View source for Dodecahedron

{{Short description|Polyhedron with 12 faces}}

{| class="wikitable floatright" width=320
|+ Common dodecahedra
|- style="text-align:center;"
!colspan=4|I{{sub|h}}, order 120
|- style="text-align:center; vertical-align:bottom;"
|[[Regular dodecahedron|Regular-]]
|[[Small stellated dodecahedron|Small stellated-]]
|[[Great dodecahedron|Great-]]
|[[Great stellated dodecahedron|Great stellated-]]
|- style="text-align:center; vertical-align:bottom;"
|[[File:Dodecahedron.png|80px]]
|[[File:Small stellated dodecahedron.png|80px]]
|[[File:Great dodecahedron.png|80px]]
|[[File:Great stellated dodecahedron.png|80px]]
|-
!T{{sub|h}}, order 24
!T, order 12
!O{{sub|h}}, order 48
!Johnson (J{{sub|84}})
|- style="text-align:center; vertical-align:bottom;"
|[[Pyritohedron]]
|Tetartoid
|[[Rhombic dodecahedron|Rhombic-]]
|Triangular-
|- style="text-align:center; vertical-align:bottom;"
|[[File:Pyritohedron.png|x80px]]
|[[File:Tetartoid.png|x80px]]
|[[File:Rhombicdodecahedron.jpg|x80px]]
|[[File:snub_disphenoid.png|x80px]]
|- align=center
!colspan=2|D{{sub|4h}}, order 16
!colspan=2|D{{sub|3h}}, order 12
|- align=center
|Rhombo-hexagonal-
|[[Rhombic dodecahedron#Parallelohedron|Rhombo-square-]]
|[[Trapezo-rhombic dodecahedron|Trapezo-rhombic-]]
|Rhombo-triangular-
|- align=center
|[[File:Rhombo-hexagonal dodecahedron.png|80px]]
|[[File:Squared rhombic dodecahedron.png|80px]]
|[[File:Trapezo-rhombic dodecahedron.png|80px]]
|[[File:Triangular square dodecahedron.png|80px]]
|}

In [[Geometry|geometry]], a '''dodecahedron''' ({{etymology|grc|''{{wiktgrc|δωδεκάεδρον}}'' ({{grc-transl|δωδεκάεδρον}})|}}; {{etymology||''{{wiktgrc|δώδεκα}}'' ({{grc-transl|δώδεκα}})|twelve||''{{wiktgrc|ἕδρα}}'' ({{grc-transl|ἕδρα}})|base, seat, face}}) or '''duodecahedron'''<ref>1908 Chambers's Twentieth Century Dictionary of the English Language, 1913 Webster's Revised Unabridged Dictionary</ref> is any [[Polyhedron|polyhedron]]  with twelve flat faces. The most familiar dodecahedron is the [[Regular dodecahedron|regular dodecahedron]] with regular pentagons as faces, which is a [[Platonic solid]]. There are also three [[Kepler–Poinsot polyhedron|regular star dodecahedra]], which are constructed as [[Stellation|stellation]]s of the convex form. All of these have [[Icosahedral symmetry|icosahedral symmetry]], order 120.

Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular:
The pyritohedron, a common crystal form in [[Chemistry:Pyrite|pyrite]], has pyritohedral symmetry, while the tetartoid has [[Tetrahedral symmetry|tetrahedral symmetry]].

The [[Rhombic dodecahedron|rhombic dodecahedron]] can be seen as a limiting case of the pyritohedron, and it has [[Octahedral symmetry|octahedral symmetry]]. The [[Elongated dodecahedron|elongated dodecahedron]] and [[Trapezo-rhombic dodecahedron|trapezo-rhombic dodecahedron]] variations, along with the rhombic dodecahedra, are space-filling. There are numerous other dodecahedra.

