page_title: Tetracontagon
section_title: Symmetry
hierarchical_section_title: Tetracontagon / Symmetry
caption_attribution_description: English: en:tetracontagon symmetries
context_page_description: In geometry, a tetracontagon or tessaracontagon is a forty-sided polygon or 40-gon. The sum of any tetracontagon's interior angles is 6840 degrees.
context_section_description: The regular tetracontagon has Dih₄₀ dihedral symmetry, order 80, represented by 40 lines of reflection. Dih₄₀ has 7 dihedral subgroups: (Dih₂₀, Dih₁₀, Dih₅), and (Dih₈, Dih₄, Dih₂, Dih₁). It also has eight more cyclic symmetries as subgroups: (Z₄₀, Z₂₀, Z₁₀, Z₅), and (Z₈, Z₄, 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 tetracontagons. Only the g40 subgroup has no degrees of freedom but can seen as directed edges.
caption: The symmetries of a regular tetracontagon. Light blue lines show subgroups of index 2. The left and right subgraphs are positionally related by index 5 subgroups.
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