Editing 5-cell (section)

===Isometries===
There are many lower symmetry forms of the 5-cell, including these found as uniform polytope [[vertex figure]]s:
{| class=wikitable width=600
!Symmetry
![3,3,3]<BR>Order 120
![3,3,1]<BR>Order 24
![3,2,1]<BR>Order 12
![3,1,1]<BR>Order 6
!~[5,2]<sup>+</sup><BR>Order 10
|- align=center
!Name
| Regular 5-cell
| Tetrahedral [[polyhedral pyramid|pyramid]]
| 
| Triangular pyramidal pyramid
|
|- align=center
![[Schläfli symbol|Schläfli]]
| {3,3,3}
| {3,3}∨(&nbsp;)
| {3}∨{&nbsp;}
| {3}∨(&nbsp;)∨(&nbsp;)
| 
|- align=center valign=top
!valign=center|Example<BR>Vertex<BR>figure
|[[File:5-simplex verf.png|120px]]<BR>[[5-simplex]]
|[[File:Truncated 5-simplex verf.png|120px]]<BR>[[Truncated 5-simplex]]
|[[File:Bitruncated 5-simplex verf.png|120px]]<BR>[[Bitruncated 5-simplex]]
|[[File:Canitruncated 5-simplex verf.png|120px]]<BR>[[Cantitruncated 5-simplex]]
|[[File:Omnitruncated 4-simplex honeycomb verf.png|120px]]<BR>[[Omnitruncated 4-simplex honeycomb]]
|}

The '''tetrahedral pyramid''' is a special case of a '''5-cell''', a [[polyhedral pyramid]], constructed as a regular [[tetrahedron]] base in a 3-space [[hyperplane]], and an [[Apex (geometry)|apex]] point ''above'' the hyperplane. The four ''sides'' of the pyramid are made of [[triangular pyramid]] cells.

Many [[uniform 5-polytope]]s have '''tetrahedral pyramid''' [[vertex figure]]s with [[Schläfli symbol]]s ( )∨{3,3}.
{| class=wikitable
|+ Symmetry [3,3,1], order 24
|-
![[Schlegel diagram|Schlegel<BR>diagram]]
|[[File:5-cell prism verf.png|100px]]
|[[File:Tesseractic prism verf.png|100px]]
|[[File:120-cell prism verf.png|100px]]
|[[File:Truncated 5-simplex verf.png|100px]]
|[[File:Truncated 5-cube verf.png|100px]]
|[[File:Truncated 24-cell honeycomb verf.png|100px]]
|-
!Name<BR>[[Coxeter diagram|Coxeter]]
![[5-cell prism|{ }×{3,3,3}]]<BR>{{CDD|node_1|2|node_1|3|node|3|node|3|node}}
![[Tesseractic prism|{ }×{4,3,3}]]<BR>{{CDD|node_1|2|node_1|4|node|3|node|3|node}}
![[120-cell prism|{ }×{5,3,3}]]<BR>{{CDD|node_1|2|node_1|5|node|3|node|3|node}}
![[Truncated 5-simplex|t{3,3,3,3}]]<BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node}}
![[Truncated 5-cube|t{4,3,3,3}]]<BR>{{CDD|node_1|4|node_1|3|node|3|node|3|node}}
![[Truncated 24-cell honeycomb|t{3,4,3,3}]]<BR>{{CDD|node_1|3|node_1|4|node|3|node|3|node}}
|}

Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a [[uniform polytope]] is represented by removing the ringed nodes of the Coxeter diagram.
{| class=wikitable
!Symmetry
!colspan=2|[3,2,1], order 12
!colspan=2|[3,1,1], order 6
![2<sup>+</sup>,4,1], order 8
![2,1,1], order 4
|- align=center
![[Schläfli symbol|Schläfli]]
|colspan=2|{3}∨{ &nbsp;}||colspan=2|{3}∨(&nbsp;)∨(&nbsp;)||colspan=2|{&nbsp;}∨{&nbsp;}∨(&nbsp;)
|-
![[Schlegel diagram|Schlegel<BR>diagram]]
|[[File:Bitruncated 5-simplex verf.png|100px]]
|[[File:Bitruncated penteract verf.png|100px]]
|[[File:Canitruncated 5-simplex verf.png|100px]]
|[[File:Canitruncated 5-cube verf.png|100px]]
|[[File:Bicanitruncated 5-simplex verf.png|100px]]
|[[File:Bicanitruncated 5-cube verf.png|100px]]
|-
!Name<BR>[[Coxeter diagram|Coxeter]]
![[bitruncated 5-simplex|t<sub>12</sub>α<sub>5</sub>]]<BR>{{CDD|node|3|node_1|3|node_1|3|node|3|node}}
![[bitruncated 5-cube|t<sub>12</sub>γ<sub>5</sub>]]<BR>{{CDD|node|4|node_1|3|node_1|3|node|3|node}}
![[Cantitruncated 5-simplex|t<sub>012</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node|3|node}}
![[Cantitruncated 5-cube|t<sub>012</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node|3|node}}
![[Bicantitruncated 5-simplex|t<sub>123</sub>α<sub>5</sub>]]<BR>{{CDD|node|3|node_1|3|node_1|3|node_1|3|node}}
![[Bicantitruncated 5-cube|t<sub>123</sub>γ<sub>5</sub>]]<BR>{{CDD|node|4|node_1|3|node_1|3|node_1|3|node}}
|}

{| class=wikitable
!Symmetry
!colspan=3|[2,1,1], order 2
![2<sup>+</sup>,1,1], order 2
![ ]<sup>+</sup>, order 1
|- align=center
![[Schläfli symbol|Schläfli]]
|colspan=3|{&nbsp;}∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)||colspan=2|(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)
|-
![[Schlegel diagram|Schlegel<BR>diagram]]
|[[File:Runcicantitruncated 5-simplex verf.png|100px]]
|[[File:Runcicantitruncated 5-cube verf.png|100px]]
|[[File:Runcicantitruncated 5-orthoplex verf.png|100px]]
|[[File:Omnitruncated 5-simplex verf.png|100px]]
|[[File:Omnitruncated 5-cube verf.png|100px]]
|-
!Name<BR>[[Coxeter diagram|Coxeter]]
![[Runcicantitruncated 5-simplex|t<sub>0123</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node}}
![[Runcicantitruncated 5-cube|t<sub>0123</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node}}
![[Runcicantitruncated 5-orthoplex|t<sub>0123</sub>β<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|4|node}}
![[Omnitruncated 5-simplex|t<sub>01234</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
![[Omnitruncated 5-cube|t<sub>01234</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1}}
|}