While the regular dodecahedron shares many features with other Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across the figure that return to the original point without crossing over any other corner.<ref name="cross">{{Cite journal |url=https://www.tandfonline.com/doi/abs/10.1080/10586458.2020.1712564 |title=Platonic Solids and High Genus Covers of Lattice Surfaces |doi=10.1080/10586458.2020.1712564 |journal=Experimental Mathematics |date=May 27, 2020 |language=en-US |last1=Athreya |first1=Jayadev S. |last2=Aulicino |first2=David |last3=Hooper |first3=W. Patrick|volume=31 |issue=3 |pages=847–877 |arxiv=1811.04131 |s2cid=119318080 }}</ref>

==Regular dodecahedron==
{{Main|Regular dodecahedron}}
The convex regular dodecahedron is one of the five regular [[Platonic solid]]s and can be represented by its [[Schläfli symbol]] {5, 3}.

The [[Dual polyhedron|dual polyhedron]] is the regular [[Icosahedron|icosahedron]] {3, 5}, having five equilateral triangles around each vertex.

{| class=wikitable align=center
|+ Four kinds of regular dodecahedra
|- align=center
|[[File:Dodecahedron.png|x150px]]<br>Convex [[Regular dodecahedron|regular dodecahedron]]
|[[File:Small stellated dodecahedron.png|x150px]]<br>[[Small stellated dodecahedron]]
|[[File:Great dodecahedron.png|x150px]]<br>[[Great dodecahedron]]
|[[File:Great stellated dodecahedron.png|x150px]]<br>[[Great stellated dodecahedron]]
|}
The convex regular dodecahedron also has three [[Stellation|stellation]]s, all of which are regular star dodecahedra. They form three of the four Kepler–Poinsot polyhedra. They are the [[Small stellated dodecahedron|small stellated dodecahedron]] {5/2, 5}, the [[Great dodecahedron|great dodecahedron]] {5, 5/2}, and the [[Great stellated dodecahedron|great stellated dodecahedron]] {5/2, 3}. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the [[Great icosahedron|great icosahedron]] {3, 5/2}. All of these regular star dodecahedra have regular pentagonal or [[Pentagram|pentagram]]mic faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of the same [[Abstract polytope|abstract regular polyhedron]]; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron.

==Other pentagonal dodecahedra==
In [[Physics:Crystallography|crystallography]], two important dodecahedra can occur as crystal forms in some symmetry classes of the [[Cubic crystal system|cubic crystal system]] that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with pyritohedral symmetry, and the tetartoid with [[Tetrahedral symmetry|tetrahedral symmetry]]:

===Pyritohedron===
{| class="wikitable floatright" style="width:260px;"
|-
! style="background:#e7dcc3;" colspan="2"|Pyritohedron
|-
| style="text-align:center;" colspan="2"|[[File:Polyhedron pyritohedron transparent max.png|250px]]<br><small>(See [[c:File:Polyhedron pyritohedron transparent max.gif|here]] for a rotating model.)</small>
|-
| style="background:#e7dcc3;"|Face polygon||[[Pentagon|isosceles pentagon]]
|-
| style="background:#e7dcc3;"|Coxeter diagrams||{{CDD|node|4|node_fh|3|node_fh}}<br>{{CDD|node_fh|3|node_fh|3|node_fh}}
|-
| style="background:#e7dcc3;"|[[Face (geometry)|Faces]]||12
|-
| style="background:#e7dcc3;"|[[Edge (geometry)|Edges]]||30 (6 + 24)
|-
| style="background:#e7dcc3;"|[[Vertex (geometry)|Vertices]]||20 (8 + 12)
|-
| style="background:#e7dcc3;"|[[List of spherical symmetry groups#Polyhedral sym|Symmetry group]]||T<sub>h</sub>, [4,3<sup>+</sup>], (3*2), order 24
|-
| style="background:#e7dcc3;"|[[Point groups in three dimensions#Rotation groups|Rotation group]]||[[Tetrahedral symmetry|T]], [3,3]<sup>+</sup>, (332), order 12
|-
| style="background:#e7dcc3;"|[[Dual polyhedron]]||Pseudoicosahedron
|-
| style="background:#e7dcc3;"|Properties||face transitive
|- align=center
|colspan=2|[[Net (polyhedron)|Net]]<br>[[File:Pyritohedron flat.png|150px]]
|}

