Editing Cuboctahedron (section)

== Configuration matrix ==
{{more citations needed section|date=March 2025}}
The cuboctahedron can be represented as a [[Configuration (polytope)|configuration matrix]] with elements grouped by symmetry transitivity classes. A configuration matrix is a [[Matrix (mathematics)|matrix]] in which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. The [[Main diagonal|diagonal]] of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element.

The cuboctahedron has 1 transitivity class of 12 vertices, 1 class of 24 edges, and 2 classes of faces: 8 triangular and 6 square; each element in a matrix's diagonal.<ref>{{cite web | url=https://www.bendwavy.org/klitzing/incmats/co.htm | title=Co }}</ref> The 24 edges can be seen in 4 central hexagons.

With [[octahedral symmetry]] ([[orbifold notation|orbifold]] 432), the squares have the 4-fold symmetry, triangles the 3-fold symmetry, and vertices the 2-fold symmetry. With [[tetrahedral symmetry]] (orbifold 332) the 24 vertices split into 2 edge classes, and the 8 triangles split into 2 face classes. The square symmetry is reduced to 2-fold.
{| class=wikitable
!colspan=2|Octahedral symmetry (432)||colspan=2|Tetrahedral symmetry (332)
|-
|[[File:cuboctahedron colored.svg|240px]]
| valign=top|
{| class=wikitable
|+ Configuration
|-
|(432)||style="background-color:#3CB44B; color: #000000"|v<sub>1</sub>||style="background-color:#FF00FF;"|e<sub>1</sub>||style="background-color:#0000FF; color: #E0E0E0"|f<sub>1</sub>||style="background-color:#FF0000;"|f<sub>2</sub>
|- align=right
|style="background-color:#3CB44B; color:#000000"|v<sub>1 (Z<sub>2</sub>)</sub>||style="background-color:#E0F0FF"|12|||4|||2|||2
|- align=right
|align=left style="background-color:#FF00FF;"|e<sub>1</sub>|||2||style="background-color:#f0FFE0"|24|||1|||1
|- align=right
|align=left style="background-color:#0000FF; color:#E0E0E0"|f<sub>1 (Z<sub>3</sub>)</sub>|||3|||3||style="background-color:#FFFFE0"|8||style="background-color:#FFFFE0"|*
|- align=right
|align=left style="background-color:#FF0000;"|f<sub>2 (Z<sub>4</sub>)</sub>|||4|||4||style="background-color:#FFFFE0"|*||style="background-color:#FFFFE0"|6
|}
|[[File:Cuboctahedron-tetrahedral colored.svg|240px]]
|
{| class=wikitable
|+ Configuration
|-
|(332)||style="background-color:#D51D5D;"|v<sub>1</sub>||style="background-color:#3CB44B;"|e<sub>1</sub>||style="background-color:#E6194B;"|e<sub>2</sub>||style="background-color:#4363D8;"|f<sub>1</sub>||style="background-color:#FFE119;"|f<sub>2</sub>||style="background-color:#F58231;"|f<sub>3</sub>
|- align=right
|align=left style="background-color:#D51D5D;"|v<sub>1</sub>||style="background-color:#E0F0FF"|12|||2|||2|||1|||1|||2
|- align=right
|align=left style="background-color:#3CB44B;"|e<sub>1</sub>|||2||style="background-color:#f0FFE0"|12||style="background-color:#f0FFE0"|*|||1|||0|||1
|- align=right
|align=left style="background-color:#E6194B;"|e<sub>2</sub>|||2||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|12|||0|||1|||1
|- align=right
|align=left style="background-color:#4363D8;"|f<sub>1 (Z<sub>3</sub>)</sub>|||3|||3|||0||style="background-color:#FFFFE0"|4||style="background-color:#FFFFE0"|*||style="background-color:#FFFFE0"|*
|- align=right
|align=left style="background-color:#FFE119;"|f<sub>2 (Z<sub>3</sub>)</sub>|||3|||0|||3||style="background-color:#FFFFE0"|*||style="background-color:#FFFFE0"|4||style="background-color:#FFFFE0"|*
|- align=right
|align=left style="background-color:#F58231;"|f<sub>3 (Z<sub>2</sub>)</sub>|||4|||2|||2||style="background-color:#FFFFE0"|*||style="background-color:#FFFFE0"|*||style="background-color:#FFFFE0"|6
|}
|}