120-gon

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Regular 120-gon
Regular polygon 120.svg
A regular 120-gon
Type Regular polygon
Edges and vertices 120
Schläfli symbol {120}, t{60}, tt{30}, ttt{15}
Coxeter diagram CDel node 1.pngCDel 12.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel 0x.pngCDel node 1.png
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[edit]

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.

Constructible[edit]

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[edit]

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[edit]

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-gon rhombic dissection.svg 120-gon rhombic dissection2.svg

120-gram[edit]

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

References[edit]

  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