# 120-gon

Regular 120-gon
A regular 120-gon
TypeRegular polygon
Edges and vertices120
Schläfli symbol, t, tt, ttt
Coxeter diagram
Symmetry groupDihedral (D120), order 2×120
Internal angle (degrees)177°
Dual polygonSelf
PropertiesConvex, 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 and also can be constructed as a truncated hexacontagon, t, or a twice-truncated triacontagon, tt, or a thrice-truncated pentadecagon, ttt.

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)

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

${\displaystyle r={\frac }t\cot {\frac {\pi }}}$

The circumradius of a regular 120-gon is

${\displaystyle R={\frac }t\csc {\frac {\pi }}}$

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 , , , , , , , , , , , , , , and , as well as 44 compound star figures with the same vertex configuration.

 Picture Interior angle Picture Interior angle 177° 159° 147° 141° 129° 123° 111° 93° 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