Editing Disdyakis dodecahedron

{{Short description|Catalan solid with 48 faces}}
{| class=wikitable align=right width="250"
!bgcolor=#e7dcc3 colspan=2|Disdyakis dodecahedron
|-
|align=center colspan=2|[[Image:disdyakisdodecahedron.jpg|240px|Disdyakis dodecahedron]]<br>([[:image:disdyakisdodecahedron.gif|rotating]] and [[:File:Disdyakis dodecahedron.stl|3D]] model)
|-
|bgcolor=#e7dcc3|Type||[[Catalan solid]]
|-
|bgcolor=#e7dcc3|[[Conway polyhedron notation|Conway notation]]||mC
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_f1|4|node_f1|3|node_f1}}
|-
|bgcolor=#e7dcc3|Face polygon||[[File:DU11 facets.png|60px]]<br>[[scalene triangle]]
|-
|bgcolor=#e7dcc3|Faces||48
|-
|bgcolor=#e7dcc3|Edges||72
|-
|bgcolor=#e7dcc3|Vertices||26 = 6 + 8 + 12
|-
|bgcolor=#e7dcc3|[[Face configuration]]||V4.6.8
|-
|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||''O''<sub>''h''</sub>, B<sub>3</sub>, [4,3], *432
|-
|bgcolor=#e7dcc3|[[Dihedral angle]]||155° 4' 56"<br><math>\arccos(-\frac{71 + 12\sqrt{2}}{97})</math>
|-
|bgcolor=#e7dcc3|[[Dual polyhedron]] || [[File:Polyhedron great rhombi 6-8 max.png|70px]]<br>[[truncated cuboctahedron]]
|-
|bgcolor=#e7dcc3|Properties||convex, [[face-transitive]]
|-
|align=center colspan=2|[[File:Disdyakis 12 net.svg|200px|Disdyakis dodecahedron]]<br>[[Net (polyhedron)|net]]
|}
In [[geometry]], a '''disdyakis dodecahedron''', (also '''hexoctahedron''',<ref>{{cite web |url=https://etc.usf.edu/clipart/keyword/forms |title = Keyword: "forms" {{!}} ClipArt ETC}}</ref> '''hexakis octahedron''', '''octakis cube''', '''octakis hexahedron''', '''kisrhombic dodecahedron'''<ref>Conway, Symmetries of things, p.284</ref>) or '''d48''', is a [[Catalan solid]] with 48 faces and the dual to the [[Archimedean solid|Archimedean]] [[truncated cuboctahedron]]. As such it is [[face-transitive]] but with irregular face polygons. It resembles an augmented [[rhombic dodecahedron]]. Replacing each face of the rhombic dodecahedron with a flat pyramid results in the [[Kleetope]] of the rhombic dodecahedron, which looks almost like the disdyakis dodecahedron, and is [[topology|topologically]] equivalent to it.{{Efn|Despite their resemblance, no subset of the disdyakis dodecahedron's vertices forms a rhombic dodecahedron (see [[#Cartesian coordinates]]), and therefore, the former is not the Kleetope of the latter. The "rhombic" bases of the pyramids of the disdyakis dodecahedron are in fact not even planar; for example, the vertices of one such rhombus are (a, 0, 0), (0, a, 0), (c, c, c), (c, c, -c) (again, see [[#Cartesian coordinates]] for the values of a and c), with diagonal midpoints (√2)×(a, a, 0) and (c, c, 0), which do not coincide.}} The net of the [[rhombic dodecahedral pyramid]] also shares the same topology.

==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]]
|}

==Cartesian coordinates==                                                                                                                                    
Let <math> ~ a = \frac{1}{1 + 2 \sqrt{2}} ~ {\color{Gray} \approx 0.261}, ~~ b = \frac{1}{2 + 3 \sqrt{2}} ~ {\color{Gray} \approx 0.160}, ~~ c = \frac{1}{3 + 3 \sqrt{2}} ~ {\color{Gray} \approx 0.138}</math>.<br> 
Then the [[Cartesian coordinates]] for the vertices of a disdyakis dodecahedron centered at the origin are:

{{color|#eb2424|●}} &nbsp; [[permutation]]s of (±'''a''', 0, 0) &nbsp; <small>(vertices of an octahedron)</small><br>
{{color|#3061d6|●}} &nbsp; permutations of (±'''b''', ±'''b''', 0) &nbsp; <small>(vertices of a [[cuboctahedron]])</small><br>
{{color|#f9b900|●}} &nbsp; (±'''c''', ±'''c''', ±'''c''') &nbsp; <small>(vertices of a cube)</small>

