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.
The algebraic construction groups the nontrivial 257th roots of unity into Gaussian periods. Starting from
their sum ,
seven successive halvings give periods containing 128, 64, 32, 16, 8, 4, and 2 terms.
Each splitting requires a quadratic equation
over the field already obtained. The final two-term periods are , giving the vertex coordinates through a sequence
of quadratic extensions (Gottlieb 1999).
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.Gottlieb, C. "The Simple and
Straightforward Construction of the Regular 257-gon." Math. Intelligencer21,
31-37, 1999. https://doi.org/10.1007/BF03024829.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.Trott, M. " à la Gauss."
§1.10.2 in The
Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5.
https://www.mathematicaguidebooks.org/.
and 
are 128-degree algebraic
numbers.
and simplicity
566.
,
seven successive halvings give periods containing 128, 64, 32, 16, 8, 4, and 2 terms.
Each splitting requires a quadratic equation
over the field already obtained. The final two-term periods are 
, giving the vertex coordinates through a sequence
of quadratic extensions (Gottlieb 1999).
, 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.Trott, M. "
à la Gauss."
§1.10.2 in