A '''pyritohedron''' is a dodecahedron with pyritohedral (T<sub>h</sub>) symmetry. Like the [[Regular dodecahedron|regular dodecahedron]], it has twelve identical [[Pentagon|pentagon]]al faces, with three meeting in each of the 20 vertices (see figure).<ref>[http://www.galleries.com/minerals/property/crystal.htm#dodecahe Crystal Habit]. Galleries.com. Retrieved on 2016-12-02.</ref> However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of [[Rotational symmetry|rotational symmetry]] are three mutually perpendicular twofold axes and four threefold axes.

Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral [[Chemistry:Pyrite|pyrite]], and it may be an inspiration for the discovery of the regular [[Platonic solid]] form. The true regular dodecahedron can occur as a shape for [[Physics:Quasicrystal|quasicrystal]]s (such as [[Physics:Holmium–magnesium–zinc quasicrystal|holmium–magnesium–zinc quasicrystal]]) with [[Icosahedral symmetry|icosahedral symmetry]], which includes true fivefold rotation axes.

[[File:Modell eines Kristalls des Minerals Pyrit (Eisernes Kreuz) -Krantz 375- (2), crop.jpg|190px|thumb|Dual positions in pyrite crystal models]]

====Crystal pyrite====
The name ''crystal pyrite'' comes from one of the two common [[Physics:Crystal habit|crystal habit]]s shown by [[Chemistry:Pyrite|pyrite]] (the other one being the [[Cube|cube]]). In pyritohedral pyrite, the faces have a [[Physics:Miller index|Miller index]] of (210), which means that the [[Dihedral angle|dihedral angle]] is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for the face of a perfect crystal (which is rarely found in nature).

<math>\text{Height} = \frac{\sqrt{5}}{2} \cdot \text{Long side}</math>

<math>\text{Width} = \frac{4}{3} \cdot \text{Long side}</math>

<math>\text{Short sides} = \sqrt{\frac{7}{12}} \cdot \text{Long side}</math>

{| <!-- Table prevents the next headline from crawling up. {{clear}} would push it down to the end of the infobox. -->
|
{{multiple image
 | align = left  | total_width = 320
 | image1 = Pyrite-184681.jpg
 | image2 = Pyrite-193871_angles.jpg
 | footer = Natural pyrite (with face angles on the right)
}}
|}

====Cartesian coordinates====
The eight vertices of a cube have the coordinates (±1, ±1, ±1).

The coordinates of the 12 additional vertices are
<big>(</big>0, ±(1 + ''h''), ±(1 − ''h''<sup>2</sup>)<big>)</big>, 
<big>(</big>±(1 + ''h''), ±(1 − ''h''<sup>2</sup>), 0<big>)</big> and
<big>(</big>±(1 − ''h''<sup>2</sup>), 0, ±(1 + ''h'')<big>)</big>.

''h'' is the height of the [[Wedge (geometry)|wedge]]<nowiki>-shaped</nowiki> "roof" above the faces of that cube with edge length 2.

An important case is ''h'' = {{sfrac|1|2}} (a quarter of the cube edge length) for perfect natural pyrite (also the pyritohedron in the [[Weaire–Phelan structure]]).

Another one is ''h'' = {{sfrac|1|[[Golden ratio|φ]]}} = 0.618... for the [[Regular dodecahedron|regular dodecahedron]]. See section ''Geometric freedom'' for other cases.

Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in the compound of two dodecahedra.

