This page uses content from Wikipedia and is licensed under CC BYSA.
Regular 360gon  

A regular 360gon


Type  Regular polygon 
Edges and vertices  360 
Schläfli symbol  {360}, t{180}, tt{90}, ttt{45} 
Coxeter diagram  
Symmetry group  Dihedral (D_{360}), order 2×360 
Internal angle (degrees)  179° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a 360gon (triacosihexacontagon or triacosihexecontagon) is a polygon with 360 sides. The sum of any 360gon's interior angles is 64440 degrees.
A regular 360gon is represented by Schläfli symbol {360} and also can be constructed as a truncated 180gon, t{180}, or a twicetruncated enneacontagon, tt{90}, or a threefoldtruncated tetracontapentagon, ttt{45}.
One interior angle in a regular 360gon is 179°, meaning that one exterior angle would be 1°.
The area of a regular 360gon is (with t = edge length)
and its inradius is
The circumradius of a regular 120gon is
Since 360 = 2^{3} × 3^{2} × 5, a regular 360gon is not constructible using a compass and straightedge,^{[1]} but is constructible if the use of an angle trisector is allowed.^{[2]}
The regular 360gon has Dih_{360} dihedral symmetry, order 720, represented by 360 lines of reflection. Dih_{360} has 23 dihedral subgroups: (Dih_{180}, Dih_{90}, Dih_{45}), (Dih_{120}, Dih_{60}, Dih_{30}, Dih_{15}), (Dih_{72}, Dih_{36}, Dih_{18}, Dih_{9}), (Dih_{40}, Dih_{20}, Dih_{10}, Dih_{5}), (Dih_{24}, Dih_{12}, Dih_{6}, Dih_{3}), and (Dih_{8}, Dih_{4}, Dih_{2}, Dih_{1}). And 24 more cyclic symmetries: (Z_{360}, Z_{180}, Z_{90}, Z_{45}), (Z_{120}, Z_{60}, Z_{30}, Z_{15}), (Z_{72}, Z_{36}, Z_{18}, Z_{9}), (Z_{40}, Z_{20}, Z_{10}, Z_{5}), (Z_{24}, Z_{12}, Z_{6}, Z_{3}), and (Z_{8}, Z_{4}, Z_{2},Z_{1}), with Z_{n} representing π/n radian rotational symmetry.
These 48 symmetries are related to 66 distinct symmetries on the 360gon. 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 360gons. Only the g360 symmetry has no degrees of freedom but can seen as directed edges.
Coxeter states that every zonogon (a 2mgon whose opposite sides are parallel and of equal length) can be dissected into m(m1)/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 360gon, m=180, and it can be divided into 16110: 90 squares and 89 sets of 180 rhombs. This decomposition is based on a Petrie polygon projection of a 180cube.
A 360gram is a 360sided 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 360grams show moiré patterns.
Regular star polygons  

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}.