Editing Dodecahedron (section)

==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===
{| 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 diagram]]s||{{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]]||[[Pyritohedral symmetry|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 symmetry|pyritohedral]] (T<sub>h</sub>) symmetry. Like the [[regular dodecahedron]], it has twelve identical [[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]] 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 [[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 [[quasicrystal]]s (such as [[holmium–magnesium–zinc quasicrystal]]) with [[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 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'']]
}}
|
|}

{| 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===
{| 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]] (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.

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>[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]] 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|[[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]].
(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}}

===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]]