View source for Dodecahedron

{{Short description|Polyhedron with 12 faces}}
{{About-distinguish|the three-dimensional shape|Roman dodecahedron}}

{| 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 dodecahedron|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 dodecahedron|Rhombo-hexagonal]]
|[[Rhombic dodecahedron#Parallelohedron|Rhombo-square]]
|[[Trapezo-rhombic dodecahedron|Trapezo-rhombic]]
|[[Triangular-rhombic dodecahedron|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]], a '''dodecahedron'''{{efn|{{ucfirst:{{etymology|grc|''{{Wikt-lang|grc|δωδεκάεδρον}}'' ({{grc-transl|δωδεκάεδρον}})|}}; {{etymology||''{{Wikt-lang|grc|δώδεκα}}'' ({{grc-transl|δώδεκα}})|twelve||''{{Wikt-lang|grc|ἕδρα}}'' ({{grc-transl|ἕδρα}})|base, seat, face}}){{r|davidson}}}}.}} or '''duodecahedron'''<ref>1913 Webster's Revised Unabridged Dictionary</ref> is any [[polyhedron]]  with twelve flat [[Face (geometry)|faces]]. The most familiar dodecahedron is the [[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]]s of the convex form. All of these have [[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|pyritohedron]], a common crystal form in [[pyrite]], has [[pyritohedral symmetry]], while the [[#Tetartoid|tetartoid]] has [[tetrahedral symmetry]].

The [[rhombic dodecahedron]] can be seen as a limiting case of the pyritohedron, and it has [[octahedral symmetry]]. The [[elongated dodecahedron]] and [[trapezo-rhombic dodecahedron]] variations, along with the rhombic dodecahedra, are [[space-filling polyhedra|space-filling]]. There are numerous [[#Other dodecahedra|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.{{r|aah}}

==Regular dodecahedron==
{{Main|Regular dodecahedron}}
The [[regular dodecahedron]] is a convex polyhedron with regular pentagonal faces, three meeting at each vertex. It has 12 faces, 30 edges, and 20 vertices.{{r|sutton}} It is one of the five regular [[Platonic solid]]s, named after [[Plato]] who described them and considered the other four to symbolize the [[classical element]]s; he assigned the regular dodecahedron to the cosmos.{{r|cromwell}} Its [[Dual polyhedron|dual]] is the [[regular icosahedron]].{{r|erickson}}

{{multiple image
 | image1 = Dodecahedron.png
 | image2 = Small stellated dodecahedron.png
 | image3 = Great dodecahedron.png
 | image4 = Great stellated dodecahedron.png
 | footer = Left to right: Regular dodecahedron, small stellated dodecahedron, great dodecahedron, great stellated dodecahedron
 | total_width = 600
 | align = center
}}

The regular dodecahedron has three [[stellation]]s, all of which are regular star dodecahedra. They form three of the four [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]]. They are the [[small stellated dodecahedron]], the [[great dodecahedron]], and the [[great stellated dodecahedron]].{{sfnp|Cromwell|1997|p=[https://books.google.com/books?id=OJowej1QWpoC&pg=PA265 265&ndash;266]}} The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the [[great icosahedron]]. All of these regular star dodecahedra have regular pentagonal or [[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.

While the three star forms satisfy the requirements for regularity and have twelve faces, and therefore any of them could also be referred to as a "regular dodecahedron", this specific term is exclusively reserved for the first, convex form. A fifth shape which could be classified as a regular dodecahedron, namely the [[dodecagon]]al [[hosohedron]], is also not referred to as such, in part because it exists only as a [[spherical polyhedron]] and is degenerate in Euclidean space.

