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| {{Even polygon db|Even polygon stat table|p12}}
| | Andreas could be the name my parents gave me and Films it. Guam is where my home is and I have everything i need correct. My day job is a regular control and order filler but I've already gotten another one particular particular. To play golf is among the list of things Enjoy most. He's been working on his website for a few days now. Look it over here: http://m.scalarenergypendants.com/<br><br>My web blog :: [http://m.scalarenergypendants.com/ Quantum Science Pendant] |
| In [[geometry]], a '''dodecagon''' is any [[polygon]] with [[12 (number)|twelve]] sides and twelve [[angle]]s.
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| ==Regular dodecagon==
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| A [[regular polygon|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}.
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| The [[area]] of a regular dodecagon with side ''a'' is given by:
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| :<math>\begin{align} A & = 3 \cot\left(\frac{\pi}{12} \right) a^2 =
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| 3 \left(2+\sqrt{3} \right) a^2 \\
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| & \simeq 11.19615242\,a^2.
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| \end{align}</math>
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| Or, if ''R'' is the radius of the [[circumscribe]]d circle,<ref>See also [[József Kürschák|Kürschák]]'s geometric proof on [http://demonstrations.wolfram.com/KurschaksDodecagon/ the Wolfram Demonstration Project]</ref>
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| :<math>A = 6 \sin\left(\frac{\pi}{6}\right) R^2 = 3 R^2.</math>
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| And, if ''r'' is the radius of the [[inscribe]]d circle,
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| :<math>\begin{align} A & = 12 \tan\left(\frac{\pi}{12}\right) r^2 =
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| 12 \left(2-\sqrt{3} \right) r^2 \\
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| & \simeq 3.2153903\,r^2.
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| \end{align}</math>
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| A simple formula for area (given the two measurements) is: <math>\scriptstyle A\,=\,3ad</math> where ''d'' is the distance between parallel sides.
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| Length ''d'' is the height of the dodecahedron when it sits on a side as base, and the diameter of the inscribed circle.
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| By simple trigonometry, <math>\scriptstyle d\,=\,a(1\,+\,2cos{30^\circ}\,+\,2cos{60^\circ})</math>.
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| == Uses ==
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| A regular dodecagon can fill a plane vertex with other regular polygons:
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| {| class=wikitable
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| |[[File:3.12.12 vertex.png|120px]]<BR>3.12.12
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| |[[File:4.6.12 vertex.png|120px]]<BR>4.6.12
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| |[[File:3.3.4.12 vertex.png|120px]]<BR>3.3.4.12
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| |[[File:3.4.3.12 vertex.png|120px]]<BR>3.4.3.12
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| |}
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| ==Dodecagon construction==
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| A regular dodecagon is [[constructible polygon|constructible]] using [[compass and straightedge]]:
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| [[File:Regular Dodecagon Inscribed in a Circle.gif]]<br>Construction of a regular dodecagon
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| ==Occurrence==
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| ===Tiling===
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| Here are 3 example [[Tiling by regular polygons|periodic plane tilings]] that use dodecagons:
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| {| width=640 class="wikitable"
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| |[[Image:Tile 3bb.svg|205px|Tile 3bb.svg]]<br>[[Truncated hexagonal tiling|Semiregular tiling 3.12.12]]
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| |[[Image:Tile 46b.svg|205px]]<br>[[Great rhombitrihexagonal tiling|Semiregular tiling: 4.6.12]]
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| |[[Image:Dem3343tbc.gif|205px]]<br>A [[Tilings of regular polygons#Other edge-to-edge tilings|demiregular tiling]]:<br>3.3.4.12 & 3.3.3.3.3.3
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| |}
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| ===Pattern blocks===
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| [[File:Wooden pattern blocks dodecagon.JPG||thumb|Dodecagon made with [[pattern blocks]]]]
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| One of the ways the [[mathematical manipulative]] [[pattern blocks]] are used is in creating a number of different dodecagons.<ref>"Doin' Da' Dodeca'" on [http://mathforum.org/berman/patternblocks/classics/responses/chal2/doin_da_dodeca.html mathforum.org]</ref>
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| ===Petrie polygons===
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| The regular dodecagon is the [[Petrie polygon]] for many higher dimensional polytopes, seen as [[orthogonal projection]]s in [[Coxeter plane]]s, including:
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| {| class=wikitable width=540
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| |- align=center valign=top
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| !valign=center|A<sub>11</sub>
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| |[[File:11-simplex_t0.svg|100px]]<br>[[11-simplex]]
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| |[[File:11-simplex_t1.svg|100px]]<br>[[Rectified 11-simplex]]
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| |[[File:11-simplex_t2.svg|100px]]<br>[[Birectified 11-simplex]]
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| |[[File:11-simplex_t3.svg|100px]]<br>[[Trirectified 11-simplex]]
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| |[[File:11-simplex_t4.svg|100px]]<br>[[Quadrirectified 11-simplex]]
