257 is a Fermat prime, and the 257-gon is therefore a constructible polygon using compass and straightedge, as proved by Gauss. An illustration of the 257-gon is not included here, since its 257 segments so closely resemble a circle.

The values of cos(pi/257) and cos(2pi/257) are 128-degree algebraic numbers.

Richelot and Schwendenwein found constructions for the 257-gon in 1832 (Coxeter 1969). DeTemple (1991) gives a construction using 150 circles (24 of which are Carlyle circles) which has geometrography symbol 94S_1+47S_2+275C_1+0C_2+150C_3 and simplicity 566.

See also

65537-gon, Constructible Polygon, Fermat Prime, Heptadecagon, Pentagon, Trigonometry Angles

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Bachmann, P. Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie. Leipzig, Germany: Teubner, 1872.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.Klein, F. "The Construction of the Regular Polygon of 17 Sides." Part I, Ch. 4 in "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 24-41, 1980.Pascal, E. "Sulla costruzione del poligono regolare di 257 lati." Rendiconto dell Accad. della scienze fisiche e matemat. sezione della Soc. a reale di Napoli, Ser. 2 1, 33-39, 1887.Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964.Richelot, F. J. "De resolutione algebraica aequationis X^(257)=1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata." J. reine angew. Math. 9, 1-26, 146-161, 209-230, and 337-358, 1832.Strommer, J. "Konstruktion des regulären 257-Ecks mit Lineal und Streckenübertrager." Acta Math. Hungar. 70, 259-292, 1996.Trott, M. "cos(2pi/257) à la Gauss." Mathematica Educ. Res. 4, 31-36, 1995.

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Weisstein, Eric W. "257-gon." From MathWorld--A Wolfram Web Resource.

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