==Other pentagonal dodecahedra==
In [[crystallography]], two important dodecahedra can occur as crystal forms in some [[crystallographic point groups|symmetry classes]] of the [[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]]:

===Pyritohedron===
[[File:Polyhedron pyritohedron transparent max.gif|thumb|Pyritohedron]]
A '''pyritohedron''' (or pentagonal dodecahedron) is a dodecahedron with [[pyritohedral symmetry|pyritohedral]] symmetry T<sub>h</sub>. Like the [[regular dodecahedron]], it has twelve identical [[pentagon]]al faces, with three meeting in each of the 20 vertices. 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]] are three mutually perpendicular twofold axes and four threefold axes.{{r|hs}}

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

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

====Crystal pyrite====
The name ''crystal pyrite'' comes from one of the two common [[crystal habit]]s shown by [[pyrite]] (the other one being the [[cube]]). In pyritohedral pyrite, the faces have a [[Miller index]] of (210), which means that the [[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]]. See section ''[[#Geometric freedom|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/[[Golden Ratio|''φ'']]
}}
|
|}

The body of the dodecahedron can also be described as an intersection of twelve half-spaces,
: <math>\{ (x,y,z) \in \mathbb{R}^3 \mid \| (x,y,z) \|_D \le 1 \}</math>
where
: <math>
\| (x,y,z) \|_D 
= 
\max \Big\{
|\alpha x + \beta z|,\ 
|\alpha x - \beta z|,\ 
|\alpha y + \beta x|,\ 
|\alpha y - \beta x|,\ 
|\alpha z + \beta y|,\ 
|\alpha z - \beta y|
\Big\}

</math>

and

<math>
\alpha = \frac{3-\sqrt{5}}{2}, \qquad 
\beta = \frac{\sqrt{5}-1}{2} = \frac{1}{\varphi}.
</math>
{| 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]] at one limit of collinear edges, and a [[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]] where all edges and angles are equal again, and the faces have been distorted into regular [[pentagram (geometry)|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]], with regular [[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]] 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]] is a degenerate case with the 6 crossedges reduced to length zero.
|[[File:exo-dodecahedron.png|120px]]<BR>Self-intersecting equilateral dodecahedron
|}

===Tetartoid===
[[File:Tetartoid perspective.gif|thumb|Tetartoid]]
A '''tetartoid''' (also '''tetragonal pentagonal dodecahedron''', '''pentagon-tritetrahedron''', and '''tetrahedric pentagon dodecahedron''') is a dodecahedron with chiral [[tetrahedral symmetry]] (T). Like the [[regular dodecahedron]], it has twelve identical [[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.

[[File:Cobaltite-d05-67a.jpg|143px|thumb|[[Cobaltite]]]]
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 [[cobaltite]] can have this symmetry form.<ref>[https://www.galleries.com/minerals/property/crystal.htm#dodecahe Crystal Habit]. Galleries.com. Retrieved on 2016-12-02.</ref>

Abstractions sharing the solid's [[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
}}
|}

{| 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]].
(The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of the [[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 [[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]]:
:(''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]] is a tetartoid with more than the required symmetry. The [[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]] to [[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}}

==Rhombic dodecahedron==
{{main article|Rhombic dodecahedron}}
[[File:Rhombicdodecahedron.jpg|160px|thumb|Rhombic dodecahedron]]
The [[rhombic dodecahedron]] is a [[Catalan solid]] with twelve rhombic faces and octahedral symmetry. It is dual to the [[cuboctahedron]], an [[Archimedean solid]],{{r|williams}} and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space with centrally symmetric faces, a [[zonohedron]].{{r|eppstein}}

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

The rhombic dodecahedron has several [[stellation]]s, the [[First stellation of rhombic dodecahedron|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]], i.e. the diagonals are in the ratio of the [[golden ratio]]. It is also a [[zonohedron]] and was described by [[Stanko Bilinski|Bilinski]] in 1960.<ref>Hafner, I. and Zitko, T. [https://www.mi.sanu.ac.rs/vismath/hafner2/IntrodRhombic.html Introduction to golden rhombic polyhedra]. Faculty of Electrical Engineering, [[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.{{r|lrk}}

