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.
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 and simplicity
566.
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. Monthly98, 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. 21, 33-39, 1887.Rademacher, H. Lectures
on Elementary Number Theory. New York: Blaisdell, 1964.Richelot,
F. J. "De resolutione algebraica aequationis , 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. " à la Gauss."
Mathematica Educ. Res.4, 31-36, 1995.
and 
are 128-degree algebraic
numbers.
and simplicity
566.
, 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. "
à la Gauss."
Mathematica Educ. Res. 4, 31-36, 1995.