Editing Polytope compound

{{short description|3D shape made of polyhedra sharing a common center}}
{{more sources needed|date=February 2025}}

In [[geometry]], a '''polyhedral compound''' is a figure that is composed of several [[polyhedra]] sharing a common [[Centroid|centre]].  They are the three-dimensional analogs of [[star polygon#Regular compounds|polygonal compound]]s such as the [[hexagram]].

The outer [[Vertex (geometry)|vertices]] of a compound can be connected to form a [[convex polyhedron]] called its [[convex hull]]. A compound is a [[faceting]] of its convex hull.{{citation needed|date=July 2022}}

Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be used as the core for a set of [[stellation]]s.

== 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>

== 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>

== Uniform compounds ==
{{main|Uniform polyhedron compound}}

In 1976 John Skilling published ''Uniform Compounds of Uniform Polyhedra'' which enumerated 75 compounds (including 6 as infinite [[Prism (geometry)|prismatic]] sets of compounds, #20-#25) made from uniform polyhedra with rotational symmetry. (Every vertex is [[vertex-transitive]] and every vertex is transitive with every other vertex.) This list includes the five regular compounds above. [https://archive.today/20070928154042/http://www.interocitors.com/polyhedra/UCs/UniformCompounds.html]

The 75 uniform compounds are listed in the Table below. Most are shown singularly colored by each polyhedron element. Some chiral pairs of face groups are colored by symmetry of the faces within each polyhedron.

