Tetradecagon

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Regular tetradecagon
Regular polygon 14 annotated.svg
A regular tetradecagon
Type Regular polygon
Edges and vertices 14
Schläfli symbol {14}, t{7}
Coxeter diagram CDel node 1.pngCDel 14.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node 1.png
Symmetry group Dihedral (D14), order 2×14
Internal angle (degrees)154+2/7°
Dual polygon Self
Properties Convex, cyclic, equilateral, isogonal, isotoxal

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

Contents

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

Construction

As 14 = 2 × 7, a regular tetradecagon cannot be constructed using a compass and straightedge. [1] However, it is constructible using neusis with use of the angle trisector, [2] or with a marked ruler, [3] as shown in the following two examples.

Tetradecagon with given circumcircle:
An animation (1 min 47 s) from a neusis construction with radius of circumcircle
O
A
-
=
6
{\displaystyle {\overline {OA}}=6}
,
according to Andrew M. Gleason, based on the angle trisection by means of the Tomahawk., pause at the end of 25 s 01-Tetradecagon-Tomahawk.gif
Tetradecagon with given circumcircle:
An animation (1 min 47 s) from a neusis construction with radius of circumcircle ,
according to Andrew M. Gleason, based on the angle trisection by means of the Tomahawk., pause at the end of 25 s
Tetradecagon with given side length:
An animation (1 min 20 s) from a neusis construction with marked ruler, according to David Johnson Leisk (Crockett Johnson) for the heptagon, pause at the end of 30 s. 01-Vierzehneck-nach Johnson.gif
Tetradecagon with given side length:
An animation (1 min 20 s) from a neusis construction with marked ruler, according to David Johnson Leisk (Crockett Johnson) for the heptagon, pause at the end of 30 s.

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

Approximated Tetradecagon Inscribed in a Circle.gif
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 (3 min 16 s), pause at the end of 25 s 01-Tetradecagon-Animation.gif
Regular tetradecagon, approximation construction as an animation (3 min 16 s), pause at the end of 25 s

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

  • Constructed side length of the tetradecagon in GeoGebra (display max 15 decimal places)
  • Side length of the tetradecagon
  • Absolute error of the constructed side length
Up to the max. displayed 15 decimal places is the absolute error
  • Constructed central angle of the tetradecagon in GeoGebra (display significant 13 decimal places)
  • Central angle of the tetradecagon
  • Absolute error of the constructed central angle
Up to the indicated significant 13 decimal places is the absolute error

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 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. Symmetries of tetradecagon.png
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. [4] 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 tetradecagon constructed by seven 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.

Dissection

14-cube t0 A13.svg
14-cube projection
14-gon rhombic dissection-size2.svg
84 rhomb dissection

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. [5] In particular this is true for regular polygons 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, with 21 of 672 faces. The list OEIS:  A006245 defines the number of solutions as 24698, including up to 14-fold rotations and chiral forms in reflection.

Dissection into 21 rhombs
7-cube graph.svg 14-gon-dissection.svg 14-gon-dissection-star.svg 14-gon rhombic dissection2.svg 14-gon rhombic dissectionx.svg 14-gon-dissection-random.svg

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. [6]

The flag of Malaysia, featuring a fourteen-pointed star Flag of Malaysia.svg
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 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.

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.

Compounds and star polygons
n1234567
FormRegularCompoundStar polygonCompoundStar polygonCompound
Image Regular polygon 14.svg
{14/1} = {14}
CDel node 1.pngCDel 14.pngCDel node.png
Regular star figure 2(7,1).svg
{14/2} = 2{7}
CDel node h3.pngCDel 14.pngCDel node.png
Regular star polygon 14-3.svg
{14/3}
CDel node 1.pngCDel 14.pngCDel rat.pngCDel 3x.pngCDel node.png
Regular star figure 2(7,2).svg
{14/4} = 2{7/2}
CDel node h3.pngCDel 14.pngCDel rat.pngCDel 2x.pngCDel node.png
Regular star polygon 14-5.svg
{14/5}
CDel node 1.pngCDel 14.pngCDel rat.pngCDel 5.pngCDel node.png
Regular star figure 2(7,3).svg
{14/6} = 2{7/3}
CDel node h3.pngCDel 14.pngCDel rat.pngCDel 3x.pngCDel node.png
Regular star figure 7(2,1).svg
{14/7} or 7{2}
Internal angle≈154.286°≈128.571°≈102.857°≈77.1429°≈51.4286°≈25.7143°

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 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}. [7]

Isotoxal forms

An isotoxal polygon can be labeled as {pα} with outer most internal angle α, and a star polygon {(p/q)α}, with q is a winding number, and gcd(p,q)=1, q<p. Isotoxal tetradecagons have p=7, and since 7 is prime all solutions, q=1..6, are polygons.

