257-gon

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
A regular 257-gon
Type	Regular polygon
Edges and vertices	257
Schläfli symbol	{257}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D257), order 2×257
Internal angle (degrees)	≈178.599°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

Regular 257-gon

The area of a regular 257-gon is (with t = edge length)

${\displaystyle A={\frac {257}{4}}t^{2}\cot {\frac {\pi }{257}}\approx 5255.751t^{2}.}$

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 ${\displaystyle \cos {\frac {\pi }{257}}}$ and ${\displaystyle \cos {\frac {2\pi }{257}}}$ 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)^[1] and Friedrich Julius Richelot (1832).^[2] Another method involves the use of 150 circles, 24 being Carlyle circles: this method is pictured below. One of these Carlyle circles solves the quadratic equation x² + x − 64 = 0.^[3]

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  Step 1
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  Step 2
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  Step 3
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  Step 4
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  Step 5
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  Step 6
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  Step 7
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  Step 8
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  Step 9

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 ${\displaystyle \left\lfloor {\frac {257}{2}}\right\rfloor =128}$.

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

1. Magnus Georg Paucker (1822). "Das regelmäßige Zweyhundersiebenundfunfzig-Eck im Kreise". Jahresverhandlungen der Kurländischen Gesellschaft für Literatur und Kunst (in German). 2: 188. Retrieved 8. December 2015.
2. Friedrich Julius Richelot (1832). "De resolutione algebraica aequationis x²⁵⁷ = 1, ..." Journal für die reine und angewandte Mathematik (in Latin). 9: 1–26, 146–161, 209–230, 337–358. Retrieved 8. December 2015.
3. DeTemple, Duane W. (Feb 1991). "Carlyle circles and Lemoine simplicity of polygon constructions" (PDF). The American Mathematical Monthly. 98 (2): 97–108. doi:10.2307/2323939. JSTOR 2323939. Archived from the original (PDF) on 2015-12-21. Retrieved 6 November 2011.

External links

• Weisstein, Eric W. "257-gon". MathWorld.
• 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. ISBN 978-0486242972
• 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)