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[[File:Schlegel half-solid truncated 120-cell.png|320px|thumb|[[Schlegel diagram]] for the [[truncated 120-cell]] with [[Tetrahedron|tetrahedral]] cells visible. This [[perspective projection]] makes edges look smaller towards the center of the projection.]]
== little hope of survival ==
[[File:120-cell t01 H3.svg|thumb|320px|[[orthographic projection]] of the truncated 120-cell, in the H<sub>3</sub> [[Coxeter plane]] (D<sub>10</sub> symmetry). Only vertices and edges are drawn.]]
In [[geometry]], a '''[[Uniform polytope|uniform]] polychoron''' (plural: '''uniform polychora''') is a [[polychoron]] (4-[[polytope]]) which is [[vertex-transitive]] and whose cells are [[uniform polyhedron|uniform polyhedra]].


This article contains the complete list of 47 non-prismatic convex uniform polychora, and describes three sets of convex prismatic forms, two being infinite.
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相关的主题文章：
<ul>
 
  <li>[http://www.ytzjj.com/bbs/home.php?mod=space&uid=32217 http://www.ytzjj.com/bbs/home.php?mod=space&uid=32217]</li>
 
  <li>[http://www.xinjilsy.com/home.php?mod=space&uid=13433 http://www.xinjilsy.com/home.php?mod=space&uid=13433]</li>
 
  <li>[http://www.xggh.org/bust-up/blog.cgi http://www.xggh.org/bust-up/blog.cgi]</li>
 
</ul>


== History of discovery ==
== again tremor. ==
* '''[[Regular polytope]]s''': (convex faces)
** '''1852''': [[Ludwig Schläfli]] proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 6 regular polytopes in 4 [[dimension]]s and only 3 in 5 or more dimensions.
* '''[[Schläfli-Hess polychoron|Regular star-polychora]]''' ([[star polyhedron]] cells and/or [[vertex figure]]s)
** '''1852''': [[Ludwig Schläfli]] also found 4 of the 10 regular star polychora, discounting 6 with cells or vertex figures [[small stellated dodecahedron|{5/2,5}]] and [[great dodecahedron|{5,5/2}]].
** '''1883''': [[Edmund Hess]] completed the list of 10 of the nonconvex regular polychora, in his book (in German) ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder'' [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001].
* '''Convex [[semiregular polytope]]s''': (Various definitions before Coxeter's '''uniform''' category)
** '''1900''': [[Thorold Gosset]] enumerated the list of nonprismatic semiregular convex polytopes with regular cells ([[Platonic solid]]s) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimensions''.<ref>[[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900</ref>
** '''1910''': [[Alicia Boole Stott]], in her publication ''Geometrical deduction of semiregular from regular polytopes and space fillings'', expanded the definition by also allowing [[Archimedean solid]] and [[Prism (geometry)|prism]] cells. This construction enumerated 45 semiregular polychora.<ref>http://dissertations.ub.rug.nl/FILES/faculties/science/2007/i.polo.blanco/c5.pdf</ref>
** '''1911''': [[Pieter Hendrik Schoute]] published ''Analytic treatment of the polytopes regularly derived from the regular polytopes'', followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on [[5-cell]], [[8-cell]]/[[16-cell]], and [[24-cell]].
** '''1912''': [[E. L. Elte]] independently expanded on Gosset's list with the publication ''The Semiregular Polytopes of the Hyperspaces'', polytopes with one or two types of semiregular facets.<ref>{{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912 | isbn = 1-4181-7968-X}} [http://www.amazon.com/Semiregular-Polytopes-Hyperspaces-Emanuel-Lodewijk/dp/141817968X]</ref>
* '''Convex uniform polytopes''':
** '''1940''': The search was expanded systematically by [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]] in his publication ''Regular and Semi-Regular Polytopes''.
** '''Convex uniform polychora''':
*** '''1965''': The complete list of convex forms was finally done by [[John Horton Conway]] and [[Michael Guy (computer scientist)|Michael Guy]], in their publication ''Four-Dimensional Archimedean Polytopes'', established by computer analysis, adding only one non-Wythoffian convex polychoron, the [[grand antiprism]].
*** '''1966''' [[Norman Johnson (mathematician)|N.W. Johnson]] completes his Ph.D. dissertation ''The Theory of Uniform Polytopes and Honeycombs'' under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher
*** '''1997''': A complete enumeration of the names and elements of the convex uniform polychora is given online by [[George Olshevsky]].<ref>{{PolyCell | urlname = uniform.html| title = Uniform Polytopes in Four Dimensions}}</ref>
*** '''2004''': A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, ''Vierdimensionale Archimedische Polytope''.<ref>[http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope] {{de icon}}</ref>
* '''Nonregular uniform star polychora''': (similar to the [[List of uniform polyhedra#Nonconvex forms with convex faces|nonconvex uniform polyhedra]])
** '''Ongoing''': Thousands of nonconvex uniform polychora are known, but mostly unpublished.  The list is presumed not to be complete, and there is no estimate of how long the complete list will be, although 1849 convex and nonconvex uniform polychora are currently known.  Participating researchers include [[Jonathan Bowers]],<ref>[http://www.polytope.net/hedrondude/polychora.htm Uniform Polychora and Other Four Dimensional Shapes]</ref> [[George Olshevsky]] and [[Norman Johnson (mathematician)|Norman Johnson]].<ref>[http://www.mit.edu/~hlb/Associahedron/program.pdf] CONVEX AND ABSTRACT POLYTOPES Workshop (2005), N.Johnson — Uniform Polychora abstract</ref>


== Regular polychora ==
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相关的主题文章：
The uniform polychora include two special subsets, which satisfy additional requirements:
<ul>
* The 16 [[List of regular polytopes#Four dimensional regular polytopes|regular polychora]], with the property that all cells, faces, edges, and vertices are congruent:
 
** 6 [[convex regular 4-polytope]]s: [[5-cell]], [[8-cell]], [[16-cell]], [[24-cell]], [[120-cell]], and [[600-cell]].
  <li>[http://cash062.com/home.php?mod=space&uid=521184 http://cash062.com/home.php?mod=space&uid=521184]</li>
** 10 [[Schläfli-Hess polychoron|Schläfli-Hess polychora]].
 
 
  <li>[http://yksgxh.com/home.php?mod=space&uid=505567 http://yksgxh.com/home.php?mod=space&uid=505567]</li>
==Convex uniform polychora==
 
 
  <li>[http://maximum.room.ne.jp/bbs/sawa/yybbs.cgi http://maximum.room.ne.jp/bbs/sawa/yybbs.cgi]</li>
=== Enumeration ===
 
 
</ul>
There are 64 convex uniform polychora, including the 6 regular convex polychora, and excluding the infinite sets of the [[duoprism]]s and the antiprismatic hyperprisms.
* 5 are polyhedral prisms based on the [[Platonic solid]]s (1 overlap with regular since a cubic hyperprism is a [[tesseract]])
* 13 are polyhedral prisms based on the [[Archimedean solid]]s
* 9 are in the self-dual regular A<sub>4</sub> [3,3,3] group ([[5-cell]]) family.
* 9 are in the self-dual regular F<sub>4</sub> [3,4,3] group ([[24-cell]]) family. (Excluding snub 24-cell)
* 15 are in the regular BC<sub>4</sub> [3,3,4] group ([[tesseract]]/[[16-cell]]) family (3 overlap with 24-cell family)
* 15 are in the regular H<sub>4</sub> [3,3,5] group ([[120-cell]]/[[600-cell]]) family.
* 1 special snub form in the [3,4,3] group ([[24-cell]]) family.
* 1 special non-Wythoffian polychoron, the grand antiprism.
* TOTAL: 68 &minus; 4 = 64
 
These 64 uniform polychora are indexed below by [[George Olshevsky]]. Repeated symmetry forms are indexed in brackets.
 
In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms:
* Set of [[uniform antiprismatic hyperprism|uniform antiprismatic prism]]s - sr{p,2}×{&nbsp;} - Polyhedral prisms of two [[antiprisms]].
* Set of uniform [[duoprism]]s - {p}×{q} - A product of two polygons.
 
=== The A<sub>4</sub> family ===
 
The 5-cell has [[Pentachoric symmetry|''diploid pentachoric'' [3,3,3] symmetry]], of [[Symmetry order|order]] 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.
 
Facets (cells) are given, grouped in their Coxeter diagram locations by removing specified nodes.
 
{| class="wikitable"
|+ [3,3,3] uniform polytopes
|-
!rowspan=2| #
!rowspan=2| Johnson Name<br> Bowers name (and acronym)
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=4 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br>{{CDD|node|3|node|3|node|2|2}}<br>(5)
! Pos. 2<br>{{CDD|node|3|node|2|2|2|node}}<br>(10)
! Pos. 1<br>{{CDD|node|2|2|2|node|3|node}}<br>(10)
! Pos. 0<br>{{CDD|2|2|node|3|node|3|node}}<br>(5)
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#f0e0e0" align=center
!1
|align=center|[[5-cell]]<br>Pentachoron (pen)
|[[File:5-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node|3|node|3|node}}<br>{3,3,3}
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
|
|
| 5
| 10
| 10
| 5
|- BGCOLOR="#f0e0e0" align=center
!2
|align=center|[[rectified 5-cell]]<br>Rectified pentachoron (rap)
|[[File:Rectified 5-cell verf.png|60px]]
|align=center|{{CDD|node|3|node_1|3|node|3|node}}<br>r{3,3,3}
|(3)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|
|
|(2)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 10
| 30
| 30
| 10
|- BGCOLOR="#f0e0e0" align=center
!3
|align=center|[[truncated 5-cell]]<br>Truncated pentachoron (tip)
|[[File:Truncated 5-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node_1|3|node|3|node}}<br>t{3,3,3}
|(3)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
|
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 10
| 30
| 40
| 20
|- BGCOLOR="#f0e0e0" align=center
!4
|align=center|[[cantellated 5-cell]]<br>Small rhombated pentachoron (srip)
|[[File:Cantellated 5-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node|3|node_1|3|node}}<br>rr{3,3,3}
|(2)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|
|(2)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
| 20
| 80
| 90
| 30
|- BGCOLOR="#f0e0e0" align=center
!7
|align=center|[[cantitruncated 5-cell]]<br>Great rhombated pentachoron (grip)
|[[File:Cantitruncated 5-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node_1|3|node_1|3|node}}<br>tr{3,3,3}
|(2)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
|(1)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
| 20
| 80
| 120
| 60
|- BGCOLOR="#f0e0e0" align=center
!8
|align=center|[[runcitruncated 5-cell]]<br>Prismatotrhombated pentachoron (prip)
|[[File:Runcitruncated 5-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node_1|3|node|3|node_1}}<br>t<sub>0,1,3</sub>{3,3,3}
|(1)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|(2)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|(1)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
| 30
| 120
| 150
| 60
|}
 
{| class="wikitable"
|+ <nowiki>[[3,3,3]]</nowiki> uniform polytopes
|-
!rowspan=2| #
!rowspan=2| Johnson Name<br> Bowers name (and acronym)
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<BR>{{CDD|node_c1|3|node_c2|3|node_c2|3|node_c1}}<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=3 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3-0<br>{{CDD|node|3|node|3|node|2|2}}<br>(10)
! Pos. 1-2<br>{{CDD|node|3|node|2|2|2|node}}<br>(20)
!Alt
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#e0f0e0" align=center
!5
|align=center|*[[runcinated 5-cell]]<br>Small prismated decachoron (spid)
|[[File:Runcinated 5-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{3,3,3}
|(2)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|(6)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|
| 30
| 70
| 60
| 20
|- BGCOLOR="#e0f0e0" align=center
!6
|align=center|*[[bitruncated 5-cell]]<br>Decachoron (deca)
|[[File:Bitruncated 5-cell verf.png|60px]]
|align=center|{{CDD|node|3|node_1|3|node_1|3|node}}<br>2t{3,3,3}
|(4)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
|
| 10
| 40
| 60
| 30
|- BGCOLOR="#e0f0e0" align=center
!9
|align=center|*[[omnitruncated 5-cell]]<br>Great prismated decachoron (gippid)
|[[File:Omnitruncated 5-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,3,3}
|(2)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|(2)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|
| 30
| 150
| 240
| 120
|- BGCOLOR="#d0f0f0"  align=center
![[#Nonuniform alternations|Nonuniform]]||[[full snub 5-cell]]<ref>http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm</ref>
|[[File:Snub 5-cell verf.png|60px]]
| align=center|{{CDD|node_h|3|node_h|3|node_h|3|node_h}}<br>ht<sub>0,1,2,3</sub>{4,3,3}
|[[File:Snub tetrahedron.png|30px]] (2)<br>[[Snub tetrahedron|(3.3.3.3.3)]]
|[[File:octahedron.png|30px]] (2)<br>[[square antiprism|(3.3.3.3)]]
|[[File:tetrahedron.png|30px]] (4)<br>[[tetrahedron|(3.3.3)]]
| 90
| 300
| 270
| 60
|}
 
The three polychora forms marked with an [[asterisk]],'''*''', have the higher [[Pentachoric symmetry|extended pentachoric symmetry]], of order 240, [<span/>[3,3,3]<span/>] because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual. There is one small index subgroup [3,3,3]<sup>+</sup>, order 60, or its doubling [<span/>[3,3,3]<span/>]<sup>+</sup>, order 120, defining a [[full snub 5-cell]] which is listed for completeness, but is not uniform.
 
==== Graphs ====
 
Three [[Coxeter plane]] [[Orthographic projection|2D projections]] are given, for the A<sub>4</sub>, A<sub>3</sub>, A<sub>2</sub> [[Coxeter group]]s, showing symmetry order 5,4,3, and doubled on even A<sub>k</sub> orders to 10,4,6 for symmetric Coxeter diagrams.
 
The 3D picture are drawn as [[Schlegel diagram]] projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.
 
{| class="wikitable"
!rowspan=2| #
!rowspan=2| Johnson Name<br> Bowers name (and acronym)
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=3|[[Coxeter plane]] graphs
!colspan=2|[[Schlegel diagram|Schlegel<br>diagram]]
|-
!A<sub>4</sub><br>[5]
!A<sub>3</sub><br>[4]
!A<sub>2</sub><br>[3]
!Tetrahedron<br>centered
!Dual tetrahedron<br>centered
|- BGCOLOR="#f0e0e0" align=center
!1
||[[5-cell]]<br>Pentachoron (pen)
||{{CDD|node_1|3|node|3|node|3|node}}<br>{3,3,3}
|[[File:4-simplex t0.svg|80px]]
|[[File:4-simplex t0 A3.svg|80px]]
|[[File:4-simplex t0 A2.svg|80px]]
|[[File:Schlegel wireframe 5-cell.png|80px]]
|- BGCOLOR="#f0e0e0" align=center
!2
||[[rectified 5-cell]]<br>Rectified pentachoron (rap)
||{{CDD|node|3|node_1|3|node|3|node}}<br>r{3,3,3}
|[[File:4-simplex t1.svg|80px]]
|[[File:4-simplex t1 A3.svg|80px]]
|[[File:4-simplex t1 A2.svg|80px]]
|[[File:Schlegel half-solid rectified 5-cell.png|80px]]
|- BGCOLOR="#f0e0e0" align=center
!3
||[[truncated 5-cell]]<br>Truncated pentachoron (tip)
||{{CDD|node_1|3|node_1|3|node|3|node}}<br>t{3,3,3}
|[[File:4-simplex t01.svg|80px]]
|[[File:4-simplex t01 A3.svg|80px]]
|[[File:4-simplex t01 A2.svg|80px]]
|[[File:Schlegel half-solid truncated pentachoron.png|80px]]
|- BGCOLOR="#f0e0e0" align=center
!4
||[[cantellated 5-cell]]<br>Small rhombated pentachoron (srip)
||{{CDD|node_1|3|node|3|node_1|3|node}}<br>rr{3,3,3}
|[[File:4-simplex t02.svg|80px]]
|[[File:4-simplex t02 A3.svg|80px]]
|[[File:4-simplex t02 A2.svg|80px]]
|[[File:Schlegel half-solid cantellated 5-cell.png|80px]]
|- BGCOLOR="#e0f0e0" align=center
!5
||*[[runcinated 5-cell]]<br>Small prismatodecachoron (spid)
||{{CDD|node_1|3|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{3,3,3}
|[[File:4-simplex t03.svg|80px]]
|[[File:4-simplex t03 A3.svg|80px]]
|[[File:4-simplex t03 A2.svg|80px]]
|colspan=2|[[File:Schlegel half-solid runcinated 5-cell.png|80px]]
|- BGCOLOR="#e0f0e0" align=center
!6
||*[[bitruncated 5-cell]]<br>Decachoron (deca)
||{{CDD|node|3|node_1|3|node_1|3|node}}<br>2t{3,3,3}
|[[File:4-simplex t12.svg|80px]]
|[[File:4-simplex t12 A3.svg|80px]]
|[[File:4-simplex t12 A2.svg|80px]]
|colspan=2|[[File:Schlegel half-solid bitruncated 5-cell.png|80px]]
|- BGCOLOR="#f0e0e0" align=center
!7
||[[cantitruncated 5-cell]]<br>Great rhombated pentachoron (grip)
||{{CDD|node_1|3|node_1|3|node_1|3|node}}<BR>tr{3,3,3}
|[[File:4-simplex t012.svg|80px]]
|[[File:4-simplex t012 A3.svg|80px]]
|[[File:4-simplex t012 A2.svg|80px]]
|[[File:Schlegel half-solid cantitruncated 5-cell.png|80px]]
|- BGCOLOR="#f0e0e0" align=center
!8
||[[runcitruncated 5-cell]]<br>Prismatotrhombated pentachoron (prip)
||{{CDD|node_1|3|node_1|3|node|3|node_1}}<br>t<sub>0,1,3</sub>{3,3,3}
|[[File:4-simplex t013.svg|80px]]
|[[File:4-simplex t013 A3.svg|80px]]
|[[File:4-simplex t013 A2.svg|80px]]
|[[File:Schlegel half-solid runcitruncated 5-cell.png|80px]]
|- BGCOLOR="#e0f0e0" align=center
!9
||*[[omnitruncated 5-cell]]<br>Great prismatodecachoron (gippid)
||{{CDD|node_1|3|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,3,3}
|[[File:4-simplex t0123.svg|80px]]
|[[File:4-simplex t0123 A3.svg|80px]]
|[[File:4-simplex t0123 A2.svg|80px]]
|colspan=2|[[File:Schlegel half-solid omnitruncated 5-cell.png|80px]]
|}
 
==== Coordinates ====
 
The coordinates of uniform 4-polytopes with pentachoric symmetry can be generated as permutations of simple integers in 5-space, all in hyperplanes with normal vector (1,1,1,1,1). The A<sub>4</sub> [[Coxeter group]] is [[palindrome|palindromic]], so repeated polytopes exist in pairs of dual configurations. There are 3 symmetric positions, and 6 pairs making the total 15 permutations of one or more rings. All 15 are listed here in order of [[binary arithmetic]] for clarity of the coordinate generation from the rings in each corresponding Coxeter diagram diagram.
 
The number of vertices can be deduced here from the [[permutation]]s of the number of coordinates, peaking at 5 [[factorial]] for the omnitruncated form with 5 unique coordinate values.
 
