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}
Coxeter diagram CDel node 1.pngCDel 12.pngCDel node.png
Symmetry group 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.

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

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.gif
A demiregular tiling:
3.3.4.12 & 3.3.3.3.3.3

Pattern blocks[edit]

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]

Petrie polygons[edit]

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

A11 11-simplex t0.svg
11-simplex
11-simplex t1.svg
Rectified 11-simplex
11-simplex t2.svg
Birectified 11-simplex
11-simplex t3.svg
Trirectified 11-simplex
11-simplex t4.svg
Quadrirectified 11-simplex
11-simplex t5.svg
Quintirectified 11-simplex
BC6 6-cube t5.svg
6-orthoplex
6-cube t4.svg
Rectified 6-orthoplex
6-cube t3.svg
Birectified 6-orthoplex
6-cube t2.svg
Birectified 6-cube
6-cube t1.svg
Rectified 6-cube
6-cube t0.svg
6-cube
D7 7-cube t6 B6.svg
t5(141)
7-cube t5 B6.svg
t4(141)
7-cube t4 B6.svg
t3(141)
7-cube t3 B6.svg
t2(141)
7-demicube t1 D7.svg
t1(141)
7-demicube t0 D7.svg
t0(141)
E6 E6 graph.svg
t0(221)
Up 2 21 t1 E6.svg
t1(221)
Up 2 21 t2 E6.svg
t1(122)
Gosset 1 22 polytope.png
t0(122)
F4 24-cell t0 F4.svg
24-cell
24-cell t1 F4.svg
Rectified 24-cell
24-cell h01 F4.svg
Snub 24-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

External links[edit]