{|
|
{{multiple image |align=left |total_width=440
| image1 = Polyhedron pyritohedron from yellow max.png
| image2 = Polyhedron pyritohedron from red max.png
| image3 = Polyhedron pyritohedron from blue max.png
| footer = Orthographic projections of the pyritohedron with ''h'' = 1/2
}}
|
{{multiple image |align=left |total_width=278
| image1 = Polyhedron pyritohedron max.png
| image2 = Polyhedron 12 pyritohedral max.png
| footer = Heights 1/2 and 1/''φ''
}}
|
|}

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="2"| Animations
|- style="background-color: white;"
|style="width: 350px;"| [[File:Endo-dodecahedron honeycomb.gif|200px]]
|style="width: 350px;"| [[File:Pyritohedron animation.gif|200px]]
|-
| [[Honeycomb (geometry)|Honeycomb]] of alternating convex and concave pyritohedra with heights between ±{{sfrac|1|[[Golden ratio|φ]]}}
| Heights between 0 (cube)<br>and 1 (rhombic dodecahedron)
|}

====Geometric freedom====
The pyritohedron has a geometric degree of freedom with [[Limiting case (mathematics)|limiting case]]s of a cubic [[Convex hull|convex hull]] at one limit of collinear edges, and a [[Rhombic dodecahedron|rhombic dodecahedron]] as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal.

It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. The ''endo-dodecahedron'' is concave and equilateral; it can tessellate space with the convex regular dodecahedron. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, and on to the regular [[Great stellated dodecahedron|great stellated dodecahedron]] where all edges and angles are equal again, and the faces have been distorted into regular pentagrams. On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces.

{| class="wikitable collapsible collapsed"
!colspan="8"| Special cases of the pyritohedron
|-
|colspan="8"| Versions with equal absolute values and opposing signs form a honeycomb together. (Compare [[:File:Endo-dodecahedron honeycomb.gif|this animation]].)<br>The ratio shown is that of edge lengths, namely those in a set of 24 (touching cube vertices) to those in a set of 6 (corresponding to cube faces).
|-
! Ratio
!1 : 1
!0 : 1
!1 : 1
!2 : 1
!1 : 1
!0 : 1
!1 : 1
|-
!rowspan="2"| ''h''
! −{{sfrac|{{radic|5}} + 1|2}}
!rowspan="2"| −1
! {{sfrac|−{{radic|5}} + 1|2}}
!rowspan="2"| 0
! {{sfrac|{{radic|5}} − 1|2}}
!rowspan="2"| 1
! {{sfrac|{{radic|5}} + 1|2}}
|-
! −1.618...
! −0.618...
! 0.618...
! 1.618...
|- style="text-align: center; vertical-align: top;"
!style="vertical-align: middle;"| Image
|[[File:Great stellated dodecahedron.png|120px]]<br>Regular star, [[Great stellated dodecahedron|great stellated dodecahedron]], with regular [[Pentagram|pentagram]] faces
|[[File:Degenerate-pyritohedron.png|120px]]<BR>Degenerate, 12 vertices in the center
|[[File:Concave pyritohedral dodecahedron.png|120px]]<br>The concave equilateral dodecahedron, called an ''endo-dodecahedron''. {{clarify|date=October 2020 |reason=Image should be replaced by one with the specified height.}}
|[[File:Pyritohedron cube.png|120px]]<br>A [[Cube|cube]] can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions.
|[[File:Dodecahedron.png|120px]]<br>A regular dodecahedron is an intermediate case with equal edge lengths.
|[[File:Rhombicdodecahedron.jpg|120px]]<br>A [[Rhombic dodecahedron|rhombic dodecahedron]] is a degenerate case with the 6 crossedges reduced to length zero.
|[[File:exo-dodecahedron.png|120px]]<BR>Self-intersecting equilateral dodecahedron
|}

