The 65537-gon is a constructible polygon using compass and straightedge
because 65537 is a Fermat prime , as proved by Gauss.
The 65537-gon has so many sides that it is, for all intents and purposes, indistinguishable
from a circle using any reasonable printing or display
methods.
The values algebraic
numbers of degree 32768.
Hermes spent 10 years on the construction of the 65537-gon at Königsberg around 1900. After the Second World War, his manuscripts were moved to the Mathematical Institute in Göttingen, where they can now be viewed (Coxeter 1969).
DeTemple (1991) notes that a geometric construction
can be done using 1332 or fewer Carlyle circles .
See also 257-gon ,
Constructible Polygon ,
Heptadecagon ,
Pentagon ,
Trigonometry Angles
Explore with Wolfram|Alpha
References Bold, B. Famous Problems of Geometry and How to Solve Them. Coxeter, H. S. M. Introduction
to Geometry, 2nd ed. DeTemple, D. W.
"Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions."
Amer. Math. Monthly 98 , 97-108, 1991. Dickson, L. E.
"Constructions with Ruler and Compasses; Regular Polygons." Ch. 8
in Monographs
on Topics of Modern Mathematics Relevant to the Elementary Field Dixon, R. Mathographics. Hermes, J. "Ueber die Teilung
des Kreises in 65537 gleiche Teile." Nachr. Königl. Gesellsch. Wissensch.
Göttingen, Math.-Phys. Klasse , pp. 170-186, 1894. History
of Mathematics Project. "Hermes's Suitcase of Göttingen." https://www.history-of-mathematics.org/artifacts/hermes-suitcase-of-gottingen . Trott,
M. "Mathematica Educ. Res. 4 , 31-36, 1995. Trott,
M. "The
Mathematica GuideBook for Symbolics. https://www.mathematicaguidebooks.org/ . Referenced
on Wolfram|Alpha 65537-gon
Cite this as:
Weisstein, Eric W. "65537-gon." From MathWorld https://mathworld.wolfram.com/65537-gon.html
Subject classifications
and 
are algebraic
numbers of degree 32768.
à la Gauss." Mathematica Educ. Res. 4, 31-36, 1995.Trott,
M. "
à la Gauss." §1.10.2 in