Shape That Has 9 Sides

Shape with nine sides

Regular enneagon (nonagon)
A regular enneagon (nonagon)
Blazon	Regular polygon
Edges and vertices	9
Schläfli symbol	{ix}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D9), guild ii×ix
Internal angle (degrees)	140°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a nonagon () or enneagon () is a ix-sided polygon or ix-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, "ix" + γωνον (from γωνία = "corner")), and is arguably more than right,^[i] though less mutual than "nonagon".

Regular nonagon [edit]

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

${\displaystyle A={\frac {ix}{4}}a^{2}\cot {\frac {\pi }{9}}=(9/two)ar=9r^{ii}\tan(\pi /9)}$

${\displaystyle =(ix/2)R^{two}\sin(ii\pi /9)\simeq half-dozen.18182\,a^{ii},}$

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

${\displaystyle r=(a/2)\cot(\pi /ix)}$

and where R is the radius of its circumscribed circle:

${\displaystyle R={\sqrt {(a/2)^{2}+r^{ii}}}=r\sec(\pi /9).}$

Construction [edit]

Although a regular nonagon is not constructible with compass and straightedge (as 9 = 3ⁱⁱ, which is non a product of distinct Fermat primes), there are very former methods of construction that produce very shut approximations.^[ii]

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

Nonagon, an animation from a neusis structure based on the angle trisection 120° by means of the Tomahawk, at the terminate 10 south break

Symmetry [edit]

Symmetries of a regular enneagon. Vertices are colored past their symmetry positions. Blue mirrors are fatigued through vertices, and royal mirrors are drawn through edge. Gyration orders are given in the center.

The regular enneagon has Dih₉ symmetry, lodge 18. In that location are 2 subgroup dihedral symmetries: Dih_iii and Dih₁, and 3 cyclic group symmetries: Z₉, Z₃, and Z₁.

These 6 symmetries can exist seen in 6 distinct symmetries on the enneagon. John Conway labels these by a letter and group guild.^[4] Total 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 cavalcade are labeled every bit 1000 for their cardinal gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g9 subgroup has no degrees of liberty but can seen equally directed edges.

Tilings [edit]

The regular enneagon tin can tessellate the euclidean tiling with gaps. These gaps can exist 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.

Graphs [edit]

The Grand₉ consummate graph is often drawn equally 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.

Popular civilisation references [edit]

• They Might Exist Giants have a song entitled "Nonagon" on their children'southward album Here Come up the 123s. It refers to both an attendee at a party at which "everybody in the political party is a many-sided polygon" and a trip the light fantastic toe they perform at this party.^[five]
• Slipknot's logo is likewise a version of a nonagon, being a 9-pointed star made of 3 triangles, referring to the ix members.
• Male monarch Gizzard & the Lizard Wizard have an anthology titled 'Nonagon Infinity', the album art featuring a nonagonal complete graph. The album consists of nine songs and repeats cyclically.

Architecture [edit]

Temples of the Baháʼí Religion, called Baháʼí Houses of Worship, are required to exist nonagonal.

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

Garsų Gaudyklė in Lithuania.

Meet also [edit]

• Enneagram (nonagram)
• Trisection of the angle 60°, Proximity construction

References [edit]

1. ^ Eric Westward. Weisstein. "Nonagon". > MathWorld--A Wolfram Web Resource . Retrieved 24 October 2018.
2. ^ J. 50. Berggren, "Episodes in the Mathematics of Medieval Islam", p. 82 - 85 Springer-Verlag New York, Inc. 1st edition 1986, retrieved on 11 December 2015.
3. ^ 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.
4. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter xx, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
5. ^ TMBW.net

External links [edit]

• Properties of a Nonagon (with interactive animation)

Shape That Has 9 Sides,

Source: https://en.wikipedia.org/wiki/Nonagon

Posted by: rosenbergequed1960.blogspot.com

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