Hexacontagon

Regular hexacontagon
Regular hexacontagon
	A regular hexacontagon
Type	Regular polygon
Edges and vertices	60
Schläfli symbol	{60}, t{30}, tt{15}
Coxeter–Dynkin diagrams	;
Symmetry group	Dihedral (D60), order 2×60
Internal angle (degrees)	174°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

In geometry, a hexacontagon or hexecontagon or 60-gon is a sixty-sided polygon.^[1]^[2] The sum of any hexacontagon's interior angles is 10440 degrees.

Regular hexacontagon properties

A regular hexacontagon is represented by Schläfli symbol {60} and also can be constructed as a truncated triacontagon, t{30}, or a twice-truncated pentadecagon, tt{15}. A truncated hexacontagon, t{60}, is a 120-gon, {120}.

One interior angle in a regular hexacontagon is 174°, meaning that one exterior angle would be 6°.

The area of a regular hexacontagon is (with t = edge length)

A=15t^{2}\cot {\frac {\pi }{60}}

and its inradius is

r={\frac {1}{2}}t\cot {\frac {\pi }{60}}

The circumradius of a regular hexacontagon is

R={\frac {1}{2}}t\csc {\frac {\pi }{60}}

This means that the trigonometric functions of π/60 can be expressed in radicals.

Constructible

Since 60 = 2² × 3 × 5, a regular hexacontagon is constructible using a compass and straightedge.^[3] As a truncated triacontagon, it can be constructed by an edge-bisection of a regular triacontagon.

Symmetry

The regular hexacontagon has Dih₆₀ dihedral symmetry, order 120, represented by 60 lines of reflection. Dih₆₀ has 11 dihedral subgroups: (Dih₃₀, Dih₁₅), (Dih₂₀, Dih₁₀, Dih₅), (Dih₁₂, Dih₆, Dih₃), and (Dih₄, Dih₂, Dih₁). And 12 more cyclic symmetries: (Z₆₀, Z₃₀, Z₁₅), (Z₂₀, Z₁₀, Z₅), (Z₁₂, Z₆, Z₃), and (Z₄, Z₂, Z₁), with Z_n representing π/n radian rotational symmetry.

These 24 symmetries are related to 32 distinct symmetries on the hexacontagon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^[4] He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedom in defining irregular hexacontagons. Only the g60 symmetry has no degrees of freedom but can seen as directed edges.

Dissection

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. ^[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hexacontagon, m=30, and it can be divided into 435: 15 squares and 14 sets of 30 rhombs. This decomposition is based on a Petrie polygon projection of a 30-cube.

Examples

Hexacontagram

A hexacontagram is a 60-sided star polygon. There are 7 regular forms given by Schläfli symbols {60/7}, {60/11}, {60/13}, {60/17}, {60/19}, {60/23}, and {60/29}, as well as 22 compound star figures with the same vertex configuration.

Regular star polygons {60/k}
Picture	{60/7}	{60/11}	{60/13}	{60/17}	{60/19}	{60/23}	{60/29}
Interior angle	138°	114°	102°	78°	66°	42°	6°

References

^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 78, ISBN 9781438109572.
^ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
^ Constructible Polygon
^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

[1] Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 78, ISBN 9781438109572.

[2] The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298

[3] Constructible Polygon

[4] John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)

[5] Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

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