* 1-19: Miscellaneous (4,5,6,9,17 are the 5 ''regular compounds'')
{|
|-
|[[Image:UC01-6 tetrahedra.png|100px]]
|[[Image:UC02-12 tetrahedra.png|100px]]
|[[Image:UC03-6 tetrahedra.png|100px]]
|[[Image:UC04-2 tetrahedra.png|100px]]
|[[Image:UC05-5 tetrahedra.png|100px]]
|[[Image:UC06-10 tetrahedra.png|100px]]
|-
|[[Image:UC07-6 cubes.png|100px]]
|[[Image:UC08-3 cubes.png|100px]]
|[[Image:UC09-5 cubes.png|100px]]
|[[Image:UC10-4 octahedra.png|100px]]
|[[Image:UC11-8 octahedra.png|100px]]
|[[Image:UC12-4 octahedra.png|100px]]
|-
|[[Image:UC13-20 octahedra.png|100px]]
|[[Image:UC14-20 octahedra.png|100px]]
|[[Image:UC15-10 octahedra.png|100px]]
|[[Image:UC16-10 octahedra.png|100px]]
|[[Image:UC17-5 octahedra.png|100px]]
|[[Image:UC18-5 tetrahemihexahedron.png|100px]]
|-
|[[Image:UC19-20 tetrahemihexahedron.png|100px]]
|}
* 20-25: Prism symmetry embedded in [[Dihedral symmetry in three dimensions|prism symmetry]],
{|
|[[Image:UC20-2k n-m-gonal prisms.png|100px]]
|[[Image:UC21-k n-m-gonal prisms.png|100px]]
|[[Image:UC22-2k n-m-gonal antiprisms.png|100px]]
|[[Image:UC23-k n-m-gonal antiprisms.png|100px]]
|[[Image:UC24-2k n-m-gonal antiprisms.png|100px]]
|[[Image:UC25-k n-m-gonal antiprisms.png|100px]]
|}
* 26-45: Prism symmetry embedded in [[Octahedral symmetry|octahedral]] or [[icosahedral symmetry]],
{|
|[[Image:UC26-12 pentagonal antiprisms.png|100px]]
|[[Image:UC27-6 pentagonal antiprisms.png|100px]]
|[[Image:UC28-12 pentagrammic crossed antiprisms.png|100px]]
|[[Image:UC29-6 pentagrammic crossed antiprisms.png|100px]]
|[[Image:UC30-4 triangular prisms.png|100px]]
|[[Image:UC31-8 triangular prisms.png|100px]]
|-
|[[Image:UC32-10 triangular prisms.png|100px]]
|[[Image:UC33-20 triangular prisms.png|100px]]
|[[Image:UC34-6 pentagonal prisms.png|100px]]
|[[Image:UC35-12 pentagonal prisms.png|100px]]
|[[Image:UC36-6 pentagrammic prisms.png|100px]]
|[[Image:UC37-12 pentagrammic prisms.png|100px]]
|-
|[[Image:UC38-4 hexagonal prisms.png|100px]]
|[[Image:UC39-10 hexagonal prisms.png|100px]]
|[[Image:UC40-6 decagonal prisms.png|100px]]
|[[Image:UC41-6 decagrammic prisms.png|100px]]
|[[Image:UC42-3 square antiprisms.png|100px]]
|[[Image:UC43-6 square antiprisms.png|100px]]
|-
|[[Image:UC44-6 pentagrammic antiprisms.png|100px]]
|[[Image:UC45-12 pentagrammic antiprisms.png|100px]]
|}
* 46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,
{|
|[[Image:UC46-2 icosahedra.png|100px]]
|[[Image:UC47-5 icosahedra.png|100px]]
|[[Image:UC48-2 great dodecahedra.png|100px]]
|[[Image:UC49-5 great dodecahedra.png|100px]]
|[[Image:UC50-2 small stellated dodecahedra.png|100px]]
|[[Image:UC51-5 small stellated dodecahedra.png|100px]]
|-
|[[Image:UC52-2 great icosahedra.png|100px]]
|[[Image:UC53-5 great icosahedra.png|100px]]
|[[Image:UC54-2 truncated tetrahedra.png|100px]]
|[[Image:UC55-5 truncated tetrahedra.png|100px]]
|[[Image:UC56-10 truncated tetrahedra.png|100px]]
|[[Image:UC57-5 truncated cubes.png|100px]]
|-
|[[Image:UC58-5 quasitruncated hexahedra.png|100px]]
|[[Image:UC59-5 cuboctahedra.png|100px]]
|[[Image:UC60-5 cubohemioctahedra.png|100px]]
|[[Image:UC61-5 octahemioctahedra.png|100px]]
|[[Image:UC62-5 rhombicuboctahedra.png|100px]]
|[[Image:UC63-5 small rhombihexahedra.png|100px]]
|-
|[[Image:UC64-5 small cubicuboctahedra.png|100px]]
|[[Image:UC65-5 great cubicuboctahedra.png|100px]]
|[[Image:UC66-5 great rhombihexahedra.png|100px]]
|[[Image:UC67-5 great rhombicuboctahedra.png|100px]]
|}
* 68-75: [[enantiomorph]] pairs
{|
|[[Image:UC68-2 snub cubes.png|100px]]
|[[Image:UC69-2 snub dodecahedra.png|100px]]
|[[Image:UC70-2 great snub icosidodecahedra.png|100px]]
|[[Image:UC71-2 great inverted snub icosidodecahedra.png|100px]]
|[[Image:UC72-2 great retrosnub icosidodecahedra.png|100px]]
|[[Image:UC73-2 snub dodecadodecahedra.png|100px]]
|-
|[[Image:UC74-2 inverted snub dodecadodecahedra.png|100px]]
|[[Image:UC75-2 snub icosidodecadodecahedra.png|100px]]
|}

== Other compounds ==
{| class=wikitable align=right width=400
|[[File:Compound of 4 cubes.png|200px]]
|[[File:Compound of 4 octahedra.png|200px]]
|-
|colspan=2|The compound of four cubes (left) is neither a regular compound, nor a dual compound, nor a uniform compound. Its dual, the compound of four octahedra (right), is a uniform compound.
|}
* [[Compound of three octahedra]]
* [[Compound of four cubes]]

Two polyhedra that are compounds but have their elements rigidly locked into place are the [[small complex icosidodecahedron]] (compound of [[icosahedron]] and [[great dodecahedron]]) and the [[great complex icosidodecahedron]] (compound of [[small stellated dodecahedron]] and [[great icosahedron]]). If the definition of a [[uniform polyhedron]] is generalised, they are uniform.

The section for enantiomorph pairs in Skilling's list does not contain the compound of two [[great snub dodecicosidodecahedron|great snub dodecicosidodecahedra]], as the [[pentagram (geometry)|pentagram]] faces would coincide. Removing the coincident faces results in the [[compound of twenty octahedra]].

