Dodecagon

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Regular dodecagon
Regular polygon 12 annotated.svg
A regular dodecagon
Type Regular polygon
Edges and vertices 12
Schläfli symbol {12}, t{6}, tt{3}
Coxeter–Dynkin diagrams CDel node 1.pngCDel 12.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.png
Symmetry group Dihedral (D12), order 2×12
Internal angle (degrees)150°
Properties Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a dodecagon or 12-gon is any twelve-sided polygon.

Contents

Regular dodecagon

Three squares of sides R can be cut and rearranged into a dodecagon of circumradius R, yielding a proof without words that its area is 3R Fit 3 unit squares in unit circle.svg
Three squares of sides R can be cut and rearranged into a dodecagon of circumradius R, yielding a proof without words that its area is 3R

A regular 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 {12} and can be constructed as a truncated hexagon, t{6}, or a twice-truncated triangle, tt{3}. The internal angle at each vertex of a regular dodecagon is 150°.

Area

The area of a regular dodecagon of side length a is given by:

And in terms of the apothem r (see also inscribed figure), the area is:

In terms of the circumradius R, the area is: [1]

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:

This can be verified with the trigonometric relationship:

Perimeter

The perimeter of a regular dodecagon in terms of circumradius is: [2]

The perimeter in terms of apothem is:

This coefficient is double the coefficient found in the apothem equation for area. [3]

Dodecagon construction

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

Regular Dodecagon Inscribed in a Circle.gif
Construction of a regular dodecagon at a given circumcircle
01-Zwolfeck-Seite-gegeben Animation.gif
Construction of a regular dodecagon
at a given side length, animation. (The construction is very similar to that of octagon at a given side length.)

Dissection

12-cube 60 rhomb dissection
12-cube t0 A11.svg 12-gon rhombic dissection-size2.svg 12-gon rhombic dissection2-size2.svg
12-gon rhombic dissection3-size2.svg 12-gon rhombic dissection4-size2.svg 12-gon rhombic dissection5-size2.svg
Isotoxal dodecagon Isotoxal 12-gon rhombic dissection-size2.svg
Isotoxal dodecagon

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. [4] 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 A006245 defines the number of solutions as 908, including up to 12-fold rotations and chiral forms in reflection.

Dissection into 15 rhombs
6-cube graph.svg
6-cube
Rhombic dissected dodecagon.svg Rhombic dissected dodecagon2.svg Rhombic dissected dodecagon3.svg Rhombic dissected dodecagon4.svg Rhombic dissected dodecagon5.svg
Rhombic dissected dodecagon12.svg Rhombic dissected dodecagon6.svg Rhombic dissected dodecagon7.svg Rhombic dissected dodecagon8.svg Rhombic dissected dodecagon13.svg Rhombic dissected dodecagon15.svg

One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons. [5] They are related to the rhombic dissections, with 3 60° rhombi merged into hexagons, half-hexagon trapezoids, or divided into 2 equilateral triangles.

Other regular dissections
Hexagonal cupola flat.png Dissected dodecagon.svg Dissected dodecagon2.svg
Socolar tiling
Wooden pattern blocks dodecagon.JPG
Pattern blocks

Symmetry

The symmetries of a regular dodecagon as shown with colors on edges and vertices. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal, diasymmetry) with mirror lines through vertices, p with mirror lines through edges (perpendicular, persymmetry) i with mirror lines through both vertices and edges (isosymmetry), and g for rotational (gyrosymmetry). a1 labels asymmetry. These lower symmetries allows degrees of freedoms in defining irregular dodecagons. Regular dodecagon symmetries4.png
The symmetries of a regular dodecagon as shown with colors on edges and vertices. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal, diasymmetry) with mirror lines through vertices, p with mirror lines through edges (perpendicular, persymmetry) i with mirror lines through both vertices and edges (isosymmetry), and g for rotational (gyrosymmetry). a1 labels asymmetry. These lower symmetries allows degrees of freedoms in defining irregular dodecagons.

