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| A '''dodecahedron''' is any [[polyhedron]] with twelve faces, but usually a '''regular dodecahedron''' is meant: a [[Platonic solid]] composed of twelve regular [[pentagon]]al faces, with three meeting at each vertex. It has twenty (20) vertices and thirty (30) edges. Its [[dual polyhedron]] is the [[icosahedron]]. To the ancient [[Greeks]], the dodecahedron was a [[Platonic solid#History|symbol of the universe]]. If one were to make every one of the Platonic solids with edges of the same length, the dodecahedron would be the largest.
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| == Area and volume ==
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| The area ''A'' and the [[volume]] ''V'' of a regular dodecahedron of edge length ''a'' are:
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| :<math>A = 3\sqrt{25+10\sqrt{5}} a^2 \approx 20.64572a^2</math>
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| :<math>V = \frac{1}{4} (15+7\sqrt{5}) a^3 \approx 7.66311896a^3</math>
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| == Cartesian coordinates ==
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| The following [[Cartesian coordinates]] define the vertices of a dodecahedron centered at the origin:
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| : (±1, ±1, ±1)
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| : (0, ±1/φ, ±φ)
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| : (±1/φ, ±φ, 0)
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| : (±φ, 0, ±1/φ)
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| where [[Golden ratio|φ]] = (1+√5)/2 is the [[golden ratio]] (also written τ). The side length is 2/φ = √5−1. The containing sphere has a radius of √3.
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| The [[dihedral angle]] of a dodecahedron is 2arctan(φ) or approximately 116.565 degrees.
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| == Geometric relations ==
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| The ''regular dodecahedron'' is the third in an infinite set of [[truncated trapezohedron|truncated trapezohedra]] which can be constructed by truncating the two axial vertices of a [[pentagonal trapezohedron]].
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| The [[stellation]]s of the dodecahedron make up three of the four [[Kepler-Poinsot polyhedra]].
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| A [[Rectification (geometry)|rectified]] dodecahedron forms an [[icosidodecahedron]].
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| The regular dodecahedron has 120 symmetries, forming the group <math>A_5\times Z_2</math>.
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| === Vertex arrangement ===
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| The dodecahedron shares its [[vertex arrangement]] with four nonconvex [[uniform polyhedron]]s and three [[Polyhedral compound#Uniform compounds|uniform compounds]].
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| Five [[cube]]s fit within, with their edges as diagonals of the dodecahedron's faces, and together these make up the regular [[polyhedral compound]] of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a dodecahedron.
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| {| class="prettytable" width=500
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| | [[Image:Great stellated dodecahedron.png|125px]]<BR>[[Great stellated dodecahedron]]
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| | [[Image:Small ditrigonal icosidodecahedron.png|125px]]<BR>[[Small ditrigonal icosidodecahedron]]
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| | [[Image:Ditrigonal dodecadodecahedron.png|125px]]<BR>[[Ditrigonal dodecadodecahedron]]
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| | [[Image:Great ditrigonal icosidodecahedron.png|125px]]<BR>[[Great ditrigonal icosidodecahedron]]
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| |-
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| | [[Image:Compound of five cubes.png|125px]]<BR>[[Compound of five cubes]]
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| | [[Image:Compound of five tetrahedra.png|125px]]<BR>[[Compound of five tetrahedra]]
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| | [[Image:Compound of ten tetrahedra.png|125px]]<BR>[[Compound of ten tetrahedra]]
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| |}
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| ===Icosahedron vs dodecahedron===
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| When a dodecahedron is inscribed in a [[sphere]], it occupies more of the sphere's volume (66.49%) than an [[icosahedron]] inscribed in the same sphere (60.54%).
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| A regular dodecahedron with edge length 1 has more than three and a half times the volume of an [[icosahedron]] with the same length edges (7.663... compared with 2.181...).
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| == Stellations ==
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| The 3 [[stellation]]s of the dodecahedron are all regular ([[List_of_regular_polytopes#Three_dimensions_2|nonconvex]]) polyhedra: ([[Kepler-Poinsot polyhedron|Kepler-Poinsot polyhedra polyhedra]])
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| {| class="prettytable" width=500
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| !
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| !0
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| !1
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| !2
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| !3
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| |-
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| |Stellation
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| | [[Image:Dodecahedron.png|125px]]<BR>''Dodecahedron''
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| | [[Image:Small stellated dodecahedron.png|125px]]<BR>[[Small stellated dodecahedron]]
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| | [[Image:Great dodecahedron.png|125px]]<BR>[[Great dodecahedron]]
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| | [[Image:Great stellated dodecahedron.png|125px]]<BR>[[Great stellated dodecahedron]]
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| |-
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| |Facet diagram
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| |[[Image:Zeroth stellation of dodecahedron facets.png|125px]]
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| |[[Image:First stellation of dodecahedron facets.png|125px]]
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| |[[Image:Second stellation of dodecahedron facets.png|125px]]
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| |[[Image:Third stellation of dodecahedron facets.png|125px]]
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| |}
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| == Other dodecahedra ==
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| The term dodecahedron is also used for other [[polyhedron|polyhedra]] with twelve faces, most notably the [[rhombic dodecahedron]] which is dual to the [[cuboctahedron]] (an [[Archimedean solid]]) and occurs in nature as a crystal form. The [[Platonic solid]] dodecahedron can be called a ''pentagonal dodecahedron'' or a ''regular dodecahedron'' to distinguish it. The [[pyritohedron]] is an irregular pentagonal dodecahedron.