===Tetartoid===
{| class="wikitable floatright" style="width:260px;"
|-
! style="background:#e7dcc3;" colspan="2"|Tetartoid<br>Tetragonal pentagonal dodecahedron
|-
| style="text-align:center;" colspan="2"|[[File:Tetartoid perspective.png|250px]]<br><small>(See [[c:File:Tetartoid perspective.gif|here]] for a rotating model.)</small>
|-
| style="background:#e7dcc3;"|Face polygon||[[Pentagon|irregular pentagon]]
|-
| style="background:#e7dcc3;"|[[Conway polyhedron notation|Conway notation]]||gT
|-
| style="background:#e7dcc3;"|[[Face (geometry)|Faces]]||12
|-
| style="background:#e7dcc3;"|[[Edge (geometry)|Edges]]||30 (6+12+12)
|-
| style="background:#e7dcc3;"|[[Vertex (geometry)|Vertices]]||20 (4+4+12)
|-
| style="background:#e7dcc3;"|[[List of spherical symmetry groups#Polyhedral sym|Symmetry group]]||[[Tetrahedral symmetry|T]], [3,3]<sup>+</sup>, (332), order 12
<!--|-
|bgcolor=#e7dcc3|[[Dual polyhedron]]||Pseudoicosahedron-->
|-
| style="background:#e7dcc3;"|Properties||[[Convex set|convex]], face transitive
<!--|- align=center
|colspan=2|[[Net (polyhedron)|Net]]<BR>[[File:Pyritohedron flat.png|200px]]-->
|}

A '''tetartoid''' (also '''tetragonal pentagonal dodecahedron''', '''pentagon-tritetrahedron''', and '''tetrahedric pentagon dodecahedron''') is a dodecahedron with chiral [[Tetrahedral symmetry|tetrahedral symmetry]] (T). Like the [[Regular dodecahedron|regular dodecahedron]], it has twelve identical [[Pentagon|pentagon]]al faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes.

Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry.<ref>Dutch, Steve. [https://www.uwgb.edu/dutchs/symmetry/xlforms.htm The 48 Special Crystal Forms] {{Webarchive|url=https://web.archive.org/web/20130918103121/https://www.uwgb.edu/dutchs/symmetry/xlforms.htm |date=2013-09-18 }}. Natural and Applied Sciences, University of Wisconsin-Green Bay, U.S.</ref> The mineral [[Chemistry:Cobaltite|cobaltite]] can have this symmetry form.<ref>[http://www.galleries.com/minerals/property/crystal.htm#dodecahe Crystal Habit]. Galleries.com. Retrieved on 2016-12-02.</ref>

Abstractions sharing the solid's [[Topology|topology]] and symmetry can be created from the cube and the tetrahedron. In the cube each face is bisected by a slanted edge. In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center. (In [[Conway polyhedron notation]] this is a gyro tetrahedron.)

{|
|
{{multiple image |align=left |total_width=440
| image1 = Tetartoid from red.png
| image2 = Tetartoid from green.png
| image3 = Tetartoid from yellow.png
| footer = Orthographic projections from 2- and 3-fold axes
}}
|
{{multiple image |align=left |total_width=300
| image1 = Tetartoid cube.png
| image2 = Tetartoid tetrahedron.png
| footer = Cubic and tetrahedral form
}}
|
[[File:Cobaltite-d05-67a.jpg|143px|thumb|[[Chemistry:Cobaltite|Cobaltite]]]]
|}

{| class="wikitable collapsible collapsed"
! Relationship to the dyakis dodecahedron
|-
|style="width: 760px;"|
A tetartoid can be created by enlarging 12 of the 24 faces of a [[Dyakis dodecahedron|dyakis dodecahedron]].
(The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of the [[Disdyakis dodecahedron|disdyakis dodecahedron]].)
<!--start inner table-->
{|
|
{{multiple image |align=left |total_width=550
| image1 = Tetartoid dark vertical (with traces of dyakis 12).png
| image2 = Disdyakis 12 untruncated to dyakis 12 vertical.png
| image3 = Tetartoid light vertical (with traces of dyakis 12).png
| footer = [[Chirality|Chiral]] tetartoids based on the dyakis dodecahedron in the middle
}}
|
[[File:Crystal model of tetartoid around dyakis dodecahedron (mirrored).jpg|thumb|right|155px|Crystal model]]
|}<!--end inner table-->
The [[Physics:Crystal model|crystal model]] on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core. Therefore, the edges between the blue faces are covered by the red skeleton edges.
|}

