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}
Coxeter diagram 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°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a dodecagon is any polygon with twelve sides and twelve angles.

A regular dodecagon has Schläfli symbol {12} and can be constructed as a quasiregular truncated hexagon, t{6}, which alternates two types of edges.

Regular dodecagon[edit]

A regular dodecagon has all sides of equal length and all angles equal to 150°. It has 12 lines of symmetry and rotational symmetry of order 12. Its Schläfli symbol is {12}.

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

\begin{align} A & = 3 \cot\left(\frac{\pi}{12} \right) a^2 = 
                     3 \left(2+\sqrt{3} \right) a^2 \\
                 & \simeq 11.19615242\,a^2.
 \end{align}

Or, if R is the radius of the circumscribed circle,[1]

A = 6 \sin\left(\frac{\pi}{6}\right) R^2 = 3 R^2.

And, if r is the radius of the inscribed circle,

\begin{align} A & = 12 \tan\left(\frac{\pi}{12}\right) r^2 = 
                     12 \left(2-\sqrt{3} \right) r^2 \\
                 & \simeq 3.2153903\,r^2.
 \end{align}

A simple formula for area (given the two measurements) is: \scriptstyle A\,=\,3ad where d is the distance between parallel sides.

Length d is the height of the dodecagon when it sits on a side as base, and the diameter of the inscribed circle.

By simple trigonometry, \scriptstyle d\,=\,a(1\,+\,2cos{30^\circ}\,+\,2cos{60^\circ}).

The perimeter for an inscribed dodecagon of radius 1 is 12√(2 - √3), or approximately 6.21165708246. [2]

The perimeter for a circumscribed dodecagon of radius 1 is 24(2 – √3), or approximately 6.43078061835. Interestingly, this is double the value of the area of the inscribed dodecagon of radius 1. [3]

With respect to the above-listed equations for area and perimeter, when the radius of the inscribed dodecagon is 1, note that the area of the inscribed dodecagon is 12(2 – √3) and the perimeter of this same inscribed dodecagon is 12√(2 - √3).

Uses[edit]

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

3.12.12 vertex.png
3.12.12
4.6.12 vertex.png
4.6.12
3.3.4.12 vertex.png
3.3.4.12
3.4.3.12 vertex.png
3.4.3.12

Dodecagon construction[edit]

A regular dodecagon is constructible using compass and straightedge:

Regular Dodecagon Inscribed in a Circle.gif
Construction of a regular dodecagon

Dissection[edit]

Hexagonal cupola flat.png
A regular dodecagon can be dissected into a central hexagon, and alternating triangles and squares
Wooden pattern blocks dodecagon.JPG
Regular dodecagon made with pattern blocks

One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons.[4]

Occurrence[edit]

Tiling[edit]

Here are 3 example periodic plane tilings that use dodecagons:

Tile 3bb.svg
Semiregular tiling 3.12.12
Tile 46b.svg
Semiregular tiling: 4.6.12
Dem3343tbc.png
A demiregular tiling:
3.3.4.12 & 3.3.3.3.3.3

Related figures[edit]

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.

n 1 2 3 4 5 6
Form Polygon Compounds Star polygon Compound
Image Regular polygon 12.svg
{12/1} = {12}
Regular star figure 2(6,1).svg
{12/2} or 2{6}
Regular star figure 3(4,1).svg
{12/3} or 3{4}
Regular star figure 4(3,1).svg
{12/4} or 4{3}
Regular star polygon 12-5.svg
{12/5}
Regular star figure 6(2,1).svg
{12/6} or 6{2}

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}.[5]

Vertex-transitive truncations of the hexagon
Quasiregular Isogonal Quasiregular
Regular polygon truncation 6 1.svg
t{6}={12}
Regular polygon truncation 6 2.svg Regular polygon truncation 6 3.svg Regular polygon truncation 6 4.svg
t{6/5}={12/5}

Petrie polygons[edit]

The regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes, including:

E6 F4 2G2 (4D)
E6 graph.svg
221
Gosset 1 22 polytope.png
122
24-cell t0 F4.svg
24-cell
24-cell h01 F4.svg
Snub 24-cell
6-6 duopyramid ortho-3.png
6-6 duopyramid
6-6 duoprism ortho-3.png
6-6 duoprism
A11 D7 B6
11-simplex t0.svg
11-simplex
7-cube t6 B6.svg
(411)
7-demicube t0 D7.svg
141
6-cube t5.svg
6-orthoplex
6-cube t0.svg
6-cube

It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell.

Examples in use[edit]

In block capitals, the letters E, H and X (and I in a slab serif font) have dodecagonal outlines.

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

Regular dodecagonal coins include:

See also[edit]

Notes[edit]

  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 [1]
  3. ^ Elements of geometry by John Playfair, William Wallace, John Davidsons, (1814) Bell & Bradfute, p. 243 [2]
  4. ^ "Doin' Da' Dodeca'" on mathforum.org
  5. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

External links[edit]