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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). De Temple (1991) gives a construction using 150 Circles (24 of which are
Carlyle Circles) which has Geometrography symbol
and
Simplicity 566.
See also 65537-gon, Constructible Polygon, Fermat Prime, Heptadecagon, Pentagon
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.
De Temple, D. W. ``Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions.'' Amer. Math. Monthly
98, 97-108, 1991.
Dixon, R. Mathographics. New York: Dover, p. 53, 1991.
Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964.