View source for 360-gon

{{Regular polygon db|Regular polygon stat table|p360}}
In [[Geometry|geometry]], a '''360-gon''' (triacosiahexe(a)contagon) is a [[Polygon|polygon]] with 360 sides. The sum of any 360-gon's interior angles is 64440 degrees.

==Regular 360-gon properties==
A ''[[Regular polygon|regular]] 360-gon'' is represented by [[Schläfli symbol]] {360} and also can be constructed as a [[Truncation (geometry)|truncated]] 180-gon, t{180}, or a twice-truncated [[Enneacontagon|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|area]] of a regular 360-gon is (with {{nowrap|''t'' {{=}} edge length}})
:<math>A = 90t^2 \cot \frac{\pi}{360}</math>

and its inradius is
:<math>r = \frac{1}{2}t \cot \frac{\pi}{360}</math>

The circumradius of a regular 120-gon is
:<math>R = \frac{1}{2}t \csc \frac{\pi}{360}</math>

Since 360 = 2<sup>3</sup> × 3<sup>2</sup> × 5, a regular 360-gon is not [[Constructible polygon|constructible]] using a compass and straightedge,<ref>[http://mathworld.wolfram.com/ConstructiblePolygon.html Constructible Polygon]</ref> but is constructible if the use of an angle trisector is allowed.<ref name=Eekhoff>{{cite web|url=http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf |title=Archived copy |accessdate=2015-02-19 |url-status=dead |archiveurl=https://web.archive.org/web/20150714082609/http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf |archivedate=2015-07-14 }}</ref>

== Symmetry==
[[File:Symmetries_of_360-gon.png|thumb|320px|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 Dih<sub>360</sub> dihedral symmetry, order 720, represented by 360 lines of reflection. Dih<sub>360</sub> has 23 dihedral subgroups: (Dih<sub>180</sub>, Dih<sub>90</sub>, Dih<sub>45</sub>), (Dih<sub>120</sub>, Dih<sub>60</sub>, Dih<sub>30</sub>, Dih<sub>15</sub>), (Dih<sub>72</sub>, Dih<sub>36</sub>, Dih<sub>18</sub>, Dih<sub>9</sub>), (Dih<sub>40</sub>, Dih<sub>20</sub>, Dih<sub>10</sub>, Dih<sub>5</sub>), (Dih<sub>24</sub>, Dih<sub>12</sub>, Dih<sub>6</sub>, Dih<sub>3</sub>), and (Dih<sub>8</sub>, Dih<sub>4</sub>, Dih<sub>2</sub>, Dih<sub>1</sub>). And 24 more [[Cyclic group|cyclic]] symmetries: (Z<sub>360</sub>, Z<sub>180</sub>, Z<sub>90</sub>, Z<sub>45</sub>), (Z<sub>120</sub>, Z<sub>60</sub>, Z<sub>30</sub>, Z<sub>15</sub>), (Z<sub>72</sub>, Z<sub>36</sub>, Z<sub>18</sub>, Z<sub>9</sub>), (Z<sub>40</sub>, Z<sub>20</sub>, Z<sub>10</sub>, Z<sub>5</sub>), (Z<sub>24</sub>, Z<sub>12</sub>, Z<sub>6</sub>, Z<sub>3</sub>), and (Z<sub>8</sub>, Z<sub>4</sub>, Z<sub>2</sub>,Z<sub>1</sub>), with Z<sub>n</sub> representing &pi;/''n'' radian rotational symmetry.

These 48 symmetries are related to 66 distinct symmetries on the 360-gon. [[Biography:John Horton Conway|John Conway]] labels these lower symmetries with a letter and order of the symmetry follows the letter.<ref>John H. Conway, Heidi Burgiel, [[Biography:Chaim Goodman-Strauss|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)</ref>  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.
{{clear}}

== Dissection==
Coxeter states that every [[Zonogon|zonogon]] (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into ''m''(''m''-1)/2 parallelograms.<ref>Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141</ref>
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the ''regular 360-gon'', ''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 ''180-cube''.
{| class=wikitable
|+ Examples
|[[File:360-gon rhombic dissection.svg|240px]]
|[[File:360-gon rhombic dissection2.svg|240px]]
|}
{{-}}

