Editing Polytope compound (section)

== Dual compounds ==

{{multiple image
 | align = right  | direction=vertical
 | image1 = Dual compound truncated 4b max.png | caption1=[[Truncated tetrahedron]] (light) and [[triakis tetrahedron]] (dark)
 | image2 = Dual compound snub 6-8 right max.png |caption2 = [[Snub cube]] (light) and [[pentagonal icositetrahedron]] (dark)
 | image3 = Dual compound 12-20 max.png |caption3 = [[Icosidodecahedron]] (light) and [[rhombic triacontahedron]] (dark)
 | footer = Dual compounds of [[Archimedean solid|Archimedean]] and [[Catalan solid]]s
}}

A '''dual''' compound is composed of a polyhedron and its dual, arranged reciprocally about a common [[midsphere]], such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five dual compounds of the regular polyhedra.

The core is the [[Rectification (geometry)|rectification]] of both solids. The hull is the dual of this rectification, and its rhombic faces have the intersecting edges of the two solids as diagonals (and have their four alternate vertices). For the convex solids, this is the [[convex hull]].

{| class="wikitable"
! Dual compound
! Picture
! Hull
! Core
![[List of spherical symmetry groups|Symmetry group]]
|-
| Two [[tetrahedron|tetrahedra]]<br><small>([[Compound of two tetrahedra]], [[stellated octahedron]])</small>
| [[Image:Dual compound 4 max.png|100px]]
|[[Cube]]
|[[Octahedron]]
|style="text-align:center"|*432<br>[4,3]<br>''O''<sub>''h''</sub>
|-
| [[Cube]] and [[octahedron]]<br><small>([[Compound of cube and octahedron]])</small>
| [[Image:Dual compound 8 max.png|100px]]
| [[Rhombic dodecahedron]]
| [[Cuboctahedron]]
|style="text-align:center"|*432<br>[4,3]<br>''O''<sub>''h''</sub>
|-
| [[Regular dodecahedron|Dodecahedron]] and [[Regular icosahedron|icosahedron]]<br><small>([[Compound of dodecahedron and icosahedron]])</small>
| [[Image:Dual compound 20 max.png|100px]]
| [[Rhombic triacontahedron]]
| [[Icosidodecahedron]]
|style="text-align:center"|*532<br>[5,3]<br>''I''<sub>''h''</sub>
|-
| [[Small stellated dodecahedron]] and [[great dodecahedron]]<br><small>([[Compound of small stellated dodecahedron and great dodecahedron|Compound of sD and gD]])</small>
| [[File:Skeleton pair Gr12 and dual, size m (crop), thick.png|100px]]
| [[Medial rhombic triacontahedron]]<br><small>(Convex: [[regular icosahedron|Icosahedron]])</small>
| [[Dodecadodecahedron]]<br><small>(Convex: [[regular dodecahedron|Dodecahedron]])</small>
|style="text-align:center"|*532<br>[5,3]<br>''I''<sub>''h''</sub>
|-
| [[Great icosahedron]] and [[great stellated dodecahedron]]<br><small>([[Compound of great icosahedron and great stellated dodecahedron|Compound of gI and gsD]])</small>
| [[File:Skeleton pair Gr20 and dual, size s, thick.png|100px]]
| [[Great rhombic triacontahedron]]<br><small>(Convex: [[regular dodecahedron|Dodecahedron]])</small>
| [[Great icosidodecahedron]]<br><small>(Convex: [[Icosadodecahedron]])</small>
|style="text-align:center"|*532<br>[5,3]<br>''I''<sub>''h''</sub>
|}

The tetrahedron is self-dual, so the dual compound of a tetrahedron with its dual is the regular [[stellated octahedron]].

The octahedral and icosahedral dual compounds are the first stellations of the [[cuboctahedron]] and [[icosidodecahedron]], respectively.

The small stellated dodecahedral (or great dodecahedral) dual compound has the great dodecahedron completely interior to the small stellated dodecahedron.<ref>{{cite web | url=https://mathworld.wolfram.com/GreatDodecahedron-SmallStellatedDodecahedronCompound.html | title=Great Dodecahedron-Small Stellated Dodecahedron Compound }}</ref>