{|class="wikitable"
|+ Pentachora truncations in 5-space:
!#
!Base point
!Name<br>(symmetric name)
![[Coxeter-Dynkin diagram|Coxeter diagram]]
!Vertices
|- BGCOLOR="#f0e0e0"
|1
|(0, 0, 0, 0, 1)
|[[5-cell]]
|{{CDD|node|3|node|3|node|3|node_1}}
|5
|- BGCOLOR="#f0e0e0"
|2
|(0, 0, 0, 1, 1)
|[[Rectified 5-cell]]
|{{CDD|node|3|node|3|node_1|3|node}}
|10
|- BGCOLOR="#f0e0e0"
|3
|(0, 0, 0, 1, 2)
|[[Truncated 5-cell]]
|{{CDD|node|3|node|3|node_1|3|node_1}}
|20
|- BGCOLOR="#e0e0f0"
|4
|(0, 0, 1, 1, 1)
|Birectified 5-cell<br>([[rectified 5-cell]])
|{{CDD|node|3|node_1|3|node|3|node}}
|10
|- BGCOLOR="#f0e0e0"
|5
|(0, 0, 1, 1, 2)
|[[Cantellated 5-cell]]
|{{CDD|node|3|node_1|3|node|3|node_1}}
|30
|- BGCOLOR="#e0f0e0"
|6
|(0, 0, 1, 2, 2)
|[[Bitruncated 5-cell]]
|{{CDD|node|3|node_1|3|node_1|3|node}}
|30
|- BGCOLOR="#f0e0e0"
|7
|(0, 0, 1, 2, 3)
|[[Cantitruncated 5-cell]]
|{{CDD|node|3|node_1|3|node_1|3|node_1}}
|60
|- BGCOLOR="#e0e0f0"
|8
|(0, 1, 1, 1, 1)
|Trirectified 5-cell<br>([[5-cell]])
|{{CDD|node_1|3|node|3|node|3|node}}
|5
|- BGCOLOR="#e0f0e0"
|9
|(0, 1, 1, 1, 2)
|[[Runcinated 5-cell]]
|{{CDD|node_1|3|node|3|node|3|node_1}}
|20
|- BGCOLOR="#e0e0f0"
|10
|(0, 1, 1, 2, 2)
|Bicantellated 5-cell<br>([[cantellated 5-cell]])
|{{CDD|node_1|3|node|3|node_1|3|node}}
|30
|- BGCOLOR="#f0e0e0"
|11
|(0, 1, 1, 2, 3)
|[[Runcitruncated 5-cell]]
|{{CDD|node_1|3|node|3|node_1|3|node_1}}
|60
|- BGCOLOR="#e0e0f0"
|12
|(0, 1, 2, 2, 2)
|Tritruncated 5-cell<br>([[truncated 5-cell]])
|{{CDD|node_1|3|node_1|3|node|3|node}}
|20
|- BGCOLOR="#e0e0f0"
|13
|(0, 1, 2, 2, 3)
|Runcicantellated 5-cell<br>([[runcitruncated 5-cell]])
|{{CDD|node_1|3|node_1|3|node|3|node_1}}
|60
|- BGCOLOR="#e0e0f0"
|14
|(0, 1, 2, 3, 3)
|Bicantitruncated 5-cell<br>([[cantitruncated 5-cell]])
|{{CDD|node_1|3|node_1|3|node_1|3|node}}
|60
|- BGCOLOR="#e0f0e0"
|15
|(0, 1, 2, 3, 4)
|[[Omnitruncated 5-cell]]
|{{CDD|node_1|3|node_1|3|node_1|3|node_1}}
|120
|}
 
===The BC<sub>4</sub> family===
 
This family has [[Hexadecachoric symmetry|''diploid hexadecachoric'' symmetry]], [4,3,3], of [[Symmetry order|order]] 24*16=384: 4!=24 permutations of the four axes, 2<sup>4</sup>=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform polychora which are also repeated in other families, [1<sup>+</sup>,4,3,3], [4,(3,3)<sup>+</sup>], and [4,3,3]<sup>+</sup>, all order 192.
 
==== Tesseract truncations ====
 
{| class="wikitable"
!rowspan=2|#
!rowspan=2|Johnson Name<br>(Bowers style acronym)
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br>{{CDD|node|4|node|3|node|2|2}}<br>(8)
! Pos. 2<br>{{CDD|node|4|node|2|2|2|node}}<br>(24)
! Pos. 1<br>{{CDD|node|2|2|2|node|3|node}}<br>(32)
! Pos. 0<br>{{CDD|2|2|node|3|node|3|node}}<br>(16)
!Alt
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#f0e0e0" align=center
!10
|align=center|[[tesseract]] or (tes)<BR>8-cell
|[[File:8-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node|3|node|3|node}}<br>{4,3,3}
|(4)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|
|
|
|
| 8
| 24
| 32
| 16
|- BGCOLOR="#f0e0e0" align=center
!11
|align=center|[[Rectified tesseract]] (rit)
|[[File:Rectified 8-cell verf.png|60px]]
| align=center|{{CDD|node|4|node_1|3|node|3|node}}<br>r{4,3,3}
|(3)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|
|
|(2)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
| 24
| 88
| 96
| 32
|- BGCOLOR="#f0e0e0" align=center
!13
|align=center|[[Truncated tesseract]] (tat)
|[[File:Truncated 8-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node_1|3|node|3|node}}<br>t{4,3,3}
|(3)<br>[[File:truncated hexahedron.png|30px]]<br>[[truncated cube|(3.8.8)]]
|
|
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
| 24
| 88
| 128
| 64
|- BGCOLOR="#f0e0e0" align=center
!14
|align=center|[[Cantellated tesseract]] (srit)
|[[File:Cantellated 8-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node|3|node_1|3|node}}<br>rr{4,3,3}
|(1)<br>[[File:small rhombicuboctahedron.png|30px]]<br>[[small rhombicuboctahedron|(3.4.4.4)]]
|
|(2)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|
| 56
| 248
| 288
| 96
|- BGCOLOR="#e0f0e0" align=center
!15
|align=center|'''[[Runcinated tesseract]]'''<br>(also '''runcinated 16-cell''') (sidpith)
|[[File:Runcinated 8-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{4,3,3}
|(1)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|(3)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|(3)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
| 80
| 208
| 192
| 64
|- BGCOLOR="#e0f0e0" align=center
!16
|align=center|'''[[Bitruncated tesseract]]'''<br>(also '''bitruncated 16-cell''') (tah)
|[[File:Bitruncated 8-cell verf.png|60px]]
| align=center|{{CDD|node|4|node_1|3|node_1|3|node}}<br>2t{4,3,3}
|(2)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
|
|(2)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
| 24
| 120
| 192
| 96
|- BGCOLOR="#f0e0e0" align=center
!18
|align=center|[[Cantitruncated tesseract]] (grit)
|[[File:Cantitruncated 8-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node_1|3|node_1|3|node}}<br>tr{4,3,3}
|(2)<br>[[File:great rhombicuboctahedron.png|30px]]<br>[[truncated cuboctahedron|(4.6.8)]]
|
|(1)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
| 56
| 248
| 384
| 192
|- BGCOLOR="#f0e0e0" align=center
!19
|align=center|[[Runcitruncated tesseract]] (proh)
|[[File:Runcitruncated 8-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node_1|3|node|3|node_1}}<br>t<sub>0,1,3</sub>{4,3,3}
|(1)<br>[[File:truncated hexahedron.png|30px]]<br>[[truncated cube|(3.8.8)]]
|(2)<br>[[File:octagonal prism.png|30px]]<br>[[octagonal prism|(4.4.8)]]
|(1)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|
| 80
| 368
| 480
| 192
|- BGCOLOR="#e0f0e0" align=center
!21
|align=center|'''[[Omnitruncated tesseract]]'''<br>(also '''omnitruncated 16-cell''') (gidpith)
|[[File:Omnitruncated 8-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,3,4}
|(1)<br>[[File:great rhombicuboctahedron.png|30px]]<br>[[truncated cuboctahedron|(4.6.8)]]
|(1)<br>[[File:octagonal prism.png|30px]]<br>[[octagonal prism|(4.4.8)]]
|(1)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|(1)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
| 80
| 464
| 768
| 384
 
|-  BGCOLOR="#d0f0f0" align=center
!12
|align=center|Demitesseract<br>[[16-cell]] (hex)
|[[File:16-cell verf.png|60px]]
| align=center|{{CDD|node_h1|4|node|3|node|3|node}} = {{CDD|nodes_10ru|split2|node|3|node}}<br>h{4,3,3}
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
|
|
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 16
| 32
| 24
| 8
 
|-  BGCOLOR="#d0f0f0" align=center
![17]
|align=center|[[Cantic tesseract]]
|[[File:Truncated demitesseract verf.png|60px]]
| align=center|{{CDD|node_h1|4|node|3|node_1|3|node}} = {{CDD|nodes_10ru|split2|node_1|3|node}}<br>h<sub>2</sub>{4,3,3}
|(4)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(6.6.3)]]
|
|
|(1)<br>[[File:octahedron.png|30px]]<br>[[Octahedron|(3.3.3.3)]]
|
| 24 
| 96
| 120
| 48
 
|-  BGCOLOR="#d0f0f0" align=center
![11]
|align=center|[[Runcic tesseract]]
|[[File:Cantellated demitesseract verf.png|60px]]
| align=center|{{CDD|node_h1|4|node|3|node|3|node_1}} = {{CDD|nodes_10ru|split2|node|3|node_1}}<br>h<sub>3</sub>{4,3,3}
|(3)<br>[[File:Cuboctahedron.png|30px]]<br>[[Cuboctahedron|(3.4.3.4)]]
|
|
|(2)<br>[[File:Tetrahedron.png|30px]]<br>[[Tetrahedron|(3.3.3)]]
|
| 24
| 88
| 96
| 32
 
|-  BGCOLOR="#d0f0f0" align=center
![16]
|align=center|[[Runcicantic tesseract]]
|[[File:Cantitruncated demitesseract verf.png|60px]]
| align=center|{{CDD|node_h1|4|node|3|node_1|3|node_1}} = {{CDD|nodes_10ru|split2|node_1|3|node_1}}<br>h<sub>2,3</sub>{4,3,3}
|(2)<br>[[File:Truncated octahedron.png|30px]]<br>[[Truncated octahedron|(3.4.3.4)]]
|
|
|(2)<br>[[File:Truncated tetrahedron.png|30px]]<br>[[Truncated tetrahedron|(3.6.6)]]
|
| 24
| 96
| 96
| 24
 
|- BGCOLOR="#d0f0f0"  align=center
![[#Nonuniform alternations|Nonuniform]]
||[[full snub tesseract]]<ref>http://www.bendwavy.org/klitzing/incmats/s3s3s4s.htm</ref><br>(Same as the ''full snub 16-cell'')
|[[File:Snub tesseract verf.png|60px]]
| align=center|{{CDD|node_h|4|node_h|3|node_h|3|node_h}}<br>ht<sub>0,1,2,3</sub>{4,3,3}
|(1)<br>[[File:Snub hexahedron.png|30px]]<br>[[Snub cube|(3.3.3.3.4)]]
|(1)<br>[[File:square antiprism.png|30px]]<br>[[square antiprism|(3.3.3.4)]]
|(1)<br>[[File:octahedron.png|30px]]<br>[[triangular antiprism|(3.3.3.3)]]
|(1)<br>[[File:Snub tetrahedron.png|30px]]<br>[[Snub tetrahedron|(3.3.3.3.3)]]
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 272
| 944
| 864
| 192
|}
 
==== 16-cell truncations ====
 
{| class="wikitable"
!rowspan=2|#
!rowspan=2|Johnson Name (Bowers style acronym)
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br>{{CDD|node|4|node|3|node}}<br>(8)
! Pos. 2<br>{{CDD|node|4|node|2|node}}<br>(24)
! Pos. 1<br>{{CDD|node|2|node|3|node}}<br>(32)
! Pos. 0<br>{{CDD|node|3|node|3|node}}<br>(16)
!Alt
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#e0e0f0" align=center
![12]
|align=center|[[16-cell]] (hex)
|[[File:16-cell verf.png|60px]]
| align=center|{{CDD|node|4|node|3|node|3|node_1}}<br>{3,3,4}
|
|
|
|(8)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
| 16
| 32
| 24
| 8
|- BGCOLOR="#e0e0f0" align=center
![22]
|align=center|*rectified 16-cell<br>(Same as '''[[24-cell]]''') (ico)
|[[File:Rectified 16-cell verf.png|60px]]
| align=center|{{CDD|node|4|node|3|node_1|3|node}}<br>r{3,3,4}
|(2)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|
|
|(4)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|
| 24
| 96
| 96
| 24
|- BGCOLOR="#e0e0f0" align=center
!17
|align=center|[[truncated 16-cell]] (thex)
|[[File:Truncated 16-cell verf.png|60px]]
| align=center|{{CDD|node|4|node|3|node_1|3|node_1}}<br>t{3,3,4}
|(1)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|
|
|(4)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
| 24
| 96
| 120
| 48
|- BGCOLOR="#e0e0f0" align=center
![23]
|align=center|*cantellated 16-cell<br>(Same as '''[[rectified 24-cell]]''') (rico)
|[[File:Cantellated 16-cell verf.png|60px]]
| align=center|{{CDD|node|4|node_1|3|node|3|node_1}}<br>rr{3,3,4}
|(1)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|(2)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|
|(2)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|
| 48
| 240
| 288
| 96
|- BGCOLOR="#e0f0e0" align=center
![15]
|align=center|'''[[runcinated tesseract|runcinated 16-cell]]'''<br>(also '''runcinated 8-cell''') (sidpith)
|[[File:Runcinated 8-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{3,3,4}
|(1)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|(3)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|(3)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
| 80
| 208
| 192
| 64
|- BGCOLOR="#e0f0e0" align=center
![16]
|align=center|'''[[bitruncated tesseract|bitruncated 16-cell]]'''<br>(also '''bitruncated 8-cell''') (tah)
|[[File:Bitruncated 8-cell verf.png|60px]]
| align=center|{{CDD|node|4|node_1|3|node_1|3|node}}<br>2t{3,3,4}
|(2)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
|
|(2)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
| 24
| 120
| 192
| 96
|- BGCOLOR="#e0e0f0" align=center
![24]
|align=center|*cantitruncated 16-cell<br>(Same as '''[[truncated 24-cell]]''') (tico)
|[[File:Cantitruncated 16-cell verf.png|60px]]
| align=center|{{CDD|node|4|node_1|3|node_1|3|node_1}}<br>tr{3,3,4}
|(1)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|(1)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|
|(2)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
| 48
| 240
| 384
| 192
|- BGCOLOR="#e0e0f0" align=center
!20
|align=center|[[runcitruncated 16-cell]] (prit)
|[[File:Runcitruncated 16-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node|3|node_1|3|node_1}}<br>t<sub>0,1,3</sub>{3,3,4}
|(1)<br>[[File:small rhombicuboctahedron.png|30px]]<br>[[rhombicuboctahedron|(3.4.4.4)]]
|(1)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|(2)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|(1)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
| 80
| 368
| 480
| 192
|- BGCOLOR="#e0f0e0" align=center
![21]
|align=center|'''[[omnitruncated tesseract|omnitruncated 16-cell]]'''<br>(also '''omnitruncated 8-cell''') (gidpith)
|[[File:Omnitruncated 8-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,3,4}
|(1)<br>[[File:great rhombicuboctahedron.png|30px]]<br>[[truncated cuboctahedron|(4.6.8)]]
|(1)<br>[[File:octagonal prism.png|30px]]<br>[[octagonal prism|(4.4.8)]]
|(1)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|(1)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
| 80
| 464
| 768
| 384
|- BGCOLOR="#d0f0f0" align=center
![31]
|align=center|''alternated cantitruncated 16-cell''<br>(Same as the [[snub 24-cell]]) (sadi)
|[[File:Snub 24-cell verf.png|60px]]
| align=center|{{CDD|node|4|node_h|3|node_h|3|node_h}}<br>sr{3,3,4}
|(1)<br>[[File:Snub tetrahedron.png|30px]]<br>[[icosahedron|(3.3.3.3.3)]]
|
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|(2)<br>[[File:Snub tetrahedron.png|30px]]<br>[[icosahedron|(3.3.3.3.3)]]
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 144
| 480
| 432
| 96
|- BGCOLOR="#d0f0f0" align=center
!Nonuniform
|Runcic snub rectified 16-cell
|[[File:Runcic snub rectified 16-cell verf.png|60px]]
| align=center|{{CDD|node_1|4|node_h|3|node_h|3|node_h}}<br>sr<sub>3</sub>{3,3,4}
|(1)<BR>[[File:small rhombicuboctahedron.png|30px]]<br>[[rhombicuboctahedron|(3.4.4.4)]]
|(2)<BR>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<BR>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|(1)<BR>[[File:Snub tetrahedron.png|30px]]<br>[[icosahedron|(3.3.3.3.3)]]
|(2)<BR>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|176
|656
|672
|192
|}
 
:(*) Just as rectifying the [[tetrahedron]] produces the [[octahedron]], rectifying the 16-cell produces the 24-cell, the regular member of the following family.
 
The ''snub 24-cell'' is repeat to this family for completeness. It is an alternation of the ''cantitruncated 16-cell'' or ''truncated 24-cell'', with the half symmetry group [(3,3)<sup>+</sup>,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices.
 
==== Graphs ====
 
The pictures are drawn as [[Schlegel diagram]] perspective projections, centered on the cell at pos. 3, with a consistent orientation, and the 16 cells at position 0 are shown solid, alternately colored.
 
{| class="wikitable"
!rowspan=2|#
!rowspan=2|Johnson Name<br>(Bowers style acronym)
!colspan=5| [[Coxeter plane]] projections
!colspan=2| [[Schlegel diagram|Schlegel<br>diagrams]]
|-
!F<sub>4</sub><br>[12/3]
!B<sub>4</sub><br>[8]
!B<sub>3</sub><br>[6]
!B<sub>2</sub><br>[4]
!A<sub>3</sub><br>[4]
!Cube<br>centered
!Tetrahedron<br>centered
|- BGCOLOR="#f0e0e0"
!10||8-cell<br>or [[tesseract]] (tes)
|[[File:4-cube t0 F4.svg|80px]]
|[[File:4-cube t0.svg|80px]]
|[[File:4-cube t0 B3.svg|80px]]
|[[File:4-cube t0 B2.svg|80px]]
|[[File:4-cube t0 A3.svg|80px]]
|[[File:Schlegel wireframe 8-cell.png|80px]]
|
|- BGCOLOR="#f0e0e0"
!11||[[rectified tesseract|rectified 8-cell]] (rit)
|[[File:4-cube t1 F4.svg|80px]]
|[[File:4-cube t1.svg|80px]]
|[[File:4-cube t1 B3.svg|80px]]
|[[File:4-cube t1 B2.svg|80px]]
|[[File:4-cube t1 A3.svg|80px]]
|[[File:Schlegel half-solid rectified 8-cell.png|80px]]
|
|- BGCOLOR="#e0e0f0"
!12||[[16-cell]] (hex)
|[[File:4-cube t3 F4.svg|80px]]
|[[File:4-cube t3.svg|80px]]
|[[File:4-cube t3 B3.svg|80px]]
|[[File:4-cube t3 B2.svg|80px]]
|[[File:4-cube t3 A3.svg|80px]]
|
|[[File:Schlegel wireframe 16-cell.png|80px]]
|- BGCOLOR="#f0e0e0"
!13||[[truncated tesseract|truncated 8-cell]] (tat)
|[[File:4-cube t01 F4.svg|80px]]
|[[File:4-cube t01.svg|80px]]
|[[File:4-cube t01 B3.svg|80px]]
|[[File:4-cube t01 B2.svg|80px]]
|[[File:4-cube t01 A3.svg|80px]]
|[[File:Schlegel half-solid truncated tesseract.png|80px]]
|
|- BGCOLOR="#f0e0e0"
!14||[[cantellated tesseract|cantellated 8-cell]] (srit)
|[[File:4-cube t02 F4.svg|80px]]
|[[File:4-cube t02.svg|80px]]
|[[File:4-cube t02 B3.svg|80px]]
|[[File:4-cube t02 B2.svg|80px]]
|[[File:4-cube t02 A3.svg|80px]]
|[[File:Schlegel half-solid cantellated 8-cell.png|80px]]
|
|- BGCOLOR="#e0f0e0"
!15||'''[[runcinated tesseract|runcinated 8-cell]]'''<br>(also '''runcinated 16-cell''') (sidpith)
|[[File:4-cube t03 F4.svg|80px]]
|[[File:4-cube t03.svg|80px]]
|[[File:4-cube t03 B3.svg|80px]]
|[[File:4-cube t03 B2.svg|80px]]
|[[File:4-cube t03 A3.svg|80px]]
|[[File:Schlegel half-solid runcinated 8-cell.png|80px]]
|[[File:Schlegel half-solid runcinated 16-cell.png|80px]]
|- BGCOLOR="#e0f0e0"
!16||'''[[bitruncated tesseract|bitruncated 8-cell]]'''<br>(also '''bitruncated 16-cell''') (tah)
|[[File:4-cube t12 F4.svg|80px]]
|[[File:4-cube t12.svg|80px]]
|[[File:4-cube t12 B3.svg|80px]]
|[[File:4-cube t12 B2.svg|80px]]
|[[File:4-cube t12 A3.svg|80px]]
| [[File:Schlegel half-solid bitruncated 8-cell.png|80px]]
|[[File:Schlegel half-solid bitruncated 16-cell.png|80px]]
|- BGCOLOR="#e0e0f0"
!17||[[truncated 16-cell]] (thex)
|[[File:4-cube t23 F4.svg|80px]]
|[[File:4-cube t23.svg|80px]]
|[[File:4-cube t23 B3.svg|80px]]
|[[File:4-cube t23 B2.svg|80px]]
|[[File:4-cube t23 A3.svg|80px]]
|
|[[File:Schlegel half-solid truncated 16-cell.png|80px]]
|- BGCOLOR="#f0e0e0"
!18||[[cantitruncated tesseract|cantitruncated 8-cell]] (grit)
|[[File:4-cube t012 F4.svg|80px]]
|[[File:4-cube t012.svg|80px]]
|[[File:4-cube t012 B3.svg|80px]]
|[[File:4-cube t012 B2.svg|80px]]
|[[File:4-cube t012 A3.svg|80px]]
| [[File:Schlegel half-solid cantitruncated 8-cell.png|80px]]
|
|- BGCOLOR="#f0e0e0"
!19||[[runcitruncated tesseract|runcitruncated 8-cell]] (proh)
|[[File:4-cube t013 F4.svg|80px]]
|[[File:4-cube t013.svg|80px]]
|[[File:4-cube t013 B3.svg|80px]]
|[[File:4-cube t013 B2.svg|80px]]
|[[File:4-cube t013 A3.svg|80px]]
| [[File:Schlegel half-solid runcitruncated 8-cell.png|80px]]
|
|- BGCOLOR="#e0e0f0"
!20||[[runcitruncated 16-cell]] (prit)
|[[File:4-cube t023 F4.svg|80px]]
|[[File:4-cube t023.svg|80px]]
|[[File:4-cube t023 B3.svg|80px]]
|[[File:4-cube t023 B2.svg|80px]]
|[[File:4-cube t023 A3.svg|80px]]
|
| [[File:Schlegel half-solid runcitruncated 16-cell.png|80px]]
|- BGCOLOR="#e0f0e0"
!21||'''[[omnitruncated tesseract|omnitruncated 8-cell]]'''<br>(also '''omnitruncated 16-cell''') (gidpith)
|[[File:4-cube t0123 F4.svg|80px]]
|[[File:4-cube t0123.svg|80px]]
|[[File:4-cube t0123 B3.svg|80px]]
|[[File:4-cube t0123 B2.svg|80px]]
|[[File:4-cube t0123 A3.svg|80px]]
|[[File:Schlegel half-solid omnitruncated 8-cell.png|80px]]
|[[File:Schlegel half-solid omnitruncated 16-cell.png|80px]]
|- BGCOLOR="#e0e0f0"
![22]||*rectified 16-cell<br>(Same as '''[[24-cell]]''') (ico)
|[[File:4-cube t2 F4.svg|80px]]
|[[File:4-cube t2.svg|80px]]
|[[File:4-cube t2 B3.svg|80px]]
|[[File:4-cube t2 B2.svg|80px]]
|[[File:4-cube t2 A3.svg|80px]]
|
|[[File:Schlegel half-solid rectified 16-cell.png|80px]]
|- BGCOLOR="#e0e0f0"
![23]||*cantellated 16-cell<br>(Same as '''[[rectified 24-cell]]''') (rico)
|[[File:4-cube t13 F4.svg|80px]]
|[[File:4-cube t13.svg|80px]]
|[[File:4-cube t13 B3.svg|80px]]
|[[File:4-cube t13 B2.svg|80px]]
|[[File:4-cube t13 A3.svg|80px]]
|
| [[File:Schlegel half-solid cantellated 16-cell.png|80px]]
|- BGCOLOR="#e0e0f0"
![24]||*cantitruncated 16-cell<br>(Same as '''[[truncated 24-cell]]''') (tico)
|[[File:4-cube t123 F4.svg|80px]]
|[[File:4-cube t123.svg|80px]]
|[[File:4-cube t123 B3.svg|80px]]
|[[File:4-cube t123 B2.svg|80px]]
|[[File:4-cube t123 A3.svg|80px]]
|
|[[File:Schlegel half-solid cantitruncated 16-cell.png|80px]]
|- BGCOLOR="#d0f0f0"
![31]||''alternated cantitruncated 16-cell''<br>(Same as the [[snub 24-cell]]) (sadi)
|[[File:24-cell h01 F4.svg|80px]]
|[[File:24-cell h01 B4.svg|80px]]
|[[File:24-cell h01 B3.svg|80px]]
|[[File:24-cell h01 B2.svg|80px]]
|
|
|[[File:Schlegel half-solid alternated cantitruncated 16-cell.png|80px]]
|}
 
