View source for Tetradecagon

{{short description|Polygon with 14 edges}}
{{Regular polygon db|Regular polygon stat table|p14}}
In [[Geometry|geometry]], a '''tetradecagon''' or '''tetrakaidecagon''' or 14-gon is a fourteen-sided [[Polygon|polygon]].

==Regular tetradecagon==

A ''[[Regular polygon|regular]] tetradecagon'' has [[Schläfli symbol]] {14} and can be constructed as a quasiregular [[Truncation (geometry)|truncated]] [[Heptagon|heptagon]], t{7}, which alternates two types of edges.

The [[Area|area]] of a [[Regular polygon|regular]] tetradecagon of side length ''a'' is given by
:<math>\begin{align}A &= \frac{14}{4}a^2\cot\frac{\pi}{14}=\frac{14}{4}a^2\left(\frac{\sqrt{7}+4\sqrt{7}\cos\left({\frac{2}{3}\arctan{\frac{\sqrt{3}}{9}}}\right)}{3}\right)\\
&\simeq 15.3345a^2\end{align}</math>

===Construction===
As 14 = 2 × 7, a regular tetradecagon cannot be [[Constructible polygon|constructed]] using a compass and straightedge.<ref>{{cite journal|last=Wantzel|first=Pierre|title=Recherches sur les moyens de Reconnaître si un Problème de géométrie peau se résoudre avec la règle et le compas|journal=Journal de Mathématiques|date=1837|pages=366–372|url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1837_1_2_A31_0.pdf}}</ref> However, it is constructible using [[Neusis construction|neusis]] with use of the angle trisector,<ref name="Gleason">{{cite journal|last=Gleason|first=Andrew Mattei|title=Angle trisection, the heptagon, p. 186 (Fig.1) –187 |journal=The American Mathematical Monthly|date=March 1988|volume=95|issue=3 |pages=185–194|url=http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf#|archiveurl=https://web.archive.org/web/20160202015324/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf|doi= 10.2307/2323624|archive-date=2016-02-02 |url-status=dead}}</ref> or with a marked ruler,<ref name="Johnson">[http://mathworld.wolfram.com/Heptagon.html Weisstein, Eric W. "Heptagon." From MathWorld, A Wolfram Web Resource.]</ref> as shown in the following two examples.

[[File:01-Tetradecagon-Tomahawk.gif|left|600px|thumb|Tetradecagon with ''given circumcircle'':<br />
An animation (1 min 47 s) from a neusis construction with radius of circumcircle <math>\overline{OA} = 6</math>,<br />
according to [[Biography:Andrew M. Gleason|Andrew M. Gleason]],<ref name="Gleason"/> based on the [[Angle trisection|angle trisection]] by means of the [[Tomahawk (geometry)|Tomahawk.]], pause at the end of 25 s]]

[[File:01-Vierzehneck-nach Johnson.gif|left|500px|thumb|Tetradecagon with ''given side length'':<br />
An animation (1 min 20 s) from a neusis construction with marked ruler, according to David Johnson Leisk (Crockett Johnson)<ref name="Johnson"/> for the heptagon, pause at the end of 30 s.]]
{{clear}}

The animation below gives an approximation of about 0.05° on the center angle:

[[File:Approximated_Tetradecagon_Inscribed_in_a_Circle.gif]]<br>Construction of an approximated regular tetradecagon

Another possible animation of an approximate construction, also possible with using straightedge and compass.
[[File:01-Tetradecagon-Animation.gif|thumb|700px|left|Regular tetradecagon, approximation construction as an animation (3 min 16 s), pause at the end of 25 s]]
{{clear}}

Based on the unit circle r = 1 [unit of length]
* Constructed side length of the tetradecagon in [[Software:GeoGebra|GeoGebra]] (display max 15 decimal places) <math> a = 0.445041867912629\; [unit\;of\;length]</math>
* Side length of the tetradecagon <math> a_{target} = 2 \cdot \sin\left(\frac{180^\circ}{14} \right) =0.445041867912629\ldots\; [unit\;of\;length]</math>
* Absolute error of the constructed side length
: Up to the max. displayed 15 decimal places is the absolute error <math> F_a =  a - a_{target} = 0.0\; [unit\;of\;length]</math> 
* Constructed central angle of the tetradecagon in GeoGebra (display significant 13 decimal places) <math> \mu = 25.7142857142857^\circ</math>
* Central angle of the tetradecagon <math> \mu_{target} = \frac{360^\circ}{14} = 25.7142857142857\ldots^\circ</math>
* Absolute error of the constructed central angle
: Up to the indicated significant 13 decimal places is the absolute error <math> F_\mu = \mu - \mu_{target} = 0^\circ </math>

Example to illustrate the error
* At a circumscribed circle radius  '''r = 1 billion km''' (the light needed for this distance about 55 minutes), the absolute error of the 1st side would be '''< 1&nbsp;mm'''.

