Editing Nonagon (section)

== Symmetry ==
[[File:Regular enneagon symmetries.png|thumb|200px|Symmetries of a regular enneagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.]]

The ''regular enneagon'' has [[dihedral symmetry|Dih<sub>9</sub> symmetry]], order 18. There are 2 subgroup dihedral symmetries: Dih<sub>3</sub> and Dih<sub>1</sub>, and 3 [[cyclic group]] symmetries: Z<sub>9</sub>, Z<sub>3</sub>, and Z<sub>1</sub>.

These 6 symmetries can be seen in 6 distinct symmetries on the enneagon. [[John Horton Conway|John Conway]] labels these by a letter and group order.<ref>John H. Conway, Heidi Burgiel, [[Chaim Goodman-Strauss]], (2008) The Symmetries of Things, {{isbn|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)</ref> Full symmetry of the regular form is '''r18''' 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 '''g9''' subgroup has no degrees of freedom but can be seen as [[directed edge]]s.