====Cartesian coordinates====
The following points are vertices of a tetartoid pentagon under [[Tetrahedral symmetry|tetrahedral symmetry]]:
:(''a'', ''b'', ''c''); (−''a'', −''b'', ''c''); (−{{sfrac|''n''|''d''<sub>1</sub>}}, −{{sfrac|''n''|''d''<sub>1</sub>}}, {{sfrac|''n''|''d''<sub>1</sub>}}); (−''c'', −''a'', ''b''); (−{{sfrac|''n''|''d''<sub>2</sub>}}, {{sfrac|''n''|''d''<sub>2</sub>}}, {{sfrac|''n''|''d''<sub>2</sub>}}),
under the following conditions:<ref>[http://demonstrations.wolfram.com/TheTetartoid/ The Tetartoid]. Demonstrations.wolfram.com. Retrieved on 2016-12-02.</ref>
:{{nowrap|1=0 ≤ ''a'' ≤ ''b'' ≤ ''c''}},
:''n'' = ''a''<sup>2</sup>''c'' − ''bc''<sup>2</sup>,
:''d''<sub>1</sub> = ''a''<sup>2</sup> − ''ab'' + ''b''<sup>2</sup> + ''ac'' − 2''bc'',
:''d''<sub>2</sub> = ''a''<sup>2</sup> + ''ab'' + ''b''<sup>2</sup> − ''ac'' − 2''bc'',
:{{nowrap|1=''nd''<sub>1</sub>''d''<sub>2</sub> ≠ 0}}.

====Geometric freedom====

The [[Regular dodecahedron|regular dodecahedron]] is a tetartoid with more than the required symmetry. The [[Triakis tetrahedron|triakis tetrahedron]] is a degenerate case with 12 zero-length edges. (In terms of the colors used above this means, that the white vertices and green edges are absorbed by the green vertices.)

{| class="wikitable collapsible collapsed"
!colspan="8"| Tetartoid variations from [[Regular dodecahedron|regular dodecahedron]] to [[Triakis tetrahedron|triakis tetrahedron]]
|- style="background-color: white;"
|[[File:Dodecahedron.png|140px]]
|[[File:Tetartoid-010.png|150px]]
|[[File:Tetartoid-020.png|150px]]
|[[File:Tetartoid-040.png|150px]]
|[[File:Tetartoid-060.png|150px]]
|[[File:Tetartoid-080.png|150px]]
|[[File:Tetartoid-095.png|150px]]
|[[File:Triakistetrahedron.jpg|100px]]
|}
{{Clear}}

===Dual of triangular gyrobianticupola===
A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedron constructed from two triangular anticupola connected base-to-base, called a ''triangular gyrobianticupola.'' It has D<sub>3d</sub> symmetry, order 12. It has 2 sets of 3 identical pentagons on the top and bottom, connected 6 pentagons around the sides which alternate upwards and downwards. This form has a hexagonal cross-section and identical copies can be connected as a partial hexagonal honeycomb, but all vertices will not match.
:[[File:Dual_triangular_gyrobianticupola.png|160px]]

==Rhombic dodecahedron==
[[File:Rhombicdodecahedron.jpg|160px|thumb|Rhombic dodecahedron]]
The ''[[Rhombic dodecahedron|rhombic dodecahedron]]'' is a [[Zonohedron|zonohedron]] with twelve rhombic faces and octahedral symmetry. It is dual to the [[Quasiregular polyhedron|quasiregular]] [[Cuboctahedron|cuboctahedron]] (an [[Archimedean solid]]) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space.

The ''rhombic dodecahedron'' can be seen as a degenerate [[Pyritohedron|pyritohedron]] where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.

The rhombic dodecahedron has several [[Stellation|stellation]]s, the first of which is also a [[Rhombic dodecahedron#Parallelohedron|parallelohedral spacefiller]].