==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, notably distinct dodecahedra (excluding pentagonal and rhombic forms) include:
<gallery widths=180 heights=180>
File:Decagonal prism.png | [[Decagonal prism]], a [[Prism (geometry)|prism]] consisting of ten squares and two decagonal bases. Its symmetry group is [[dihedral symmetry|D<sub>10h</sub>]] symmetry of order 40.{{r|johnson}}
File:Pentagonal antiprism.png | [[Pentagonal antiprism]]: an [[antiprism]] consisting of ten equilateral triangles and two pentagonal bases. It is [[dihedral symmetry|D<sub>5d</sub>]] symmetry of order 20.{{r|johnson}}
File:Pentagonal cupola.png | [[Pentagonal cupola]]: a [[Johnson solid]] with five triangles, five squares, one pentagon, and one decagon. It is [[cyclic symmetry|C<sub>5v</sub>]] symmetry of order 10.{{r|berman|johnson}}
File:Snub disphenoid.png | [[Snub disphenoid]]: both Johnson solid and [[deltahedron]], consisting of twelve equilateral triangles. It is [[dihedral symmetry|D<sub>2d</sub>]] of order 8, the same symmetries as a [[tetragonal disphenoid]].{{r|johnson|cundy}} Its dual is [[elongated gyrobifastigium]], an [[octahedron]].
File:Elongated square dipyramid.png | [[Elongated square dipyramid]], a Johnson solid. Obtained by augmenting two opposite faces of a [[cube]] by [[equilateral square pyramid]]s, the resulting polyhedron has eight triangles and four squares. It is [[dihedral symmetry|D<sub>4h</sub>]] symmetry of order 16.{{r|berman|johnson}}
File:Metabidiminished icosahedron.png | [[Metabidiminished icosahedron]]: a Johnson solid obtained by removing two [[pentagonal pyramid]]s from a [[regular icosahedron]], resulting in ten triangles and two pentagons. It is [[cyclic symmetry|C<sub>2v</sub>]] symmetry of order 4.{{r|berman|johnson}}
File:Hexagonale bipiramide.png | [[Hexagonal bipyramid]]: a [[bipyramid]] with twelve isosceles [[triangle]]s, obtained by attaching two [[hexagonal pyramid]]s base-to-base.{{r|aarts}} If both affixed pyramids have [[Regular polygon|regular bases]] and apices perpendicular to the center of their base, the bipyramid has [[dihedral symmetry|D<sub>6h</sub>]] symmetry of order 24.{{r|fsz}} Like any other bipyramids, the hexagonal bipyramid is [[face-transitive]] and the dual of [[hexagonal prism]].{{r|cpsb|sibley}}
File:Hexagonal trapezohedron.png | [[Hexagonal trapezohedron]]: a [[trapezohedron]] with twelve [[kite (geometry)|kites]]. It is face-transitive, the dual of [[hexagonal antiprism]], and has [[dihedral symmetry|D<sub>6d</sub>]] symmetry of order 24.
File:Triakistetrahedron.jpg | [[Triakis tetrahedron]]: a [[Catalan solid]] and the [[Kleetope]] of a [[regular tetrahedron]]. Obtained by affixing four triangular pyramids onto each face of the regular tetrahedron, the resulting polyhedron has twelve isosceles triangles.{{r|williams|bpv}} It is face-transitive and [[Rupert property]].{{r|koca|fred}} It is the dual of [[truncated tetrahedron]], shares the same symmetry as [[tetrahedral symmetry|T<sub>d</sub>]] symmetry of order 24.{{r|williams}}
File:Hendecagonal pyramid.svg | Hendecagonal pyramid: the twelve isosceles triangles and one regular [[hendecagon]] faces in a pyramid. It has [[cyclic symmetry|C<sub>11v</sub>]] symmetry of order 11.{{r|alexandroff}}
File:Trapezo-rhombic dodecahedron.png | [[Trapezo-rhombic dodecahedron]]: a polyhedron with six rhombi, six [[trapezoid]]s. It is the dual of [[triangular orthobicupola]], both have [[dihedral symmetry|D<sub>3h</sub>]] symmetry of order 12. It is known for [[Space-filling polyhedron|tessellating]] by translating a copy of itself.
File:Rhombo-hexagonal dodecahedron.png | [[Rhombo-hexagonal dodecahedron]] (also known as "elongated dodecahedron"): a polyhedron with eight rhombi and four equilateral [[hexagon]]s. it has [[dihedral symmetry|D<sub>4h</sub>]] symmetry of order 16. It is one of five [[Federov polyhedra]] or parallelohedra, generated to create its [[Honeycomb (geometry)|honeycomb]].{{r|alexandrov}}
</gallery>