==== Coordinates ====
 
The tesseractic family of polychora are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform polychoron. All coordinates correspond with uniform polychora of edge length 2.
 
{|class="wikitable"
|+Coordinates for uniform polychora in Tesseract/16-cell family
|-
!#
!Base point
!Johnson Name<br>Bowers Name (Bowers style acronym)
![[Coxeter-Dynkin diagram|Coxeter diagram]]
|- BGCOLOR="#f0e0e0"
|1
|(0,0,0,1)√2
|[[16-cell]]<br>Hexadecachoron (hex)
|{{CDD|node|4|node|3|node|3|node_1}}
|- BGCOLOR="#f0e0e0"
|2
|(0,0,1,1)√2
|[[Rectified 16-cell]]<br>Icositetrachoron (ico)
|{{CDD|node|4|node|3|node_1|3|node}}
|- BGCOLOR="#f0e0e0"
|3
|(0,0,1,2)√2
|[[Truncated 16-cell]]<br>Truncated hexadecachoron (thex)
|{{CDD|node|4|node|3|node_1|3|node_1}}
|- BGCOLOR="#e0e0f0"
|4
|(0,1,1,1)√2
|[[Rectified tesseract]] (birectified 16-cell)<br>Rectified tesseract (rit)
|{{CDD|node|4|node_1|3|node|3|node}}
|-BGCOLOR="#f0e0e0"
|5
|(0,1,1,2)√2
|[[Cantellated 16-cell]]<br>Rectified icositetrachoron (rico)
|{{CDD|node|4|node_1|3|node|3|node_1}}
|- BGCOLOR="#e0f0e0"
|6
|(0,1,2,2)√2
|[[Bitruncated 16-cell]]<br>Tesseractihexadecachoron (tah)
|{{CDD|node|4|node_1|3|node_1|3|node}}
|- BGCOLOR="#f0e0e0"
|7
|(0,1,2,3)√2
|[[cantitruncated 16-cell]]<br>Truncated icositetrachoron (tico)
|{{CDD|node|4|node_1|3|node_1|3|node_1}}
|- BGCOLOR="#e0e0f0"
|8
|(1,1,1,1)
|[[Tesseract]]<br>Tesseract (tes)
|{{CDD|node_1|4|node|3|node|3|node}}
|- BGCOLOR="#e0f0e0"
|9
|(1,1,1,1) + (0,0,0,1)√2
|[[Runcinated tesseract]] (runcinated 16-cell)<br>Small disprismatotesseractihexadecachoron (sidpith)
|{{CDD|node_1|4|node|3|node|3|node_1}}
|- BGCOLOR="#e0e0f0"
|10
|(1,1,1,1) + (0,0,1,1)√2
|[[Cantellated tesseract]]<br>Small rhombated tesseract (srit)
|{{CDD|node_1|4|node|3|node_1|3|node}}
|- BGCOLOR="#f0e0e0"
|11
|(1,1,1,1) + (0,0,1,2)√2
|[[Runcitruncated 16-cell]]<br>Prismatorhombated tesseract (prit)
|{{CDD|node_1|4|node|3|node_1|3|node_1}}
|- BGCOLOR="#e0e0f0"
|12
|(1,1,1,1) + (0,1,1,1)√2
|[[Truncated tesseract]]<br>Truncated tesseract (tat)
|{{CDD|node_1|4|node_1|3|node|3|node}}
|- BGCOLOR="#e0e0f0"
|13
|(1,1,1,1) + (0,1,1,2)√2
|[[Runcitruncated tesseract]] (runcicantellated 16-cell)<br>Prismatorhombated hexadecachoron (proh)
|{{CDD|node_1|4|node_1|3|node|3|node_1}}
|- BGCOLOR="#e0e0f0"
|14
|(1,1,1,1) + (0,1,2,2)√2
|[[Cantitruncated tesseract]]<br>Great rhombated tesseract (grit)
|{{CDD|node_1|4|node_1|3|node_1|3|node}}
|- BGCOLOR="#e0f0e0"
|15
|(1,1,1,1) + (0,1,2,3)√2
|[[Omnitruncated 16-cell]] (omnitruncated tesseract)<br>Great disprismatotesseractihexadecachoron (gidpith)
|{{CDD|node_1|4|node_1|3|node_1|3|node_1}}
|}
 
===The F<sub>4</sub> family===
 
This family has [[Icositetrachoric symmetry|''diploid icositetrachoric'' symmetry]], [3,4,3], of [[Symmetry order|order]] 24*48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There are 3 small index subgroups, with the first two isomorphic pairs generating uniform polychora which are also repeated in other families, [3<sup>+</sup>,4,3], [3,4,3<sup>+</sup>], and [3,4,3]<sup>+</sup>, all order 576.
 
{| class="wikitable"
|+ [3,4,3] uniform polychora
|-
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|[[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br>{{CDD|node|3|node|4|node|2|2}}<br>(24)
! Pos. 2<br>{{CDD|node|3|node|2|2|node}}<br>(96)
! Pos. 1<br>{{CDD|node|2|2|2|node|3|node}}<br>(96)
! Pos. 0<br>{{CDD|2|2|node|4|node|3|node}}<br>(24)
!Alt
! Cells
! Faces
! Edges
! Vertices
|-  BGCOLOR="#f0e0e0" align=center
!22
|align=center|[[24-cell]]<br>(Same as '''rectified 16-cell''') (ico)
|[[File:24 cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node|4|node|3|node}}<br>{3,4,3}
|(6)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|
|
|
|
| 24
| 96
| 96
| 24
|- BGCOLOR="#f0e0e0" align=center
!23
|align=center|[[rectified 24-cell]]<br>(Same as '''cantellated 16-cell''') (rico)
|[[File:Rectified 24-cell verf.png|60px]]
|align=center|{{CDD|node|3|node_1|4|node|3|node}}<br>r{3,4,3}
|(3)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|
|
|(2)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|
| 48
| 240
| 288
| 96
|- BGCOLOR="#f0e0e0" align=center
!24
|align=center|[[truncated 24-cell]]<br>(Same as '''cantitruncated 16-cell''') (tico)
|[[File:truncated 24-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node_1|4|node|3|node}}<br>t{3,4,3}
|(3)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
|
|(1)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|
| 48
| 240
| 384
| 192
|- BGCOLOR="#f0e0e0" align=center
!25
|align=center|[[cantellated 24-cell]] (srico)
|[[File:Cantellated 24-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node|4|node_1|3|node}}<br>rr{3,4,3}
|(2)<br>[[File:small rhombicuboctahedron.png|30px]]<br>[[small rhombicuboctahedron|(3.4.4.4)]]
|
|(2)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|
| 144
| 720
| 864
| 288
|- BGCOLOR="#f0e0e0" align=center
!28
|align=center|[[cantitruncated 24-cell]] (grico)
|[[File:cantitruncated 24-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node_1|4|node_1|3|node}}<br>tr{3,4,3}
|(2)<br>[[File:great rhombicuboctahedron.png|30px]]<br>[[truncated cuboctahedron|(4.6.8)]]
|
|(1)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:truncated hexahedron.png|30px]]<br>[[truncated cube|(3.8.8)]]
|
| 144
| 720
| 1152
| 576
|- BGCOLOR="#f0e0e0" align=center
!29
|align=center|[[runcitruncated 24-cell]] (prico)
|[[File:runcitruncated 24-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node_1|4|node|3|node_1}}<br>t<sub>0,1,3</sub>{3,4,3}
|(1)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|(2)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|(1)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:small rhombicuboctahedron.png|30px]]<br>[[small rhombicuboctahedron|(3.4.4.4)]]
|
| 240
| 1104
| 1440
| 576
|- BGCOLOR="#d0f0f0" align=center
!31
|align=center|†[[snub 24-cell]] (sadi)
|[[File:Snub 24-cell verf.png|60px]]
|align=center|{{CDD|node_h|3|node_h|4|node|3|node}}<br>s{3,4,3}
|(3)<br>[[File:icosahedron.png|30px]]<br>[[icosahedron|(3.3.3.3.3)]]
|
|
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 144
| 480
| 432
| 96
|- BGCOLOR="#d0f0f0" align=center
![[#Nonuniform alternations|Nonuniform]]
|[[Runcic snub 24-cell]] (prissi)
|[[File:Runcic snub 24-cell verf.png|60px]]
|align=center|{{CDD|node_h|3|node_h|4|node|3|node_1}}<br>s<sub>3</sub>{3,4,3}
|(1)<BR>[[File:icosahedron.png|30px]]<br>[[icosahedron|(3.3.3.3.3)]]
|(2)<BR>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|
|(1)<BR>[[File:Truncated tetrahedron.png|30px]]<br>[[Truncated tetrahedron|(3.6.6)]]
|(3)<BR>[[File:Triangular cupola.png|30px]]<BR>[[Triangular cupola|Tricup]]
|240
|960
|1008
|288
|- BGCOLOR="#d0f0f0" align=center
![25]
|Cantic 24-cell<BR>(Same as [[cantellated 24-cell]]) (srico)
|[[File:Cantic snub 24-cell verf.png|60px]]
|{{CDD|node_h|3|node_h|4|node_1|3|node}}<br>s<sub>2</sub>{3,4,3}
|(2)<br>[[File:small rhombicuboctahedron.png|30px]]<br>[[small rhombicuboctahedron|(3.4.4.4)]]
|
|
|(1)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|(2)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
| 144
| 720
| 864
| 288
|- BGCOLOR="#d0f0f0" align=center
![29]
|Runcicantic 24-cell<BR>(Same as [[runcitruncated 24-cell]]) (prico)
|[[File:Runcicantic snub 24-cell verf.png|60px]]
|{{CDD|node_h|3|node_h|4|node_1|3|node_1}}<br>s<sub>2,3</sub>{3,4,3}
|(1)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
|(1)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:small rhombicuboctahedron.png|30px]]<br>[[small rhombicuboctahedron|(3.4.4.4)]]
|(2)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
| 240
| 1104
| 1440
| 576
|}
 
: (†) The snub 24-cell here, despite its common name, is not analogous to the [[snub cube]]; rather, is derived by an [[Alternation (geometry)|alternation]] of the truncated 24-cell. Its [[symmetry number]] is only 576, (the ''ionic diminished icositetrachoric'' group, [3<sup>+</sup>,4,3]).
 
{| class="wikitable"
|+ <nowiki>[[3,4,3]]</nowiki> uniform polychora
|-
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|[[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<BR>{{CDD|node_c1|3|node_c2|4|node_c2|3|node_c1}}<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=3 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3-0<br>{{CDD|node|3|node|4|node|2|2}}<BR>{{CDD|2|2|node|4|node|3|node}}<br>(48)
! Pos. 2-1<br>{{CDD|node|3|node|2|2|node}}<BR>{{CDD|node|2|2|node|3|node}}<br>(192)
!Alt
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#e0f0e0" align=center
!26
|align=center|*[[runcinated 24-cell]] (spic)
|[[File:runcinated 24-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node|4|node|3|node_1}}<br>t<sub>0,3</sub>{3,4,3}
|(2)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|(6)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|
| 240
| 672
| 576
| 144
|- BGCOLOR="#e0f0e0" align=center
!27
|align=center|*[[bitruncated 24-cell]] (cont)
|[[File:bitruncated 24-cell verf.png|60px]]
|align=center|{{CDD|node|3|node_1|4|node_1|3|node}}<br>2t{3,4,3}
|(4)<br>[[File:truncated hexahedron.png|30px]]<br>[[truncated cube|(3.8.8)]]
|
|
| 48
| 336
| 576
| 288
|- BGCOLOR="#e0f0e0" align=center
!30
|align=center|*[[omnitruncated 24-cell]] (gippic)
|[[File:omnitruncated 24-cell verf.png|60px]]
|align=center|{{CDD|node_1|3|node_1|4|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,4,3}
|(2)<br>[[File:great rhombicuboctahedron.png|30px]]<br>[[truncated cuboctahedron|(4.6.8)]]
|(2)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|
| 240
| 1392
| 2304
| 1152
|- BGCOLOR="#d0f0f0" align=center
![[#Nonuniform alternations|Nonuniform]]
|align=center|[[Full snub 24-cell]]<ref>http://www.bendwavy.org/klitzing/incmats/s3s4s3s.htm</ref>
|[[File:Full snub 24-cell verf.png|60px]]
|align=center|{{CDD|node_h|3|node_h|4|node_h|3|node_h}}<br>ht<sub>0,1,2,3</sub>{3,4,3}
|(2)<br>[[File:snub hexahedron.png|30px]]<br>[[snub cube|(3.3.3.3.4)]]
|(2)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 816
| 2832
| 2592
| 576
|}
 
: (*) Like the 5-cell, the 24-cell is self-dual, and so the three asterisked forms have twice as many symmetries, bringing their total to 2304 ([[Icositetrachoric symmetry|extended icositetrachoric symmetry]] [<span/>[3,4,3]<span/>]).
 
==== Graphs ====
 
{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name<br>[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>[[Schläfli symbol]]
!colspan=4|Graph<br>
!colspan=2|[[Schlegel diagram|Schlegel<br>diagram]]
!Orthogonal<br>Projection
|-
!F<sub>4</sub><br>[12]
!B<sub>4</sub><br>[8]
!B<sub>3</sub><br>[6]
!B<sub>2</sub><br>[4]
!Octahedron<br>centered
!Dual octahedron<br>centered
!Octahedron<br>centered
|- BGCOLOR="#f0e0e0" align=center
!22
||[[24-cell]] (ico)<br>(rectified 16-cell)<br>{{CDD|node_1|3|node|4|node|3|node}}<br>{3,4,3}
|[[File:24-cell t0 F4.svg|80px]]
|[[File:24-cell t0 B4.svg|80px]]
|[[File:24-cell t0 B3.svg|80px]]
|[[File:24-cell t0 B2.svg|80px]]
|[[File:Schlegel wireframe 24-cell.png|80px]]
|- BGCOLOR="#f0e0e0" align=center
!23
||[[rectified 24-cell]] (rico)<br>(cantellated 16-cell)<br>{{CDD|node|3|node_1|4|node|3|node}}<br>t<sub>1</sub>{3,4,3}
|[[File:24-cell t1 F4.svg|80px]]
|[[File:24-cell t1 B4.svg|80px]]
|[[File:24-cell t1 B3.svg|80px]]
|[[File:24-cell t1 B2.svg|80px]]
| [[File:Schlegel half-solid cantellated 16-cell.png|80px]]
|- BGCOLOR="#f0e0e0"  align=center
!24
||[[truncated 24-cell]] (tico)<br>(cantitruncated 16-cell)<br>{{CDD|node_1|3|node_1|4|node|3|node}}<br>t<sub>0,1</sub>{3,4,3}
|[[File:24-cell t01 F4.svg|80px]]
|[[File:24-cell t01 B4.svg|80px]]
|[[File:24-cell t01 B3.svg|80px]]
|[[File:24-cell t01 B2.svg|80px]]
|[[File:Schlegel half-solid truncated 24-cell.png|80px]]
|- BGCOLOR="#f0e0e0" align=center
!25
||[[cantellated 24-cell]] (srico)<br>{{CDD|node_1|3|node|4|node_1|3|node}}<br>t<sub>0,2</sub>{3,4,3}
|[[File:24-cell t02 F4.svg|80px]]
|[[File:24-cell t02 B4.svg|80px]]
|[[File:24-cell t02 B3.svg|80px]]
|[[File:24-cell t02 B2.svg|80px]]
|[[File:Cantel 24cell1.png|80px]]
|- BGCOLOR="#e0f0e0"  align=center
!26
||*[[runcinated 24-cell]] (spic)<br>{{CDD|node_1|3|node|4|node|3|node_1}}<br>t<sub>0,3</sub>{3,4,3}
|[[File:24-cell t03 F4.svg|80px]]
|[[File:24-cell t03 B4.svg|80px]]
|[[File:24-cell t03 B3.svg|80px]]
|[[File:24-cell t03 B2.svg|80px]]
|colspan=2|[[File:Runcinated 24-cell Schlegel halfsolid.png|80px]]
|- BGCOLOR="#e0f0e0" align=center
!27
||*[[bitruncated 24-cell]] (cont)<br>{{CDD|node|3|node_1|4|node_1|3|node}}<br>t<sub>1,2</sub>{3,4,3}
|[[File:24-cell t12 F4.svg|80px]]
|[[File:24-cell t12 B4.svg|80px]]
|[[File:24-cell t12 B3.svg|80px]]
|[[File:24-cell t12 B2.svg|80px]]
|colspan=2|[[File:Bitruncated 24-cell Schlegel halfsolid.png|80px]]
|- BGCOLOR="#f0e0e0" align=center
!28
||[[cantitruncated 24-cell]] (grico)<br>{{CDD|node_1|3|node_1|4|node_1|3|node}}<br>t<sub>0,1,2</sub>{3,4,3}
|[[File:24-cell t012 F4.svg|80px]]
|[[File:24-cell t012 B4.svg|80px]]
|[[File:24-cell t012 B3.svg|80px]]
|[[File:24-cell t012 B2.svg|80px]]
|[[File:Cantitruncated 24-cell schlegel halfsolid.png|80px]]
|- BGCOLOR="#f0e0e0" align=center 
!29
||[[runcitruncated 24-cell]] (prico)<br>{{CDD|node_1|3|node_1|4|node|3|node_1}}<br>t<sub>0,1,3</sub>{3,4,3}
|[[File:24-cell t013 F4.svg|80px]]
|[[File:24-cell t013 B4.svg|80px]]
|[[File:24-cell t013 B3.svg|80px]]
|[[File:24-cell t013 B2.svg|80px]]
|[[File:Runcitruncated 24-cell.png|80px]]
|- BGCOLOR="#e0f0e0" align=center
!30
||*[[omnitruncated 24-cell]] (gippic)<br>{{CDD|node_1|3|node_1|4|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,4,3}
|[[File:24-cell t0123 F4.svg|80px]]
|[[File:24-cell t0123 B4.svg|80px]]
|[[File:24-cell t0123 B3.svg|80px]]
|[[File:24-cell t0123 B2.svg|80px]]
|colspan=2| [[File:Omnitruncated 24-cell.png|80px]]
|- BGCOLOR="#d0f0f0" align=center
!31
||[[snub 24-cell]] (sadi)<br>{{CDD|node_h|3|node_h|4|node|3|node}}<br>s{3,4,3}
|[[File:24-cell h01 F4.svg|80px]]
|[[File:24-cell h01 B4.svg|80px]]
|[[File:24-cell h01 B3.svg|80px]]
|[[File:24-cell h01 B2.svg|80px]]
|[[File:Schlegel half-solid alternated cantitruncated 16-cell.png|80px]]
|
|[[File:Ortho solid 969-uniform polychoron 343-snub.png|80px]]
|- BGCOLOR="#d0f0f0"  align=center
! -
|[[Runcic snub 24-cell]]<br>{{CDD|node_h|3|node_h|4|node|3|node_1}}<br>s<sub>3</sub>{3,4,3}
|
|
|
|
|
|
|[[File:Runcic snub 24-cell.png|80px]]
|}
 
==== Coordinates ====
 
Vertex coordinates for all 15 forms are given below, including dual configurations from the two regular 24-cells. (The dual configurations are named in bold.) Active rings in the first and second nodes generate points in the first column. Active rings in the third and fourth nodes generate the points in the second column. The sum of each of these points are then permutated by coordinate positions, and sign combinations. This generates all vertex coordinates. Edge lengths are 2.
 