For details, see: [https://de.wikibooks.org/wiki/Mathematik:_Schulmathematik:_Planimetrie:_Polygonkonstruktionen:_Vierzehneck Wikibooks: Tetradecagon, construction description (German)]
{{clear}}

== Symmetry ==
[[File:Symmetries_of_tetradecagon.png|thumb|320px|Symmetries of a regular tetradecagon. 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 tetradecagon'' has Dih<sub>14</sub> symmetry, order 28. There are 3 subgroup dihedral symmetries: Dih<sub>7</sub>, Dih<sub>2</sub>, and Dih<sub>1</sub>, and 4 [[Cyclic group|cyclic group]] symmetries: Z<sub>14</sub>, Z<sub>7</sub>, Z<sub>2</sub>, and Z<sub>1</sub>.

These 8 symmetries can be seen in 10 distinct symmetries on the tetradecagon, a larger number because the lines of reflections can either pass through vertices or edges. [[Biography:John Horton Conway|John Conway]] labels these by a letter and group order.<ref>John H. Conway, Heidi Burgiel, [[Biography:Chaim Goodman-Strauss|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 '''r28''' 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 '''g14''' subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular tetradecagons are '''d14''', an [[Isogonal figure|isogonal]] tetradecagon constructed by seven mirrors which can alternate long and short edges, and '''p14''', an [[Isotoxal figure|isotoxal]] tetradecagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are [[Dual polygon|duals]] of each other and have half the symmetry order of the regular tetradecagon.

{{-}}

== Dissection ==
{| class=wikitable align=right
|- align=center
|[[File:14-cube t0 A13.svg|160px]]<BR>14-cube projection
|[[File:14-gon rhombic dissection-size2.svg|160px]]<BR>84 rhomb dissection
|}
Coxeter states that every [[Zonogon|zonogon]] (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into ''m''(''m''-1)/2 parallelograms.<ref>Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141</ref>
In particular this is true for [[Regular polygon|regular polygon]]s with evenly many sides, in which case the parallelograms are all rhombi. For the ''regular tetradecagon'', ''m''=7, and it can be divided into 21: 3 sets of 7 rhombs. This decomposition is based on a [[Petrie polygon]] projection of a [[7-cube|7-cube]], with 21 of 672 faces. The list {{OEIS2C|1=A006245}} defines the number of solutions as 24698, including up to 14-fold rotations and chiral forms in reflection.
{| class=wikitable
|+ Dissection into 21 rhombs
|- align=center valign=top
|[[File:7-cube_graph.svg|160px]]
|[[File:14-gon-dissection.svg|160px]]
|[[File:14-gon-dissection-star.svg|160px]]
|[[File:14-gon rhombic dissection2.svg|160px]]
|[[File:14-gon rhombic dissectionx.svg|160px]]
|[[File:14-gon-dissection-random.svg|160px]]
|}

==Numismatic use==
The regular tetradecagon is used as the shape of some commemorative gold and silver Malaysian coins, the number of sides representing the 14 states of the Malaysian Federation.<ref>''The Numismatist'', Volume 96, Issues 7-12, Page 1409, American Numismatic Association, 1983.</ref>

==Related figures==
[[File:Flag of Malaysia.svg|thumb|alt=The flag of Malaysia|The flag of Malaysia, featuring a fourteen-pointed star]]

A '''tetradecagram''' is a 14-sided star polygon, represented by symbol {14/n}. There are two regular [[Star polygon|star polygon]]s: {14/3} and {14/5}, using the same vertices, but connecting every third or fifth points. There are also three compounds: {14/2} is reduced to 2{7} as two [[Heptagon|heptagon]]s, while {14/4} and {14/6} are reduced to 2{7/2} and 2{7/3} as two different [[Heptagram|heptagram]]s, and finally {14/7} is reduced to seven [[Digon|digon]]s.

