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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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[[File:Docdodec.JPG|right|thumb|[[Meglos]] and the Dodecahedron. ([[TV]]: ''[[Meglos (TV story)|Meglos]]'')]]
A '''dodecahedron''' was a twelve-sided regular {{w|polyhedron}}, with each side a pentagon.


== Area and volume ==
The [[Tigellan]]s used a Dodecahedron as a power source. The [[Deon]]s, a religious group, believed it to be a gift from their [[god]], [[Ti]].


The area ''A'' and the [[volume]] ''V'' of a regular dodecahedron of edge length ''a'' are:
[[File:The Screens of Zolfa-Thura.jpg|left|thumb|The Dodecahedron atop the [[Screens of Zolfa-Thura]]. ([[TV]]: ''[[Meglos (TV story)|Meglos]]'')]]
:<math>A = 3\sqrt{25+10\sqrt{5}} a^2 \approx 20.64572a^2</math>
In [[1980]], the [[Zolfa-Thuran]], [[Meglos]], wished to steal it and use its power to conquer and destroy worlds. Under the guise of the [[Fourth Doctor]], Meglos compressed the Dodecahedron and stole it. He took it back to [[Zolfa-Thura]].


:<math>V = \frac{1}{4} (15+7\sqrt{5}) a^3 \approx 7.66311896a^3</math>
Using the [[Screens of Zolfa-Thura]], Meglos was able to boost the Dodecahedron's power. He planned to destroy Tigella with it but the Doctor reverted the calculations so it destroyed Zolfa-Thura. ([[TV]]: ''[[Meglos (TV story)|Meglos]]'')


== Cartesian coordinates ==
{{NameSort}}
The following [[Cartesian coordinates]] define the vertices of a dodecahedron centered at the origin:
: (±1, ±1, ±1)
: (0, ±1/&phi;, ±&phi;)
: (±1/&phi;, ±&phi;, 0)
: (±&phi;, 0, ±1/&phi;)
where [[Golden ratio|&phi;]] = (1+√5)/2 is the [[golden ratio]] (also written τ). The side length is 2/φ = √5&minus;1. The containing sphere has a radius of √3.


The [[dihedral angle]] of a dodecahedron is 2arctan(φ) or approximately 116.565 degrees.
[[Category:Alien artefacts]]
 
[[Category:Geometry from the real world]]
== Geometric relations ==   
[[Category:Superweapons]]
 
[[Category:Power sources]]
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]].
[[Category:Stolen property]]
 
[[Category:Sacred or religious artefacts]]
The [[stellation]]s of the dodecahedron make up three of the four [[Kepler-Poinsot polyhedra]].
 
A [[Rectification (geometry)|rectified]] dodecahedron forms an [[icosidodecahedron]].
 
The regular dodecahedron has 120 symmetries, forming the group <math>A_5\times Z_2</math>.
 
=== Vertex arrangement ===
 
The dodecahedron shares its [[vertex arrangement]] with four nonconvex [[uniform polyhedron]]s and three [[Polyhedral compound#Uniform compounds|uniform compounds]].
 
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.
 
{| class="prettytable" width=500
| [[Image:Great stellated dodecahedron.png|125px]]<BR>[[Great stellated dodecahedron]]
| [[Image:Small ditrigonal icosidodecahedron.png|125px]]<BR>[[Small ditrigonal icosidodecahedron]]
| [[Image:Ditrigonal dodecadodecahedron.png|125px]]<BR>[[Ditrigonal dodecadodecahedron]]
| [[Image:Great ditrigonal icosidodecahedron.png|125px]]<BR>[[Great ditrigonal icosidodecahedron]]
|-
| [[Image:Compound of five cubes.png|125px]]<BR>[[Compound of five cubes]]
| [[Image:Compound of five tetrahedra.png|125px]]<BR>[[Compound of five tetrahedra]]
| [[Image:Compound of ten tetrahedra.png|125px]]<BR>[[Compound of ten tetrahedra]]
|}
 
===Icosahedron vs dodecahedron===
 
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%).
 
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...).
 
 
== Stellations ==
 
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]])
 
{| class="prettytable" width=500
!
!0
!1
!2
!3
|-
|Stellation
| [[Image:Dodecahedron.png|125px]]<BR>''Dodecahedron''
| [[Image:Small stellated dodecahedron.png|125px]]<BR>[[Small stellated dodecahedron]]
| [[Image:Great dodecahedron.png|125px]]<BR>[[Great dodecahedron]]
| [[Image:Great stellated dodecahedron.png|125px]]<BR>[[Great stellated dodecahedron]]
|-
|Facet diagram
|[[Image:Zeroth stellation of dodecahedron facets.png|125px]]
|[[Image:First stellation of dodecahedron facets.png|125px]]
|[[Image:Second stellation of dodecahedron facets.png|125px]]
|[[Image:Third stellation of dodecahedron facets.png|125px]]
|}
 
== Other dodecahedra ==
 
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.
 
