360-gon

Regular 360-gon
A regular 360-gon
Type	Regular polygon
Edges and vertices	360
Schläfli symbol	{360}, t{180}, tt{90}, ttt{45}
Coxeter diagram
Symmetry group	Dihedral (D360), order 2×360
Internal angle (degrees)	179°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a 360-gon (triacosihexacontagon or triacosihexecontagon) is a polygon with 360 sides. The sum of any 360-gon's interior angles is 64440 degrees.

Regular 360-gon properties

A regular 360-gon is represented by Schläfli symbol {360} and also can be constructed as a truncated 180-gon, t{180}, or a twice-truncated enneacontagon, tt{90}, or a thrice-truncated tetracontapentagon, ttt{45}.

One interior angle in a regular 360-gon is 179°, meaning that one exterior angle would be 1°.

The area of a regular 360-gon is (with t = edge length)

${\displaystyle A=90t^{2}\cot {\frac {\pi }{360}}}$

and its inradius is

${\displaystyle r={\frac {1}{2}}t\cot {\frac {\pi }{360}}}$

The circumradius of a regular 120-gon is

${\displaystyle R={\frac {1}{2}}t\csc {\frac {\pi }{360}}}$

Since 360 = 2³ × 3² × 5, a regular 360-gon is not constructible using a compass and straightedge,^[1] but is constructible if the use of an angle trisector is allowed.^[2]

Symmetry

The symmetries of a regular 360-gon. Symmetries are related as index 2 subgroups in each box. The 6 boxes are related as 3 and 5 index subgroups.

The regular 360-gon has Dih₃₆₀ dihedral symmetry, order 720, represented by 360 lines of reflection. Dih₃₆₀ has 23 dihedral subgroups: (Dih₁₈₀, Dih₉₀, Dih₄₅), (Dih₁₂₀, Dih₆₀, Dih₃₀, Dih₁₅), (Dih₇₂, Dih₃₆, Dih₁₈, Dih₉), (Dih₄₀, Dih₂₀, Dih₁₀, Dih₅), (Dih₂₄, Dih₁₂, Dih₆, Dih₃), and (Dih₈, Dih₄, Dih₂, Dih₁). And 24 more cyclic symmetries: (Z₃₆₀, Z₁₈₀, Z₉₀, Z₄₅), (Z₁₂₀, Z₆₀, Z₃₀, Z₁₅), (Z₇₂, Z₃₆, Z₁₈, Z₉), (Z₄₀, Z₂₀, Z₁₀, Z₅), (Z₂₄, Z₁₂, Z₆, Z₃), and (Z₈, Z₄, Z₂,Z₁), with Z_n representing π/n radian rotational symmetry.

These 48 symmmetries are related to 66 distinct symmetries on the 360-gon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^[3] Full symmetry is r720 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 freedom in defining irregular 360-gons. Only the g360 symmetry has no degrees of freedom but can seen as directed edges.

360-gram

A 360-gram is a 360-sided star polygon. There are 47 regular forms given by Schläfli symbols {360/7}, {360/11}, {360/13}, {360/17}, {360/19}, {360/23}, {360/29}, {360/31}, {360/37}, {360/41}, {360/43}, {360/47}, {360/49}, {360/53}, {360/59}, {360/61}, {360/67}, {360/71}, {360/73}, {360/77}, {360/79}, {360/83}, {360/89}, {360/91}, {360/97}, {360/101}, {360/103}, {360/107}, {360/109}, {360/113}, {360/119}, {360/121}, {360/127}, {360/131}, {360/133}, {360/137}, {360/139}, {360/143}, {360/149}, {360/151}, {360/157}, {360/161}, {360/163}, {360/167}, {360/169}, {360/173}, and {360/179}, as well as 132 compound star figures with the same vertex configuration. Many of the more intricate 360-grams show moiré patterns.

Regular star polygons {360/k}  Picture {360} {360/7} {360/11} {360/13} {360/17} {360/19} {360/23} {360/29} Interior angle 179° 173° 169° 167° 163° 161° 157° 151° Picture {360/31} {360/37} {360/41} {360/43} {360/47} {360/49} {360/53} {360/59} Interior angle 149° 143° 139° 137° 133° 131° 127° 121° Picture {360/61} {360/67} {360/71} {360/73} {360/77} {360/79} {360/83} {360/89} Interior angle 119° 113° 109° 107° 103° 101° 97° 91° Picture {360/91} {360/97} {360/101} {360/103} {360/107} {360/109} {360/113} {360/119} Interior angle 89° 83° 79° 77° 73° 71° 67° 61° Picture {360/121} {360/127} {360/131} {360/133} {360/137} {360/139} {360/143} {360/149} Interior angle 59° 53° 49° 47° 43° 41° 37° 31° Picture {360/151} {360/157} {360/161} {360/163} {360/167} {360/169} {360/173} {360/179} Interior angle 29° 23° 19° 17° 13° 11° 7° 1°

The regular convex and star polygons whose interior angles are some integer number of degrees are precisely those whose numbers of sides are integer divisors of 360 that are not unity, i.e. {2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360}.

References

1. Constructible Polygon
2. http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf
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)