page_title: Decagon
section_title: Symmetry
hierarchical_section_title: Decagon / Symmetry
caption_attribution_description: English: en:Decagon symmetryJohn 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.
context_page_description: In geometry, a decagon is a ten-sided polygon or 10-gon.
context_section_description: The regular decagon has Dih₁₀ symmetry, order 20. There are 3 subgroup dihedral symmetries: Dih₅, Dih₂, and Dih₁, and 4 cyclic group symmetries: Z₁₀, Z₅, Z₂, and Z₁.
  These 8 symmetries can be seen in 10 distinct symmetries on the decagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order. Full symmetry of the regular form is r20 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
  Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g10 subgroup has no degrees of freedom but can seen as directed edges.
  The highest symmetry irregular decagons are d10, an isogonal decagon constructed by five mirrors which can alternate long and short edges, and p10, an isotoxal decagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular decagon.
caption: Symmetries of a regular decagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edges. Gyration orders are given in the center.
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