The only exception is the snub 24-cell, which is generated by ''half'' of the coordinate permutations, only an even number of coordinate swaps. φ=(√5+1)/2.
 
{|class="wikitable"
|+ 24-cell family coordinates
!Base point(s)<br>t(0,1)
!Base point(s)<br>t(2,3)
![[Schläfli symbol]]
!Name<br>
![[Coxeter-Dynkin diagram|Coxeter diagram]]
|-
!colspan=4|&nbsp;
|- BGCOLOR="#f0e0e0" align=center
|
|(0,0,1,1)√2
|t<sub>0</sub>{3,4,3}
|[[24-cell]]
|{{CDD|node|3|node|4|node|3|node_1}}
|- BGCOLOR="#f0e0e0" align=center
|
|(0,1,1,2)√2
|t<sub>1</sub>{3,4,3}
|[[Rectified 24-cell]]
|{{CDD|node|3|node|4|node_1|3|node}}
|- BGCOLOR="#f0e0e0" align=center
|
|(0,1,2,3)√2
|t<sub>0,1</sub>{3,4,3}
|[[Truncated 24-cell]]
|{{CDD|node|3|node|4|node_1|3|node_1}}
|- BGCOLOR="#f0e0e0" align=center
|
|(0,1,φ,φ+1)√2
|s{3,4,3}
|[[Snub 24-cell]]
|{{CDD|node|3|node|4|node_h|3|node_h}}
|-
!colspan=4|&nbsp;
|- BGCOLOR="#e0e0f0" align=center
|(0,2,2,2)<br>(1,1,1,3)
|
|t<sub>2</sub>{3,4,3}
|Birectified 24-cell<br>([[Rectified 24-cell]])
|{{CDD|node|3|node_1|4|node|3|node}}
|- BGCOLOR="#f0e0e0" align=center
|(0,2,2,2) +<br>(1,1,1,3) +
|(0,0,1,1)√2<br>"
|t<sub>0,2</sub>{3,4,3}
|[[Cantellated 24-cell]]
|{{CDD|node|3|node_1|4|node|3|node_1}}
|- BGCOLOR="#e0f0e0" align=center
|(0,2,2,2) +<br>(1,1,1,3) +
|(0,1,1,2)√2<br>"
|t<sub>1,2</sub>{3,4,3}
|[[Bitruncated 24-cell]]
|{{CDD|node|3|node_1|4|node_1|3|node}}
|- BGCOLOR="#f0e0e0" align=center
|(0,2,2,2) +<br>(1,1,1,3) +
|(0,1,2,3)√2<br>"
|t<sub>0,1,2</sub>{3,4,3}
|[[Cantitruncated 24-cell]]
|{{CDD|node|3|node_1|4|node_1|3|node_1}}
|-
!colspan=4|&nbsp;
|- BGCOLOR="#e0e0f0" align=center
|(0,0,0,2)<br>(1,1,1,1)
|
|t<sub>3</sub>{3,4,3}
|Trirectified 24-cell<br>([[24-cell]])
|{{CDD|node_1|3|node|4|node|3|node}}
|- BGCOLOR="#e0f0e0" align=center
|(0,0,0,2) +<br>(1,1,1,1) +
|(0,0,1,1)√2<br>"
|t<sub>0,3</sub>{3,4,3}
|[[Runcinated 24-cell]]
|{{CDD|node_1|3|node|4|node|3|node_1}}
|- BGCOLOR="#e0e0f0" align=center
|(0,0,0,2) +<br>(1,1,1,1) +
|(0,1,1,2)√2<br>"
|t<sub>1,3</sub>{3,4,3}
|bicantellated 24-cell<br>([[Cantellated 24-cell]])
|{{CDD|node_1|4|node|3|node_1|3|node}}
|- BGCOLOR="#f0e0e0" align=center
|(0,0,0,2) +<br>(1,1,1,1) +
|(0,1,2,3)√2<br>"
|t<sub>0,1,3</sub>{3,4,3}
|[[Runcitruncated 24-cell]]
|{{CDD|node_1|3|node|4|node_1|3|node_1}}
<!--|- BGCOLOR="#f0e0e0" align=center *** NEED TO VERIFY
|(0,0,0,2) +<br>(1,1,1,1) +
|(0,1,φ,φ+1)√2<br>"
|s<sub>3</sub>{3,4,3}
|[[Runcic snub 24-cell]]
|{{CDD|node_1|3|node|4|node_h|3|node_h}}
-->
|-
!colspan=4|&nbsp;
|- BGCOLOR="#e0e0f0" align=center
|(1,1,1,5)<br>(1,3,3,3)<br>(2,2,2,4)
|
|t<sub>2,3</sub>{3,4,3}
|Tritruncated 24-cell<br>([[Truncated 24-cell]])
|{{CDD|node_1|3|node_1|4|node|3|node}}
|- BGCOLOR="#e0e0f0" align=center
|(1,1,1,5) +<br>(1,3,3,3) +<br>(2,2,2,4) +
|(0,0,1,1)√2<br>"<br>"
|t<sub>0,2,3</sub>{3,4,3}
|Runcicantellated 24-cell<br>([[Runcitruncated 24-cell]])
|{{CDD|node_1|3|node_1|4|node|3|node_1}}
|- BGCOLOR="#e0e0f0" align=center
|(1,1,1,5) +<br>(1,3,3,3) +<br>(2,2,2,4) +
|(0,1,1,2)√2<br>"<br>"
|t<sub>1,2,3</sub>{3,4,3}
|Bicantitruncated 24-cell<br>([[Cantitruncated 24-cell]])
|{{CDD|node_1|3|node_1|4|node_1|3|node}}
|- BGCOLOR="#e0f0e0" align=center
|(1,1,1,5) +<br>(1,3,3,3) +<br>(2,2,2,4) +
|(0,1,2,3)√2<br>"<br>"
|t<sub>0,1,2,3</sub>{3,4,3}
|[[Omnitruncated 24-cell]]
|{{CDD|node_1|3|node_1|4|node_1|3|node_1}}
|}
 
===The H<sub>4</sub> family===
 
This family has [[Hexacosichoric symmetry|''diploid hexacosichoric'' symmetry]], [5,3,3], of [[Symmetry order|order]] 120*120=24*600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra. There is one small index subgroups [5,3,3]<sup>+</sup>, all order 7200.
 
==== 120-cell truncations ====
 
{| class="wikitable"
!rowspan=2|#
!rowspan=2|Johnson Name<br>(Bowers style Acronym)
!rowspan=2|[[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br>{{CDD|node|5|node|3|node|2}}<br>(120)
! Pos. 2<br>{{CDD|node|5|node|2|node}}<br>(720)
! Pos. 1<br>{{CDD|node|2|2|node|3|node}}<br>(1200)
! Pos. 0<br>{{CDD|2|node|3|node|3|node}}<br>(600)
!Alt
! Cells
! Faces
! Edges
! Vertices
|-  BGCOLOR="#f0e0e0" align=center
!32
![[120-cell]] (hi)
|[[File:120-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node|3|node|3|node}}<br>{5,3,3}
|(4)<br>[[File:dodecahedron.png|30px]]<br>[[dodecahedron|(5.5.5)]]
|
|
|
|
| 120
| 720
| 1200
| 600
|- BGCOLOR="#f0e0e0"  align=center
!33
![[rectified 120-cell]] (rahi)
|[[File:Rectified 120-cell verf.png|60px]]
| align=center|{{CDD|node|5|node_1|3|node|3|node}}<br>r{5,3,3}
|(3)<br>[[File:icosidodecahedron.png|30px]]<br>[[icosidodecahedron|(3.5.3.5)]]
|
|
|(2)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
| 720
| 3120
| 3600
| 1200
|- BGCOLOR="#f0e0e0"  align=center
!36
![[truncated 120-cell]] (thi)
|[[File:Truncated 120-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node_1|3|node|3|node}}<br>t{5,3,3}
|(3)<br>[[File:truncated dodecahedron.png|30px]]<br>[[truncated dodecahedron|(3.10.10)]]
|
|
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
| 720
| 3120
| 4800
| 2400
|- BGCOLOR="#f0e0e0"  align=center
!37
![[cantellated 120-cell]] (srahi)
|[[File:Cantellated 120-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node|3|node_1|3|node}}<br>rr{5,3,3}
|(1)<br>[[File:small rhombicosidodecahedron.png|30px]]<br>[[small rhombicosidodecahedron|(3.4.5.4)]]
|
|(2)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|
| 1920
| 9120
| 10800
| 3600
|- BGCOLOR="#e0f0e0" align=center
!38
|align=center|'''[[runcinated 120-cell]]'''<br>(also '''runcinated 600-cell''') (sidpixhi)
|[[File:Runcinated 120-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{5,3,3}
|(1)<br>[[File:dodecahedron.png|30px]]<br>[[dodecahedron|(5.5.5)]]
|(3)<br>[[File:pentagonal prism.png|30px]]<br>[[pentagonal prism|(4.4.5)]]
|(3)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
| 2640
| 7440
| 7200
| 2400
|- BGCOLOR="#e0f0e0" align=center
!39
|align=center|'''[[bitruncated 120-cell]]'''<br>(also '''bitruncated 600-cell''') (xhi)
|[[File:Bitruncated 120-cell verf.png|60px]]
| align=center|{{CDD|node|5|node_1|3|node_1|3|node}}<br>2t{5,3,3}
|(2)<br>[[File:truncated icosahedron.png|30px]]<br>[[truncated icosahedron|(5.6.6)]]
|
|
|(2)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
| 720
| 4320
| 7200
| 3600
|- BGCOLOR="#f0e0e0"  align=center
!42
![[cantitruncated 120-cell]] (grahi)
|[[File:Cantitruncated 120-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node_1|3|node_1|3|node}}<br>tr{5,3,3}
|(2)<br>[[File:great rhombicosidodecahedron.png|30px]]<br>[[truncated icosidodecahedron|(4.6.10)]]
|
|(1)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
| 1920
| 9120
| 14400
| 7200
|- BGCOLOR="#f0e0e0" align=center
!43
![[runcitruncated 120-cell]] (prix)
|[[File:Runcitruncated 120-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node_1|3|node|3|node_1}}<br>t<sub>0,1,3</sub>{5,3,3}
|(1)<br>[[File:truncated dodecahedron.png|30px]]<br>[[truncated dodecahedron|(3.10.10)]]
|(2)<br>[[File:Decagonal prism.png|30px]]<br>[[Decagonal prism|(4.4.10)]]
|(1)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|
| 2640
| 13440
| 18000
| 7200
|- BGCOLOR="#e0f0e0" align=center
!46
|align=center|'''[[omnitruncated 120-cell]]'''<br>(also '''omnitruncated 600-cell''') (gidpixhi)
|[[File:Omnitruncated 120-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{5,3,3}
|(1)<br>[[File:great rhombicosidodecahedron.png|30px]]<br>[[truncated icosidodecahedron|(4.6.10)]]
|(1)<br>[[File:Decagonal prism.png|30px]]<br>[[Decagonal prism|(4.4.10)]]
|(1)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|(1)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
| 2640
| 17040
| 28800
| 14400
|- BGCOLOR="#d0f0f0" align=center
![[#Nonuniform alternations|Nonuniform]]||[[full snub 120-cell]]<ref>http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm</ref><br>(Same as the ''full snub 600-cell'')
|[[File:Snub 120-cell verf.png|60px]]
| align=center|{{CDD|node_h|5|node_h|3|node_h|3|node_h}}<br>ht<sub>0,1,2,3</sub>{5,3,3}
|[[File:Snub dodecahedron cw.png|30px]] (1)<br>[[Snub cube|(3.3.3.3.5)]]
|[[File:pentagonal antiprism.png|30px]] (1)<br>[[pentagonal antiprism|(3.3.3.5)]]
|[[File:octahedron.png|30px]] (1)<br>[[triangular antiprism|(3.3.3.3)]]
|[[File:Snub tetrahedron.png|30px]] (1)<br>[[Snub tetrahedron|(3.3.3.3.3)]]
|[[File:tetrahedron.png|30px]] (4)<br>[[tetrahedron|(3.3.3)]]
| 9840
| 35040
| 32400
| 7200
|}
 
==== 600-cell truncations ====
 
{| class="wikitable"
!rowspan=2|#
!rowspan=2|Johnson Name<br>(Bowers style acronym)
!rowspan=2|[[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=4 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br>{{CDD|node|5|node|3|node}}<br>(120)
! Pos. 2<br>{{CDD|node|5|node|2|node}}<br>(720)
! Pos. 1<br>{{CDD|node|2|node|3|node}}<br>(1200)
! Pos. 0<br>{{CDD|node|3|node|3|node}}<br>(600)
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#e0e0f0" align=center
!35
|[[600-cell]] (ex)
|[[File:600-cell verf.png|60px]]
| align=center|{{CDD|node|5|node|3|node|3|node_1}}<br>{3,3,5}
|
|
|
|(20)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 600
| 1200
| 720
| 120
|- BGCOLOR="#e0e0f0" align=center
!34
|[[rectified 600-cell]] (rox)
|[[File:Rectified 600-cell verf.png|60px]]
| align=center|{{CDD|node|5|node|3|node_1|3|node}}<br>r{3,3,5}
|(2)<br>[[File:icosahedron.png|30px]]<br>[[icosahedron|(3.3.3.3.3)]]
|
|
|(5)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
| 720
| 3600
| 3600
| 720
|- BGCOLOR="#e0e0f0" align=center
!41
|[[truncated 600-cell]] (tex)
|[[File:Truncated 600-cell verf.png|60px]]
| align=center|{{CDD|node|5|node|3|node_1|3|node_1}}<br>t{3,3,5}
|(1)<br>[[File:icosahedron.png|30px]]<br>[[icosahedron|(3.3.3.3.3)]]
|
|
|(5)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
| 720
| 3600
| 4320
| 1440
|-  BGCOLOR="#e0e0f0" align=center
!40
|[[cantellated 600-cell]] (srix)
|[[File:Cantellated 600-cell verf.png|60px]]
| align=center|{{CDD|node|5|node_1|3|node|3|node_1}}<br>rr{3,3,5}
|(1)<br>[[File:icosidodecahedron.png|30px]]<br>[[icosidodecahedron|(3.5.3.5)]]
|(2)<br>[[File:pentagonal prism.png|30px]]<br>[[pentagonal prism|(4.4.5)]]
|
|(1)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
| 1440
| 8640
| 10800
| 3600
|- BGCOLOR="#e0f0e0" align=center
![38]
|align=center|'''[[runcinated 600-cell]]'''<br>(also '''runcinated 120-cell''') (sidpixhi)
|[[File:Runcinated 120-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{3,3,5}
|(1)<br>[[File:dodecahedron.png|30px]]<br>[[dodecahedron|(5.5.5)]]
|(3)<br>[[File:pentagonal prism.png|30px]]<br>[[pentagonal prism|(4.4.5)]]
|(3)<br>[[File:triangular prism.png|30px]]<br>[[triangular prism|(3.4.4)]]
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 2640
| 7440
| 7200
| 2400
|- BGCOLOR="#e0f0e0" align=center
![39]
|align=center|'''[[bitruncated 600-cell]]'''<br>(also '''bitruncated 120-cell''') (xhi)
|[[File:Bitruncated 120-cell verf.png|60px]]
| align=center|{{CDD|node|5|node_1|3|node_1|3|node}}<br>2t{3,3,5}
|(2)<br>[[File:truncated icosahedron.png|30px]]<br>[[truncated icosahedron|(5.6.6)]]
|
|
|(2)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
| 720
| 4320
| 7200
| 3600
|- BGCOLOR="#e0e0f0" align=center
!45
|[[cantitruncated 600-cell]] (grix)
|[[File:Cantitruncated 600-cell verf.png|60px]]
| align=center|{{CDD|node|5|node_1|3|node_1|3|node_1}}<br>tr{3,3,5}
|(1)<br>[[File:truncated icosahedron.png|30px]]<br>[[truncated icosahedron|(5.6.6)]]
|(1)<br>[[File:pentagonal prism.png|30px]]<br>[[pentagonal prism|(4.4.5)]]
|
|(2)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
| 1440
| 8640
| 14400
| 7200
|- BGCOLOR="#e0e0f0" align=center
!44
|[[runcitruncated 600-cell]] (prahi)
|[[File:Runcitruncated 600-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node|3|node_1|3|node_1}}<br>t<sub>0,1,3</sub>{3,3,5}
|(1)<br>[[File:small rhombicosidodecahedron.png|30px]]<br>[[small rhombicosidodecahedron|(3.4.5.4)]]
|(1)<br>[[File:pentagonal prism.png|30px]]<br>[[pentagonal prism|(4.4.5)]]
|(2)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|(1)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
| 2640
| 13440
| 18000
| 7200
|- BGCOLOR="#e0f0e0" align=center
![46]
|align=center|'''[[omnitruncated 600-cell]]'''<br>(also '''omnitruncated 120-cell''') (gidpixhi)
|[[File:Omnitruncated 120-cell verf.png|60px]]
| align=center|{{CDD|node_1|5|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,3,5}
|(1)<br>[[File:great rhombicosidodecahedron.png|30px]]<br>[[truncated icosidodecahedron|(4.6.10)]]
|(1)<br>[[File:Decagonal prism.png|30px]]<br>[[Decagonal prism|(4.4.10)]]
|(1)<br>[[File:hexagonal prism.png|30px]]<br>[[hexagonal prism|(4.4.6)]]
|(1)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
| 2640
| 17040
| 28800
| 14400
|}
 
