The name nonagon is a prefixhybrid 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 Greekenneagonon (εννεα, "nine" + γωνον (from γωνία = "corner")), and is arguably more correct,[1] though less common than "nonagon".
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.[2]
It can be also constructed using neusis, or by allowing the use of an angle trisector.
Nonagon, an animation from a neusis construction based on the angle trisection 120° by means of the Tomahawk, at the end 10 s breakNonagon, a neusis construction based on a hexagon with trisection of the angle according to Archimedes[3]
Symmetryedit
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 Dih9 symmetry, order 18. There are 2 subgroup dihedral symmetries: Dih3 and Dih1, and 3 cyclic group symmetries: Z9, Z3, and Z1.
These 6 symmetries can be seen in 6 distinct symmetries on the enneagon. John Conway labels these by a letter and group order.[4] 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.
Tilingsedit
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.
Graphsedit
The K9complete 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.
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.[5]
Slipknot's logo is also a version of a nonagon, being a nine-pointed star made of three triangles, referring to the nine members.
Trisection of the angle 60°, Proximity construction
Referencesedit
^Eric W. Weisstein. "Nonagon". > MathWorld--A Wolfram Web Resource. Retrieved 24 October 2018.
^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.
^Ernst Bindel, Helmut von Kügelgen. "KLASSISCHE PROBLEME DES GRIECHISCHENALTERTUMS IM MATHEMATIKUNTERRICHT DER OBERSTUFE" (PDF). Erziehungskunst. Bund der Freien Waldorfschulen Deutschlands. pp. 234–237.Retrieved on 14 July 2019.
^John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
Properties of a Nonagon (with interactive animation)
December 12, 2023
nonagon, geometry, nonagon, enneagon, nine, sided, polygon, regular, enneagon, nonagon, regular, enneagon, nonagon, typeregular, polygonedges, vertices9schläfli, symbol, coxeter, dynkin, diagramssymmetry, groupdihedral, order, 9internal, angle, degrees, proper. In geometry a nonagon ˈ n ɒ n e ɡ ɒ n or enneagon ˈ ɛ n i e ɡ ɒ n is a nine sided polygon or 9 gon Regular enneagon nonagon A regular enneagon nonagon TypeRegular polygonEdges and vertices9Schlafli symbol 9 Coxeter Dynkin diagramsSymmetry groupDihedral D9 order 2 9Internal angle degrees 140 PropertiesConvex cyclic equilateral isogonal isotoxalDual polygonSelfThe 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 ennea nine gwnon from gwnia corner and is arguably more correct 1 though less common than nonagon Contents 1 Regular nonagon 2 Construction 3 Symmetry 4 Tilings 5 Graphs 6 Pop culture references 7 Architecture 8 See also 9 References 10 External linksRegular nonagon editA regular nonagon is represented by Schlafli symbol 9 and has internal angles of 140 The area of a regular nonagon of side length a is given by A 9 4 a 2 cot p 9 9 2 a r 9 r 2 tan p 9 displaystyle A frac 9 4 a 2 cot frac pi 9 9 2 ar 9r 2 tan pi 9 nbsp 9 2 R 2 sin 2 p 9 6 18182 a 2 displaystyle 9 2 R 2 sin 2 pi 9 simeq 6 18182 a 2 nbsp dd dd where the radius r of the inscribed circle of the regular nonagon is r a 2 cot p 9 displaystyle r a 2 cot pi 9 nbsp and where R is the radius of its circumscribed circle R a 2 2 r 2 r sec p 9 a 2 csc p 9 displaystyle R sqrt a 2 2 r 2 r sec pi 9 a 2 csc pi 9 nbsp Construction editAlthough 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 2 It can be also constructed using neusis or by allowing the use of an angle trisector nbsp Nonagon an animation from a neusis construction based on the angle trisection 120 by means of the Tomahawk at the end 10 s break nbsp Nonagon a neusis construction based on a hexagon with trisection of the angle according to Archimedes 3 Symmetry edit nbsp 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 Dih9 symmetry order 18 There are 2 subgroup dihedral symmetries Dih3 and Dih1 and 3 cyclic group symmetries Z9 Z3 and Z1 These 6 symmetries can be seen in 6 distinct symmetries on the enneagon John Conway labels these by a letter and group order 4 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 Tilings editThe 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 nbsp Graphs editThe K9 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 nbsp 8 simplex 8D Pop culture references editThey 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 5 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 amp 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 nbsp Garsu GaudykleArchitecture editTemples of the Bahaʼi Faith called Bahaʼi Houses of Worship are required to be nonagonal The U S Steel Tower is an irregular nonagon Garsu Gaudykle in Lithuania See also editEnneagram nonagram Trisection of the angle 60 Proximity constructionReferences edit Eric W Weisstein Nonagon gt MathWorld A Wolfram Web Resource Retrieved 24 October 2018 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 Ernst Bindel Helmut von Kugelgen KLASSISCHE PROBLEME DES GRIECHISCHENALTERTUMS IM MATHEMATIKUNTERRICHT DER OBERSTUFE PDF Erziehungskunst Bund der Freien Waldorfschulen Deutschlands pp 234 237 Retrieved on 14 July 2019 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 netExternal links editProperties of a Nonagon with interactive animation Retrieved from https en wikipedia org w index php title Nonagon amp oldid 1186890742, wikipedia, wiki, book, books, library,
