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

Last updated December 09, 2021

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°

Related Research Articles

In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.

In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.

In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon. The sum of any pentacontagon's interior angles is 8640 degrees.

In geometry, a hectogon or hecatontagon or 100-gon is a hundred-sided polygon. The sum of all hectogon's interior angles are 17640 degrees.

In geometry, an octacontagon is an eighty-sided polygon. The sum of any octacontagon's interior angles is 14040 degrees.

In geometry, a tetracontagon or tessaracontagon is a forty-sided polygon or 40-gon. The sum of any tetracontagon's interior angles is 6840 degrees.

In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.

In geometry, a heptacontagon or 70-gon is a seventy-sided polygon. The sum of any heptacontagon's interior angles is 12240 degrees.

In geometry, an enneacontagon or enenecontagon or 90-gon is a ninety-sided polygon. The sum of any enneacontagon's interior angles is 15840 degrees.

In geometry, a triacontadigon or 32-gon is a thirty-two-sided polygon. In Greek, the prefix triaconta- means 30 and di- means 2. The sum of any triacontadigon's interior angles is 5400 degrees.

In geometry, a tetracontadigon or 42-gon is a forty-two-sided polygon. The sum of any tetracontadigon's interior angles is 7200 degrees.

In geometry, a hexacontatetragon or 64-gon is a sixty-four-sided polygon. The sum of any hexacontatetragon's interior angles is 11160 degrees.

In geometry, a tetracontaoctagon or 48-gon is a forty-eight sided polygon. The sum of any tetracontaoctagon's interior angles is 8280 degrees.

In geometry, an enneacontahexagon or enneacontakaihexagon or 96-gon is a ninety-six-sided polygon. The sum of any enneacontahexagon's interior angles is 16920 degrees.

In geometry, a 120-gon is a polygon with 120 sides. The sum of any 120-gon's interior angles is 21240 degrees.

In geometry, a 360-gon is a polygon with 360 sides. The sum of any 360-gon's interior angles is 64440 degrees.

In geometry, an icosidigon or 22-gon is a twenty-two-sided polygon. The sum of any icosidigon's interior angles is 3600 degrees.

In geometry, an icosihexagon or 26-gon is a twenty-six-sided polygon. The sum of any icosihexagon's interior angles are 4320°.

In geometry, an icosioctagon or 28-gon is a twenty eight sided polygon. The sum of any icosioctagon's interior angles is 4680 degrees.

In geometry, a triacontatetragon or triacontakaitetragon is a thirty-four-sided polygon or 34-gon. The sum of any triacontatetragon's interior angles is 5760 degrees.

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

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[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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