{{Clear}}

== 4-polytope compounds ==
{| class=wikitable align=right
|+ Orthogonal projections
|[[File:Regular compound 75 tesseracts.png|200px]]
|[[File:Regular compound 75 16-cells.png|200px]]
|-
!75 [[tesseract|{4,3,3}]]
!75 [[16-cell|{3,3,4}]]
|}

In 4-dimensions, there are a large number of regular compounds of regular polytopes. [[Coxeter]] lists a few of these in his book [[Regular Polytopes (book)|Regular Polytopes]].<ref name="cox">Regular polytopes, Table VII, p. 305</ref> [[Peter McMullen|McMullen]] added six in his paper ''New Regular Compounds of 4-Polytopes''.<ref name="McMullen">McMullen, Peter (2018), ''New Regular Compounds of 4-Polytopes'', New Trends in Intuitive Geometry, 27: 307–320</ref>

'''Self-duals:'''
{| class=wikitable
!Compound 
!Constituent
!Symmetry
|-
| 120 [[5-cell]]s || [[5-cell]] || [5,3,3], order 14400<ref name="cox"/>
|-
| 120 [[5-cell]]s<sup>(var)</sup> || [[5-cell]] || order 1200<ref name="McMullen"/>
|-
| 720 [[5-cell]]s || [[5-cell]] || [5,3,3], order 14400<ref name="cox"/>
|-
| 5 [[24-cell]]s || [[24-cell]] || [5,3,3], order 14400<ref name="cox"/>
|}

'''Dual pairs:'''
{| class=wikitable
!Compound 1
!Compound 2
!Symmetry
|-
|3 [[16-cell]]s<ref>{{KlitzingPolytopes|..//incmats/stico.htm|Uniform compound|stellated icositetrachoron}}</ref>||3 [[tesseract]]s||[3,4,3], order 1152<ref name="cox"/>
|-
|15 [[16-cell]]s||15 [[tesseract]]s||[5,3,3], order 14400<ref name="cox"/>
|-
|75 [[16-cell]]s||75 [[tesseract]]s||[5,3,3], order 14400<ref name="cox"/>
|-
|75 [[16-cell]]s<sup>(var)</sup>||75 [[tesseract]]s<sup>(var)</sup>||order 600<ref name="McMullen"/>
|-
|300 [[16-cell]]s||300 [[tesseract]]s||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/>
|-
|600 [[16-cell]]s||600 [[tesseract]]s||[5,3,3], order 14400<ref name="cox"/>
|-
|25 [[24-cell]]s||25 [[24-cell]]s||[5,3,3], order 14400<ref name="cox"/>
|}
Uniform compounds and duals with convex 4-polytopes:
{| class=wikitable
!Compound 1<br>[[Vertex-transitive]]
!Compound 2<br>[[Cell-transitive]]
!Symmetry
|-
|2 [[16-cell]]s<ref>{{KlitzingPolytopes|..//incmats/haddet.htm|Uniform compound|demidistesseract}}</ref>||2 [[tesseract]]s||[4,3,3], order 384<ref name="cox"/>
|-
|100 [[24-cell]]s||100 [[24-cell]]s||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/>
|-
|200 [[24-cell]]s||200 [[24-cell]]s||[5,3,3], order 14400<ref name="cox"/>
|-
|5 [[600-cell]]s||5 [[120-cell]]s||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/>
|-
|10 [[600-cell]]s||10 [[120-cell]]s||[5,3,3], order 14400<ref name="cox"/>
|-
|25 [[24-cell]]s<sup>(var)</sup>||25 [[24-cell]]s<sup>(var)</sup>||order 600<ref name="McMullen"/>
|}
The superscript (var) in the tables above indicates that the labeled compounds are distinct from the other compounds with the same number of constituents.

=== Compounds with regular star 4-polytopes ===
'''Self-dual star compounds:'''
{| class=wikitable
!Compound 
!Symmetry
|-
| 5 [[Great 120-cell|{5,5/2,5}]] || [5,3,3]<sup>+</sup>, order 7200<ref name="cox"/>
|-
| 10 [[Great 120-cell|{5,5/2,5}]] || [5,3,3], order 14400<ref name="cox"/>
|-
| 5 [[Grand stellated 120-cell|{5/2,5,5/2}]] || [5,3,3]<sup>+</sup>, order 7200<ref name="cox"/>
|-
| 10 [[Grand stellated 120-cell|{5/2,5,5/2}]] || [5,3,3], order 14400<ref name="cox"/>
|}