==Practical usage==
[[Armand Spitz]] used a dodecahedron as the "globe" equivalent for his [[Planetarium projector|Digital Dome planetarium projector]],{{r|ley}} based upon a suggestion from [[Albert Einstein]].

==Notes==
{{notelist}}

==References==
{{reflist|30em|refs=

<ref name="aah">{{cite journal
 | last1 = Athreya | first1 = Jayadev S.
 | last2 = Aulicino | first2 = David
 | last3 = Hooper | first3 = W. Patrick
 | title = Platonic Solids and High Genus Covers of Lattice Surfaces
 | doi = 10.1080/10586458.2020.1712564
 | journal = [[Experimental Mathematics]]
 | year = 2020
 | volume = 31 | issue = 3 | pages = 847–877
 | arxiv = 1811.04131
 | s2cid = 119318080
}}</ref>

<ref name="aarts">{{cite book
 | last = Aarts | first = J. M.
 | year = 2008
 | title = Plane and Solid Geometry
 | publisher = Springer
 | page = 303
 | url = https://books.google.com/books?id=1ctGAAAAQBAJ&pg=PA303
 | isbn = 978-0-387-78241-6
 | doi = 10.1007/978-0-387-78241-6
}}</ref>

<ref name="alexandroff">{{cite book
 | last = Alexandroff | first = Paul
 | year = 2012
 | title = An Introduction to the Theory of Groups
 | url = https://books.google.com/books?id=DPrDAgAAQBAJ&pg=PA48
 | page = 48
 | publisher = Dover Publications
 | isbn = 978-0-486-48813-4
}}</ref>

<ref name=alexandrov>{{cite book
 | last = Alexandrov | first = A. D. | author-link = Aleksandr Danilovich Aleksandrov
 | contribution = 8.1 Parallelohedra
 | pages = 349–359
 | publisher = Springer
 | title = Convex Polyhedra
 | title-link = Convex Polyhedra (book)
 | year = 2005
}}</ref>

<ref name="berman">{{cite journal
 | last = Berman | first = Martin
 | year = 1971
 | title = Regular-faced convex polyhedra
 | journal = Journal of the Franklin Institute
 | volume = 291
 | issue = 5
 | pages = 329–352
 | doi = 10.1016/0016-0032(71)90071-8
 | mr = 290245
}}</ref>

<ref name="bpv">{{cite book
 | last1 = Brigaglia | first1 = Aldo
 | last2 = Palladino | first2 = Nicla
 | last3 = Vaccaro | first3 = Maria Alessandra
 | editor1-last = Emmer | editor1-first = Michele
 | editor2-last = Abate | editor2-first = Marco
 | contribution = Historical notes on star geometry in mathematics, art and nature
 | doi = 10.1007/978-3-319-93949-0_17
 | pages = 197–211
 | publisher = Springer International Publishing
 | title = Imagine Math 6: Between Culture and Mathematics
 | year = 2018| isbn = 978-3-319-93948-3
 | hdl = 10447/325250
 | hdl-access = free
}}</ref>

<ref name="cf">{{cite journal
 | last1 = Canfield | first1 = Paul C.
 | last2 = Fisher | first2 = Ian R.
 | title = High-temperature solution growth of intermetallic single crystals and quasicrystals
 | journal = Journal of Crystal Growth
 | publisher = Elsevier BV
 | volume = 225 | issue = 2–4 | year = 2001 | pages = 155–161
 | issn = 0022-0248
 | doi = 10.1016/s0022-0248(01)00827-2
 | bibcode = 2001JCrGr.225..155C
}}</ref>

