# Octacontagon

Regular octacontagon
A regular octacontagon
TypeRegular polygon
Edges and vertices80
Schläfli symbol{80}, t{40}, tt{20}, ttt{10}, tttt{5}
Coxeter diagram
Symmetry groupDihedral (D80), order 2×80
Internal angle (degrees)175.5°
Dual polygonSelf
PropertiesConvex, 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 175​12°, meaning that one exterior angle would be 4​12°.

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

{\displaystyle {\begin{aligned}A=20t^{2}\cot {\frac {\pi }{80}}=\cot {\frac {\pi }{40}}+{\sqrt {\cot ^{2}{\frac {\pi }{40}}+1}}=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)^{2}+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)+\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)\right)^{2}+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)^{2}+{\binom {2}{1}}\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)+\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)^{2}\right)+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(\left(11+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)+{\binom {2}{1}}\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)+\left(12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)\right)+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(23+8{\sqrt {5}}+2\cdot {\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}+{\binom {2}{1}}\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)\right)+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {24+8{\sqrt {5}}+2\cdot {\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}+{\binom {2}{1}}\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)}}\right)t^{2}\end{aligned}}}

${\displaystyle r={\frac {1}{2}}t\cot {\frac {\pi }{80}}}$

The circumradius of a regular octacontagon is

{\displaystyle {\begin{aligned}R={\frac {1}{2}}t\csc {\frac {\pi }{80}}={\frac {\sqrt {\left(\cot {\tfrac {\pi }{80}}\right)^{2}+1}}{2}}t\end{aligned}}}

### 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:

${\displaystyle \sin {\frac {\pi }{80}}=\sin 2.25^{\circ }={\frac {1}{8}}(1+{\sqrt {5}})\left(-{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}-{\sqrt {(2+{\sqrt {2}})\left(2-{\sqrt {2+{\sqrt {2}}}}\right)}}\right)}$
${\displaystyle +{\frac {1}{4}}{\sqrt {{\frac {1}{2}}(5-{\sqrt {5}})}}\left({\sqrt {(2+{\sqrt {2}})\left(2+{\sqrt {2+{\sqrt {2}}}}\right)}}-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}\right)}$
${\displaystyle \cos {\frac {\pi }{80}}=\cos 2.25^{\circ }={\sqrt {{\frac {1}{2}}+{\frac {1}{4}}{\sqrt {{\frac {1}{2}}\left(4+{\sqrt {2\left(4+{\sqrt {2(5+{\sqrt {5}})}}\right)}}\right)}}}}}$

## 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. ^
4. ^ Constructible Polygon
5. ^ The Symmetries of Things, Chapter 20
6. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141