120-gon

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

Regular 120-gon
A regular 120-gon
Type	Regular polygon
Edges and vertices	120
Schläfli symbol	{120}, t{60}, tt{30}, ttt{15}
Coxeter diagram
Symmetry group	Dihedral (D120), order 2×120
Internal angle (degrees)	177°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

Alternative names include dodecacontagon and hecatonicosagon.[1]

Regular 120-gon properties

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

One interior angle in a regular 120-gon is 177°, meaning that one exterior angle would be 3°.

The area of a regular 120-gon is (with t = edge length)

${\displaystyle A=30t^{2}\cot {\frac {\pi }{120}}}$

and its inradius is

${\displaystyle r={\frac {1}{2}}t\cot {\frac {\pi }{120}}}$

The circumradius of a regular 120-gon is

${\displaystyle R={\frac {1}{2}}t\csc {\frac {\pi }{120}}}$

This means that the trigonometric functions of π/120 can be expressed in radicals.

Constructible

Since 120 = 2³ × 3 × 5, a regular 120-gon is constructible using a compass and straightedge.[2] As a truncated hexacontagon, it can be constructed by an edge-bisection of a regular hexacontagon.

Symmetry

The symmetries of a regular 120-gon. Symmetries are related as index 2 subgroups in each box. The 4 boxes are related as 3 and 5 index subgroups.

The regular 120-gon has Dih₁₂₀ dihedral symmetry, order 240, represented by 120 lines of reflection. Dih₁₂₀ has 15 dihedral subgroups: (Dih₆₀, Dih₃₀, Dih₁₅), (Dih₄₀, Dih₂₀, Dih₁₀, Dih₅), (Dih₂₄, Dih₁₂, Dih₆, Dih₃), and (Dih₈, Dih₄, Dih₂, Dih₁). And 16 more cyclic symmetries: (Z₁₂₀, Z₆₀, Z₃₀, Z₁₅), (Z₄₀, Z₂₀, Z₁₀, Z₅), (Z₂₄, Z₁₂, Z₆, Z₃), and (Z₈, Z₄, Z₂, Z₁), with Z_n representing π/n radian rotational symmetry.

These 32 symmetries are related to 44 distinct symmetries on the 120-gon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[3] 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 120-gons. Only the g120 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.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular 120-gon, m=60, and it can be divided into 1770: 30 squares and 29 sets of 60 rhombs. This decomposition is based on a Petrie polygon projection of a 60-cube.

Examples

120-gram

A 120-gram is a 120-sided star polygon. There are 15 regular forms given by Schläfli symbols {120/7}, {120/11}, {120/13}, {120/17}, {120/19}, {120/23}, {120/29}, {120/31}, {120/37}, {120/41}, {120/43}, {120/47}, {120/49}, {120/53}, and {120/59}, as well as 44 compound star figures with the same vertex configuration.

Regular star polygons {120/k}

Picture	{120}	{120/7}	{120/11}	{120/13}	{120/17}	{120/19}	{120/23}	{120/29}
Interior angle	177°	159°	147°	141°	129°	123°	111°	93°
Picture	{120/31}	{120/37}	{120/41}	{120/43}	{120/47}	{120/49}	{120/53}	{120/59}
Interior angle	87°	69°	57°	51°	39°	33°	21°	3°

References

1. Norman Johnson, Geometries and Transformations (2018), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5 full polychoric groups
2. Constructible Polygon
3. 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)
4. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141