Editing Dodecahedron (section)

===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
|}