==== Graphs ====
{| class="wikitable"
!rowspan=2|#
!rowspan=2|Johnson Name<br>(Bowers style Acronym)
!colspan=6|[[Coxeter plane]] projections
!colspan=2|[[Schlegel diagram]]s
|-
!F4<br>[12]||[20]||H4<br>[30]||H3<br>[10]||A3<br>[4]||A2<br>[3]
!Dodecahedron<br>centered
!Tetrahedron<br>centered
|- BGCOLOR="#f0e0e0" align=center
!32
||[[120-cell]] (hi)
||[[File:120-cell t0 F4.svg|80px]]||[[File:120-cell t0 p20.svg|80px]]||[[File:120-cell graph H4.svg|80px]]||[[File:120-cell t0 H3.svg|80px]]||[[File:120-cell t0 A3.svg|80px]]||[[File:120-cell t0 A2.svg|80px]]
||[[File:Schlegel wireframe 120-cell.png|80px]]
|
|- BGCOLOR="#f0e0e0"  align=center
!33
||[[rectified 120-cell]] (rahi)||[[File:120-cell t1 F4.svg|80px]]||[[File:120-cell t1 p20.svg|80px]]||[[File:120-cell t1 H4.svg|80px]]||[[File:120-cell t1 H3.svg|80px]]||[[File:120-cell t1 A3.svg|80px]]||[[File:120-cell t1 A2.svg|80px]]
||[[File:Rectified 120-cell schlegel halfsolid.png|80px]]
|
|- BGCOLOR="#e0e0f0" align=center
!34
||[[rectified 600-cell]] (rox)
||[[File:600-cell t1 F4.svg|80px]]||[[File:600-cell t1 p20.svg|80px]]||[[File:600-cell t1 H4.svg|80px]]||[[File:600-cell t1 H3.svg|80px]]||[[File:600-cell t1.svg|80px]]||[[File:600-cell t1 A2.svg|80px]]
|[[File:Rectified 600-cell schlegel halfsolid.png|80px]]
||
|- BGCOLOR="#e0e0f0" align=center
!35
||[[600-cell]] (ex)
||[[File:600-cell t0 F4.svg|80px]]||[[File:600-cell t0 p20.svg|80px]]||[[File:600-cell graph H4.svg|80px]]||[[File:600-cell t0 H3.svg|80px]]||[[File:600-cell t0.svg|80px]]||[[File:600-cell t0 A2.svg|80px]]
|[[File:Schlegel wireframe 600-cell vertex-centered.png|80px]]
||[[File:Stereographic polytope 600cell.png|80px]]
|- BGCOLOR="#f0e0e0"  align=center
!36
||[[truncated 120-cell]] (thi)||[[File:120-cell t01 F4.svg|80px]]||[[File:120-cell t01 p20.svg|80px]]||[[File:120-cell t01 H4.svg|80px]]||[[File:120-cell t01 H3.svg|80px]]||[[File:120-cell t01 A3.svg|80px]]||[[File:120-cell t01 A2.svg|80px]]
||[[File:Schlegel half-solid truncated 120-cell.png|80px]]
|
|- BGCOLOR="#f0e0e0"  align=center
!37
||[[cantellated 120-cell]] (srahi)
|| || || || [[File:120-cell t02 H3.png|80px]]||||[[File:120-cell t02 B3.png|80px]]
||[[File:Cantellated 120 cell center.png|80px]]
|
|- BGCOLOR="#e0f0e0" align=center
!38
||'''[[runcinated 120-cell]]'''<br>(also '''runcinated 600-cell''') (sidpixhi)
|| || || || [[File:120-cell t03 H3.png|80px]]||||[[File:120-cell t03 B3.png|80px]]
||[[File:runcinated 120-cell.png|80px]]
|
|- BGCOLOR="#e0f0e0" align=center
!39
||'''[[bitruncated 120-cell]]'''<br>(also '''bitruncated 600-cell''') (xhi)
|| || || || [[File:120-cell t12 H3.png|80px]]||[[File:120-cell t12 A3.png|80px]]||[[File:120-cell t12 B3.png|80px]]
||[[File:Bitruncated 120-cell schlegel halfsolid.png|80px]]
|
|-  BGCOLOR="#e0e0f0" align=center
!40
||[[cantellated 600-cell]] (srix)
||[[File:600-cell t02 F4.svg|80px]]||[[File:600-cell t02 p20.svg|80px]]||[[File:600-cell t02 H4.svg|80px]]||[[File:600-cell t02 H3.svg|80px]]||[[File:600-cell t02 B2.svg|80px]]||[[File:600-cell t02 B3.svg|80px]]
|
||[[File:Cantellated 600 cell center.png|80px]]
|- BGCOLOR="#e0e0f0" align=center
!41
||[[truncated 600-cell]] (tex)
||[[File:600-cell t01 F4.svg|80px]]||[[File:600-cell t01 p20.svg|80px]]||[[File:600-cell t01 H4.svg|80px]]||[[File:600-cell t01 H3.svg|80px]]||[[File:600-cell t01.svg|80px]]||[[File:600-cell t01 A2.svg|80px]]
|
||[[File:Schlegel half-solid truncated 600-cell.png|80px]]
|- BGCOLOR="#f0e0e0"  align=center
!42
||[[cantitruncated 120-cell]] (grahi)
|| || || || [[File:120-cell t012 H3.png|80px]]||||[[File:120-cell t012 B3.png|80px]]
|| [[File:Cantitruncated 120-cell.png|80px]]
|
|- BGCOLOR="#f0e0e0" align=center
!43
||[[runcitruncated 120-cell]] (prix)
|| || || || [[File:120-cell t013 H3.png|80px]]||||[[File:120-cell t013 B3.png|80px]]
||[[File:Runcitruncated 120-cell.png|80px]]
|
|- BGCOLOR="#e0e0f0" align=center
!44
||[[runcitruncated 600-cell]] (prahi)
|| || || || [[File:120-cell t023 H3.png|80px]]||||[[File:120-cell t023 B3.png|80px]]
|
|| [[File:Runcitruncated 600-cell.png|80px]]
|- BGCOLOR="#e0e0f0" align=center
!45
||[[cantitruncated 600-cell]] (grix)
|| || || || [[File:120-cell t123 H3.png|80px]]||||[[File:120-cell t123 B3.png|80px]]
|
||[[File:Cantitruncated 600-cell.png|80px]]
|- BGCOLOR="#e0f0e0" align=center
!46
||'''[[omnitruncated 120-cell]]'''<br>(also omnitruncated 600-cell) (gidpixhi)
|| || || || [[File:120-cell t0123 H3.png|80px]]||||[[File:120-cell t0123 B3.png|80px]]
||[[File:Omnitruncated 120-cell wireframe.png|80px]]
|
|}
 
=== The D<sub>4</sub> family ===
 
This [[Demitesseractic symmetry|demitesseract family]], [3<sup>1,1,1</sup>], introduces no new uniform polychora, but it is worthy to repeat these alternative constructions. This family has [[Symmetry order|order]] 12*16=192: 4!/2=12 permutations of the four axes, half as alternated, 2<sup>4</sup>=16 for reflection in each axis. There is one small index subgroups that generating uniform polychora, [3<sup>1,1,1</sup>]<sup>+</sup>, order 96.
 
{| class="wikitable"
|+ [3<sup>1,1,1</sup>] uniform polychora
|-
!rowspan=2|#
!rowspan=2|Johnson Name (Bowers style acronym)
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>[[File:CD B4 nodes.png]]
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 0<br>{{CDD|node|3|node|3|node}}<br>(8)
! Pos. 2<br>{{CDD|nodes|2|node}}<br>(24)
! Pos. 1<br>{{CDD|nodes|split2|node}}<br>(8)
! Pos. 3<br>{{CDD|node|3|node|3|node}}<br>(8)
! Pos. Alt<br>(96)
!3
!2
!1
!0
|- align=center
![12]
| align=center|demitesseract<BR>Half tesseract<br>(Same as '''[[16-cell]]''') (hex)
|[[File:16-cell verf.png|60px]]
| align=center|{{CDD|nodes_10ru|split2|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node}}<BR>h{4,3,4}
|
|
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
| 16
| 32
| 24
| 8
|- align=center
![17]
| align=center|Cantic tesseract<br>(Same as '''[[truncated 16-cell]]''') (thex)
|[[File:Truncated demitesseract verf.png|60px]]
| align=center|{{CDD|nodes_10ru|split2|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node}}<BR>h<sub>2</sub>{4,3,3}
|(1)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|
|(2)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|(2)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
| 24
| 96
| 120
| 48
|-  align=center
![11]
| align=center|Runcic tesseract<br>(Same as '''[[rectified tesseract]]''') (rit)
|[[File:Cantellated demitesseract verf.png|60px]]
| align=center|{{CDD|nodes_10ru|split2|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1}}<BR>h<sub>3</sub>{4,3,3}
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|(3)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|
| 24
| 88
| 96
| 32
|- align=center
![16]
| align=center|Runcicantic tesseract<br>(Same as '''[[bitruncated tesseract]]''') (tah)
|[[File:Cantitruncated demitesseract verf.png|60px]]
| align=center|{{CDD|nodes_10ru|split2|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1}}<BR>h<sub>2,3</sub>{4,3,3}
|(1)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|
|(1)<br>[[File:truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|(2)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|
| 24
| 96
| 96
| 24
|}
 
When the 3 bifurcated branch nodes are identically ringed, the symmetry can be increased by 6, as [3[3<sup>1,1,1</sup>]] = [3,4,3], and thus these polytopes are repeated from the [[24-cell]] family.
 
{| class="wikitable"
|+ [3[3<sup>1,1,1</sup>]] uniform polychora
|-
!rowspan=2|#
!rowspan=2|Johnson Name (Bowers style acronym)
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<BR>{{CDD|nodeab_c1|split2|node_c2|3|node_c1}} = {{CDD|node|4|node_c1|3|node_c2|3|node_c1}}<BR>{{CDD|node_c2|3|node_c1|4|node|3|node}} = {{CDD|node_c2|splitsplit1|branch3_c1|node_c1}}
!colspan=3 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 0,1,3<br>{{CDD|node|3|node|3|node}}<br>(24)
! Pos. 2<br>{{CDD|nodes|2|node}}<br>(24)
! Pos. Alt<br>(96)
!3
!2
!1
!0
|-  align=center
![22]
| align=center|rectified 16-cell)<br>(Same as '''[[24-cell]]''') (ico)
|[[File:Rectified demitesseract verf.png|60px]]
|align=center|{{CDD|nodes|split2|node_1|3|node}} = {{CDD|node|4|node|3|node_1|3|node}} = {{CDD|node_1|3|node|4|node|3|node}} = {{CDD|node_1|splitsplit1|branch3|node}}<BR>{3<sup>1,1,1</sup>} = r{3,3,4} = {3,4,3}
|(6)<br>[[File:octahedron.png|30px]]<br>[[octahedron|(3.3.3.3)]]
|
|
| 48
| 240
| 288
| 96
|- align=center
![23]
| align=center|Cantellated 16-cell<br>(Same as '''[[rectified 24-cell]]''') (rico)
|[[File:Runcicantellated demitesseract verf.png|60px]]
|align=center|{{CDD|nodes_11|split2|node|3|node_1}} = {{CDD|node|4|node_1|3|node|3|node_1}} = {{CDD|node|3|node_1|4|node|3|node}} = {{CDD|node|splitsplit1|branch3_11|node_1}}<BR>r{3<sup>1,1,1</sup>} = rr{3,3,4} = r{3,4,3}
|(3)<br>[[File:cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|(2)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|
| 24
| 120
| 192
| 96
|- align=center
![24]
| align=center|Cantitruncated 16-cell<br>(Same as '''[[truncated 24-cell]]''') (tico)
|[[File:Omnitruncated demitesseract verf.png|60px]]
|align=center|{{CDD|nodes_11|split2|node_1|3|node_1}} = {{CDD|node|4|node_1|3|node_1|3|node_1}} = {{CDD|node_1|3|node_1|4|node|3|node}} = {{CDD|node_1|splitsplit1|branch3_11|node_1}}<BR>t{3<sup>1,1,1</sup>} = tr{3,3,4} = t{3,4,3}
|(3)<br>[[File:truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|(1)<br>[[File:hexahedron.png|30px]]<br>[[cube|(4.4.4)]]
|
| 48
| 240
| 384
| 192
|- BGCOLOR="#d0f0f0" align=center
![31]
|align=center|[[snub 24-cell]] (sadi)
|[[File:Snub 24-cell verf.png|60px]]
|align=center|{{CDD|nodes_hh|split2|node_h|3|node_h}} = {{CDD|node|4|node_h|3|node_h|3|node_h}} = {{CDD|node_h|3|node_h|4|node|3|node}} = {{CDD|node_h|splitsplit1|branch3_hh|node_h}}<BR>s{3<sup>1,1,1</sup>} = sr{3,3,4} = s{3,4,3}
|(3)<br>[[File:Snub tetrahedron.png|30px]]<br>[[icosahedron|(3.3.3.3.3)]]
|(1)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|(4)<br>[[File:tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
| 144
| 480
| 432
| 96
|}
 
Here again the ''snub 24-cell'', with the symmetry group [3<sup>1,1,1</sup>]<sup>+</sup> this time, represents an alternated truncation of the truncated 24-cell creating 96 new tetrahedra at the position of the deleted vertices. In contrast to its appearance within former groups as partly snubbed polychoron, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the [[snub cube]] and the [[snub dodecahedron]].
 
==== Graphs ====
{| class="wikitable"
!rowspan=2|#
!rowspan=2|Johnson Name (Bowers style acronym)<br>[[Coxeter-Dynkin diagram|Coxeter diagram]]
!colspan=3|[[Coxeter plane]] projections
!colspan=2| [[Schlegel diagram]]s
!colspan=1|Parallel<br>3D
|-
!B<sub>4</sub><br>[8/2]
!D<sub>4</sub><br>[6]
!D<sub>3</sub><br>[2]
!Cube<br>centered
!Tetrahedron<br>centered
!D<sub>4</sub><br>[6]
|- align=center
![12]
||demitesseract<br>(Same as [[16-cell]]) (hex)<br>{{CDD|nodes_10ru|split2|node|3|node}} or {{CDD|node_1|3|node|split1|nodes}}<P>h{4,3,3} = {3,3<sup>1,1</sup>}
|[[File:4-demicube t0 B4.svg|80px]]
|[[File:4-demicube t0 D4.svg|80px]]
|[[File:4-demicube t0 D3.svg|80px]]
|
|[[File:Schlegel wireframe 16-cell.png|80px]]
|- align=center
![17]
||[[truncated demitesseract]] (thex)<br>{{CDD|nodes_10ru|split2|node_1|3|node}} or {{CDD|node_1|3|node_1|split1|nodes}}<P>h<sub>2</sub>{4,3,3} = t{3,3<sup>1,1</sup>}
|[[File:4-demicube t01 B4.svg|80px]]
|[[File:4-demicube t01 D4.svg|80px]]
|[[File:4-demicube t01 D3.svg|80px]]
|
|[[File:Schlegel half-solid truncated 16-cell.png|80px]]
|-  align=center
![11]
||birectified demitesseract<br>(Same as [[rectified tesseract]]) (rit)<br>{{CDD|nodes_10ru|split2|node|3|node_1}} or {{CDD|node|3|node|split1|nodes_11}}<P>h<sub>3</sub>{4,3,3} = 2r{3,3<sup>1,1</sup>}
|[[File:4-demicube t02 B4.svg|80px]]
|[[File:4-demicube t02 D4.svg|80px]]
|[[File:4-demicube t02 D3.svg|80px]]
|[[File:Schlegel half-solid rectified 8-cell.png|80px]]
|
|- align=center
![16]
||bitruncated demitesseract<br>(Same as [[bitruncated tesseract]]) (tah)<br>{{CDD|nodes_10ru|split2|node_1|3|node_1}} or {{CDD|node|3|node_1|split1|nodes_11}}<P>h<sub>2,3</sub>{4,3,3} = 2t{3,3<sup>1,1</sup>}
|[[File:4-demicube t012 B4.svg|80px]]
|[[File:4-demicube t012 D4.svg|80px]]
|[[File:4-demicube t012 D3.svg|80px]]
|
| [[File:Schlegel half-solid bitruncated 16-cell.png|80px]]
|-  align=center
![22]
||rectified demitesseract<br>(Same as '''[[24-cell]]''') (ico)<br>{{CDD|nodes|split2|node_1|3|node}}<P>{3<sup>1,1,1</sup>}
|[[File:4-cube t2.svg|80px]]
|[[File:4-demicube t1 D4.svg|80px]]
|[[File:4-demicube t1 D3.svg|80px]]
|
|[[File:Schlegel wireframe 24-cell.png|80px]]
|- align=center
![23]
||Cantellated demitesseract<br>(Same as '''[[rectified 24-cell]]''') (rico)<br>{{CDD|nodes_11|split2|node|3|node_1}}<P>r{3<sup>1,1,1</sup>}
|[[File:4-cube t02.svg|80px]]
|[[File:4-demicube t023 D4.svg|80px]]
|[[File:4-demicube t023 D3.svg|80px]]
|
| [[File:Schlegel half-solid cantellated 16-cell.png|80px]]
|- align=center
![24]
||cantitruncated demitesseract<br>(Same as '''[[truncated 24-cell]]''') (tico)<br>{{CDD|nodes_11|split2|node_1|3|node_1}}<P>t{3<sup>1,1,1</sup>}
|[[File:4-cube t012.svg|80px]]
|[[File:4-demicube t0123 D4.svg|80px]]
|[[File:4-demicube t0123 D3.svg|80px]]
|
|[[File:Schlegel half-solid truncated 24-cell.png|80px]]
|- BGCOLOR="#d0f0f0" align=center
![31]
||Snub demitesseract<br>([[snub 24-cell]]) (sadi)<br>{{CDD|nodes_hh|split2|node_h|3|node_h}}<P>s{3<sup>1,1,1</sup>}
|[[File:24-cell h01 F4.svg|80px]]
|[[File:24-cell h01 B3.svg|80px]]
|[[File:24-cell h01 B2.svg|80px]]
|
|
|[[File:Ortho solid 969-uniform polychoron 343-snub.png|80px]]
|}
 
==== Coordinates ====
 
The ''base point'' can generate the coordinates of the polytope by taking all coordinate permutations and sign combinations. The edges' length will be √2. Some polytopes have two possible generator points. Points are prefixed by ''Even'' to imply only an even count of sign permutations should be included.
 
{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name(s)
!rowspan=2|Base point
!rowspan=2|Johnson and Bowers Names
!colspan=3|[[Coxeter diagram]]s
|-
!D<sub>4</sub>
!B<sub>4</sub>
!F<sub>4</sub>
|- align=center
![12]
!t<sub>3</sub>γ<sub>4</sub> = β<sub>4</sub>
|(0,0,0,2)
||[[16-cell]]
||{{CDD|nodes|split2|node|3|node_1}}
||{{CDD|node|4|node|3|node|3|node_1}}
|
|-  align=center
![22]
!t<sub>2</sub>γ<sub>4</sub> = t<sub>1</sub>β<sub>4</sub>
|(0,0,2,2)
||[[Rectified 16-cell]]
||{{CDD|nodes|split2|node_1|3|node}}
||{{CDD|node|4|node|3|node_1|3|node}}
||{{CDD|node_1|3|node|4|node|3|node}}
|- align=center
![17]
!t<sub>2,3</sub>γ<sub>4</sub> = t<sub>0,1</sub>β<sub>4</sub>
|(0,0,2,4)
||[[Truncated 16-cell]]
||{{CDD|nodes|split2|node_1|3|node_1}}
||{{CDD|node|4|node|3|node_1|3|node_1}}
|
|-  align=center
![11]
!t<sub>1</sub>γ<sub>4</sub> = t<sub>2</sub>β<sub>4</sub>
|(0,2,2,2)
||[[Cantellated 16-cell]]
||{{CDD|nodes_11|split2|node|3|node}}
||{{CDD|node|4|node_1|3|node|3|node}}
|
|- align=center
![23]
!t<sub>1,3</sub>γ<sub>4</sub> = t<sub>0,2</sub>β<sub>4</sub>
|(0,2,2,4)
||[[Cantellated 16-cell]]
||{{CDD|nodes_11|split2|node|3|node_1}}
||{{CDD|node|4|node_1|3|node|3|node_1}}
||{{CDD|node|3|node_1|4|node|3|node}}
|- align=center
![16]
!t<sub>1,2</sub>γ<sub>4</sub> = t<sub>1,2</sub>β<sub>4</sub>
|(0,2,4,4)
||[[Bitruncated 16-cell]]
||{{CDD|nodes_11|split2|node_1|3|node}}
||{{CDD|node|4|node_1|3|node_1|3|node}}
|
|- align=center
![24]
!t<sub>1,2,3</sub>γ = t<sub>0,1,2</sub>β<sub>4</sub>
|(0,2,4,6)
|[[Cantitruncated 16-cell]]
||{{CDD|nodes_11|split2|node_1|3|node_1}}
||{{CDD|node|4|node_1|3|node_1|3|node_1}}
||{{CDD|node_1|3|node_1|4|node|3|node}}
|- align=center
![12]
!hγ<sub>4</sub>
|Even (1,1,1,1)
|[[16-cell]]
||{{CDD|nodes_10ru|split2|node|3|node}}
||{{CDD|node_h1|4|node|3|node|3|node}}
|
|- align=center
![17]
!h<sub>2</sub>γ<sub>4</sub>
|Even (1,1,3,3)
|[[Truncated 16-cell]]
||{{CDD|nodes_10ru|split2|node_1|3|node}}
|{{CDD|node_h1|4|node|3|node_1|3|node}}
|
|-  align=center
![11]
!h<sub>3</sub>γ<sub>4</sub>
|Even (1,1,1,3)
|[[Cantellated 16-cell]]
||{{CDD|nodes_10ru|split2|node|3|node_1}}
|{{CDD|node_h1|4|node|3|node|3|node_1}}
|
|- align=center
![16]
!h<sub>2,3</sub>γ<sub>4</sub>
|Even (1,3,3,3)
|[[Cantitruncated 16-cell]]
||{{CDD|nodes_10ru|split2|node_1|3|node_1}}
|{{CDD|node_h1|4|node|3|node_1|3|node_1}}
|
|- BGCOLOR="#d0f0f0" align=center
|'''[31]'''
|s{3<sup>1,1,1</sup>}
|(0,1,φ,φ+1)/√2
||[[Snub 24-cell]]
||{{CDD|nodes_hh|split2|node_h|3|node_h}}
||{{CDD|node|4|node_h|3|node_h|3|node_h}}
||{{CDD|node_h|3|node_h|4|node|3|node}}
|}
 
===The grand antiprism===
There is one non-Wythoffian uniform convex polychoron, known as the [[grand antiprism]], consisting of 20 [[pentagonal antiprism]]s forming two perpendicular rings joined by 300 [[tetrahedron|tetrahedra]].  It is loosely analogous to the three-dimensional [[antiprism]]s, which consist of two parallel [[polygon]]s joined by a band of [[triangle]]s. Unlike them, however, the grand antiprism is not a member of an infinite family of uniform polytopes.
 