Another important rhombic dodecahedron, the [[Bilinski dodecahedron]], has twelve faces congruent to those of the [[Rhombic triacontahedron|rhombic triacontahedron]], i.e. the diagonals are in the ratio of the [[Golden ratio|golden ratio]]. It is also a [[Zonohedron|zonohedron]] and was described by [[Biography:Stanko Bilinski|Bilinski]] in 1960.<ref>Hafner, I. and Zitko, T. [http://www.mi.sanu.ac.rs/vismath/hafner2/IntrodRhombic.html Introduction to golden rhombic polyhedra]. Faculty of Electrical Engineering, [[Organization:University of Ljubljana|University of Ljubljana]], Slovenia.</ref> This figure is another spacefiller, and can also occur in non-periodic [[Honeycomb (geometry)|spacefilling]]s along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.<ref>{{cite journal|url=http://met.iisc.ernet.in/~lord/webfiles/tcq.html |author1=Lord, E. A. |author2=Ranganathan, S. |author3=Kulkarni, U. D. |title=Tilings, coverings, clusters and quasicrystals|journal=Curr. Sci. |volume=78 |year=2000|pages= 64–72}}</ref>

==Other dodecahedra==
There are 6,384,634 topologically distinct ''convex'' dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20.<ref>[http://www.numericana.com/data/polycount.htm Counting polyhedra]. Numericana.com (2001-12-31). Retrieved on 2016-12-02.</ref> (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)

Topologically distinct dodecahedra (excluding pentagonal and rhombic forms)
*Uniform polyhedra:
**[[Decagonal prism]] – 10 squares, 2 decagons, D<sub>10h</sub> symmetry, order 40.
**[[Pentagonal antiprism]] – 10 equilateral triangles, 2 pentagons, D<sub>5d</sub> symmetry, order 20
*[[Johnson solid]]s (regular faced):
**[[Pentagonal cupola]] – 5 triangles, 5 squares, 1 pentagon, 1 decagon, C<sub>5v</sub> symmetry, order 10
**[[Snub disphenoid]] – 12 triangles, D<sub>2d</sub>, order 8
**Elongated square dipyramid – 8 triangles and 4 squares, D<sub>4h</sub> symmetry, order 16
**[[Metabidiminished icosahedron]] – 10 triangles and 2 pentagons, C<sub>2v</sub> symmetry, order 4
*Congruent irregular faced: (face-transitive)
**[[Hexagonal bipyramid]] – 12 isosceles [[Triangle|triangle]]s, dual of [[Hexagonal prism|hexagonal prism]], D<sub>6h</sub> symmetry, order 24
**[[Hexagonal trapezohedron]] – 12 [[Kite (geometry)|kites]], dual of [[Hexagonal antiprism|hexagonal antiprism]], D<sub>6d</sub> symmetry, order 24
**[[Triakis tetrahedron]] – 12 isosceles triangles, dual of [[Truncated tetrahedron|truncated tetrahedron]], [[Tetrahedral symmetry|T<sub>d</sub>]] symmetry, order 24
*Other less regular faced:
**Hendecagonal [[Pyramid (geometry)|pyramid]] – 11 isosceles triangles and 1 regular [[Hendecagon|hendecagon]], C<sub>11v</sub>, order 11
**[[Trapezo-rhombic dodecahedron]] – 6 rhombi, 6 [[Trapezoid|trapezoid]]s – dual of [[Triangular orthobicupola|triangular orthobicupola]], D<sub>3h</sub> symmetry, order 12
**Rhombo-hexagonal dodecahedron or ''elongated Dodecahedron'' – 8 rhombi and 4 equilateral [[Hexagon|hexagon]]s, D<sub>4h</sub> symmetry, order 16
**[[Truncated trapezohedron|Truncated pentagonal trapezohedron]], D<sub>5d</sub>, order 20, topologically equivalent to regular dodecahedron

==Practical usage==
Armand Spitz used a dodecahedron as the "globe" equivalent for his [[Philosophy:Planetarium projector|Digital Dome planetarium projector]],<ref name="ley196502">{{Cite magazine
 |last=Ley
 |first=Willy
 |date=February 1965
 |title=Forerunners of the Planetarium
 |department=For Your Information
 |url=https://archive.org/stream/Galaxy_v23n02_1964-12#page/n93/mode/2up
 |magazine=Galaxy Science Fiction
 |pages=87–98
}}</ref> based upon a suggestion from [[Biography:Albert Einstein|Albert Einstein]].

