In geometry, a 257-gon is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900°.
Regular 257-gon The area of a regular 257-gon is (with )
A whole regular 257-gon is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 24 parts per million.
Construction The regular 257-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 257 is a Fermat prime, being of the form 22n + 1 (in this case n = 3). Thus, the values and are 128-degree algebraic numbers, and like all constructible numbers they can be written using square roots and no higher-order roots.
Although it was known to Gauss by 1801 that the regular 257-gon was constructible, the first explicit constructions of a regular 257-gon were given by Magnus Georg Paucker (1822) Retrieved 8. December 2015. and Friedrich Julius Richelot (1832). Retrieved 8. December 2015. Another method involves the use of 150 circles, 24 being Carlyle circles: this method is pictured below, along with a full construction showing all steps. One of these Carlyle circles solves the quadratic equation x2 + x − 64 = 0.
Symmetry The regular 257-gon has Dih257 symmetry, order 514. Since 257 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z257, and Z1.
257-gram A 257-gram is a 257-sided star polygon. As 257 is prime, there are 127 regular forms generated by Schläfli symbols {257/n} for all integers 2 ≤ n ≤ 128 as .
Below is a view of {257/128}, with 257 nearly radial edges, with its star vertex internal angles 180°/257 (~0.7°).
See also 17-gon
References
External links
Robert Dixon Mathographics. New York: Dover, p. 53, 1991. Benjamin Bold, Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982. H. S. M. Coxeter Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Chapter 2, Regular polygons Leonard Eugene Dickson 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. 257-gon, exact construction the 1st side using the quadratrix according of Hippias as an additional aid (German)