geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

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

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

Regular tetradecagon

A '' regular tetradecagon'' has Schläfli symbol 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 "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than '' hepta-'', a Greek-derived nu ...

, t, which alternates two types of edges. The

area Area is the quantity that expresses the extent of a region on the plane or on a curved 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 ope ...

of a regular 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. However, it is constructible using neusis 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 group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

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 active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

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 seen as

directed edge In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pai ...

s. 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 ''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, P ...

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 and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

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. Examples A regular polygon is a zonogon if and ...

(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 direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

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 ...

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 co ...

, 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. The federation, federal constitutional monarchy consists of States and federal territories of Malaysia, thirteen states and three federal territories, separated by the South China Sea into two r ...

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 polygons: 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 drawn with seven straight strokes. The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek suffix ''-gram''. The ''-gram'' suffix derives from ''γρ� ...

s, and finally is reduced to seven

digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...

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 ( ms, Jalur Gemilang), is composed of a field of 14 alternating red and white stripes along the fly and a blue canton bearing a crescent and a 14-point star known as the ''Bintan ...

, which incorporates a yellow tetradecagram in the top-right corner, representing the unity of the thirteen states with the federal government. 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 in ...

) 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 ( he, ברנקו גרונבאום; 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 around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...

, 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 wer ...

s, including:

References

External links

* {{Polygons Polygons by the number of sides