View source for Nonagon

{{Short description|Shape with nine sides}}
{{Regular polygon db|Regular polygon stat table|p9}}
In [[geometry]], a '''nonagon''' ({{IPAc-en|ˈ|n|ɒ|n|ə|g|ɒ|n}}) or '''enneagon''' ({{IPAc-en|ˈ|ɛ|n|i|ə|ɡ|ɒ|n}}) is a nine-sided [[polygon]] or 9-gon.
The name ''nonagon'' is a [[prefix]] [[Hybrid word|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 language|Greek]] ''enneagonon'' (εννεα, "nine" + γωνον (from γωνία = "corner")), and is arguably more correct,<ref>{{mathworld|title=Nonagon |id=Nonagon}}</ref> though less common than "nonagon".
== Regular nonagon ==
A ''[[regular polygon|regular]] nonagon'' is represented by [[Schläfli symbol]] {9} and has [[internal angle]]s of 140°. The area of a regular nonagon of side length ''a'' is given by
:<math>A = \frac{9}{4}a^2\cot\frac{\pi}{9}=(9/2)ar = 9r^2\tan(\pi/9) </math>
:::<math>= (9/2)R^2\sin(2\pi/9)\simeq6.18182\,a^2,</math>
where the radius ''r'' of the [[inscribed circle]] of the regular nonagon is
:<math>r=(a/2)\cot(\pi/9)</math>
and where ''R'' is the radius of its [[circumscribed circle]]:
:<math>R = \sqrt{(a/2)^2 + r^2 }=r\sec(\pi/9)=(a/2)\csc(\pi/9).</math>
==Construction==
Although a regular nonagon is not [[constructible polygon|constructible]] with [[compass and straightedge]] (as 9 = 3<sup>2</sup>, which is not a product of distinct [[Fermat prime]]s), there are very old methods of construction that produce very close approximations.<ref>J. L. Berggren, [https://books.google.com/books?id=MPTxBwAAQBAJ&dq=%22The+Construction+of+the+Regular+Nonagon%22&pg=PA82 "Episodes in the Mathematics of Medieval Islam", p. 82 - 85] Springer-Verlag New York, Inc. 1st edition 1986, retrieved on 11 December 2015.</ref>
It can be also constructed using [[neusis construction|neusis]], or by allowing the use of an [[Angle trisection|angle trisector]].
[[File:01-Neuneck Tomahawk Animation.gif|350px|left|thumb|Nonagon, an animation from a neusis construction based on the angle trisection 120° by means of the [[Tomahawk (geometry)|Tomahawk]], at the end 10 s break]]
[[File:01 Neuneck-Archimedes.gif|350px|left|thumb|Nonagon, a neusis construction based on a hexagon with  [[Angle trisection#With a marked ruler|trisection of the angle according to Archimedes]]<ref>{{cite web|title=KLASSISCHE PROBLEME DES GRIECHISCHENALTERTUMS IM MATHEMATIKUNTERRICHT DER OBERSTUFE|periodical=Erziehungskunst|publisher=Bund der Freien Waldorfschulen Deutschlands|url=https://www.erziehungskunst.de/fileadmin/archiv_alt/1960-1967/1965_07_08_Jg_29.pdf#page=46&zoom=auto,-18,592|format=PDF|last=Ernst Bindel, Helmut von Kügelgen|pages=234–237}}Retrieved on 14 July 2019.</ref>]]
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== 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.
== Tilings ==
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;*;[2]) with H representing *632 hexagonal symmetry in the plane.
:[[File:Conway tiling dKH.png|320px]]
==Graphs==
The K<sub>9</sub> [[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]].
{| class=wikitable
|- align=center
|[[File:8-simplex t0.svg|150px]]<br>[[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.<ref>[http://tmbw.net/wiki/Lyrics:Nonagon TMBW.net]</ref>
*[[Slipknot (band)|Slipknot]]'s logo is also a version of a nonagon, being a nine-pointed star made of three triangles, referring to the nine members.
*[[King Gizzard & the Lizard Wizard]] have an album titled '[[Nonagon Infinity]]', the album art featuring a nonagonal complete graph. The album consists of nine songs and repeats cyclically.
[[File:Garsų Gaudyklė, Gintaro ilanka, Neringa, Litva 02.jpg|thumb|200x200px|[[Garsų Gaudyklė]]]]
==Architecture==
Temples of the [[Baháʼí Faith]], called [[Baháʼí House of Worship|Baháʼí Houses of Worship]], are required to be nonagonal.
The [[U.S. Steel Tower]] is an irregular nonagon.
[[Garsų Gaudyklė]] in Lithuania.
[[Palmanova]] in Italy.
==See also==
*[[Enneagram (geometry)|Enneagram]] (nonagram)
*[[Commons:File:01- Dreiteilung des Winkels 60°.svg|Trisection of the angle 60°, Proximity construction]]
==References==
{{Reflist}}
==External links==
*[http://www.mathopenref.com/nonagon.html Properties of a Nonagon] (with interactive animation)
{{Polygons}}
[[Category:Polygons by the number of sides]]
[[Category:9 (number)]]
[[Category:Elementary shapes]]