257-gon

Regular 257-gon
Regular 257-gon
	A regular 257-gon
Type	Regular polygon
Edgesandvertices	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

Ingeometry,a257-gonis apolygonwith 257 sides. The sum of the interior angles of any non-self-intersecting257-gon is 45,900°.

Regular 257-gon

The area of aregular257-gon is (witht= edge length)

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

A whole regular 257-gon is not visually discernible from acircle,and its perimeter differs from that of thecircumscribed circleby about 24parts per million.

Construction

The regular 257-gon (one with all sides equal and all angles equal) is of interest for being aconstructible polygon:that is, it can beconstructed using a compass and an unmarked straightedge.This is because 257 is aFermat prime,being of the form 2^2ⁿ+ 1 (in this casen= 3). Thus, the values $\cos {\frac {\pi }{257}}$ and $\cos {\frac {2\pi }{257}}$ are 128-degreealgebraic numbers,and like allconstructible numbersthey can be written usingsquare rootsand no higher-order roots.

Although it was known toGaussby 1801 that the regular 257-gon was constructible, the first explicit constructions of a regular 257-gon were given byMagnus Georg Paucker(1822)^[1]andFriedrich Julius Richelot(1832).^[2]Another method involves the use of 150 circles, 24 beingCarlyle circles:this method is pictured below. One of these Carlyle circles solves thequadratic equationx²+x− 64 = 0.^[3]

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

Symmetry

Theregular 257-gonhasDih₂₅₇symmetry,order 514. Since 257 is aprime numberthere is one subgroup with dihedral symmetry: Dih₁,and 2cyclic groupsymmetries: Z₂₅₇,and Z₁.

257-gram

A 257-gram is a 257-sidedstar polygon.As 257 is prime, there are 127 regular forms generated bySchläfli symbols{257/n} for allintegers2 ≤n≤ 128 as $\left\lfloor {\frac {257}{2}}\right\rfloor =128$ .

Below is a view of {257/128}, with 257 nearly radial edges, with its star vertexinternal angles180°/257 (~0.7°).

References

^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.
^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.
^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 fromthe original(PDF)on 2015-12-21.Retrieved6 November2011.

External links

Weisstein, Eric W."257-gon".MathWorld.
Robert DixonMathographics.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. CoxeterIntroduction to Geometry,2nd ed. New York: Wiley, 1969. Chapter 2, Regular polygons
Leonard Eugene DicksonConstructions 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)

[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.

[DeTemple-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 fromthe original(PDF)on 2015-12-21.Retrieved6 November2011.

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