Isotoxal tetradecagon.svg
{7α}
Intersecting isotoxal tetradecagon.svg
{(7/2)α}
Intersecting isotoxal tetradecagon3.svg
{(7/3)α}
Intersecting isotoxal tetradecagon4.svg
{(7/4)α}
Intersecting isotoxal tetradecagon5.svg
{(7/5)α}
Intersecting isotoxal tetradecagon6.svg
{(7/6)α}

Petrie polygons

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

Related Research Articles

Octagon polygon shape with eight sides

In geometry, an octagon is an eight-sided polygon or 8-gon.

Decagon shape with ten sides

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

Heptagon shape with seven sides

In geometry, a heptagon is a seven-sided polygon or 7-gon.

Enneadecagon Polygon with 19 edges

In geometry an enneadecagon or enneakaidecagon or 19-gon is a nineteen-sided polygon.

Tridecagon Polygon with 13 edges

In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon.

Pentacontagon Polygon with 50 edges

In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon. The sum of any pentacontagon's interior angles is 8640 degrees.

Pentadecagon Polygon with 15 edges

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.

Hexadecagon Polygon with 16 edges

In mathematics, a hexadecagon is a sixteen-sided polygon.

Hexacontagon Polygon with 60 edges

In geometry, a hexacontagon or hexecontagon or 60-gon is a sixty-sided polygon. The sum of any hexacontagon's interior angles is 10440 degrees.

Icositetragon Polygon with 24 edges

In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.

Heptacontagon Polygon with 70 edges

In geometry, a heptacontagon or 70-gon is a seventy-sided polygon. The sum of any heptacontagon's interior angles is 12240 degrees.

Enneacontagon Polygon with 90 edges

In geometry, an enneacontagon or enenecontagon or 90-gon is a ninety-sided polygon. The sum of any enneacontagon's interior angles is 15840 degrees.

Tetracontadigon Polygon with 42 edges

In geometry, a tetracontadigon or 42-gon is a forty-two-sided polygon. The sum of any tetracontadigon's interior angles is 7200 degrees.

Hexacontatetragon Polygon with 64 edges

In geometry, a hexacontatetragon or 64-gon is a sixty-four-sided polygon. The sum of any hexacontatetragon's interior angles is 11160 degrees.

360-gon

In geometry, a 360-gon is a polygon with 360 sides. The sum of any 360-gon's interior angles is 64440 degrees.

Icosidigon Polygon with 22 edges

In geometry, an icosidigon or 22-gon is a twenty-two-sided polygon. The sum of any icosidigon's interior angles is 3600 degrees.

Icositrigon

In geometry, an icositrigon or 23-gon is a 23-sided polygon. The icositrigon has the distinction of being the smallest regular polygon that is not neusis constructible.

Icosihexagon Polygon with 26 edges

In geometry, an icosihexagon or 26-gon is a twenty-six-sided polygon. The sum of any icosihexagon's interior angles are 4320°.

Icosioctagon Polygon with 28 edges

In geometry, an icosioctagon or 28-gon is a twenty eight sided polygon. The sum of any icosioctagon's interior angles is 4680 degrees.

Triacontatetragon Polygon with 34 edges

In geometry, a triacontatetragon or triacontakaitetragon is a thirty-four-sided polygon or 34-gon. The sum of any triacontatetragon's interior angles is 5760 degrees.

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" (PDF). Journal de Mathématiques: 366–372.
  2. 1 2 Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, p. 186 (Fig.1) –187" (PDF). The American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. Archived from the original (PDF) on 2016-02-02.
  3. 1 2 Weisstein, Eric W. "Heptagon." From MathWorld, A Wolfram Web Resource.
  4. 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)
  5. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  6. The Numismatist, Volume 96, Issues 7-12, Page 1409, American Numismatic Association, 1983.
  7. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum