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65537-gon


65537 is the largest known Fermat prime, and the 65537-gon is therefore a constructible polygon using compass and straightedge, 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 cos(pi/65537) and cos(2pi/65537) are 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

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References

Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.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 (Ed. J. W. A. Young). New York: Dover, pp. 352-386, 1955.Dixon, R. Mathographics. New York: Dover, p. 53, 1991.Hermes, J. "Ueber die Teilung des Kreises in 65537 gleiche Teile." Nachr. Königl. Gesellsch. Wissensch. Göttingen, Math.-Phys. Klasse, pp. 170-186, 1894.Trott, M. "cos(2pi/257) à la Gauss." Mathematica Educ. Res. 4, 31-36, 1995.Trott, M. "cos(2pi/257) à la Gauss." §1.10.2 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, pp. 312-321, 2006. http://www.mathematicaguidebooks.org/.

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65537-gon

Cite this as:

Weisstein, Eric W. "65537-gon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/65537-gon.html

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