<ref name="cpsb">{{cite book
 | last1 = Chang | first1 = Ch.
 | last2 = Patzer | first2 = A. B. C.
 | last3 = Sülzle | first3 = D.
 | last4 = Hauer | first4 = H.
 | contribution = Onion-Like Inorganic Fullerenes from a Polyhedral Perspective
 | contribution-url = https://books.google.com/books?id=BLZZEAAAQBAJ&pg=RA3-SA15-PA4
 | year = 
 | editor-last = Sattler | editor-first = Klaus D.
 | title = 21st Century Nanoscience: A Handbook
 | page = 15-4
 | publisher = Taylor & Francis
}}</ref>

<ref name="cundy">{{cite journal
 | last = Cundy | first = H. Martyn | author-link = Martyn Cundy
 | doi = 10.2307/3608204
 | journal = [[The Mathematical Gazette]]
 | mr = 0051525
 | pages = 263–266
 | title = Deltahedra
 | volume = 36
 | year = 1952| issue = 318 | jstor = 3608204 | s2cid = 250435684
}}</ref>

<ref name="cromwell">{{cite book
 | last = Cromwell | first = Peter R.
 | year = 1997
 | title = Polyhedra
 | publisher = Cambridge University Press
 | url = https://books.google.com/books?id=OJowej1QWpoC&pg=PA53
 | page = 53
 | isbn = 978-0-521-55432-9
}}</ref>

<ref name="davidson">{{cite book
 | year = 1908
 | title = Chambers's Twentieth Century Dictionary of the English Language
 | publisher = W. & R. Chambers
 | url = https://archive.org/details/chambersstwentie00daviiala/page/274
 | page = 274
}}</ref>

<ref name="eppstein">{{cite journal
 | last = Eppstein | first = David | authorlink = David Eppstein
 | year = 1996
 | title = Zonohedra and zonotopes
 | journal = Mathematica in Education and Research
 | volume = 5
 | issue = 4
 | pages = 15–21
 | url = https://www.ics.uci.edu/~eppstein/junkyard/ukraine/ukraine.html
}}</ref>

<ref name="erickson">{{cite book
 | last = Erickson | first = Martin
 | year = 2011
 | title = Beautiful Mathematics
 | publisher = [[Mathematical Association of America]]
 | url = https://books.google.com/books?id=LgeP62-ZxikC&pg=PA62
 | page = 62
 | isbn = 978-1-61444-509-8
}}</ref>

<ref name="fred">{{cite journal
 | last = Fredriksson | first = Albin
 | title = Optimizing for the Rupert property
 | journal = [[The American Mathematical Monthly]]
 | pages = 255–261
 | volume = 131
 | issue = 3
 | year = 2024
 | doi = 10.1080/00029890.2023.2285200
 | arxiv = 2210.00601
}}</ref>

<ref name="fsz">{{cite book
 | last1 = Flusser | first1 = Jan
 | last2 = Suk | first2 = Tomas
 | last3 = Zitofa | first3 = Barbara
 | year = 2017
 | title = 2D and 3D Image Analysis by Moments
 | page = 126
 | url = https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126
 | publisher = John & Sons Wiley
 | isbn = 978-1-119-03935-8
}}</ref>

<ref name="hs">{{cite book
 | last1 = Hurlbut | first1 = Cornelius S.
 | last2 = Sharp | first2 = W. Edwin
 | year = 1998
 | edition = 4th
 | title = Dana's Minerals and How to Study Them
 | publisher = John Wiley & Sons
 | url = https://books.google.com/books?id=pgn5w0JPWlMC&pg=PA26
 | page = 26
 | isbn = 978-0-471-15677-2
}}</ref>

