Dodecagon
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In
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 c ...
, a dodecagon or 12-gon is any twelve-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 t ...
.


Regular dodecagon

A
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...
dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a truncated
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular hexagon'' h ...
, t, or a twice-truncated
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
, tt. The internal angle at each vertex of a regular dodecagon is 150°.


Area

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 open su ...
of a regular dodecagon of side length ''a'' is given by: :\begin A & = 3 \cot\left(\frac \right) a^2 = 3 \left(2+\sqrt \right) a^2 \\ & \simeq 11.19615242\,a^2 \end And in terms of the apothem ''r'' (see also inscribed figure), the area is: :\begin A & = 12 \tan\left(\frac\right) r^2 = 12 \left(2-\sqrt \right) r^2 \\ & \simeq 3.2153903\,r^2 \end In terms of the circumradius ''R'', the area is: :A = 6 \sin\left(\frac\right) R^2 = 3 R^2 The span ''S'' of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area (given side length and span) is: :A = 3aS This can be verified with the trigonometric relationship: :S = a(1+ 2\cos + 2\cos)


Perimeter

The perimeter of a regular dodecagon in terms of circumradius is: :\begin p & = 24R \tan\left(\frac\right) = 12R \sqrt\\ & \simeq 6.21165708246\,R \end The perimeter in terms of apothem is: :\begin p & = 24r \tan\left(\frac\right) = 24r(2-\sqrt)\\ & \simeq 6.43078061835\,r \end This coefficient is double the coefficient found in the apothem equation for area.


Dodecagon construction

As 12 = 22 × 3, regular dodecagon is constructible using
compass-and-straightedge construction 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 ...
:


Dissection

Coxeter states that every zonogon (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 polygons with evenly many sides, in which case the parallelograms are all rhombi. For the ''regular dodecagon'', ''m''=6, and it can be divided into 15: 3 squares, 6 wide 30° rhombs and 6 narrow 15° rhombs. This decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. The sequence OEIS sequence defines the number of solutions as 908, including up to 12-fold rotations and chiral forms in reflection. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons. They are related to the rhombic dissections, with 3 60° rhombi merged into hexagons, half-hexagon trapezoids, or divided into 2 equilateral triangles.


Symmetry

The ''regular dodecagon'' has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges.


Occurrence


Tiling

A regular dodecagon can fill a plane vertex with other regular polygons in 4 ways: Here are 3 example periodic plane tilings that use regular dodecagons, defined by their vertex configuration:


Skew dodecagon

A skew dodecagon is a skew polygon with 12 vertices and edges but not existing on the same plane. The interior of such an dodecagon is not generally defined. A ''skew zig-zag dodecagon'' has vertices alternating between two parallel planes. A
regular skew dodecagon The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrume ...
is
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 fa ...
with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a hexagonal antiprism with the same D5d, +,10symmetry, order 20. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons.


Petrie polygons

The regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as
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 i ...
s in
Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
s. Examples in 4 dimensions are the 24-cell, snub 24-cell,
6-6 duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, whe ...
,
6-6 duopyramid In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, whe ...
. In 6 dimensions 6-cube, 6-orthoplex, 221, 122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell.


Related figures

A
dodecagram In geometry, a dodecagram (γραμμή
Henry George Liddell, Rob ...
is a 12-sided star polygon, represented by symbol . There is one regular star polygon: , using the same vertices, but connecting every fifth point. There are also three compounds: is reduced to 2 as two
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular hexagon'' h ...
s, and is reduced to 3 as three
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
s, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams 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 fa ...
) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated hexagon is a dodecagon, t=. A quasitruncated hexagon, inverted as , is a dodecagram: t=.The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', Branko Grünbaum


Examples in use

In block capitals, the letters E, H and X (and I in a slab serif font) have dodecagonal outlines. A
cross A cross is a geometrical figure consisting of two intersecting lines or bars, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X, is termed a s ...
is a dodecagon, as is the logo for the Chevrolet automobile division. The regular dodecagon features prominently in many buildings. The
Torre del Oro The Torre del Oro ( ar, بُرْج الذَّهَب, burj aḏẖ-ḏẖahab, lit=Tower of Gold) is a dodecagonal military watchtower in Seville, southern Spain. It was erected by the Almohad Caliphate in order to control access to Seville via th ...
is a dodecagonal military watchtower in
Seville Seville (; es, Sevilla, ) is the capital and largest city of the Spanish autonomous community of Andalusia and the province of Seville. It is situated on the lower reaches of the River Guadalquivir, in the southwest of the Iberian Peninsul ...
, southern
Spain , image_flag = Bandera de España.svg , image_coat = Escudo de España (mazonado).svg , national_motto = '' Plus ultra'' ( Latin)(English: "Further Beyond") , national_anthem = (English: "Royal March") , ...
, built by the Almohad dynasty. The early thirteenth century Vera Cruz church in
Segovia Segovia ( , , ) is a city in the autonomous community of Castile and León, Spain. It is the capital and most populated municipality of the Province of Segovia. Segovia is in the Inner Plateau ('' Meseta central''), near the northern slopes o ...
, Spain is dodecagonal. Another example is the Porta di Venere (Venus' Gate), in Spello,
Italy Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the homonymous geographical ...
, built in the 1st century BC has two dodecagonal towers, called "Propertius' Towers". Regular
dodecagonal coin Although the vast majority of coins are round, coins are made in a variety of other shapes, including squares, diamonds, hexagons, heptagons, octagons, decagons, and dodecagons. They have also been struck with scalloped (wavy) edges, and with hole ...
s include: * British threepenny bit from 1937 to 1971, when it ceased to be legal tender. *
British One Pound Coin The British one pound (£1) coin A coin is a small, flat (usually depending on the country or value), round piece of metal or plastic used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in ...
, introduced in 2017. *
Australian 50-cent coin The twelve-sided Australian fifty-cent coin is the third-highest denomination coin of the Australian dollar and the largest in terms of size in circulation. It is equal in size and shape to the Cook Island $5 coin, and both remain the only ...
* Fijian 50 cents * Tongan 50-seniti, since 1974 * Solomon Islands 50 cents * Croatian 25 kuna * Romanian 5000 lei, 2001–2005 * Canadian penny, 1982–1996 * South Vietnamese 20 đồng, 1968–1975 * Zambian 50 ngwee, 1969–1992 * Malawian 50 tambala, 1986–1995 * Mexican 20 centavos, 1992-2009


See also

* Dodecagonal number * Dodecahedron – any polyhedron with 12 faces. *
Dodecagram In geometry, a dodecagram (γραμμή
Henry George Liddell, Rob ...


Notes


External links

*
Kürschak's Tile and Theorem
With interactive animation
The regular dodecagon in the classroom
usin

{{Polygons Constructible polygons Polygons by the number of sides 12 (number) Elementary shapes