==360-gram==
A 360-gram is a 360-sided [[Star polygon|star polygon]]. There are 47 regular forms given by [[Schläfli symbol]]s {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|vertex configuration]]. Many of the more intricate 360-grams show [[Moiré pattern|moiré pattern]]s.
{| class="wikitable collapsible collapsed"
!colspan=10|Regular star polygons
|- align=center
! Picture
|[[File:Regular polygon 360.svg|80px]]<br>{360}
|[[File:Star polygon 360-7.svg|80px]]<br>{360/7}
|[[File:Star polygon 360-11.svg|80px]]<br>{360/11}
|[[File:Star polygon 360-13.svg|80px]]<br>{360/13}
|[[File:Star polygon 360-17.svg|80px]]<br>{360/17}
|[[File:Star polygon 360-19.svg|80px]]<br>{360/19}
|[[File:Star polygon 360-23.svg|80px]]<br>{360/23}
|[[File:Star polygon 360-29.svg|80px]]<br>{360/29}
|- align=center
! Interior angle
|| 179°|| 173°|| 169°|| 167°|| 163°|| 161°|| 157°|| 151°
|- align=center
! Picture
|[[File:Star polygon 360-31.svg|80px]]<br>{360/31}
|[[File:Star polygon 360-37.svg|80px]]<BR>{360/37}
|[[File:Star polygon 360-41.svg|80px]]<BR>{360/41}
|[[File:Star polygon 360-43.svg|80px]]<BR>{360/43}
|[[File:Star polygon 360-47.svg|80px]]<BR>{360/47}
|[[File:Star polygon 360-49.svg|80px]]<BR>{360/49}
|[[File:Star polygon 360-53.svg|80px]]<BR>{360/53}
|[[File:Star polygon 360-59.svg|80px]]<BR>{360/59}
|- align=center
! Interior angle
||149°||143°||139°||137°||133°||131°||127°||121°
|- align=center
! Picture
|[[File:Star polygon 360-61.svg|80px]]<br>{360/61}
|[[File:Star polygon 360-67.svg|80px]]<BR>{360/67}
|[[File:Star polygon 360-71.svg|80px]]<BR>{360/71}
|[[File:Star polygon 360-73.svg|80px]]<BR>{360/73}
|[[File:Star polygon 360-77.svg|80px]]<BR>{360/77}
|[[File:Star polygon 360-79.svg|80px]]<BR>{360/79}
|[[File:Star polygon 360-83.svg|80px]]<BR>{360/83}
|[[File:Star polygon 360-89.svg|80px]]<BR>{360/89}
|- align=center
! Interior angle
|119°||113°||109°||107°||103°||101°||97°||91°
|- align=center
! Picture
|[[File:Star polygon 360-91.svg|80px]]<br>{360/91}
|[[File:Star polygon 360-97.svg|80px]]<BR>{360/97}
|[[File:Star polygon 360-101.svg|80px]]<BR>{360/101}
|[[File:Star polygon 360-103.svg|80px]]<BR>{360/103}
|[[File:Star polygon 360-107.svg|80px]]<BR>{360/107}
|[[File:Star polygon 360-109.svg|80px]]<BR>{360/109}
|[[File:Star polygon 360-113.svg|80px]]<BR>{360/113}
|[[File:Star polygon 360-119.svg|80px]]<BR>{360/119}
|- align=center
! Interior angle
||89°||83°||79°||77°||73°||71°||67°||61°
|- align=center
! Picture
|[[File:Star polygon 360-121.svg|80px]]<br>{360/121}
|[[File:Star polygon 360-127.svg|80px]]<BR>{360/127}
|[[File:Star polygon 360-131.svg|80px]]<BR>{360/131}
|[[File:Star polygon 360-133.svg|80px]]<BR>{360/133}
|[[File:Star polygon 360-137.svg|80px]]<BR>{360/137}
|[[File:Star polygon 360-139.svg|80px]]<BR>{360/139}
|[[File:Star polygon 360-143.svg|80px]]<BR>{360/143}
|[[File:Star polygon 360-149.svg|80px]]<BR>{360/149}
|- align=center
! Interior angle
||59°||53°||49°||47°||43°||41°||37°||31°
|- align=center
! Picture
|[[File:Star polygon 360-151.svg|80px]]<br>{360/151}
|[[File:Star polygon 360-157.svg|80px]]<BR>{360/157}
|[[File:Star polygon 360-161.svg|80px]]<BR>{360/161}
|[[File:Star polygon 360-163.svg|80px]]<BR>{360/163}
|[[File:Star polygon 360-167.svg|80px]]<BR>{360/167}
|[[File:Star polygon 360-169.svg|80px]]<BR>{360/169}
|[[File:Star polygon 360-173.svg|80px]]<BR>{360/173}
|[[File:Star polygon 360-179.svg|80px]]<BR>{360/179}
|- align=center
! 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. {[[Digon|2]], [[Equilateral triangle|3]], [[Square|4]], [[Pentagon|5]], [[Hexagon|6]], [[Octagon|8]], 9, [[Decagon|10]], [[Dodecagon|12]], [[Pentadecagon|15]], [[Octadecagon|18]], [[Icosagon|20]], [[Icositetragon|24]], [[Triacontagon|30]], 36, [[Tetracontagon|40]], 45, [[Hexacontagon|60]], 72, [[Enneacontagon|90]], [[120-gon|120]], 180, 360}.

==References==
{{reflist}}
{{Polygons}}
[[Category:Polygons]]

{{Sourceattribution|360-gon|1}}