Other dodecahedra include:
* [[Uniform polyhedron|Uniform polyhedra]]:
*# [[Pentagonal antiprism]] - 10 equilateral triangles, 2 pentagons
*# [[Decagonal prism]] - 10 squares, 2 decagons
* [[Johnson solid]]s (regular faced):
*# [[Pentagonal cupola]] - 5 triangles, 5 squares, 1 pentagon, 1 decagon
*# [[Snub disphenoid]] - 12 triangles
*# [[Elongated square dipyramid]] - 8 triangles and 4 squares
*# [[Metabidiminished icosahedron]] - 10 triangles and 2 pentagons
* Congruent nonregular faced: ([[face-transitive]])
*# [[Hexagonal bipyramid]] - 12 isosceles [[triangle]]s, dual of [[hexagonal prism]]
*# [[Hexagonal trapezohedron]] - 12 [[kite (geometry)|kite]]s, dual of [[hexagonal antiprism]]
*# [[Triakis tetrahedron]] - 12 isosceles [[triangle]]s, dual of [[truncated tetrahedron]]
*# [[Rhombic dodecahedron]] (mentioned above) - 12 [[rhombus|rhombi]], dual of [[cuboctahedron]]
* Other nonregular faced:
*# [[Hendecagon]]al [[pyramid (geometry)|pyramid]] - 11 isosceles triangles and 1 [[polygon|hendecagon]]
*# [[Trapezo-rhombic dodecahedron]] - 6 rhombi, 6 [[trapezoid]]s - dual of [[Triangular orthobicupola]]
*# [[Rhombo-hexagonal dodecahedron]] or ''Elongated Dodecahedron'' - 8 rhombi and 4 equilateral [[hexagon]]s.
 
 
 
==See also==
*[[:Image:Dodecahedron.gif|Spinning dodecahedron]]
*[[Truncated dodecahedron]]
*[[Snub dodecahedron]]
*[[Pentakis dodecahedron]]
*[[Hamiltonian path]]
*[[120-cell]]: a [[Convex regular 4-polytope|regular polychoron]] (4D polytope) whose surface consists of 120 dodecahedral cells.
 
==References==
<!-- See [[Wikipedia:Footnotes]] for instructions. -->
{{reflist}}
 
==External links==
{{Commonscat|Dodecahedra}}
*[http://www.software3d.com/Dodecahedron.php Paper models of the dodecahedron]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.artabus.com/calendar.php Dodecahedron calendar], and another [http://www.ii.uib.no/~arntzen/kalender/ Dodecahedron calendar]  
*[http://www.flickr.com/photos/pascalin/sets/72157594234292561/ Origami Polyhedra] - Models made with Modular Origami
*[http://polyhedra.org/poly/show/3/dodecahedron Dodecahedron] - 3-d model that works in your browser
*[http://www.korthalsaltes.com/ Paper Models of Polyhedra] Many links
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
** [[VRML]] models
*# [http://www.georgehart.com/virtual-polyhedra/vrml/dodecahedron.wrl Regular dodecahedron] regular
*# [http://www.georgehart.com/virtual-polyhedra/vrml/rhombic_dodecahedron.wrl Rhombic dodecahedron] quasiregular
*# [http://www.georgehart.com/virtual-polyhedra/vrml/decagonal_prism.wrl Decagonal prism] vertex-transitive
*# [http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_antiprism.wrl Pentagonal antiprism] vertex-transitive
*# [http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_dipyramid.wrl Hexagonal dipyramid] face-transitive
*# [http://www.georgehart.com/virtual-polyhedra/vrml/triakistetrahedron.wrl Triakis tetrahedron] face-transitive
*# [http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_trapezohedron.wrl hexagonal trapezohedron] face-transitive
*# [http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_cupola_(J5).wrl Pentagonal cupola] regular faces
* {{MathWorld | urlname=Dodecahedron | title=Dodecahedron}}
* {{MathWorld | urlname=ElongatedDodecahedron | title=Elongated Dodecahedron}}
* [http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra]
 
{{Polyhedra}}
 
[[Category:Platonic solids]]
[[Category:Regular polyhedra]]
[[Category:Polyhedra]]
 
[[az:Dodekaedr]]
[[ca:Dodecàedre]]
[[cs:Dvanáctistěn]]
[[da:Dodekaeder]]
[[de:Dodekaeder]]
[[et:Korrapärane dodekaeeder]]
[[es:Dodecaedro]]
[[eo:Dekduedro]]
[[fr:Dodécaèdre]]
[[ko:정십이면체]]
[[it:Dodecaedro]]
[[he:דודקהדרון]]
[[ht:Dodekayèd]]
[[lv:Dodekaedrs]]
[[hu:Dodekaéder]]
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[[ja:正十二面体]]
[[no:Dodekaeder]]
[[pl:Dwunastościan foremny]]
[[pt:Dodecaedro]]
[[qu:Chunka iskayniyuq uya]]
[[ru:Додекаэдр]]
[[sr:Додекаедар]]
[[fi:Dodekaedri]]
[[sv:Dodekaeder]]
[[ta:பன்னிரண்டுமுக ஐங்கோணகம்]]
[[th:ทรงสิบสองหน้า]]
[[uk:Додекаедр]]
[[zh:正十二面體]]

Latest revision as of 20:13, 25 June 2021

Dodecahedron
Meglos and the Dodecahedron. (TV: Meglos)

A dodecahedron was a twelve-sided regular polyhedron, with each side a pentagon.

The Tigellans used a Dodecahedron as a power source. The Deons, a religious group, believed it to be a gift from their god, Ti.

The Dodecahedron atop the Screens of Zolfa-Thura. (TV: Meglos)

In 1980, the Zolfa-Thuran, Meglos, wished to steal it and use its power to conquer and destroy worlds. Under the guise of the Fourth Doctor, Meglos compressed the Dodecahedron and stole it. He took it back to Zolfa-Thura.

Using the Screens of Zolfa-Thura, Meglos was able to boost the Dodecahedron's power. He planned to destroy Tigella with it but the Doctor reverted the calculations so it destroyed Zolfa-Thura. (TV: Meglos)