'''Dual pairs of compound stars:'''
{| class=wikitable
!Compound 1
!Compound 2
!Symmetry
|-
|5 {3,5,5/2}||5 {5/2,5,3}||[5,3,3]<sup>+</sup>, order 7200
|-
|10 {3,5,5/2}||10 {5/2,5,3}||[5,3,3], order 14400
|-
|5 {5,5/2,3}||5 {3,5/2,5}||[5,3,3]<sup>+</sup>, order 7200
|-
|10 {5,5/2,3}||10 {3,5/2,5}||[5,3,3], order 14400
|-
|5 {5/2,3,5}||5 {5,3,5/2}||[5,3,3]<sup>+</sup>, order 7200
|-
|10 {5/2,3,5}||10 {5,3,5/2}||[5,3,3], order 14400
|}

'''Uniform compound stars and duals''':
{| class=wikitable
!Compound 1<br>[[Vertex-transitive]]
!Compound 2<br>[[Cell-transitive]]
!Symmetry
|-
|5 [[Grand 600-cell|{3,3,5/2}]]||5 [[Great grand stellated 120-cell|{5/2,3,3}]]||[5,3,3]<sup>+</sup>, order 7200
|-
|10 [[Grand 600-cell|{3,3,5/2}]]||10 [[Great grand stellated 120-cell|{5/2,3,3}]]||[5,3,3], order 14400
|}

=== Compounds with duals ===

'''Dual positions:'''
{| class=wikitable
!Compound 
!Constituent
!Symmetry
|-
| [[Compound of two 5-cells|2 5-cell]] || [[5-cell]] || <nowiki>[[</nowiki>3,3,3]], order 240
|-
| [[Compound of two 24-cells|2 24-cell]] || [[24-cell]] || <nowiki>[[</nowiki>3,4,3]], order 2304
|-
| 1 tesseract, 1 16-cell || [[tesseract]], [[16-cell]] ||
|-
| [[Compound of 120-cell and 600-cell|1 120-cell, 1 600-cell]] || [[120-cell]], [[600-cell]] ||
|-
| [[Compound of two great 120-cells|2 great 120-cell]] || [[great 120-cell]] ||
|-
| [[Compound of two grand stellated 120-cells|2 grand stellated 120-cell]] || [[grand stellated 120-cell]] ||
|-
| [[Compound of icosahedral 120-cell and small stellated 120-cell|1 icosahedral 120-cell, 1 small stellated 120-cell]] || [[icosahedral 120-cell]], [[small stellated 120-cell]] ||
|-
| [[Compound of grand 120-cell and great stellated 120-cell|1 grand 120-cell, 1 great stellated 120-cell]] || [[grand 120-cell]], [[great stellated 120-cell]] ||
|-
| [[Compound of great grand 120-cell and great icosahedral 120-cell|1 great grand 120-cell, 1 great icosahedral 120-cell]] || [[great grand 120-cell]], [[great icosahedral 120-cell]] ||
|-
| [[Compound of great grand stellated 120-cell and grand 600-cell|1 great grand stellated 120-cell, 1 grand 600-cell]] || [[great grand stellated 120-cell]], [[grand 600-cell]] ||
|}

==Group theory==
In terms of [[group theory]], if ''G'' is the symmetry group of a polyhedral compound, and the group [[acts transitively]] on the polyhedra (so that each polyhedron can be sent to any of the others, as in uniform compounds), then if ''H'' is the [[stabilizer subgroup|stabilizer]] of a single chosen polyhedron, the polyhedra can be identified with the [[orbit space]] ''G''/''H'' – the coset ''gH'' corresponds to which polyhedron ''g'' sends the chosen polyhedron to.

==Compounds of tilings==
There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not been enumerated.