{| class="wikitable collapsible collapsed" style="text-align: left;"
!colspan="1" width=400|[[Convex hull|Convex hulls]]
|-
|Combining an octahedron, cube, and cuboctahedron to form the disdyakis dodecahedron. The convex hulls for these vertices<ref>{{cite journal
|title=Catalan Solids Derived From 3D-Root Systems and Quaternions
|first1=Mehmet
|last1=Koca
|first2=Nazife
|last2=Ozdes Koca
|first3=Ramazon
|last3=Koc
|year=2010
|journal=Journal of Mathematical Physics
|volume=51
|issue=4
|doi=10.1063/1.3356985 
|arxiv=0908.3272
}}</ref> scaled by <math>1/a</math> result in Cartesian coordinates of unit [[circumradius]], which are visualized in the figure below:
|-
|rowspan="1"|[[File:Disdyakis Dodecahedron convex hulls.svg|400px|Combining an octahedron, cube, and cuboctahedron to form the disdyakis dodecahedron]]

|-
|}

==Dimensions==
If its smallest edges have length ''a'', its surface area and volume are
:<math>\begin{align} A &= \tfrac67\sqrt{783+436\sqrt 2}\,a^2 \\ V &= \tfrac17\sqrt{3\left(2194+1513\sqrt 2\right)}a^3\end{align}</math>

The faces are scalene triangles. Their angles are <math>\arccos\biggl(\frac{1}{6}-\frac{1}{12}\sqrt{2}\biggr) ~{\color{Gray}\approx 87.201^{\circ}}</math>, <math>\arccos\biggl(\frac{3}{4}-\frac{1}{8}\sqrt{2}\biggr) ~{\color{Gray}\approx 55.024^{\circ}}</math> and <math>\arccos\biggl(\frac{1}{12}+\frac{1}{2}\sqrt{2}\biggr) ~{\color{Gray}\approx 37.773^{\circ}}</math>.

== Orthogonal projections ==
The truncated cuboctahedron and its dual, the ''disdyakis dodecahedron'' can be drawn in a number of symmetric orthogonal projective orientations. Between a polyhedron and its dual, vertices and faces are swapped in positions, and edges are perpendicular.
{| class=wikitable
|- align=center
!Projective<br>symmetry
|[4]
|[3]
|[2]
|[2]
|[2]
|[2]
|[2]<sup>+</sup>
|- align=center
!Image
|[[File:dual cube t012 B2.png|60px]]
|[[File:dual cube t012.png|60px]]
|[[File:dual cube t012 f4.png|60px]]
|[[File:dual cube t012 e46.png|60px]]
|[[File:dual cube t012 e48.png|60px]]
|[[File:dual cube t012 e68.png|60px]]
|[[File:dual cube t012 v.png|60px]]
|- align=center
!Dual<br>image
|[[File:3-cube t012 B2.svg|60px]]
|[[File:3-cube t012.svg|60px]]
|[[File:cube t012 f4.png|60px]]
|[[File:cube t012 e46.png|60px]]
|[[File:cube t012 e48.png|60px]]
|[[File:cube t012 e68.png|60px]]
|[[File:cube t012 v.png|60px]]
|}

== Related polyhedra and tilings ==
{| class=wikitable align=right width=320
|[[File:Conway polyhedron m3O.png|160px]]
|[[File:Conway polyhedron m3C.png|160px]]
|-
|colspan=2|Polyhedra similar to the disdyakis dodecahedron are duals to the [[Symmetrohedron|Bowtie octahedron and cube]], containing extra pairs triangular faces .<ref>[http://www.cgl.uwaterloo.ca/csk/papers/bridges2001.html Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons] Craig S. Kaplan</ref>
|}

The disdyakis dodecahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

{{Octahedral truncations}}

It is a polyhedra in a sequence defined by the [[face configuration]] V4.6.2''n''. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any ''n''&nbsp;≥&nbsp;7.

With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.

Each face on these domains also corresponds to the fundamental domain of a [[symmetry group]] with order 2,3,''n'' mirrors at each triangle face vertex.
{{Omnitruncated table}}

{{Omnitruncated4 table}}

==See also==
* [[First stellation of rhombic dodecahedron]]
* [[Disdyakis triacontahedron]]
* [[Kisrhombille tiling]]
* [[Great rhombihexacron]]—A uniform dual polyhedron with the same surface topology

== Notes ==
{{notelist}}

==References==
{{reflist}}
* {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{ISBN|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic dodecahedron)

==External links==
* {{Mathworld2 |urlname=DisdyakisDodecahedron |title=Disdyakis dodecahedron |urlname2=CatalanSolid |title2=Catalan solid}}
* [https://web.archive.org/web/20070701185441/http://polyhedra.org/poly/show/37/hexakis_octahedron Disdyakis Dodecahedron (Hexakis Octahedron)] Interactive Polyhedron Model

{{Catalan solids}}
{{Polyhedron navigator}}

[[Category:Catalan solids]]