Its symmetry is the ''ionic diminished Coxeter group'', [[10,2<sup>+</sup>,10]], order 400.
 
{| class="wikitable"
!rowspan=2|#
!rowspan=2| Johnson Name (Bowers style acronym)
!rowspan=2| [[Schlegel diagram|Picture]]
!rowspan=2|[[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!rowspan=2 colspan=2|Cells by type
!colspan=4|Element counts
|-
! Cells
! Faces
! Edges
! Vertices
|-
!47
![[grand antiprism]] (gap)
|[[File:Grand antiprism.png|50px]]
|[[File:Grand antiprism verf.png|50px]]
|No symbol
|300 [[File:Tetrahedron.png|20px]]<br>(''[[Tetrahedron|3.3.3]]'')
|20 [[File:Pentagonal antiprism.png|20px]]<br>(''[[Pentagonal antiprism|3.3.3.5]]'')
|320
|20 [[pentagon|{5}]]<br>700 [[triangle|{3}]]
|500
|100
|}
 
===Prismatic uniform polychora===
 
A prismatic polytope is a [[Cartesian product]] of two polytopes of lower dimension; familiar examples are the 3-dimensional [[prism (geometry)|prisms]], which are products of a [[polygon]] and a [[line segment]].  The prismatic uniform polychora consist of two infinite families:
* ''Polyhedral prisms'': products of a line segment and a uniform polyhedron.  This family is infinite because it includes prisms built on 3-dimensional prisms and [[antiprism]]s.
* ''Duoprisms'': products of two polygons.
 
==== Convex polyhedral prisms ====
The most obvious family of prismatic polychora is the ''polyhedral prisms,'' i.e. products of a polyhedron with a [[line segment]].  The cells of such a polychoron are two identical uniform polyhedra lying in parallel [[hyperplane]]s (the ''base'' cells) and a layer of prisms joining them (the ''lateral'' cells).  This family includes prisms for the 75 nonprismatic [[uniform polyhedron|uniform polyhedra]] (of which 18 are convex; one of these, the cube-prism, is listed above as the ''tesseract'').{{Citation needed|date=January 2011}}
 
There are '''18 convex polyhedral prisms''' created from 5 [[Platonic solid]]s and 13 [[Archimedean solid]]s as well as for the infinite families of three-dimensional [[prism (geometry)|prism]]s and [[antiprism]]s.{{Citation needed|date=January 2011}}  The symmetry number of a polyhedral prism is twice that of the base polyhedron.
 
==== Tetrahedral prisms: A<sub>3</sub> × A<sub>1</sub> ====
 
This [[Hexadecachoric symmetry|prismatic tetrahedral symmetry]] is [3,3,2], order 48. There are two index 2 subgroups, [(3,3)<sup>+</sup>,2] and [3,3,2]<sup>+</sup>, but the second doesn't generate a uniform polychoron.
 
{| class="wikitable"
|+ [3,3,2] uniform polychora
!rowspan=2|#
!rowspan=2| Johnson Name (Bowers style acronym)
!rowspan=2| [[Schlegel diagram|Picture]]
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!rowspan=2 colspan=3|Cells by type
!colspan=4|Element counts
|-
! Cells
! Faces
! Edges
! Vertices
|- align=center
!48
![[Tetrahedral prism]] (tepe)
|[[File:Tetrahedral prism.png|75px]]
|[[File:Tetrahedral prism verf.png|75px]]
|align=center|{{CDD|node_1|3|node|3|node|2|node_1}}<br>{3,3}×{&nbsp;}<br>t<sub>0,3</sub>{3,3,2}
|2 [[File:Tetrahedron.png|20px]]<br>[[Tetrahedron|3.3.3]]
|4 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|
|6
|8 {3}<br>6 {4}
|16
|8
|- align=center
!49
![[Truncated tetrahedral prism]] (tuttip)
|[[File:Truncated tetrahedral prism.png|75px]]
|[[File:Truncated tetrahedral prism verf.png|75px]]
|align=center|{{CDD|node_1|3|node_1|3|node|2|node_1}}<br>t{3,3}×{&nbsp;}<br>t<sub>0,1,3</sub>{3,3,2}
|2 [[File:Truncated tetrahedron.png|20px]]<br>[[Truncated tetrahedron|3.6.6]]
|4 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|4 [[File:Hexagonal prism.png|20px]]<br>[[Hexagonal prism|4.4.6]]
|10
|8 {3}<br>18 {4}<br>8 {6}
|48
|24
|}
 
{| class="wikitable"
|+ <nowiki>[[</nowiki>3,3],2] uniform polychora
!rowspan=2|#
!rowspan=2| Johnson Name (Bowers style acronym)
!rowspan=2| [[Schlegel diagram|Picture]]
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!rowspan=2 colspan=3|Cells by type
!colspan=4|Element counts
|-
! Cells
! Faces
! Edges
! Vertices
|- align=center
![51]
|align=center|''Rectified tetrahedral prism''<br>(Same as [[octahedral prism]]) (ope)
|[[File:Octahedral prism.png|75px]]
|[[File:tetratetrahedral prism verf.png|75px]]
|align=center|{{CDD|node|3|node_1|3|node|2|node_1}}<br>r{3,3}×{&nbsp;}<br>t<sub>1,3</sub>{3,3,2}
|2 [[File:Octahedron.png|20px]]<br>[[Octahedron|3.3.3.3]]
|4 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|
|6
|16 {3}<br>12 {4}
|30
|12
|- align=center
![50]
|align=center|''Cantellated tetrahedral prism''<br>(Same as [[cuboctahedral prism]]) (cope)
| [[File:Cuboctahedral prism.png|75px]]
|[[File:Cuboctahedral prism verf.png|75px]]
|align=center|{{CDD|node_1|3|node|3|node_1|2|node_1}}<br>rr{3,3}×{&nbsp;}<br>t<sub>0,2,3</sub>{3,3,2}
|2 [[File:Cuboctahedron.png|20px]]<br>[[Cuboctahedron|3.4.3.4]]
|8 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|6 [[File:Hexahedron.png|20px]]<br>[[Cube|4.4.4]]
|16
|16 {3}<br>36 {4}
|60
|24
|- align=center
![54]
|align=center|''Cantitruncated tetrahedral prism''<br>(Same as [[truncated octahedral prism]]) (tope)
|[[File:Truncated octahedral prism.png|75px]]
|[[File:Truncated octahedral prism verf.png|75px]]
|align=center|{{CDD|node_1|3|node_1|3|node_1|2|node_1}}<br>tr{3,3}×{&nbsp;}<br>t<sub>0,1,2,3</sub>{3,3,2}
|2 [[File:Truncated octahedron.png|20px]]<br>[[Truncated octahedron|4.6.6]]
|8 [[File:Hexagonal prism.png|20px]]<br>[[Hexagonal prism|6.4.4]]
|6 [[File:Hexahedron.png|20px]]<br>[[Cube|4.4.4]]
|16
|48 {4}<br>16 {6}
|96
|48
|- align=center
![59]
|align=center|''Snub tetrahedral prism''<br>(Same as [[icosahedral prism]]) (ipe)
|[[File:Icosahedral prism.png|75px]]
|[[File:snub tetrahedral prism verf.png|75px]]
|align=center|{{CDD|node_h|3|node_h|3|node_h|2|node_1}}<br>sr{3,3}×{&nbsp;}
|2 [[File:Icosahedron.png|20px]]<br>[[Icosahedron|3.3.3.3.3]]
|20 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|
|22
|40 {3}<br>30 {4}
|72
|24
|- align=center
![[#Nonuniform alternations|Nonuniform]]
![[Full snub tetrahedral antiprism]]
|
|[[File:Snub 332 verf.png|75px]]
|align=center|{{CDD|node_h|3|node_h|3|node_h|2x|node_h}}<BR><math>s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}</math>
|2 [[File:Icosahedron.png|20px]]<br>[[Icosahedron|3.3.3.3.3]]
|8 [[File:octahedron.png|20px]]<br>[[octahedron|3.3.3.3]]
|6+24 [[File:tetrahedron.png|20px]]<br>[[tetrahedron|3.3.3]]
|40
|16+96 {3}
|96
|24
|}
 
==== Octahedral prisms: BC<sub>3</sub> × A<sub>1</sub> ====
 
This [[Hexadecachoric symmetry|prismatic octahedral family symmetry]] is [4,3,2], order 96. There are 6 subgroups of index 2, order 48 that are expressed in alterated polychora below. All are either repeats from other families, or nonuniform, but included for completeness, [[Coxeter notation|aymmetries]] are [(4,3)<sup>+</sup>,2], [1<sup>+</sup>,4,3,2], [4,3,2<sup>+</sup>], [4,3<sup>+</sup>,2], [4,(3,2)<sup>+</sup>], and [4,3,2]<sup>+</sup>.
 
{| class="wikitable"
!rowspan=2| #
!rowspan=2| Johnson Name (Bowers style acronym)
!rowspan=2| [[Schlegel diagram|Picture]]
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!rowspan=2 colspan=4|Cells by type
!colspan=4|Element counts
|-
! Cells
! Faces
! Edges
! Vertices
|- align=center
![10]
|align=center|''Cubic prism''<br>(Same as '''[[tesseract]]''')<br>(Same as ''4-4 duoprism'') (tes)
|[[File:Schlegel wireframe 8-cell.png|75px]]
|[[File:cubic prism verf.png|75px]]
|align=center|{{CDD|node_1|4|node|3|node|2|node_1}}<br>{4,3}×{&nbsp;}<br>t<sub>0,3</sub>{4,3,2}
|2 [[File:Hexahedron.png|20px]]<br>[[cube|4.4.4]]
|6 [[File:Hexahedron.png|20px]]<br>[[cube|4.4.4]]
|
|
|8||24 {4}||32||16
|- align=center
!50
|align=center|[[Cuboctahedral prism]]<br>(Same as ''cantellated tetrahedral prism'') (cope)
|[[File:Cuboctahedral prism.png|75px]]
|[[File:Cuboctahedral prism verf.png|75px]]
|align=center|{{CDD|node|4|node_1|3|node|2|node_1}}<br>r{4,3}×{&nbsp;}<br>t<sub>1,3</sub>{4,3,2}
| 2 [[File:Cuboctahedron.png|20px]]<br>[[Cuboctahedron|3.4.3.4]]
| 8 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
| 6 [[File:Hexahedron.png|20px]]<br>[[Cube|4.4.4]]
|
| 16|| 16 {3}<br>36 {4}|| 60|| 24
|- align=center
!51
|align=center|'''[[Octahedral prism]]'''<br>(Same as ''rectified tetrahedral prism'')<br>(Same as ''triangular antiprismatic prism'') (ope)
|[[File:Octahedral prism.png|75px]]
|[[File:Tetratetrahedral prism verf.png|75px]]
|align=center|{{CDD|node|4|node|3|node_1|2|node_1}}<br>{3,4}×{&nbsp;}<br>t<sub>2,3</sub>{4,3,2}
|2 [[File:Octahedron.png|20px]]<br>[[Octahedron|3.3.3.3]]
|8 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|
|
|10||16 {3}<br>12 {4}||30||12
|- align=center
!52
![[Rhombicuboctahedral prism]] (sircope)
|[[File:Rhombicuboctahedral prism.png|75px]]
|[[File:Rhombicuboctahedron prism verf.png|75px]]
|align=center|{{CDD|node_1|4|node|3|node_1|2|node_1}}<br>rr{4,3}×{&nbsp;}<br>t<sub>0,2,3</sub>{4,3,2}
| 2 [[File:Small rhombicuboctahedron.png|20px]]<br>[[Rhombicuboctahedron|3.4.4.4]]
| 8 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
| 18 [[File:Hexahedron.png|20px]]<br>[[Cube|4.4.4]]
|
| 28|| 16 {3}<br>84 {4}|| 120|| 48
|- align=center
!53
![[Truncated cubic prism]] (ticcup)
|[[File:Truncated cubic prism.png|75px]]
|[[File:Truncated cubic prism verf.png|75px]]
|align=center|{{CDD|node_1|4|node_1|3|node|2|node_1}}<br>t{4,3}×{&nbsp;}<br>t<sub>0,1,3</sub>{4,3,2}
| 2 [[File:Truncated hexahedron.png|20px]]<br>[[Truncated cube|3.8.8]]
| 8 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
| 6 [[File:Octagonal prism.png|20px]]<br>[[Octagonal prism|4.4.8]]
|
| 16|| 16 {3}<br>36 {4}<br>12 {8}|| 96|| 48
|- align=center
!54
|align=center|[[Truncated octahedral prism]]<br>(Same as ''cantitruncated tetrahedral prism'') (tope)
|[[File:Truncated octahedral prism.png|75px]]
|[[File:Truncated octahedral prism verf.png|75px]]
|align=center|{{CDD|node|4|node_1|3|node_1|2|node_1}}<br>t{3,4}×{&nbsp;}<br>t<sub>1,2,3</sub>{4,3,2}
| 2 [[File:Truncated octahedron.png|20px]]<br>[[Truncated octahedron|4.6.6]]
| 6 [[File:Hexahedron.png|20px]]<br>[[Cube|4.4.4]]
| 8 [[File:Hexagonal prism.png|20px]]<br>[[Hexagonal prism|4.4.6]]
|
| 16|| 48 {4}<br>16 {6}|| 96|| 48
|- align=center
!55
![[Truncated cuboctahedral prism]] (gircope)
|[[File:Truncated cuboctahedral prism.png|75px]]
|[[File:Truncated cuboctahedral prism verf.png|75px]]
|align=center|{{CDD|node_1|4|node_1|3|node_1|2|node_1}}<br>tr{4,3}×{&nbsp;}<br>t<sub>0,1,2,3</sub>{4,3,2}
| 2 [[File:Great rhombicuboctahedron.png|20px]]<br>[[Truncated cuboctahedron|4.6.8]]
| 12 [[File:Hexahedron.png|20px]]<br>[[Cube|4.4.4]]
| 8 [[File:Hexagonal prism.png|20px]]<br>[[Hexagonal prism|4.4.6]]
| 6 [[File:Octagonal prism.png|20px]]<br>[[Octagonal prism|4.4.8]]
| 28|| 96 {4}<br>16 {6}<br>12 {8}|| 192|| 96
|- align=center
!56
![[Snub cubic prism]] (sniccup)
|[[File:Snub cubic prism.png|75px]]
|[[File:Snub cubic prism verf.png|75px]]
|align=center|{{CDD|node_h|4|node_h|3|node_h|2|node_1}}<br>sr{4,3}×{&nbsp;}
| 2 [[File:Snub hexahedron.png|20px]]<br>[[Snub cube|3.3.3.3.4]]
| 32 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
| 6 [[File:Hexahedron.png|20px]]<br>[[Cube|4.4.4]]
|
| 40|| 64 {3}<br>72 {4}|| 144|| 48
 
|- align=center
![48]
![[Tetrahedral prism]] (tepe)
|[[File:Tetrahedral prism.png|75px]]
|[[File:Tetrahedral prism verf.png|75px]]
|align=center|{{CDD|node_h1|4|node|3|node|2|node_1}}<br>h{4,3}×{&nbsp;}
|2 [[File:Tetrahedron.png|20px]]<br>[[Tetrahedron|3.3.3]]
|4 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|
|
|6||8 {3}<br>6 {4}||16||8
|- align=center
![59]
!align=center|[[Icosahedral prism]] (ipe)
|[[File:Icosahedral prism.png|75px]]
|[[File:Snub tetrahedral prism verf.png|75px]]
|align=center|{{CDD|node|4|node_h|3|node_h|2|node_1}}<br>s{3,4}×{&nbsp;}
|2 [[File:Icosahedron.png|20px]]<br>[[Icosahedron|3.3.3.3.3]]
|20 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|
|
| 22|| 40 {3}<br>30 {4}|| 72|| 24
 
|- align=center
![12]
!align=center|[[16-cell]] (hex)
|[[File:Schlegel wireframe 16-cell.png|75px]]
|[[File:16-cell verf.png|75px]]
|align=center|{{CDD|node_h|2x|node_h|4|node|3|node}}<br>s{2,4,3}
|2+6+8 [[File:tetrahedron.png|20px]]<br>[[tetrahedron|3.3.3.3]]
|
|
|
|16||32 {3}||24||8
 
|- align=center
![[#Nonuniform alternations|Nonuniform]]
![[Full snub tetrahedral antiprism]]
|
|[[File:Snub 332 verf.png|75px]]
|align=center|{{CDD|node|4|node_h|3|node_h|2x|node_h}}<br>sr{2,3,4}
|2 [[File:Icosahedron.png|20px]]<br>[[Icosahedron|3.3.3.3.3]]
|8 [[File:octahedron.png|20px]]<br>[[octahedron|3.3.3.3]]
|6+24 [[File:tetrahedron.png|20px]]<br>[[tetrahedron|3.3.3]]
|
|40||16+96 {3}||96||24
|- align=center
![[#Nonuniform alternations|Nonuniform]]
![[Full snub cubic antiprism]]
|
|[[File:Snub 432 verf.png|75px]]
|align=center|{{CDD|node_h|4|node_h|3|node_h|2x|node_h}}<BR><math>s\left\{\begin{array}{l}4\\3\\2\end{array}\right\}</math>
| 2 [[File:snub hexahedron.png|20px]]<br>[[snub cube|3.3.3.3.4]]
| 12+48 [[File:tetrahedron.png|20px]]<br>[[tetrahedron|3.3.3]]
| 8 [[File:octahedron.png|20px]]<br>[[octahedron|3.3.3.3]]
| 6 [[File:square antiprism.png|20px]]<br>[[square antiprism|3.3.3.4]]
|76||16+192 {3}<br>12 {4}||192||48
|}
 
==== Icosahedral prisms: H<sub>3</sub> × A<sub>1</sub> ====
 
This [[Hexacosichoric symmetry|prismatic icosahedral symmetry]] is [5,3,2], order 240. There are two index 2 subgroups, [(5,3)<sup>+</sup>,2] and [5,3,2]<sup>+</sup>, but the second doesn't generate a uniform polychoron.
 
