# Nonagon

Regular nonagon (enneagon) | |
---|---|

A regular nonagon (enneagon) | |

Type | Regular polygon |

Edges and vertices | 9 |

Schläfli symbol | {9} |

Coxeter diagram | |

Symmetry group |
Dihedral (D_{9}), order 2×9 |

Internal angle (degrees) | 140° |

Dual polygon | Self |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

In geometry, a **nonagon** /ˈnɒnəɡɒn/ (or **enneagon** /ˈɛniːəɡɒn/) is a nine-sided polygon or 9-gon.

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".

## Regular nonagon

A *regular nonagon* is represented by Schläfli symbol {9} and has internal angles of 140°. The area of a regular nonagon of side length *a* is given by

where the radius *r* of the inscribed circle of the regular nonagon is

and where *R* is the radius of its circumscribed circle:

## Construction

Although a regular nonagon is not constructible with compass and straightedge (as 9 = 3^{2}, which is not a product of distinct Fermat primes), there are very old methods of construction that produce very close approximations.^{[1]}

It can be also constructed using neusis, or by allowing the use of an angle trisector.

The following is an *approximate* construction of a nonagon using a straightedge and compass.

*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 w
_{3}and w_{4}.

- 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).

## Symmetry

The *regular enneagon* has Dih_{9} symmetry, order 18. There are 2 subgroup dihedral symmetries: Dih_{3} and Dih_{1}, and 3 cyclic group symmetries: Z_{9}, Z_{3}, and Z_{1}.

These 6 symmetries can be seen in 6 distinct symmetries on the enneagon. John Conway labels these by a letter and group order.^{[2]} 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.

## Graphs

The K_{9} complete graph is often drawn as a *regular enneagon* with all 36 edges connected. This graph also represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex.

8-simplex (8D) |

## 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.^{[3]} 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.

## Architecture

Temples of the Baha'i Faith are required to be nonagonal.

The U.S. Steel Tower is an irregular nonagon.

## See also

## References

- ↑ 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)
- ↑ TMBW.net

## External links

- Properties of a Nonagon (with interactive animation)