==See also==
* [[120-cell|120-cell]] – a regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells
* {{em|Braarudosphaera bigelowii}} – a dodecahedron shaped [[Biology:Coccolithophore|coccolithophore]] (a [[Biology:Unicellular organism|unicellular]] [[Earth:Phytoplankton|phytoplankton]] [[Biology:Algae|algae]])
* [[Pentakis dodecahedron]]
* [[Roman dodecahedron]]
* [[Snub dodecahedron]]
* [[Truncated dodecahedron]]

==References==
<!--See Wikipedia:Footnotes for instructions.-->
{{Reflist|30em}}

==External links==
*''Plato's Fourth Solid and the "Pyritohedron"'', by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp.&nbsp;220–226 [https://www.jstor.org/pss/3619718]
*[http://bulatov.org/polyhedra/dodeca270/index.html Stellation of Pyritohedron] VRML models and animations of Pyritohedron and its [[Stellation|stellation]]s
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3o5x – doe}}
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1bk9bWiCSjJz6LpNRYDsAu8YDBWnSMrt0ydjpIfF8jmyc682nzINN9xaGayOA9FBx396IIYMhulg2mGXcK0mAk5Rmo8qm9ut0kE1qP&name=Dodecahedron#applet Editable printable net of a dodecahedron with interactive 3D view]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[https://www.flickr.com/photos/pascalin/sets/72157594234292561/ Origami Polyhedra] – Models made with Modular Origami
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
*[http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra]
*[http://www.bodurov.com/VectorVisualizer/?vectors=-0.94/-2.885/-3.975/-1.52/-4.67/-0.94v-3.035/0/-3.975/-4.91/0/-0.94v3.975/-2.885/-0.94/1.52/-4.67/0.94v1.52/-4.67/0.94/-1.52/-4.67/-0.94v0.94/-2.885/3.975/1.52/-4.67/0.94v-3.975/-2.885/0.94/-1.52/-4.67/-0.94v-3.975/-2.885/0.94/-4.91/0/-0.94v-3.975/2.885/0.94/-4.91/0/-0.94v-3.975/2.885/0.94/-1.52/4.67/-0.94v-2.455/1.785/3.975/-3.975/2.885/0.94v-2.455/-1.785/3.975/-3.975/-2.885/0.94v-1.52/4.67/-0.94/-0.94/2.885/-3.975v4.91/0/0.94/3.975/-2.885/-0.94v3.975/2.885/-0.94/2.455/1.785/-3.975v2.455/-1.785/-3.975/3.975/-2.885/-0.94v1.52/4.67/0.94/-1.52/4.67/-0.94v3.035/0/3.975/0.94/2.885/3.975v0.94/2.885/3.975/-2.455/1.785/3.975v-2.455/1.785/3.975/-2.455/-1.785/3.975v-2.455/-1.785/3.975/0.94/-2.885/3.975v0.94/-2.885/3.975/3.035/0/3.975v2.455/1.785/-3.975/-0.94/2.885/-3.975v-0.94/2.885/-3.975/-3.035/0/-3.975v-3.035/0/-3.975/-0.94/-2.885/-3.975v-0.94/-2.885/-3.975/2.455/-1.785/-3.975v2.455/-1.785/-3.975/2.455/1.785/-3.97v3.035/0/3.975/4.91/0/0.94v4.91/0/0.94/3.975/2.885/-0.94v3.975/2.885/-0.94/1.52/4.67/0.94v1.52/4.67/0.94/0.94/2.885/3.975 Dodecahedron 3D Visualization]
*[http://www.software3d.com/Stella.php Stella: Polyhedron Navigator]: Software used to create some of the images on this page.
*[http://video.fc2.com/content/20141015mMG9QR5R How to make a dodecahedron from a Styrofoam cube]

{{Polyhedra}}
{{Convex polyhedron navigator}}
{{Polytopes}}

[[Category:Individual graphs]]
[[Category:Planar graphs]]
[[Category:Platonic solids]]

{{Sourceattribution|Dodecahedron}}