A notable application of a fourteen-pointed star is in the flag of Malaysia, which incorporates a yellow {14/6} tetradecagram in the top-right corner, representing the unity of the thirteen states with the federal government.
{| class="collapsible collapsed wikitable"
!colspan=8|Compounds and star polygons
|-
!n||1||2||3||4||5||6||7
|-
!Form
!Regular
!Compound
!Star polygon
!Compound
!Star polygon
!colspan=2|Compound
|- align=center
!Image
|BGCOLOR="#ffe0e0"|[[File:Regular_polygon_14.svg|100px]]<br>{14/1} = {14}<BR>{{CDD|node_1|14|node}}
|[[File:Regular_star_figure_2(7,1).svg|100px]]<br>{14/2} = 2{7}<BR>{{CDD|node_h3|14|node}}
|BGCOLOR="#ffe0e0"|[[File:Regular_star_polygon_14-3.svg|100px]]<br>{14/3}<BR>{{CDD|node_1|14|rat|3x|node}}
|[[File:Regular_star_figure_2(7,2).svg|100px]]<br>{14/4} = 2{7/2}<BR>{{CDD|node_h3|14|rat|2x|node}}
|BGCOLOR="#ffe0e0"|[[File:Regular_star_polygon_14-5.svg|100px]]<br>{14/5}<BR>{{CDD|node_1|14|rat|5|node}}
|[[File:Regular_star_figure_2(7,3).svg|100px]]<br>{14/6} = 2{7/3}<BR>{{CDD|node_h3|14|rat|3x|node}}
|[[File:Regular_star_figure_7(2,1).svg|100px]]<br>{14/7} or 7{2}
|- align=center
! Internal angle
| ≈154.286°
| ≈128.571°
| ≈102.857°
| ≈77.1429°
| ≈51.4286°
| ≈25.7143°
| 0°
|}

Deeper truncations of the regular heptagon and [[Heptagram|heptagram]]s can produce isogonal (vertex-transitive) intermediate tetradecagram forms with equally spaced vertices and two edge lengths. Other truncations can form double covering polygons 2{p/q}, namely: t{7/6}={14/6}=2{7/3}, t{7/4}={14/4}=2{7/2}, and t{7/2}={14/2}=2{7}.<ref>The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', [[Biography:Branko Grünbaum|Branko Grünbaum]]</ref>

{| class="wikitable collapsible collapsed"
!colspan=5|Isogonal truncations of heptagon and heptagrams
|-
!Quasiregular
!colspan=3|Isogonal
!Quasiregular<BR>Double covering
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular_polygon_truncation_7_1.svg|100px]]<BR>t{7}={14}
|[[File:regular_polygon_truncation_7_2.svg|100px]]
|[[File:regular_polygon_truncation_7_3.svg|100px]]
|[[File:regular_polygon_truncation_7_4.svg|100px]]
|BGCOLOR="#e0e0ff"|[[File:Regular star polygon 7-3.svg|100px]]<BR>{7/6}={14/6}<BR>=2{7/3}
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular_star_truncation_7-3_1.svg|100px]]<BR>t{7/3}={14/3}
|[[File:regular_star_truncation_7-3_2.svg|100px]]
|[[File:regular_star_truncation_7-3_3.svg|100px]]
|[[File:regular_star_truncation_7-3_4.svg|100px]]
|BGCOLOR="#e0e0ff"|[[File:Regular star polygon 7-2.svg|100px]]<BR>t{7/4}={14/4}<BR>=2{7/2}
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular_star_truncation_7-5_1.svg|100px]]<BR>t{7/5}={14/5}
|[[File:regular_star_truncation_7-5_2.svg|100px]]
|[[File:regular_star_truncation_7-5_3.svg|100px]]
|[[File:regular_star_truncation_7-5_4.svg|100px]]
|BGCOLOR="#e0e0ff"|[[File:Regular polygon 7.svg|100px]]<BR>t{7/2}={14/2}<BR>=2{7}
|}

===Petrie polygons===
Regular skew tetradecagons exist as [[Petrie polygon]] for many higher-dimensional polytopes, shown in these skew orthogonal projections, including:

{| class="wikitable collapsible collapsed"
!colspan=5|Petrie polygons
|-
!colspan=2|B<sub>7</sub>
!colspan=2|2I<sub>2</sub>(7) (4D)
!
|- align=center valign=center
|[[File:7-cube_t6.svg|100px]]<br>[[7-orthoplex|7-orthoplex]]
|[[File:7-cube_t0.svg|100px]]<br>[[7-cube|7-cube]]
|[[File:7-7_duopyramid_ortho.png|100px]]<BR>[[Duopyramid|7-7 duopyramid]]
|[[File:7-7 duoprism ortho-Dih7.png|100px]]<BR>[[Duoprism|7-7 duoprism]]
|
|-
!A<sub>13</sub>
!colspan=2|D<sub>8</sub>
!colspan=2|E<sub>8</sub>
|- align=center valign=center
|[[File:13-simplex_t0.svg|100px]]<br>13-simplex
|[[File:8-cube_t7_B7.svg|100px]]<br>5<sub>11</sub>
|[[File:8-demicube_t0_D8.svg|100px]]<br>1<sub>51</sub>
|[[File:4 21 t0 B7.svg|100px]]<br>4<sub>21</sub>
|[[File:2 41 t0 B7.svg|100px]]<br>2<sub>41</sub>
|}

==References==
{{reflist}}

==External links==
*{{MathWorld|title=Tetradecagon|urlname=Tetradecagon}}

{{Polygons}}

[[Category:Polygons]]

{{Sourceattribution|Tetradecagon|1}}