page_title: 360-gon
section_title: Symmetry
hierarchical_section_title: 360-gon / Symmetry
caption_attribution_description: English: en:360-gon symmetries
context_page_description: In geometry, a 360-gon is a polygon with 360 sides. The sum of any 360-gon's interior angles is 64440 degrees.
context_section_description: 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 symmetries 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.
caption: 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.
exif: {}