Editing Polytope compound (section)

== 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]] ||
|}