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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a nonagon () or enneagon () is a nine-sided
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
or 9-gon. The name ''nonagon'' is a
prefix A prefix is an affix which is placed before the stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy''. Particul ...
hybrid formation, from
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
(''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 Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
''enneagonon'' (εννεα, "nine" + γωνον (from γωνία = "corner")), and is arguably more correct, though less common than "nonagon".


Regular nonagon

A '' regular nonagon'' is represented by
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
and has
internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
s of 140°. The area of a regular nonagon of side length ''a'' is given by :A = \fraca^2\cot\frac=(9/2)ar = 9r^2\tan(\pi/9) :::= (9/2)R^2\sin(2\pi/9)\simeq6.18182\,a^2, where the radius ''r'' of the
inscribed circle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incen ...
of the regular nonagon is :r=(a/2)\cot(\pi/9) and where ''R'' is the radius of its
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
: :R = \sqrt=r\sec(\pi/9).


Construction

Although a regular nonagon is not constructible with
compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
(as 9 = 32, which is not a product of distinct
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...
s), there are very old methods of construction that produce very close approximations. It can be also constructed using
neusis In geometry, the neusis (; ; plural: grc, νεύσεις, neuseis, label=none) is a geometric construction method that was used in antiquity by Greek mathematicians. Geometric construction The neusis construction consists of fitting a lin ...
, or by allowing the use of an
angle trisector Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
.


Symmetry

The ''regular enneagon'' has Dih9 symmetry, order 18. There are 2 subgroup dihedral symmetries: Dih3 and Dih1, and 3
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
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. 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 edge In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pai ...
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;*; with H representing *632 hexagonal symmetry in the plane. :


Graphs

The K9
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...
is often drawn as a ''regular enneagon'' with all 36 edges connected. This graph also represents an
orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogona ...
of the 9 vertices and 36 edges of the
8-simplex In geometry, an 8- simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle i ...
.


Pop culture references

*
They Might Be Giants They Might Be Giants (often abbreviated as TMBG) is an American alternative rock band formed in 1982 by John Flansburgh and John Linnell. During TMBG's early years, Flansburgh and Linnell frequently performed as a duo, often accompanied by a dr ...
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.TMBW.net
/ref> * 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 ''Nonagon Infinity'' is the eighth studio album by Australian psychedelic rock band King Gizzard & the Lizard Wizard. It was released on 29 April 2016 on ATO Records. The album is designed to play as an "infinite loop" where each song segues in ...
', the album art featuring a nonagonal complete graph. The album consists of nine songs and repeats cyclically.


Architecture

Temples of the
Baháʼí Faith The Baháʼí Faith is a religion founded in the 19th century that teaches the essential worth of all religions and the unity of all people. Established by Baháʼu'lláh in the 19th century, it initially developed in Iran and parts of the ...
, called Baháʼí Houses of Worship, are required to be nonagonal. The U.S. Steel Tower is an irregular nonagon. Garsų Gaudyklė in Lithuania.


See also

*
Enneagram Enneagram is a compound word derived from the Greek neoclassical stems for "nine" (''ennea'') and something "written" or "drawn" (''gramma''). Enneagram may refer to: * Enneagram (geometry), a nine-sided star polygon with various configurations ...
(nonagram) * Trisection of the angle 60°, Proximity construction


References


External links


Properties of a Nonagon
(with interactive animation) {{Polygons Polygons by the number of sides 9 (number) Elementary shapes