# Octacontagon

Regular octacontagon
A regular octacontagon
Type Regular polygon
Edges and vertices 80
Schläfli symbol {80}, t{40}, tt{20}, ttt{10}, tttt{5}
Coxeter diagram
Symmetry group Dihedral (D80), order 2×80
Internal angle (degrees) 175.5°
Dual polygon Self
Properties Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an octacontagon (or ogdoëcontagon or 80-gon from Ancient Greek ὁγδοήκοντα, eighty[1]) is an eighty-sided polygon.[2][3] The sum of any octacontagon's interior angles is 14040 degrees.

## Regular octacontagon

A regular octacontagon is represented by Schläfli symbol {80} and can also be constructed as a truncated tetracontagon, t{40}, or a twice-truncated icosagon, tt{20}, or a thrice-truncated decagon, ttt{10}, or a four-fold-truncated pentagon, tttt{5}.

One interior angle in a regular octacontagon is 17512°, meaning that one exterior angle would be 412°.

The area of a regular octacontagon is (with t = edge length)

The circumradius of a regular octacontagon is

### Construction

Since 80 = 24 × 5, a regular octacontagon is constructible using a compass and straightedge.[4] As a truncated tetracontagon, it can be constructed by an edge-bisection of a regular tetracontagon. This means that the trigonometric functions of π/80 can be expressed in radicals:

## Symmetry

The symmetries of a regular octacontagon. Light blue lines show subgroups of index 2. The left and right subgraphs are positionally related by index 5 subgroups.

The regular octacontagon has Dih80 dihedral symmetry, order 80, represented by 80 lines of reflection. Dih40 has 9 dihedral subgroups: (Dih40, Dih20, Dih10, Dih5), and (Dih16, Dih8, Dih4, and Dih2, Dih1). It also has 10 more cyclic symmetries as subgroups: (Z80, Z40, Z20, Z10, Z5), and (Z16, Z8, Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[5] r160 represents full symmetry and a1 labels no symmetry. 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.

These lower symmetries allows degrees of freedoms in defining irregular octacontagons. Only the g80 subgroup has no degrees of freedom but can seen as directed edges.

## Dissection

80-gon with 3120 rhombs

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. [6] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octacontagon, m=40, and it can be divided into 780: 20 squares and 19 sets of 40 rhombs. This decomposition is based on a Petrie polygon projection of a 40-cube.

## Octacontagram

An octacontagram is an 80-sided star polygon. There are 15 regular forms given by Schläfli symbols {80/3}, {80/7}, {80/9}, {80/11}, {80/13}, {80/17}, {80/19}, {80/21}, {80/23}, {80/27}, {80/29}, {80/31}, {80/33}, {80/37}, and {80/39}, as well as 24 regular star figures with the same vertex configuration.

 Picture Interior angle Picture Interior angle {80/3} {80/7} {80/9} {80/11} {80/13} {80/17} {80/19} {80/21} 166.5° 148.5° 139.5° 130.5° 121.5° 103.5° 94.5° 85.5° {80/23} {80/27} {80/29} {80/31} {80/33} {80/37} {80/39} 76.5° 58.5° 49.5° 40.5° 31.5° 13.5° 4.5°

## References

1. Greek Numbers and Numerals (Ancient and Modern) by Harry Foundalis
2. Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 110, ISBN 9781438109572 .
3. Constructible Polygon
4. The Symmetries of Things, Chapter 20
5. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
• Weisstein, Eric W. "Octacontagon". MathWorld.
• Naming Polygons and Polyhedra
• ogdoacontagon