Editing Dodecahedron (section)

====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
|}