|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 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 very old methods of construction that produce very close approximations.
Example to illustrate the error, when the constructed central angle is 39.99906°:
At a circumscribed circle radius r = 100 m, the absolute error of the 1st side would be approximately 1.6 mm.
Another two animations of an approximate construction
- Downsize the angle JMK (also 60°) with four bisections of angle and make a thirds of circular arc MON with an approximate solution between bisections of angle w3 and w4.
- Straight auxiliary line g aims over the point O to the point N (virtually a ruler at the points O and N applied), between O and N, therefore no auxiliary line.
- Thus, the circular arc MON is freely accessible for the later intersection point R.
- RMK = 40.0000000052441...°
- 360° ÷ 9 = 40°
- RMK - 40° = 5.2...E-9°
- Example to illustrate the error:
- At a circumscribed circle radius
- r = 100,000 km, the absolute error of the 1st side would be approximately 8.6 mm.
See also the calculation (Berechnung, German).
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.
The regular enneagon can tessellate the euclidean tiling with gaps. These gaps can be filled with regular hexagons and isosceles triangles. In the notation of symmetrohedron this tiling is called H(*;3;*;) with H representing *632 hexagonal symmetry in the plane.
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 three triangles. King Gizzard & the Lizard Wizard have an album titled 'Nonagon Infinity', the album art featuring a nonagonal complete graph.
Temples of the Baha'i Faith are required to be nonagonal.
The U.S. Steel Tower is an irregular nonagon.
- J. L. Berggren, "Episodes in the Mathematics of Medieval Islam", p. 82 - 85 Springer-Verlag New York, Inc. 1st edition 1986, retrieved on 11 December 2015.
- 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)