Tetradecagon

Regular tetradecagon
A regular tetradecagon
Type	Regular polygon
Edges and vertices	14
Schläfli symbol	{14}, t{7}
Coxeter diagram
Symmetry group	Dihedral (D14), order 2×14
Internal angle (degrees)	≈154.2857°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a tetradecagon (or tetrakaidecagon) is a fourteen-sided polygon or 14-gon.

Regular tetradecagon

A regular tetradecagon has Schläfli symbol {14} and can be constructed as a quasiregular truncated heptagon, t{7}, which alternates two types of edges.

The area of a regular tetradecagon of side length a is given by

$A = \frac{14}{4}a^2\cot\frac{\pi}{14}\simeq 15.3345a^2$

Construction

As 14 = 2 × 7, a regular tetradecagon cannot be constructed using a compass and straightedge.^[1] However, it is constructible using neusis, as for example in the following illustration with use of the angle trisector.^[2]

An animation from a neusis construction with radius of circumcircle $\overline{OA} = 6$,
according to Andrew M. Gleason,^[2] based on the angle trisection by means of the Tomahawk.

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

Construction of an approximated regular tetradecagon

Another possible animation of an approximate construction, also possible with using straightedge and compass.

Regular tetradecagon, approximation construction as an animation

Based on the unit circle R = 1 [unit of length]

• Constructed side of tetradecagon a = 0.445041867912621...[unit of length]
• Side of the tetradecagon $a_{SHOULD} = 2 \cdot \sin\left(\frac{180^\circ}{14} \right)$ = 0.44504186791262880... [unit of length]
• Absolute error of the constructed side: $F_a = a - a_{SHOULD}$ = -7,8...E-15 [unit of length]

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 approximately -7.8 mm.

For details, see: Wikibooks: Tetradecagon, construction description (German)

Symmetry

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 Dih₁₄ symmetry, order 28. There are 3 subgroup dihedral symmetries: Dih₇, Dih₂, and Dih₁, and 4 cyclic group symmetries: Z₁₄, Z₇, Z₂, and Z₁.

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. John Conway labels these by a letter and group order.^[3] 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, a isogonal dotetradecagon constructed by five mirrors which can alternate long and short edges, and p14, an isotoxal tetradecagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular tetradecagon.

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.^[4]

Related figures

A tetradecagram is a 14-sided star polygon, represented by symbol {14/n}. There are two regular star polygons: {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 heptagons, while {14/4} and {14/6} are reduced to 2{7/2} and 2{7/3} as two different heptagrams, and finally {14/7} is reduced to seven digons.

Compounds and star polygons 

n	1	2	3	4	5	6	7
Form	Regular	Compound	Star polygon	Compound	Star polygon	Compound
Image	{14/1} = {14}	{14/2} = 2{7}	{14/3}	{14/4} = 2{7/2}	{14/5}	{14/6} = 2{7/3}	{14/7} or 7{2}
Internal angle	≈154.286°	≈128.571°	≈102.857°	≈77.1429°	≈51.4286°	≈25.7143°	0°

Deeper truncations of the regular heptagon and heptagrams can produce isogonal (vertex-transitive) intermediate tetradecagram forms with equally spaced vertices and two edge lengths. Other truncations can form double covering polyons 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}.^[5]

Isogonal truncations of heptagon and heptagrams 

Quasiregular	Isogonal			Quasiregular Double covering
t{7}={14}				{7/6}={14/6} =2{7/3}
t{7/3}={14/3}				t{7/4}={14/4} =2{7/2}
t{7/5}={14/5}				t{7/2}={14/2} =2{7}

Petrie polygons

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

Petrie polygons 

B7		2I2(7) (4D)
7-orthoplex	7-cube	7-7 duopyramid	7-7 duoprism
A13	D8		E8
13-simplex	511	151	421	241

References

1. Wantzel, Pierre (1837). "Recherches sur les moyens de Reconnaître si un Problème de géométrie peau se résoudre avec la règle et le compas". Journal de Mathématiques: 366–372.
2. Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, and the triskaidecagon p. 186 (Fig.1) –187" (PDF). The American Mathematical Monthly 95 (3): 185–194. Archived from the original (PDF) on 2016-01-31.
3. 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)
4. The Numismatist, Volume 96, Issues 7-12, Page 1409, American Numismatic Association, 1983.
5. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

External links

• Weisstein, Eric W., "Tetradecagon", MathWorld.