{| class="wikitable"
!rowspan=2|#
!rowspan=2| Johnson Name (Bowers style acronym)
!rowspan=2| [[Schlegel diagram|Picture]]
!rowspan=2| [[Vertex figure|Vertex<br>figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!rowspan=2 colspan=4|Cells by type
!colspan=4|Element counts
|-
! Cells
! Faces
! Edges
! Vertices
|- align=center
!57
![[Dodecahedral prism]] (dope)
|[[File:Dodecahedral prism.png|75px]]
|[[File:Dodecahedral prism verf.png|75px]]
|align=center|{{CDD|node_1|5|node|3|node|2|node_1}}<br>{5,3}×{&nbsp;}<br>t<sub>0,3</sub>{5,3,2}
| 2 [[File:Dodecahedron.png|20px]]<br>[[Dodecahedron|5.5.5]]
|12 [[File:Pentagonal prism.png|20px]]<br>[[Pentagonal prism|4.4.5]]
|
|
| 14
| 30 {4}<br>24 {5}
| 80
| 40
|- align=center
!58
![[Icosidodecahedral prism]] (iddip)
|[[File:Icosidodecahedral prism.png|75px]]
|[[File:Icosidodecahedral prism verf.png|75px]]
|align=center|{{CDD|node|5|node_1|3|node|2|node_1}}<br>r{5,3}×{&nbsp;}<br>t<sub>1,3</sub>{5,3,2}
|2 [[File:Icosidodecahedron.png|20px]]<br>[[Icosidodecahedron|3.5.3.5]]
|20 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|12 [[File:Pentagonal prism.png|20px]]<br>[[Pentagonal prism|4.4.5]]
|
| 34
| 40 {3}<br>60 {4}<br>24 {5}
| 150
| 60
|- align=center
!59
|align=center|[[Icosahedral prism]]<br>(same as ''snub tetrahedral prism'') (ipe)
|[[File:Icosahedral prism.png|75px]]
|[[File:Snub tetrahedral prism verf.png|75px]]
|align=center|{{CDD|node|5|node|3|node_1|2|node_1}}<br>{3,5}×{&nbsp;}<br>t<sub>2,3</sub>{5,3,2}
|2 [[File:Icosahedron.png|20px]]<br>[[Icosahedron|3.3.3.3.3]]
|20 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|
|
| 22
| 40 {3}<br>30 {4}
| 72
| 24
|- align=center
!60
![[Truncated dodecahedral prism]] (tiddip)
|[[File:Truncated dodecahedral prism.png|75px]]
|[[File:Truncated dodecahedral prism verf.png|75px]]
|align=center|{{CDD|node_1|5|node_1|3|node|2|node_1}}<br>t{5,3}×{&nbsp;}<br>t<sub>0,1,3</sub>{5,3,2}
|2 [[File:Truncated dodecahedron.png|20px]]<br>[[Truncated dodecahedron|3.10.10]]
|20 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|12 [[File:Decagonal prism.png|20px]]<br>[[Decagonal prism|4.4.5]]
|
| 34
| 40 {3}<br>90 {4}<br>24 {10}
| 240
| 120
|- align=center
!61
![[Rhombicosidodecahedral prism]] (sriddip)
|[[File:Rhombicosidodecahedral prism.png|75px]]
|[[File:Rhombicosidodecahedron prism verf.png|75px]]
|align=center|{{CDD|node_1|5|node|3|node_1|2|node_1}}<br>rr{5,3}×{&nbsp;}<br>t<sub>0,2,3</sub>{5,3,2}
|2 [[File:Small rhombicosidodecahedron.png|20px]]<br>[[rhombicosidodecahedron|3.4.5.4]]
|20 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|30 [[File:Hexahedron.png|20px]]<br>[[Cube|4.4.4]]
|12 [[File:Pentagonal prism.png|20px]]<br>[[Pentagonal prism|4.4.5]]
| 64
| 40 {3}<br>180 {4}<br>24 {5}
| 300
| 120
|- align=center
!62
![[Truncated icosahedral prism]] (tipe)
|[[File:Truncated icosahedral prism.png|75px]]
|[[File:Truncated icosahedral prism verf.png|75px]]
|align=center|{{CDD|node|5|node_1|3|node_1|2|node_1}}<br>t{3,5}×{&nbsp;}<br>t<sub>1,2,3</sub>{5,3,2}
|2 [[File:Truncated icosahedron.png|20px]]<br>[[Truncated icosahedron|5.6.6]]
|12 [[File:Pentagonal prism.png|20px]]<br>[[Pentagonal prism|4.4.5]]
|20 [[File:Hexagonal prism.png|20px]]<br>[[Hexagonal prism|4.4.6]]
|
| 34
| 90 {4}<br>24 {5}<br>40 {6}
| 240
| 120
|- align=center
!63
![[Truncated icosidodecahedral prism]] (griddip)
|[[File:Truncated icosidodecahedral prism.png|75px]]
|[[File:Truncated icosidodecahedral prism verf.png|75px]]
|align=center|{{CDD|node_1|5|node_1|3|node_1|2|node_1}}<br>tr{5,3}×{&nbsp;}<br>t<sub>0,1,2,3</sub>{5,3,2}
|2 [[File:Great rhombicosidodecahedron.png|20px]]<br>[[Truncated icosidodecahedron|4.6.10]]
|30 [[File:Hexahedron.png|20px]]<br>[[Cube|4.4.4]]
|20 [[File:Hexagonal prism.png|20px]]<br>[[Hexagonal prism|4.4.6]]
|12 [[File:Decagonal prism.png|20px]]<br>[[Decagonal prism|4.4.10]]
| 64
| 240 {4}<br>40 {6}<br>24 {10}
| 480
| 240
|- align=center
!64
![[Snub dodecahedral prism]] (sniddip)
|[[File:Snub dodecahedral prism.png|75px]]
|[[File:Snub dodecahedral prism verf.png|75px]]
|align=center|{{CDD|node_h|5|node_h|3|node_h|2|node_1}}<br>sr{5,3}×{&nbsp;}
|2 [[File:Snub dodecahedron ccw.png|20px]]<br>[[Snub dodecahedron|3.3.3.3.5]]
|80 [[File:Triangular prism.png|20px]]<br>[[Triangular prism|3.4.4]]
|12 [[File:Pentagonal prism.png|20px]]<br>[[Pentagonal prism|4.4.5]]
|
| 94
| 240 {4}<br>40 {6}<br>24 {5}
| 360
| 120
|- align=center
![[#Nonuniform alternations|Nonuniform]]
![[Full snub dodecahedral antiprism]]
|
|[[File:Snub 532 verf.png|75px]]
|align=center|{{CDD|node_h|5|node_h|3|node_h|2x|node_h}}<BR><math>s\left\{\begin{array}{l}5\\3\\2\end{array}\right\}</math>
|2 [[File:snub dodecahedron cw.png|20px]]<br>[[snub dodecahedron|3.3.3.3.5]]
|30+120 [[File:tetrahedron.png|20px]]<br>[[tetrahedron|3.3.3]]
|20 [[File:octahedron.png|20px]]<br>[[octahedron|3.3.3.3]]
|12 [[File:pentagonal antiprism.png|20px]]<br>[[pentagonal antiprism|3.3.3.5]]
|184||20+240 {3}<br>24 {5}||220||120
|}
 
==== Duoprisms: [p] × [q] ====
[[File:3-3 duoprism.png|thumb|The simplest of the duoprisms, the 3,3-duoprism, in [[Schlegel diagram]], one of 6 [[triangular prism]] cells shown.]]
The second is the infinite family of [[duoprism|uniform duoprisms]], products of two [[regular polygon]]s. A duoprism's [[Coxeter-Dynkin diagram]] is {{CDD|node_1|p|node|2|node_1|q|node}}. Its [[vertex figure]] is an [[disphenoid tetrahedron]], [[File:Pq-duoprism verf.png|75px]].
 
This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism.  The symmetry number of a duoprism whose factors are a ''p''-gon and a ''q''-gon (a "''p,q''-duoprism") is 4''pq'' if ''p''≠''q''; if the factors are both ''p''-gons, the symmetry number is 8''p''<sup>2</sup>. The tesseract can also be considered a 4,4-duoprism.
 
The elements of a ''p,q''-duoprism (''p'' ≥ 3, ''q'' ≥ 3) are:
* Cells: '''p''' ''q''-gonal prisms, '''q''' ''p''-gonal prisms
* Faces: '''pq''' squares, '''p''' ''q''-gons, '''q''' ''p''-gons
* Edges: '''2pq'''
* Vertices: '''pq'''
 
There is no uniform analogue in four dimensions to the infinite family of three-dimensional [[antiprism]]s.
 
Infinite set of '''p-q duoprism''' - {{CDD|node_1|p|node|2|node_1|q|node}} - '''p''' ''q''-gonal prisms, '''q''' ''p''-gonal prisms:
{| class=wikitable
|-
!Name
!Coxeter graph
!Cells
!Images
|- align=center
! 3-3 duoprism (triddip)
||{{CDD|node_1|3|node|2|node_1|3|node}}||3+3 triangular prisms||[[File:3-3 duoprism.png|75px]]
|- align=center
! 3-4 duoprism (tisdip)
||{{CDD|node_1|3|node|2|node_1|4|node}}||3 cubes<br>4 triangular prisms||[[File:3-4 duoprism.png|75px]] [[File:4-3 duoprism.png|75px]]
|- align=center
! 4-4 duoprism (tes)<br>(same as tesseract)
||{{CDD|node_1|4|node|2|node_1|4|node}}||4+4 cubes||[[File:4-4 duoprism.png|75px]]
|- align=center
! 3-5 duoprism (trapedip)
||{{CDD|node_1|3|node|2|node_1|5|node}}||3 pentagonal prisms<br>5 triangular prisms||[[File:5-3 duoprism.png|75px]] [[File:3-5 duoprism.png|75px]]
|- align=center
! 4-5 duoprism (squipdip)
||{{CDD|node_1|4|node|2|node_1|5|node}}||4 pentagonal prisms<br>5 cubes|| [[File:3-4 duoprism.png|75px]] [[File:4-3 duoprism.png|75px]]
|- align=center
! 5-5 duoprism (pedip)
||{{CDD|node_1|5|node|2|node_1|5|node}}||5+5 pentagonal prisms||[[File:5-5 duoprism.png|75px]]
|- align=center
! 3-6 duoprism (thiddip)
||{{CDD|node_1|3|node|2|node_1|6|node}}||3 hexagonal prisms<br>6 triangular prisms||[[File:3-6 duoprism.png|75px]] [[File:6-3 duoprism.png|75px]]
|- align=center
! 4-6 duoprism (shiddip)
||{{CDD|node_1|4|node|2|node_1|6|node}}||4 hexagonal prisms<br>6 cubes||[[File:4-6 duoprism.png|75px]] [[File:6-4 duoprism.png|75px]]
|- align=center
! 5-6 duoprism (phiddip)
||{{CDD|node_1|5|node|2|node_1|6|node}}||5 hexagonal prisms<br>6 pentagonal prisms|| [[File:5-6 duoprism.png|75px]] [[File:6-5 duoprism.png|75px]]
|- align=center
! 6-6 duoprism (hiddip)
||{{CDD|node_1|6|node|2|node_1|6|node}}||6+6 hexagonal prisms||[[File:6-6 duoprism.png|75px]]
|}
 
{| class="wikitable" width=600px
|- align=center
|[[File:3-3 duoprism.png|75px]]<br>3-3
|[[File:3-4 duoprism.png|75px]]<br>3-4
|[[File:3-5 duoprism.png|75px]]<br>3-5
|[[File:3-6 duoprism.png|75px]]<br>3-6
|[[File:3-7 duoprism.png|75px]]<br>3-7
|[[File:3-8 duoprism.png|75px]]<br>3-8
|- align=center
|[[File:4-3 duoprism.png|75px]]<br>4-3
|[[File:4-4 duoprism.png|75px]]<br>4-4
|[[File:4-5 duoprism.png|75px]]<br>4-5
|[[File:4-6 duoprism.png|75px]]<br>4-6
|[[File:4-7 duoprism.png|75px]]<br>4-7
|[[File:4-8 duoprism.png|75px]]<br>4-8
|- align=center
|[[File:5-3 duoprism.png|75px]]<br>5-3
|[[File:5-4 duoprism.png|75px]]<br>5-4
|[[File:5-5 duoprism.png|75px]]<br>5-5
|[[File:5-6 duoprism.png|75px]]<br>5-6
|[[File:5-7 duoprism.png|75px]]<br>5-7
|[[File:5-8 duoprism.png|75px]]<br>5-8
|- align=center
|[[File:6-3 duoprism.png|75px]]<br>6-3
|[[File:6-4 duoprism.png|75px]]<br>6-4
|[[File:6-5 duoprism.png|75px]]<br>6-5
|[[File:6-6 duoprism.png|75px]]<br>6-6
|[[File:6-7 duoprism.png|75px]]<br>6-7
|[[File:6-8 duoprism.png|75px]]<br>6-8
|- align=center
|[[File:7-3 duoprism.png|75px]]<br>7-3
|[[File:7-4 duoprism.png|75px]]<br>7-4
|[[File:7-5 duoprism.png|75px]]<br>7-5
|[[File:7-6 duoprism.png|75px]]<br>7-6
|[[File:7-7 duoprism.png|75px]]<br>7-7
|[[File:7-8 duoprism.png|75px]]<br>7-8
|- align=center
|[[File:8-3 duoprism.png|75px]]<br>8-3
|[[File:8-4 duoprism.png|75px]]<br>8-4
|[[File:8-5 duoprism.png|75px]]<br>8-5
|[[File:8-6 duoprism.png|75px]]<br>8-6
|[[File:8-7 duoprism.png|75px]]<br>8-7
|[[File:8-8 duoprism.png|75px]]<br>8-8
|}
 
==== Polygonal prismatic prisms: [p] × [ ] × [ ] ====
 
The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - {{CDD|node_1|p|node|2|node_1|2|node_1}} - ''p'' cubes and 4 ''p''-gonal prisms - (All are the same as '''4-p duoprism''')
{{Convex prismatic prisms}}
 
The infinite sets of '''[[uniform antiprismatic prism]]s''' are constructed from two parallel uniform [[antiprism]]s): (p≥2) - {{CDD|node_h|p|node_h|2x|node_h|2|node_1}} - 2 ''p''-gonal antiprisms, connected by 2 ''p''-gonal prisms and ''2p'' triangular prisms.
{{Convex_antiprismatic_prisms}}
 
A ''p-gonal antiprismatic prism'' has ''4p'' triangle, ''4p'' square and ''4'' p-gon faces. It has ''10p'' edges, and ''4p'' vertices.
 
=== Nonuniform alternations ===
{| class=wikitable align=right
|+ [[Vertex figure]]s
|- align=center
|[[File:Snub 5-cell verf.png|100px]]<br>ht<sub>0,1,2,3</sub>{3,3,3}<br>{{CDD|node_h|3|node_h|3|node_h|3|node_h}}
|[[File:Snub tesseract verf.png|100px]]<br>ht<sub>0,1,2,3</sub>{4,3,3}<br>{{CDD|node_h|4|node_h|3|node_h|3|node_h}}
|[[File:Snub 120-cell verf.png|100px]]<br>ht<sub>0,1,2,3</sub>{5,3,3}<br>{{CDD|node_h|5|node_h|3|node_h|3|node_h}}
|- align=center
|[[File:16-cell verf.png|100px]]<br><math>s\left\{\begin{array}{l}2\\2\\2\end{array}\right\}</math><br>{{CDD|node_h|2x|node_h|2x|node_h|2x|node_h}}
|[[File:Snub p2q verf.png|100px]]<br>ht<sub>0,1,2,3</sub>{p,2,q}<br>{{CDD|node_h|p|node_h|2x|node_h|q|node_h}}
|[[File:Full snub 24-cell verf.png|100px]]<br>ht<sub>0,1,2,3</sub>{3,4,3}<br>{{CDD|node_h|3|node_h|4|node_h|3|node_h}}
|- align=center
|[[File:Snub 332 verf.png|100px]]<br><math>s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}</math><br>{{CDD|node_h|3|node_h|3|node_h|2x|node_h}}
|[[File:Snub 432 verf.png|100px]]<br><math>s\left\{\begin{array}{l}4\\3\\2\end{array}\right\}</math><br>{{CDD|node_h|4|node_h|3|node_h|2x|node_h}}
|[[File:Snub 532 verf.png|100px]]<br><math>s\left\{\begin{array}{l}5\\3\\2\end{array}\right\}</math><br>{{CDD|node_h|5|node_h|3|node_h|2x|node_h}}
|- align=center
|[[File:Runcic snub 24-cell verf.png|100px]]<BR>s<sub>3</sub>{3,4,3}<br>{{CDD|node_h|3|node_h|4|node|3|node_1}}
|[[File:Runcic snub rectified 16-cell verf.png|100px]]<BR>sr<sub>3</sub>{3,3,4}<BR>{{CDD|node_h|3|node_h|3|node_h|4|node_1}}
|}
There are a number of [[Alternation (geometry)|alternations]] of the uniform polychora that can not be made uniform as they have too many parameters to satisfy.
 
Four [[Snub (geometry)|snubs]] are not uniform unlike their 3-dimensional analogies. Only the [[snub 24-cell]] is uniform, although it is more accurately called a ''semisnub 24-cell'' or ''snub demitesseract'' for being a full snub of the bifurcating family D<sub>4</sub> with the demitesseract as the alternated tesseract.
* [[Full snub 5-cell]] (snip), ht<sub>0,1,2,3</sub>{3,3,3}, {{CDD|node_h|3|node_h|3|node_h|3|node_h}}, 10 [[icosahedron]]s, 20 [[octahedron]]s, and 60 [[tetrahedron]]s, with [[Pentachoric symmetry|chiral extended pentachoric symmetry]], [<span/>[3,3,3]<span/>]<sup>+</sup>, order 120.
* [[Full snub tesseract]] (snet), ht<sub>0,1,2,3</sub>{4,3,3}, {{CDD|node_h|4|node_h|3|node_h|3|node_h}}, with 16 [[icosahedron]]s, 32 [[octahedra]], 24 [[square antiprism]]s, 8 [[snub cube]]s and 192 [[tetrahedron]]s, with [[Hexadecachoric symmetry|chiral hexadecachoric symmetry]], [4,3,3]<sup>+</sup>, order 192.
* The ''runcic snub rectified 16-cell'', {{CDD|node_1|4|node_h|3|node_h|3|node_h}}, has 176 cells, 656 faces, 672 edges, and 192 vertices. Its symmetry is [[4,(3,3,3)<sup>+</sup>]], order 192.
* [[Full snub 24-cell]] (snico), ht<sub>0,1,2,3</sub>{3,4,3}, {{CDD|node_h|3|node_h|4|node_h|3|node_h}}, from 48 [[snub cube]]s, 192 [[octahedron]]s, and 576 [[tetrahedron]]s, with [[Icositetrachoric symmetry|chiral extended icositetrachoric symmetry]], [<span/>[3,4,3]<span/>]<sup>+</sup>, order 1152.
* [[Runcic snub 24-cell]] (prissi), s<sub>3</sub>{3,4,3}, {{CDD|node_h|3|node_h|4|node|3|node_1}}, from 24 [[icosahedron]]s, 24 [[truncated tetrahedron]]s, 96 [[triangular prism]]s, and 96 [[triangular cupola]] in the gaps, for a total of 240 cells, 960 faces, 1008 edges, and 288 vertices. It is [[vertex-transitive]], and equilateral, but not uniform, due to the cupola. Like the [[snub 24-cell]], it has symmetry [3<sup>+</sup>,4,3], order 576.<ref>{{KlitzingPolytopes|polychora.htm|4D|s3s4o3x}}</ref>
* [[Full snub 120-cell]] (snahi), ht<sub>0,1,2,3</sub>{5,3,3}, {{CDD|node_h|5|node_h|3|node_h|3|node_h}}, 1200 [[octahedron]]s, 600 [[icosahedron]]s, 720 [[pentagonal antiprism]]s, 120 [[snub dodecahedron]]s, and 7200 [[tetrahedron]]s, with [[Hexacosichoric symmetry|chiral hexacosichoric symmetry]], [5,3,3]<sup>+</sup>, order 7200.
The polyhedral prisms {{CDD|node_1|p|node_1|q|node_1|2|node_1}}, can be alternated into {{CDD|node_h|p|node_h|q|node_h|2x|node_h}}, but do not generate uniform solutions.
# [[Full snub tetrahedral antiprism]], <math>s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}</math> {{CDD|node_h|3|node_h|3|node_h|2x|node_h}}, 2 [[icosahedron]]s connected by 6 [[tetrahedron]]s, and 8 [[octahedron]]s, with 24 tetrahedra in the alternated gaps.
# [[Full snub cubic antiprism]], <math>s\left\{\begin{array}{l}4\\3\\2\end{array}\right\}</math> {{CDD|node_h|4|node_h|3|node_h|2x|node_h}}, 2 [[snub cube]]s connected by 12 [[tetrahedron]]s, 6 [[square antiprism]]s, and 8 [[octahedron]]s, with 48 tetrahedra in the alternated gaps.
# [[Full snub dodecahedral antiprism]], <math>s\left\{\begin{array}{l}5\\3\\2\end{array}\right\}</math> {{CDD|node_h|5|node_h|3|node_h|2x|node_h}}, 2 [[snub dodecahedron]]s connected by 30 [[tetrahedron]]s, 12 [[pentagonal antiprism]]s, and 20 [[octahedron]]s, with 120 tetrahedra in the alternated gaps.
 
