm (too short).

For details, see:

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-Eck mathematik-olympiaden.de (German), with images of the documentation HERMES; retrieved on April 25, 2016
• Wikibooks 65573-Eck (German) Approximate construction of the first side in two main steps