The Euclidean and hyperbolic compound families 2 {''p'',''p''} (4 ≤ ''p'' ≤ ∞, ''p'' an integer) are analogous to the spherical [[stella octangula]], 2 {3,3}.
{| class="wikitable"
|+ A few examples of Euclidean and hyperbolic regular compounds
!Self-dual
!colspan=2|Duals
!Self-dual
|-
!2 [[square tiling|{4,4}]]
!2 [[hexagonal tiling|{6,3}]]
!2 [[triangular tiling|{3,6}]]
!2 [[infinite-order apeirogonal tiling|{∞,∞}]]
|- align=center
|[[File:Kah 4 4.png|160px]] 
|[[File:Compound 2 hexagonal tilings.svg|160px]]
|[[File:Compound 2 triangular tilings.svg|160px]]
| [[File:Infinite-order apeirogonal tiling and dual.png|160px]]

|-
!
!3 {6,3}
!3 {3,6}
!3 [[infinite-order apeirogonal tiling|{∞,∞}]]
|-
|
|[[File:Compound 3 hexagonal tilings.svg|160px]]
|[[File:Compound 3 triangular tilings.svg|160px]]
|[[File:Iii symmetry 000.png|160px]]
|}

A known family of regular Euclidean compound honeycombs in any number of dimensions is an infinite family of compounds of [[hypercubic honeycomb]]s, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs.

There are also ''dual-regular'' tiling compounds. A simple example is the E<sup>2</sup> compound of a [[hexagonal tiling]] and its dual [[triangular tiling]], which shares its edges with the [[deltoidal trihexagonal tiling]]. The Euclidean compounds of two hypercubic honeycombs are both regular and dual-regular.

== See also ==
*[[List of regular polytope compounds]]

==Footnotes==
{{reflist}}

==External links==
*[http://mathworld.wolfram.com/PolyhedronCompound.html MathWorld: Polyhedron Compound]
*[http://www.georgehart.com/virtual-polyhedra/compounds-info.html Compound polyhedra] &ndash; from Virtual Reality Polyhedra
** [http://www.georgehart.com/virtual-polyhedra/uniform-compounds-index.html Uniform Compounds of Uniform Polyhedra]
*[https://archive.today/20070928154042/http://www.interocitors.com/polyhedra/UCs/UniformCompounds.html Skilling's 75 Uniform Compounds of Uniform Polyhedra]
*[http://www.tunnissen.eu/polyh/UniformCompounds.vrml.html Skilling's Uniform Compounds of Uniform Polyhedra]
*[https://web.archive.org/web/20070102152247/http://www.uwgb.edu/dutchs/SYMMETRY/polycpd.htm Polyhedral Compounds]
*http://users.skynet.be/polyhedra.fleurent/Compounds_2/Compounds_2.htm
*[http://members.aol.com/Polycell/regs.html Compound of Small Stellated Dodecahedron and Great Dodecahedron {5/2,5}+{5,5/2}]
* {{KlitzingPolytopes|../explain/compound.htm|Compound polytopes}}

== References ==
*{{citation|first=John|last=Skilling|title=Uniform Compounds of Uniform Polyhedra|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|volume=79|pages=447–457|year=1976|issue=3 |doi=10.1017/S0305004100052440|bibcode=1976MPCPS..79..447S |mr=0397554|s2cid=123279687 }}.
*{{citation|first=Peter R.|last=Cromwell|title=Polyhedra|location=Cambridge|year=1997}}.
*{{citation|first=Magnus|last=Wenninger|author-link=Magnus Wenninger|title=Dual Models|location=Cambridge, England|publisher=Cambridge University Press|year=1983|pages=51–53}}.
*{{citation|first=Michael G.|last=Harman|title=Polyhedral Compounds|publisher=unpublished manuscript|year=1974|url=http://www.georgehart.com/virtual-polyhedra/compounds-harman.html}}.
*{{citation|first=Edmund|last=Hess|author-link=Edmund Hess|title=Zugleich Gleicheckigen und Gleichflächigen Polyeder|journal=Schriften der Gesellschaft zur Berörderung der Gasammten Naturwissenschaften zu Marburg|volume=11|year=1876|pages=5–97}}.
*{{citation|first=Luca|last=Pacioli|author-link=Luca Pacioli|title=De Divina Proportione|year=1509}}.
* ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}}
* {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7  }} p.&nbsp;87 Five regular compounds
*{{citation|first=Peter|last=McMullen|title=New Trends in Intuitive Geometry |chapter=New Regular Compounds of 4-Polytopes |series=Bolyai Society Mathematical Studies |volume=27|pages=307–320|year=2018|doi=10.1007/978-3-662-57413-3_12|isbn=978-3-662-57412-6 }}.

[[Category:Polyhedral compounds| ]]