Editing Octahedron (section)

===Uniform colorings and symmetry===
There are 3 [[uniform coloring]]s of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111.

The octahedron's [[symmetry group]] is O<sub>h</sub>, of order 48, the three dimensional [[hyperoctahedral group]].  This group's [[subgroup]]s include D<sub>3d</sub> (order 12), the symmetry group of a triangular [[antiprism]]; '''D<sub>4h</sub>''' (order 16), the symmetry group of a square [[bipyramid]]; and T<sub>d</sub> (order 24), the symmetry group of a [[Octahedron#Tetratetrahedron|rectified tetrahedron]]. These symmetries can be emphasized by different colorings of the faces.
{| class=wikitable
!Name
!Octahedron
![[Rectification (geometry)|Rectified]] [[tetrahedron]]<br>(Tetratetrahedron)
!Triangular [[antiprism]]
!Square [[bipyramid]]
!Rhombic fusil
|- align=center
!Image<br>(Face coloring)
|[[File:Uniform polyhedron-43-t2.png|100px]]<br>(1111)
|[[File:Uniform polyhedron-33-t1.svg|100px]]<br>(1212)
|[[File:Trigonal antiprism.png|100px]]<br>(1112)
|[[File:Square bipyramid.png|100px]]<br>(1111)
|[[File:Rhombic bipyramid.png|100px]]<br>(1111)
|- align=center
![[Coxeter diagram]]
|{{CDD|node_1|3|node|4|node}}
|{{CDD|node_1|3|node|4|node_h0}} = {{CDD|node_1|split1|nodes}}
|{{CDD|node_h|2x|node_h|6|node}}<br>{{CDD|node_h|2x|node_h|3|node_h}}
|{{CDD|node_f1|2x|node_f1|4|node}}
|{{CDD|node_f1|2x|node_f1|2x|node_f1}}
|- align=center
![[Schläfli symbol]]
|{3,4}
|r{3,3}
|s{2,6}<br>sr{2,3}
|ft{2,4}<br>{ } + {4}
|ftr{2,2}<br>{ } + { } + { }
|- align=center
![[Wythoff symbol]]
| 4 {{pipe}} 3 2
| 2 {{pipe}} 4 3
| 2 {{pipe}} 6 2 <br> {{pipe}} 2 3 2
| ||
|- align=center
![[List of spherical symmetry groups|Symmetry]]
|O<sub>h</sub>, [4,3], (*432)
|T<sub>d</sub>, [3,3], (*332)
|D<sub>3d</sub>, [2<sup>+</sup>,6], (2*3)<br>D<sub>3</sub>, [2,3]<sup>+</sup>, (322)
|D<sub>4h</sub>, [2,4], (*422)
|D<sub>2h</sub>, [2,2], (*222)
|- align=center
![[Group order|Order]]
|48
|24
|12<br>6
|16
|8
|}