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.

Example dodecagons by symmetry
Full symmetry dodecagon.png
r24
Hexagonal star dodecagon.png
d12
Gyrated dodecagon.png
g12
Truncated hexagon dodecagon.png
p12
Cross dodecagon.png
i8
Hexagonal star d6 dodecagon.png
d6
Twisted hexagonal star dodecagon.png
g6
Truncated triangular star dodecagon.png
p6
D4 star dodecagon.png
d4
Twisted cross dodecagon.png
g4
H-shape-dodecagon.png
p4
Twisted triangle star dodecagon.png
g3
D2 star dodecagon.png
d2
Distorted twisted cross dodecagon.png
g2
Distorted H-shape-dodecagon.png
p2
No symmetry dodecagon.png
a1

Occurrence

Tiling

A regular dodecagon can fill a plane vertex with other regular polygons in 4 ways:

Vertex type 3-12-12.svg Vertex type 4-6-12.svg Vertex type 3-3-4-12.svg Vertex type 3-4-3-12.svg
3.12.124.6.123.3.4.123.4.3.12

Here are 3 example periodic plane tilings that use regular dodecagons, defined by their vertex configuration:

1-uniform2-uniform
1-uniform n4.svg
3.12.12
1-uniform n3.svg
4.6.12
2-uniform n2.svg
3.12.12; 3.4.3.12

Skew dodecagon

A regular skew dodecagon seen as zig-zagging edges of a hexagonal antiprism. Regular skew polygon in hexagonal antiprism.png
A regular skew dodecagon seen as zig-zagging edges of a hexagonal antiprism.

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 is vertex-transitive 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, [2+,10] symmetry, order 20. The dodecagrammic antiprism, s{2,24/5} and dodecagrammic crossed-antiprism, s{2,24/7} also have regular skew dodecagons.

Petrie polygons

The regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensions are the 24-cell, snub 24-cell, 6-6 duoprism, 6-6 duopyramid. 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.

A dodecagram is a 12-sided star polygon, represented by symbol {12/n}. There is one regular star polygon: {12/5}, using the same vertices, but connecting every fifth point. There are also three compounds: {12/2} is reduced to 2{6} as two hexagons, and {12/3} is reduced to 3{4} as three squares, {12/4} is reduced to 4{3} as four triangles, and {12/6} is reduced to 6{2} as six degenerate digons.

Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated hexagon is a dodecagon, t{6}={12}. A quasitruncated hexagon, inverted as {6/5}, is a dodecagram: t{6/5}={12/5}. [7]

Examples in use

In block capitals, the letters E, H and X (and I in a slab serif font) have dodecagonal outlines. A cross is a dodecagon, as is the logo for the Chevrolet automobile division.

The Vera Cruz church in Segovia Segovia Vera Cruz.jpg
The Vera Cruz church in Segovia

The regular dodecagon features prominently in many buildings. The Torre del Oro is a dodecagonal military watchtower in Seville, southern Spain, built by the Almohad dynasty. The early thirteenth century Vera Cruz church in Segovia, Spain is dodecagonal. Another example is the Porta di Venere (Venus' Gate), in Spello, Italy, built in the 1st century BC has two dodecagonal towers, called "Propertius' Towers".

A 1942 British threepence, reverse 1942 threepence reverse.jpg
A 1942 British threepence, reverse

Regular dodecagonal coins include:

See also

Notes

  1. See also Kürschák's geometric proof on the Wolfram Demonstration Project
  2. Plane Geometry: Experiment, Classification, Discovery, Application by Clarence Addison Willis B., (1922) Blakiston's Son & Company, p. 249
  3. Elements of geometry by John Playfair, William Wallace, John Davidsons, (1814) Bell & Bradfute, p. 243
  4. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  5. "Doin' Da' Dodeca'" on mathforum.org
  6. 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)
  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

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