# 120-gon

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

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

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)

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

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

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 Zn 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.

## 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.

 Picture Interior angle Picture Interior angle {120} {120/7} {120/11} {120/13} {120/17} {120/19} {120/23} {120/29} 177° 159° 147° 141° 129° 123° 111° 93° {120/31} {120/37} {120/41} {120/43} {120/47} {120/49} {120/53} {120/59} 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