Type Regular polygon Internal angle (degrees) 179° | Edges and vertices 360 Dual polygon Self | |
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Schläfli symbol {360}, t{180}, tt{90}, ttt{45} Symmetry group Dihedral (D360), order 2×360 | ||
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.
Contents
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)
and its inradius is
The circumradius of a regular 120-gon is
Since 360 = 23 × 32 × 5, a regular 360-gon is not constructible using a compass and straightedge, but is constructible if the use of an angle trisector is allowed.
Symmetry
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 Zn 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. 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.
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}.