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
TypeRegular polygon
Edges and vertices120
Schläfli symbol{120}, t{60}, tt{30}, ttt{15}
Coxeter diagram
Symmetry groupDihedral (D120), order 2×120
Internal angle (degrees)177°
Dual polygonSelf
PropertiesConvex, 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)

and its inradius is

The circumradius of a regular 120-gon is

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

Further information: Trigonometric constants expressed in real radicals § 1.5°: regular hecatonicosagon (120-sided polygon)

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.

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°
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gollark: How helpful for research actually *is* simulating all the protein folding stuff?

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
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