page_title: Heptacontagon
section_title: Symmetry
hierarchical_section_title: Heptacontagon / Symmetry
caption_attribution_description: English: en:heptacontagon symmetries
context_page_description: In geometry, a heptacontagon or 70-gon is a seventy-sided polygon. The sum of any heptacontagon's interior angles is 12240 degrees.
  A regular heptacontagon is represented by Schläfli symbol {70} and can also be constructed as a truncated triacontapentagon, t{35}, which alternates two types of edges.
context_section_description: The regular heptacontagon has Dih₇₀ dihedral symmetry, order 140, represented by 70 lines of reflection. Dih₇₀ has 7 dihedral subgroups: Dih₃₅, (Dih₁₄, Dih₇), (Dih₁₀, Dih₅), and (Dih₂, Dih₁). It also has 8 more cyclic symmetries as subgroups: (Z₇₀, Z₃₅), (Z₁₄, Z₇), (Z₁₀, Z₅), and (Z₂, Z₁), with Zₙ representing π/n radian rotational symmetry.
  John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. 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. a1 labels no symmetry. 
  These lower symmetries allows degrees of freedoms in defining irregular heptacontagons. Only the g70 subgroup has no degrees of freedom but can seen as directed edges.
caption: The symmetries of a regular heptacontagon. Light blue lines show subgroups of index 2. The four subgraphs are positionally related by index 5 and index 7 subgroups.
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