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| |[[File:11-simplex_t5.svg|100px]]<br>[[Quintirectified 11-simplex]]
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| |- align=center valign=top
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| !valign=center|BC<sub>6</sub>
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| |[[File:6-cube_t5.svg|100px]]<br>[[6-orthoplex]]
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| |[[File:6-cube_t4.svg|100px]]<br>[[Rectified 6-orthoplex]]
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| |[[File:6-cube_t3.svg|100px]]<br>[[Birectified 6-orthoplex]]
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| |[[File:6-cube_t2.svg|100px]]<br>[[Birectified 6-cube]]
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| |[[File:6-cube_t1.svg|100px]]<br>[[Rectified 6-cube]]
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| |[[File:6-cube_t0.svg|100px]]<br>[[6-cube]]
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| |- align=center valign=top
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| !valign=center|D<sub>7</sub>
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| |[[File:7-cube_t6_B6.svg|100px]]<br>[[7-orthoplex|t<sub>5</sub>(1<sub>41</sub>)]]
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| |[[File:7-cube_t5_B6.svg|100px]]<br>[[Rectified 7-orthoplex|t<sub>4</sub>(1<sub>41</sub>)]]
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| |[[File:7-cube_t4_B6.svg|100px]]<br>[[Birectified 7-orthoplex|t<sub>3</sub>(1<sub>41</sub>)]]
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| |[[File:7-cube_t3_B6.svg|100px]]<br>[[Birectified 7-cube|t<sub>2</sub>(1<sub>41</sub>)]]
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| |[[File:7-demicube_t1_D7.svg|100px]]<br>[[Rectified 7-cube|t<sub>1</sub>(1<sub>41</sub>)]]
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| |[[File:7-demicube_t0_D7.svg|100px]]<br>[[7-demicube|t<sub>0</sub>(1<sub>41</sub>)]]
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| |- align=center valign=top
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| !valign=center|[[E6 (mathematics)|E<sub>6</sub>]]
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| |[[File:E6 graph.svg|100px]]<br>[[2 21 polytope|t<sub>0</sub>(2<sub>21</sub>)]]
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| |[[File:Up 2 21 t1 E6.svg|100px]]<br>t<sub>1</sub>(2<sub>21</sub>)
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| |[[File:Up 2 21 t2 E6.svg|100px]]<br>t<sub>1</sub>(1<sub>22</sub>)
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| |[[File:Gosset 1 22 polytope.png|100px]]<br>[[1 22 polytope|t<sub>0</sub>(1<sub>22</sub>)]]
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| |- align=center valign=top
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| !valign=center|[[F4 (mathematics)|F<sub>4</sub>]]
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| |[[File:24-cell_t0_F4.svg|100px]]<br>[[24-cell]]
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| |[[File:24-cell_t1_F4.svg|100px]]<br>[[Rectified 24-cell]]
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| |[[File:24-cell h01 F4.svg|100px]]<br>[[Snub 24-cell]]
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| |}
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| ==Examples in use==
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| In [[block capitals]], the letters [[E]], [[H]] and [[X]] (and [[I]] in a [[slab serif]] font) have dodecagonal outlines.
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| [[File:Segovia Vera Cruz.jpg|thumb|The Vera Cruz church in [[Segovia]]]]
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| The regular dodecagon features prominently in many buildings. The [[Torre del Oro]] is a dodecagonal military [[Watchtower (fortification)|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".
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| [[File:1942 threepence reverse.jpg|thumb|A 1942 British threepence, reverse]]
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| Regular dodecagonal coins include:
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| *[[Threepence (British coin)|British threepenny bit]] from 1937 to 1971, when it ceased to be legal tender.
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| *[[Australian 50-cent coin]]
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| *[[Coins of the Fijian dollar|Fijian 50 cents]]
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| *[[Tongan paʻanga|Tongan 50-seniti]], since 1974
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| *[[Solomon Islands dollar|Solomon Islands 50 cents]]
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| *[[Croatian kuna|Croatian 25 kuna]]
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| *[[Romanian leu|Romanian 5000 lei]], 2001–2005
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| *[[Penny (Canadian coin)|Canadian penny]], 1982–1996
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| *[[South Vietnamese đồng|South Vietnamese 25 đồng]], 1968–1975
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| *[[Zambian kwacha|Zambian 50 ngwee]], 1969–1992
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| *[[Malawian kwacha|Malawian 50 tambala]], 1986–1995
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| *[[Mexican peso|Mexican 20 centavos]], since 1992
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| ==See also==
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| *[[Dodecagonal number]]
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| *[[Dodecahedron]] – a regular [[polyhedron]] with 12 [[pentagon]]al faces.
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| *[[Dodecagram]]
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| ==Notes==
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| {{Reflist}}
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| ==External links==
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| *{{MathWorld |urlname=Dodecagon |title=Dodecagon}}
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| *[http://www.cut-the-knot.org/Curriculum/Geometry/KurschakTile.shtml Kürschak's Tile and Theorem]
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| *[http://www.mathopenref.com/dodecagon.html Definition and properties of a dodecagon] With interactive animation
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| {{Polygons}}
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| [[Category:Polygons]]
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