| Regular tetradecagon | |
|---|---|
 A regular tetradecagon | |
| Type | Regular polygon |
| Edges andvertices | 14 |
| Schläfli symbol | {14}, t{7} |
| Coxeter–Dynkin diagrams |     |
| Symmetry group | Dihedral (D14), order 2×14 |
| Internal angle (degrees) | 154+2/7° |
| Properties | Convex,cyclic,equilateral,isogonal,isotoxal |
| Dual polygon | Self |
Ingeometry, atetradecagon ortetrakaidecagon or 14-gon is a fourteen-sidedpolygon.
Aregular tetradecagon hasSchläfli symbol {14} and can be constructed as a quasiregulartruncatedheptagon, t{7}, which alternates two types of edges.
Thearea of aregular tetradecagon of side lengtha is given by
As 14 = 2 × 7, a regular tetradecagon cannot beconstructed using acompass and straightedge.[1] However, it is constructible usingneusis with use of theangle trisector,[2] or with a marked ruler,[3] as shown in the following two examples.
Theregular tetradecagon hasDih14 symmetry, order 28. There are 3 subgroup dihedral symmetries: Dih7, Dih2, and Dih1, and 4cyclic 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 isr28 and no symmetry is labeleda1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), andi when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled asg for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only theg14 subgroup has no degrees of freedom but can be seen asdirected edges.
The highest symmetry irregular tetradecagons ared14, anisogonal tetradecagon constructed by seven mirrors which can alternate long and short edges, andp14, anisotoxal tetradecagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms areduals of each other and have half the symmetry order of the regular tetradecagon.
 14-cube projection |  84 rhomb dissection |
Coxeter states that everyzonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected intom(m-1)/2 parallelograms.[5]In particular this is true forregular polygons with evenly many sides, in which case the parallelograms are all rhombi. For theregular tetradecagon,m=7, and it can be divided into 21: 3 sets of 7 rhombs. This decomposition is based on aPetrie polygon projection of a7-cube, with 21 of 672 faces. The listOEIS: A006245 defines the number of solutions as 24698, including up to 14-fold rotations and chiral forms in reflection.
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The regular tetradecagon is used as the shape of some commemorative gold and silverMalaysian coins, the number of sides representing the 14 states of the Malaysian Federation.[6]
Atetradecagram is a 14-sided star polygon, represented by symbol {14/n}. There are two regularstar 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 twoheptagons, while {14/4} and {14/6} are reduced to 2{7/2} and 2{7/3} as two differentheptagrams, and finally {14/7} is reduced to sevendigons.
A notable application of a fourteen-pointed star is in theflag of Malaysia, which incorporates a yellow {14/6} tetradecagram in the top-right corner, representing the unity of the thirteenstates with thefederal government.
| 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} | .svg) {14/2} = 2{7}  |  {14/3}   | .svg) {14/4} = 2{7/2}  |  {14/5}  | .svg) {14/6} = 2{7/3} | .svg) {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 andheptagrams 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]
| 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} |
Anisotoxal polygon can be labeled as {pα} with outer most internal angle α, and a star polygon {(p/q)α}, withq is awinding number, and gcd(p,q)=1,q<p. Isotoxal tetradecagons havep=7, and since 7 is prime all solutions, q=1..6, are polygons.
 {7α} |  {(7/2)α} |  {(7/3)α} |  {(7/4)α} |  {(7/5)α} |  {(7/6)α} |
Regular skew tetradecagons exist asPetrie polygon for many higher-dimensional polytopes, shown in these skeworthogonal 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 |