|Regular nonagon (enneagon)|
A regular nonagon (enneagon)
|Edges and vertices||9|
|Symmetry group||Dihedral (D9), order 2×9|
|Internal angle (degrees)||140°|
|Properties||convex, cyclic, equilateral, isogonal, isotoxal|
The name "nonagon" is a prefix hybrid formation, from Latin (nonus, "ninth" + gonon), used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century. The name "enneagon" comes from Greek enneagonon (εννεα, "nine" + γωνον (from γωνία = "corner")), and is arguably more correct, though somewhat less common than "nonagon".
where the radius r of the inscribed circle of the regular nonagon is
and where R is the radius of its circumscribed circle:
Although a regular nonagon is not constructible with compass and straightedge (as 9 = 32, which is not a product of distinct Fermat primes), there are methods of construction that produce very close approximations. It can be constructed using neusis, or by allowing the use of an angle trisector.
Another animation of an approximate construction
These 6 symmetries can be seen in 6 distinct symmetries on the enneagon. John Conway labels these by a letter and group order. 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 seen as directed edges.
Pop culture references
They Might Be Giants have a song entitled "Nonagon" on their children's album Here Come the 123s. It refers to both an attendee at a party at which "everybody in the party is a many-sided polygon" and a dance they perform at this party. Slipknot's logo is also a version of a nonagon, being a nine-pointed star made of 3 triangles.
Temples of Baha'i Faith are required to be nonagonal.
The U.S. Steel Tower is an irregular nonagon.
- 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)
- Properties of a Nonagon (with interactive animation)