<ref name="johnson">{{cite journal
 | last = Johnson | first = Norman W. | author-link = Norman W. Johnson
 | year = 1966
 | title = Convex polyhedra with regular faces
 | journal = [[Canadian Journal of Mathematics]]
 | volume = 18
 | pages = 169–200
 | doi = 10.4153/cjm-1966-021-8
 | mr = 0185507
 | s2cid = 122006114
 | zbl = 0132.14603 | doi-access = free
}}</ref>

<ref name="koca">{{cite journal
 | title = Catalan Solids Derived From 3D-Root Systems and Quaternions
 | last1 = Koca | first1 = Mehmet
 | last2 = Ozdes Koca | first2 = Nazife
 | last3 = Koc | first3 = Ramazon
 | year = 2010
 | journal = Journal of Mathematical Physics
 | volume = 51 | issue = 4 | doi = 10.1063/1.3356985 |arxiv=0908.3272
}}</ref>

<ref name="ley">{{cite magazine
 | last = Ley | first = Willy
 | date = February 1965
 | title = Forerunners of the Planetarium
 | department = For Your Information
 | url = https://archive.org/stream/Galaxy_v23n03_1965-02#page/n87/mode/2up
 | magazine = Galaxy Science Fiction
 | pages = 87–98
}}</ref>

<ref name="lrk">{{cite journal
 | url = http://met.iisc.ernet.in/~lord/webfiles/tcq.html
 | last1 = Lord | first1 = E. A.
 | last2 = Ranganathan | first2 = S.
 | last3 = Kulkarni | first3 = U. D.
 | title = Tilings, coverings, clusters and quasicrystals
 | journal = Current Science
 | volume = 78 | year = 2000 | pages = 64–72
}}</ref>

<ref name="sibley">{{cite book
 | last = Sibley | first = Thomas Q.
 | year = 2015
 | title = Thinking Geometrically: A Survey of Geometries
 | publisher = Mathematical Association of America
 | page = 53
 | url = https://books.google.com/books?id=EUh2CgAAQBAJ&pg=PA53
 | isbn =978-1-939512-08-6
}}</ref>

<ref name="stephenson">{{cite journal
 | last = Stephenson | first = Paul 
 | title = Plato's Fourth Solid and the "Pyritohedron"
 | journal = [[The Mathematical Gazette]]
 | volume = 77 | issue = 479 | year = 1993 | pages = 220&ndash;226
 | doi = 10.2307/3619718
 | jstor = 3619718
}}</ref>

<ref name="sutton">{{cite book
 | last = Sutton | first = Daud
 | title = Platonic & Archimedean Solids
 | series = Wooden Books
 | publisher = Bloomsbury Publishing USA
 | year = 2002
 | isbn = 9780802713865
 | url = https://books.google.com/books?id=vgo7bTxDmIsC&pg=PA55
 | page = 55
}}</ref>

<ref name="williams">{{cite book
 | last = Williams | first = Robert | author-link = Robert Williams (geometer)
 | year = 1979
 | title = The Geometrical Foundation of Natural Structure: A Source Book of Design
 | publisher = Dover Publications, Inc.
 | url = https://archive.org/details/geometricalfound00will/page/72
 | page = 72
 | isbn = 978-0-486-23729-9
}}</ref>

}}

==External links==
{{Commons category|Polyhedra with 12 faces}}
*''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]
*[https://bulatov.org/polyhedra/dodeca270/index.html Stellation of Pyritohedron] VRML models and animations of Pyritohedron and its [[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]
*[https://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
*[https://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]
*[https://www.software3d.com/Stella.php Stella: Polyhedron Navigator]: Software used to create some of the images on this page.
*[https://video.fc2.com/content/20141015mMG9QR5R How to make a dodecahedron from a Styrofoam cube]

{{Polyhedra}}
{{Convex polyhedron navigator}}
{{Polytopes}}
{{Authority control}}

[[Category:Individual graphs]]
[[Category:Planar graphs]]
[[Category:Platonic solids]]
[[Category:12 (number)]]