Editing Disdyakis dodecahedron (section)

==Symmetry==
It has O<sub>h</sub> [[octahedral symmetry]]. Its collective edges represent the reflection planes of the symmetry. It can also be seen in the corner and mid-edge triangulation of the regular cube and octahedron, and rhombic dodecahedron.

{|class="wikitable" style="text-align: center;"
|- style="vertical-align: top"
| [[File:Disdyakis 12.png|x120px]]<br>Disdyakis<br>dodecahedron
| [[File:Disdyakis 12 in deltoidal 24.png|x120px]]<br>[[Deltoidal icositetrahedron|Deltoidal<br>icositetrahedron]]
| [[File:Disdyakis 12 in rhombic 12.png|x120px]]<br>[[Rhombic dodecahedron|Rhombic<br>dodecahedron]]
| [[File:Disdyakis 12 in Platonic 6.png|x125px]]<br>[[Hexahedron]]
| [[File:Disdyakis 12 in Platonic 8.png|x125px]]<br>[[Octahedron]]
|}

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="4"| [[Spherical polyhedron]]
|- valign=top
| [[File:Disdyakis 12 spherical.png|170px]]
| [[File:Disdyakis 12 spherical from blue.png|170px]]
| [[File:Disdyakis 12 spherical from yellow.png|170px]]
| [[File:Disdyakis 12 spherical from red.png|170px]]
|-
| (see [[:File:Disdyakis 12 spherical.gif|rotating model]])
|colspan="3"| [[Orthographic projection]]s from 2-, 3- and 4-fold axes
|}

The edges of a spherical disdyakis dodecahedron belong to 9 [[great circle]]s. Three of them form a spherical octahedron (gray in the images below). The remaining six form three square [[hosohedron|hosohedra]] (red, green and blue in the images below). They all correspond to [[Reflection (mathematics)|mirror planes]] - the former in [[Dihedral symmetry in three dimensions|dihedral]] [2,2], and the latter in [[Tetrahedral symmetry|tetrahedral]] [3,3] symmetry. A spherical disdyakis dodecahedron can be thought of as the [[barycentric subdivision]] of the [[spherical cube]] or of the [[spherical octahedron]].<ref>{{citation
 | last1 = Langer | first1 = Joel C.
 | last2 = Singer | first2 = David A.
 | doi = 10.1007/s00032-010-0124-5
 | issue = 2
 | journal = Milan Journal of Mathematics
 | mr = 2781856
 | pages = 643–682
 | title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem
 | volume = 78
 | year = 2010}}</ref>

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="4"| [[Stereographic projection]]s
|-
|rowspan="2"| [[File:Spherical disdyakis dodecahedron RGB.png|230px]]
! 2-fold
! 3-fold
! 4-fold
|-
| [[File:Disdyakis dodecahedron stereographic d2.svg|x200px]]
| [[File:Disdyakis dodecahedron stereographic d3.svg|x200px]]
| [[File:Disdyakis dodecahedron stereographic d4.svg|x200px]]
|}