geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

Regular tetradecagon

A ''

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

tetradecagon'' has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

and can be constructed as a quasiregular truncated

heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...

, t, which alternates two types of edges. The

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of a

tetradecagon of side length ''a'' is given by :

A = \fraca^2\cot\frac \approx 15.3345a^2

Construction

As 14 = 2 × 7, a regular tetradecagon cannot be constructed using a

compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...

. However, it is constructible using

neusis In geometry, the neusis (; ; plural: ) is a geometric construction method that was used in antiquity by Greek mathematics, Greek mathematicians. Geometric construction The neusis construction consists of fitting a line element of given length ...

with use of the

angle trisector Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...

, or with a marked ruler,Weisstein, Eric W. "Heptagon." From MathWorld, A Wolfram Web Resource.
/ref> as shown in the following two examples. 01-Tetradecagon-Tomahawk

Symmetry

The ''regular tetradecagon'' has Dih₁₄ symmetry, order 28. There are 3 subgroup dihedral symmetries: Dih₇, Dih₂, and Dih₁, and 4

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

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 John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many br ...

labels these by a letter and group order. 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 be 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 In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given tw ...

tetradecagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, ...

of each other and have half the symmetry order of the regular tetradecagon.

Dissection

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

states that every

zonogon In geometry, a zonogon is a centrally-symmetric, convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations, the two-dimensional analog of a zonohedron. Ex ...

(a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into ''m''(''m''-1)/2 parallelograms. In particular this is true for

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

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 In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...

projection of a

7-cube In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , being com ...

, with 21 of 672 faces. The list defines the number of solutions as 24698, including up to 14-fold rotations and chiral forms in reflection.

Numismatic use

The regular tetradecagon is used as the shape of some commemorative gold and silver

Malaysia Malaysia is a country in Southeast Asia. Featuring the Tanjung Piai, southernmost point of continental Eurasia, it is a federation, federal constitutional monarchy consisting of States and federal territories of Malaysia, 13 states and thre ...

n coins, the number of sides representing the 14 states of the Malaysian Federation.

Related figures

A tetradecagram is a 14-sided star polygon, represented by symbol . There are two regular

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...

s: and , using the same vertices, but connecting every third or fifth points. There are also three compounds: is reduced to 2 as two

s, while and are reduced to 2 and 2 as two different

heptagram A heptagram, septagram, septegram or septogram is a seven-point star polygon, star drawn with seven straight strokes. The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek language, Greek suffix ''wikt:-gram, -gram ...

s, and finally is reduced to seven

digon In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides wou ...

s. A notable application of a fourteen-pointed star is in the

flag of Malaysia The national flag of Malaysia, also known as the Stripes of Glory (, also "Stripes of Excellence") is composed of a field of 14 alternating red and white stripes along the Flag terminology, fly and a blue Flag terminology, canton bearing a Star ...

, which incorporates a yellow tetradecagram in the top-right corner, representing the unity of the thirteen

states State most commonly refers to: * State (polity), a centralized political organization that regulates law and society within a territory **Sovereign state, a sovereign polity in international law, commonly referred to as a country **Nation state, a ...

with the

federal government A federation (also called a federal state) is an entity characterized by a political union, union of partially federated state, self-governing provinces, states, or other regions under a #Federal governments, federal government (federalism) ...

. Deeper truncations of the regular heptagon and

s can produce isogonal (

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...

) intermediate tetradecagram forms with equally spaced vertices and two edge lengths. Other truncations can form double covering polygons 2, namely: t

2, t

2, and t

2.The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'',

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentisotoxal polygon can be labeled as with outer most internal angle α, and a star polygon , with ''q'' is a

winding number In mathematics, the winding number or winding index of a closed curve in the plane (mathematics), plane around a given point (mathematics), point is an integer representing the total number of times that the curve travels counterclockwise aroun ...

, and gcd(''p'',''q'')=1, ''q''<''p''. Isotoxal tetradecagons have ''p''=7, and since 7 is prime all solutions, q=1..6, are polygons.

Petrie polygons

Regular skew tetradecagons exist as

for many higher-dimensional polytopes, shown in these skew

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ...

s, including:

References

External links

* {{Polygons Polygons by the number of sides