Editing Polytope compound (section)

== Regular compounds ==
{{Disputed-section|date=November 2023}}
A regular polyhedral compound can be defined as a compound which, like a [[regular polyhedron]], is [[vertex-transitive]], [[edge-transitive]], and [[face-transitive]]. Unlike the case of polyhedra, this is not equivalent to the [[symmetry group]] acting transitively on its [[flag (geometry)|flags]]; the compound of two tetrahedra is the only regular compound with that property. There are five regular compounds of polyhedra:

{| class="wikitable"
!Regular compound<BR>(Coxeter symbol)
!Picture
!Spherical
![[Convex hull]]
!Common core
![[List of spherical symmetry groups|Symmetry group]]
![[Subgroup]]<br>restricting<br>to one<br>constituent
!Dual-regular compound
|- align=center
| [[Compound of two tetrahedra|Two tetrahedra]]<BR>{4,3}[2{3,3}]{3,4}
| [[Image:Compound of two tetrahedra.png|100px]]|| [[Image:Spherical compound of two tetrahedra.png|100px]]
||[[Cube (geometry)|Cube]]
<ref name=":0" />
|[[Octahedron]]
|style="text-align:center"|*432<br>[4,3]<br>''O''<sub>''h''</sub>
|style="text-align:center"|*332<br>[3,3]<br>''T''<sub>''d''</sub>
|Two tetrahedra
|- align=center
| [[Compound of five tetrahedra|Five tetrahedra]]<BR>{5,3}[5{3,3}]{3,5}
| [[Image:Compound of five tetrahedra.png|100px]]|| [[Image:Spherical compound of five tetrahedra.png|100px]]
||[[Dodecahedron]]
<ref name=":0" />
|[[Icosahedron]]
<ref name=":0" />
|style="text-align:center"|532<br>[5,3]<sup>+</sup><br>''I''
|style="text-align:center"|332<br>[3,3]<sup>+</sup><br>''T''
|[[chirality (mathematics)|Chiral]] twin<br>[[chirality (mathematics)|(Enantiomorph)]]
|- align=center
| [[Compound of ten tetrahedra|Ten tetrahedra]]<BR>2{5,3}[10{3,3}]2{3,5}
| [[Image:Compound of ten tetrahedra.png|100px]]|| [[Image:Spherical compound of ten tetrahedra.png|100px]]
||Dodecahedron
<ref name=":0" />
|Icosahedron
|style="text-align:center"|*532<br>[5,3]<br>''I''<sub>''h''</sub>
|style="text-align:center"|332<br>[3,3]<br>''T''
|Ten tetrahedra
|- align=center
| [[Compound of five cubes|Five cubes]]<BR>2{5,3}[5{4,3}]
| [[Image:Compound of five cubes.png|100px]]|| [[Image:Spherical compound of five cubes.png|100px]]
||Dodecahedron
<ref name=":0" />
|[[Rhombic triacontahedron]]
<ref name=":0" />
|style="text-align:center"|*532<br>[5,3]<br>''I''<sub>''h''</sub>
|style="text-align:center"|3*2<br>[3,3]<br>''T''<sub>''h''</sub>
|Five octahedra
|- align=center
| [[Compound of five octahedra|Five octahedra]]<BR>[5{3,4}]2{3,5}
| [[Image:Compound of five octahedra.png|100px]]|| [[Image:Spherical compound of five octahedra.png|100px]]
||[[Icosidodecahedron]]
<ref name=":0" />
|Icosahedron
<ref name=":0" />
|style="text-align:center"|*532<br>[5,3]<br>''I''<sub>''h''</sub>
|style="text-align:center"|3*2<br>[3,3]<br>''T''<sub>''h''</sub>
|Five cubes
|}

Best known is the regular compound of two [[tetrahedron|tetrahedra]], often called the [[stella octangula]], a name given to it by [[Johannes Kepler|Kepler]]. The vertices of the two tetrahedra define a [[Cube (geometry)|cube]], and the intersection of the two define a regular [[octahedron]], which shares the same face-planes as the compound. Thus the compound of two tetrahedra is a [[stellation]] of the octahedron, and in fact, the only finite stellation thereof.

The regular [[compound of five tetrahedra]] comes in two [[chirality (mathematics)|enantiomorph]]ic versions, which together make up the regular compound of ten tetrahedra.<ref name=":0">{{Cite web|title=Compound Polyhedra|url=https://www.georgehart.com/virtual-polyhedra/compounds-info.html|access-date=2020-09-03|website=www.georgehart.com}}</ref> The regular compound of ten tetrahedra can also be seen as a compound of five stellae octangulae.<ref name=":0" />

Each of the regular tetrahedral compounds is self-dual or dual to its chiral twin; the regular compound of five cubes and the regular compound of five octahedra are dual to each other.

Hence, regular polyhedral compounds can also be regarded as '''dual-regular compounds'''.

Coxeter's notation for regular compounds is given in the table above, incorporating [[Schläfli symbol]]s. The material inside the square brackets, [''d''{''p'',''q''}], denotes the components of the compound: ''d'' separate {''p'',''q''}'s. The material ''before'' the square brackets denotes the vertex arrangement of the compound: ''c''{''m'',''n''}[''d''{''p'',''q''}] is a compound of ''d'' {''p'',''q''}'s sharing the vertices of {''m'',''n''} counted ''c'' times. The material ''after'' the square brackets denotes the facet arrangement of the compound: [''d''{''p'',''q''}]''e''{''s'',''t''} is a compound of ''d'' {''p'',''q''}'s sharing the faces of {''s'',''t''} counted ''e'' times. These may be combined: thus ''c''{''m'',''n''}[''d''{''p'',''q''}]''e''{''s'',''t''}  is a compound of ''d'' {''p'',''q''}'s sharing the vertices of {''m'',''n''} counted ''c'' times ''and'' the faces of {''s'',''t''} counted ''e'' times. This notation can be generalised to compounds in any number of dimensions.<ref>{{cite book | first=Harold Scott MacDonald | last=Coxeter | author-link=Harold Scott MacDonald Coxeter | title=Regular Polytopes | publisher = Dover Publications | edition=Third | oclc=798003 | date = 1973  | orig-year=1948 | isbn = 0-486-61480-8 | url=https://books.google.com/books?id=2ee7AQAAQBAJ |page=48}}</ref>