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| Other dodecahedra include:
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| * [[Uniform polyhedron|Uniform polyhedra]]:
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| *# [[Pentagonal antiprism]] - 10 equilateral triangles, 2 pentagons
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| *# [[Decagonal prism]] - 10 squares, 2 decagons
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| * [[Johnson solid]]s (regular faced):
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| *# [[Pentagonal cupola]] - 5 triangles, 5 squares, 1 pentagon, 1 decagon
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| *# [[Snub disphenoid]] - 12 triangles
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| *# [[Elongated square dipyramid]] - 8 triangles and 4 squares
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| *# [[Metabidiminished icosahedron]] - 10 triangles and 2 pentagons
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| * Congruent nonregular faced: ([[face-transitive]])
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| *# [[Hexagonal bipyramid]] - 12 isosceles [[triangle]]s, dual of [[hexagonal prism]]
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| *# [[Hexagonal trapezohedron]] - 12 [[kite (geometry)|kite]]s, dual of [[hexagonal antiprism]]
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| *# [[Triakis tetrahedron]] - 12 isosceles [[triangle]]s, dual of [[truncated tetrahedron]]
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| *# [[Rhombic dodecahedron]] (mentioned above) - 12 [[rhombus|rhombi]], dual of [[cuboctahedron]]
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| * Other nonregular faced:
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| *# [[Hendecagon]]al [[pyramid (geometry)|pyramid]] - 11 isosceles triangles and 1 [[polygon|hendecagon]]
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| *# [[Trapezo-rhombic dodecahedron]] - 6 rhombi, 6 [[trapezoid]]s - dual of [[Triangular orthobicupola]]
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| *# [[Rhombo-hexagonal dodecahedron]] or ''Elongated Dodecahedron'' - 8 rhombi and 4 equilateral [[hexagon]]s.
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| ==See also==
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| *[[:Image:Dodecahedron.gif|Spinning dodecahedron]]
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| *[[Truncated dodecahedron]]
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| *[[Snub dodecahedron]]
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| *[[Pentakis dodecahedron]]
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| *[[Hamiltonian path]]
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| *[[120-cell]]: a [[Convex regular 4-polytope|regular polychoron]] (4D polytope) whose surface consists of 120 dodecahedral cells.
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| ==References==
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| <!-- See [[Wikipedia:Footnotes]] for instructions. -->
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| {{reflist}}
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| ==External links==
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| {{Commonscat|Dodecahedra}}
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| *[http://www.software3d.com/Dodecahedron.php Paper models of the dodecahedron]
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| *[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
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| *[http://www.artabus.com/calendar.php Dodecahedron calendar], and another [http://www.ii.uib.no/~arntzen/kalender/ Dodecahedron calendar]
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| *[http://www.flickr.com/photos/pascalin/sets/72157594234292561/ Origami Polyhedra] - Models made with Modular Origami
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| *[http://polyhedra.org/poly/show/3/dodecahedron Dodecahedron] - 3-d model that works in your browser
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| *[http://www.korthalsaltes.com/ Paper Models of Polyhedra] Many links
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
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| ** [[VRML]] models
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| *# [http://www.georgehart.com/virtual-polyhedra/vrml/dodecahedron.wrl Regular dodecahedron] regular
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| *# [http://www.georgehart.com/virtual-polyhedra/vrml/rhombic_dodecahedron.wrl Rhombic dodecahedron] quasiregular
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| *# [http://www.georgehart.com/virtual-polyhedra/vrml/decagonal_prism.wrl Decagonal prism] vertex-transitive
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| *# [http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_antiprism.wrl Pentagonal antiprism] vertex-transitive
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| *# [http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_dipyramid.wrl Hexagonal dipyramid] face-transitive
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| *# [http://www.georgehart.com/virtual-polyhedra/vrml/triakistetrahedron.wrl Triakis tetrahedron] face-transitive
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| *# [http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_trapezohedron.wrl hexagonal trapezohedron] face-transitive
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| *# [http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_cupola_(J5).wrl Pentagonal cupola] regular faces
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| * {{MathWorld | urlname=Dodecahedron | title=Dodecahedron}}
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| * {{MathWorld | urlname=ElongatedDodecahedron | title=Elongated Dodecahedron}}
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| * [http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra]
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| {{Polyhedra}}
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| [[Category:Platonic solids]]
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| [[Category:Regular polyhedra]]
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| [[Category:Polyhedra]]
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