The [[duoprism]]s {{CDD|node_1|p|node_1|2|node_1|q|node_1}}, t<sub>0,1,2,3</sub>{p,2,q}, can be alternated into {{CDD|node_h|p|node_h|2x|node_h|q|node_h}}, ht<sub>0,1,2,3</sub>{p,2,q}, called [[duoantiprism]]s, which cannot be made uniform in general. The only convex uniform solution is the trivial case of p=q=2, which is a lower symmetry construction of the [[tesseract]] {{CDD|node_1|2c|node_1|2c|node_1|2c|node_1}}, t<sub>0,1,2,3</sub>{2,2,2} = t{2<sup>1,1,1</sub>}, with its alternation as the [[16-cell]], {{CDD|node_h|2x|node_h|2x|node_h|2x|node_h}}, s{2<sup>1,1,1</sub>} = <math>s\left\{\begin{array}{l}2\\2\\2\end{array}\right\}</math>.
 
=== Geometric derivations for 46 nonprismatic Wythoffian uniform polychora ===<!-- This section is linked from [[Convex uniform honeycomb]] -->
The 46 Wythoffian polychora include the six [[convex regular 4-polytope|convex regular polychora]]. The other forty can be derived from the regular polychora by geometric operations which preserve most or all of their [[symmetry|symmetries]], and therefore may be classified by the [[symmetry group]]s that they have in common.
 
{| class=wikitable align=center width=480
|[[File:Polychoron truncation chart.png|240px]]<BR>Summary chart of truncation operations
|[[File:Uniform honeycomb truncations.png|240px]]<BR>Example locations of kaleidoscopic generator point on fundamental domain.
|}
 
The geometric operations that derive the 40 uniform polychora from the regular polychora are ''truncating'' operations. A polychoron may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below.
 
The '''[[Coxeter-Dynkin diagram]]''' shows the four mirrors of the Wythoffian kaleidoscope as nodes, and the edges between the nodes are labeled by an integer showing the angle between the mirrors ([[pi|&pi;]]/''n'' [[radian]]s or 180/''n'' degrees). Circled nodes show which mirrors are active for each form; a mirror is active with respect to a vertex that does not lie on it.
 
{| class="wikitable"
!Operation
![[Schläfli symbol]]
![[Coxeter notation|Symmetry]]
![[Coxeter diagram]]
!Description
|- align=center
!Parent
|t<sub>0</sub>{p,q,r}
|rowspan=15|[p,q,r]
|{{CDD|node_1|p|node|q|node|r|node}}
|Original regular form {p,q,r}
|- align=center
![[Rectification (geometry)|Rectification]]
|t<sub>1</sub>{p,q,r}
|{{CDD|node|p|node_1|q|node|r|node}}
|Truncation operation applied until the original edges are degenerated into points.
|- align=center
!Birectification<BR>(Rectified dual)
|t<sub>2</sub>{p,q,r}
|{{CDD|node|p|node|q|node_1|r|node}}
|Face are fully truncated to points. Same as rectified dual.
|- align=center
!Trirectification<br>([[dual polytope|dual]])
|t<sub>3</sub>{p,q,r}
|{{CDD|node|p|node|q|node|r|node_1}}
|Cells are truncated to points. Regular dual {r,q,p}
|- align=center
![[Truncation (geometry)|Truncation]]
|t<sub>0,1</sub>{p,q,r}
|{{CDD|node_1|p|node_1|q|node|r|node}}
|Each vertex is cut off so that the middle of each original edge remains.  Where the vertex was, there appears a new cell, the parent's [[vertex figure]].  Each original cell is likewise truncated.
|- align=center
![[Bitruncation (geometry)|Bitruncation]]
|t<sub>1,2</sub>{p,q,r}
|{{CDD|node|p|node_1|q|node_1|r|node}}
| A truncation between a rectified form and the dual rectified form.
|- align=center
!Tritruncation
|t<sub>2,3</sub>{p,q,r}
|{{CDD|node|p|node|q|node_1|r|node_1}}
|Truncated dual {r,q,p}.
|- align=center
![[Cantellation (geometry)|Cantellation]]
|t<sub>0,2</sub>{p,q,r}
|{{CDD|node_1|p|node|q|node_1|r|node}}
| A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.
|- align=center
!Bicantellation
|t<sub>1,3</sub>{p,q,r}
|{{CDD|node|p|node_1|q|node|r|node_1}}
|Cantellated dual {r,q,p}.
|- align=center
![[Runcination (geometry)|Runcination]]<br>(or [[Expansion (geometry)|expansion]])
|t<sub>0,3</sub>{p,q,r}
|{{CDD|node_1|p|node|q|node|r|node_1}}
|A truncation applied to the cells, faces and edges; defines a progression between a regular form and the dual.
|- align=center
!Cantitruncation
|t<sub>0,1,2</sub>{p,q,r}
|{{CDD|node_1|p|node_1|q|node_1|r|node}}
|Both the ''cantellation'' and ''truncation'' operations applied together.
|- align=center
!Bicantitruncation
|t<sub>1,2,3</sub>{p,q,r}
|{{CDD|node|p|node_1|q|node_1|r|node_1}}
|Cantitruncated dual {r,q,p}.
|- align=center
|'''Runcitruncation'''
|t<sub>0,1,3</sub>{p,q,r}
|{{CDD|node_1|p|node_1|q|node|r|node_1}}
|Both the ''runcination'' and ''truncation'' operations applied together.
|- align=center
!Runcicantellation
|t<sub>0,1,3</sub>{p,q,r}
|{{CDD|node_1|p|node|q|node_1|r|node_1}}
|Runcitruncated dual {r,q,p}.
|- align=center
![[Omnitruncation (geometry)|Omnitruncation]]<br>(runcicantitruncation)
|t<sub>0,1,2,3</sub>{p,q,r}
|{{CDD|node_1|p|node_1|q|node_1|r|node_1}}
| Application of all three operators.
|- align=center
!Half
|h{2p,3,q}
|rowspan=4|[1<sup>+</sup>,2p,3,q]
|{{CDD|node_h1|2x|p|node|3|node|q|node}}
|[[Alternation (geometry)|Alternation]] of {{CDD|node_1|2x|p|node|3|node|q|node}}, same as {{CDD|labelp|branch_10r|split2|node|q|node}}
|- align=center
!Cantic
|h<sub>2</sub>{2p,3,q}
|{{CDD|node_h1|2x|p|node|3|node_1|q|node}}
|Same as {{CDD|labelp|branch_10r|split2|node_1|q|node}}
|- align=center
!Runcic
|h<sub>3</sub>{2p,3,q}
|{{CDD|node_h1|2x|p|node|3|node|q|node_1}}
|Same as {{CDD|labelp|branch_10r|split2|node|q|node_1}}
|- align=center
!Runcicantic
|h<sub>2,3</sub>{2p,3,q}
|{{CDD|node_h1|2x|p|node|3|node_1|q|node_1}}
|Same as {{CDD|labelp|branch_10r|split2|node_1|q|node_1}}
|- align=center
!Quarter
|q{2p,3,2q}
|[1<sup>+</sup>,2p,3,2r,1<sup>+</sup>]
|{{CDD|node_h1|2x|p|node|3|node|2x|q|node_h1}}
|Same as {{CDD|labelp|branch_10r|splitcross|branch_01l|labelq}}
|- align=center
!Snub
|s{p,2q,r}
|rowspan=4|[p<sup>+</sup>,2q,r]
|{{CDD|node_h|p|node_h|2x|q|node|r|node}}
|Alternated truncation
|- align=center
!Cantic snub
|s<sub>2</sub>{p,2q,r}
|{{CDD|node_h|p|node_h|2x|q|node_1|r|node}}
|Cantellated alternated truncation
|- align=center
!Runcic snub
|s<sub>3</sub>{p,2q,r}
|{{CDD|node_h|p|node_h|2x|q|node|r|node_1}}
|Runcinated alternated truncation
|- align=center
!Runcicantic snub
|s<sub>2,3</sub>{p,2q,r}
|{{CDD|node_h|p|node_h|2x|q|node_1|r|node_1}}
|Runcicantellated alternated truncation
|- align=center
!Snub rectified
|sr{p,q,2r}
|[(p,q)<sup>+</sup>,2r]
|{{CDD|node_h|p|node_h|q|node_h|2x|r|node}}
|Alternated truncated rectification
|- align=center
!
|ht<sub>0,3</sub>{2p,q,2r}
|[(2p,q,2r,2<sup>+</sup>)]
|{{CDD|node_h|2x|p|node|q|node|2x|r|node_h}}
|Alternated runcination
|- align=center
!
|ht<sub>1,2</sub>{2p,q,2r}
|[2p,q<sup>+</sup>,2r]
|{{CDD|node|2x|p|node_h|q|node_h|2x|r|node}}
|Alternated bitruncation
|- align=center
!Full snub
|ht<sub>0,1,2,3</sub>{p,q,r}
|[p,q,r]<sup>+</sup>
|{{CDD|node_h|p|node_h|q|node_h|r|node_h}}
|Alternated omnitruncation
|}
 
See also [[convex uniform honeycomb]]s, some of which illustrate these operations as applied to the regular [[cubic honeycomb]].
 
If two polytopes are [[Dual polytope|dual]]s of each other (such as the tesseract and 16-cell, or the 120-cell and 600-cell), then '''bitruncating''', '''runcinating''' or '''omnitruncating''' either produces the same figure as the same operation to the other.  Thus where only the participle appears in the table it should be understood to apply to either parent.
 
==== Summary of constructions by extended symmetry ====
The 46 uniform polychora constructed from the A<sub>4</sub>, BC<sub>4</sub>, F<sub>4</sub>, H<sub>4</sub> symmetry are given in this table by their full extended symmetry and Coxeter diagrams. Alternations are grouped by their chiral symmetry. All alternations are given, although the [[snub 24-cell]], with its 3 family of constructions is the only one that is uniform. Counts in parenthesis are either repeats or nonuniform. The Coxeter diagrams are given with subscript indices 1 through 46. The 3-3 and 4-4 duoprismatic family is included, the second for its relation to the BC<sub>4</sub> family.
{| class=wikitable
![[Coxeter group]]
![[Goursat tetrahedron#3-sphere (finite) solutions|Extended<br>symmetry]]
!colspan=2|Polychora
!Chiral<br>extended<br>symmetry
!colspan=2|Alternation honeycombs
 
|- align=center
|rowspan=2|[3,3,3]<br>{{CDD|node|3|node|3|node|3|node}}||[3,3,3]<br>{{CDD|node_c1|3|node_c2|3|node_c3|3|node_c4}}<br>(order 120)||6
| {{CDD|node_1|3|node|3|node|3|node}}<sub>1</sub> | {{CDD|node|3|node_1|3|node|3|node}}<sub>2</sub> | {{CDD|node_1|3|node_1|3|node|3|node}}<sub>3</sub><br>{{CDD|node_1|3|node|3|node_1|3|node}}<sub>4</sub> | {{CDD|node_1|3|node_1|3|node_1|3|node}}<sub>7</sub> | {{CDD|node_1|3|node_1|3|node|3|node_1}}<sub>8</sub>
|colspan=3|
|- align=center
|[2<sup>+</sup>[3,3,3]]<br>{{CDD|node_c1|3|node_c2|3|node_c2|3|node_c1}}<br>(order 240)||3
|{{CDD|node_1|3|node|3|node|3|node_1}}<sub>5</sub>| {{CDD|node|3|node_1|3|node_1|3|node}}<sub>6</sub> | {{CDD|node_1|3|node_1|3|node_1|3|node_1}}<sub>9</sub>
|[2<sup>+</sup>[3,3,3]]<sup>+</sup><br>(order 120)||(1)
|{{CDD|node_h|3|node_h|3|node_h|3|node_h}}<sub>-</sub>
 
|- align=center
|rowspan=3|[3,3<sup>1,1</sup>]<br>{{CDD|node|3|node|split1|nodes}} ||[3,3<sup>1,1</sup>]<br>{{CDD|node_c3|3|node_c4|split1|nodeab_c1-2}}<br>(order 192)||0
| (none)
|colspan=3|
|- BGCOLOR="#e0f0e0" align=center
|[1[3,3<sup>1,1</sup>]]=[4,3,3]<br>{{CDD|node_c1|3|node_c2|split1|nodeab_c3}} = {{CDD|node_c1|3|node_c2|3|node_c3|4|node}}<br>(order 384)||(4)
|{{CDD|node_1|3|node|split1|nodes}}<sub>12</sub> | {{CDD|node_1|3|node_1|split1|nodes}}<sub>17</sub> | {{CDD|node|3|node|split1|nodes_11}}<sub>11</sub> | {{CDD|node|3|node_1|split1|nodes_11}}<sub>16</sub>
|colspan=3|
|- BGCOLOR="#e0f0e0" align=center
|[3[3<sup>1,1,1</sup>]]=[3,4,3]<br>{{CDD|node_c1|3|node_c2|split1|nodeab_c1}} = {{CDD|node_c1|3|node_c2|4|node|3|node}}<br>(order 1152)||(3)
|{{CDD|node|3|node_1|split1|nodes}}<sub>22</sub> | {{CDD|node_1|3|node|split1|nodes_11}}<sub>23</sub> | {{CDD|node_1|3|node_1|split1|nodes_11}}<sub>24</sub>
|[3[3,3<sup>1,1</sup>]]<sup>+</sup><br>=[3,4,3]<sup>+</sup><br>(order 576)||(1)
|{{CDD|node_h|3|node_h|split1|nodes_hh}}<sub>31</sub>, {{CDD|node_h|3|node_h|3|node|3|node_1}}<sub>-</sub>
 
|- align=center
|rowspan=3|[4,3,3]<br>{{CDD|node|4|node|3|node|3|node}}
|BGCOLOR="#e0f0e0"|[3[1<sup>+</sup>,4,3,3]]=[3,4,3]<br>{{CDD|node|4|node_c1|3|node_c2|3|node_c1}} = {{CDD|node_c2|3|node_c1|4|node|3|node}}<br>(order 1152)||BGCOLOR="#e0f0e0"|(3)||BGCOLOR="#e0f0e0"| {{CDD|node|4|node|3|node_1|3|node}}<sub>22</sub> | {{CDD|node|4|node_1|3|node|3|node_1}}<sub>23</sub> | {{CDD|node|4|node_1|3|node_1|3|node_1}}<sub>24</sub>
|BGCOLOR="#e0f0e0" colspan=3|
|- align=center
|rowspan=2|[4,3,3]<br>{{CDD|node_c1|4|node_c2|3|node_c3|3|node_c4}}<br>(order 384)||rowspan=2|12
|rowspan=2|{{CDD|node_1|4|node|3|node|3|node}}<sub>10</sub> | {{CDD|node|4|node_1|3|node|3|node}}<sub>11</sub> | {{CDD|node|4|node|3|node|3|node_1}}<sub>12</sub> | {{CDD|node_1|4|node_1|3|node|3|node}}<sub>13</sub> | {{CDD|node_1|4|node|3|node_1|3|node}}<sub>14</sub><br>{{CDD|node_1|4|node|3|node|3|node_1}}<sub>15</sub> | {{CDD|node|4|node_1|3|node_1|3|node}}<sub>16</sub> | {{CDD|node|4|node|3|node_1|3|node_1}}<sub>17</sub> | {{CDD|node_1|4|node_1|3|node_1|3|node}}<sub>18</sub> | {{CDD|node_1|4|node_1|3|node|3|node_1}}<sub>19</sub><br>{{CDD|node_1|4|node|3|node_1|3|node_1}}<sub>20</sub> | {{CDD|node_1|4|node_1|3|node_1|3|node_1}}<sub>21</sub>
|[1<sup>+</sup>,4,3,3]<sup>+</sup><br>(order 96)||(2)
|{{CDD|node_h1|4|node|3|node|3|node}}<sub>12</sub> (= {{CDD|nodes_10r|split2|node|3|node}})<BR>{{CDD|node|4|node_h|3|node_h|3|node_h}}<sub>31</sub>
|- align=center
|[4,3,3]<sup>+</sup><br>(order 192)||(1)
|{{CDD|node_h|4|node_h|3|node_h|3|node_h}}<sub>-</sub>
 
|- align=center
|rowspan=2|[3,4,3]<br>{{CDD|node|3|node|4|node|3|node}}||[3,4,3]<br>{{CDD|node_c1|3|node_c2|4|node_c3|3|node_c4}}<br>(order 1152)||6
|{{CDD|node_1|3|node|4|node|3|node}}<sub>22</sub> | {{CDD|node|3|node_1|4|node|3|node}}<sub>23</sub> | {{CDD|node_1|3|node_1|4|node|3|node}}<sub>24</sub><br>{{CDD|node_1|3|node|4|node_1|3|node}}<sub>25</sub> | {{CDD|node_1|3|node_1|4|node_1|3|node}}<sub>28</sub> | {{CDD|node_1|3|node_1|4|node|3|node_1}}<sub>29</sub>
|[2<sup>+</sup>[3<sup>+</sup>,4,3<sup>+</sup>]]<br>(order 576)||1
|{{CDD|node_h|3|node_h|4|node|3|node}}<sub>31</sub>
|- align=center
|[2<sup>+</sup>[3,4,3]]<br>{{CDD|node_c1|3|node_c2|4|node_c2|3|node_c1}}<br>(order 2304)||3
|{{CDD|node_1|3|node|4|node|3|node_1}}<sub>26</sub> | {{CDD|node|3|node_1|4|node_1|3|node}}<sub>27</sub> | {{CDD|node_1|3|node_1|4|node_1|3|node_1}}<sub>30</sub>
|[2<sup>+</sup>[3,4,3]]<sup>+</sup><br>(order 1152)||(1)
|{{CDD|node_h|3|node_h|4|node_h|3|node_h}}<sub>-</sub>
 
|- align=center
|[5,3,3]<br>{{CDD|node|5|node|3|node|3|node}}||[5,3,3]<br>{{CDD|node_c1|5|node_c2|3|node_c3|3|node_c4}}<br>(order 14400)||15
| {{CDD|node_1|5|node|3|node|3|node}}<sub>32</sub> | {{CDD|node|5|node_1|3|node|3|node}}<sub>33</sub> | {{CDD|node|5|node|3|node_1|3|node}}<sub>34</sub> | {{CDD|node|5|node|3|node|3|node_1}}<sub>35</sub> | {{CDD|node_1|5|node_1|3|node|3|node}}<sub>36</sub><br>{{CDD|node_1|5|node|3|node_1|3|node}}<sub>37</sub> | {{CDD|node_1|5|node|3|node|3|node_1}}<sub>38</sub> | {{CDD|node|5|node_1|3|node_1|3|node}}<sub>39</sub> | {{CDD|node|5|node_1|3|node|3|node_1}}<sub>40</sub>  | {{CDD|node|5|node|3|node_1|3|node_1}}<sub>41</sub><br>{{CDD|node_1|5|node_1|3|node_1|3|node}}<sub>42</sub> | {{CDD|node_1|5|node_1|3|node|3|node_1}}<sub>43</sub> | {{CDD|node_1|5|node|3|node_1|3|node_1}}<sub>44</sub> | {{CDD|node|5|node_1|3|node_1|3|node_1}}<sub>45</sub> | {{CDD|node_1|5|node_1|3|node_1|3|node_1}}<sub>46</sub>
|[5,3,3]<sup>+</sup><br>(order 7200)||(1)
|{{CDD|node_h|5|node_h|3|node_h|3|node_h}}<sub>-</sub>
 
|- align=center
|rowspan=4|[3,2,3]<br>{{CDD|node|3|node|2|node|3|node}}||[3,2,3]<br>{{CDD|node_c1|3|node_c2|2|node_c3|3|node_c3}}<br>(order 36)||0
| (none)
|[(3,2,3]<sup>+</sup><br>(order 18)||0
| (none)
|- align=center
|[2<sup>+</sup>[3,2,3]]<br>{{CDD|node_c1|3|node_c2|2|node_c2|3|node_c1}}<br>(order 72)||0
| {{CDD|node_1|3|node|2|node|3|node_1}}
| [2<sup>+</sup>[3,2,3]]<sup>+</sup><br>(order 36)|| 0
| (none)
|- align=center
||[[3],2,3]=[6,2,3]<br>{{CDD|node_c1|3|node_c1|2|node_c2|3|node_c3}} = {{CDD|node_c1|6|node|2|node_c2|3|node_c3}}<br>(order 72)||1
| {{CDD|node_1|3|node_1|2|node_1|3|node}}
||[1[3,2,3]]=[[3],2,3]<sup>+</sup>=[6,2,3]<sup>+</sup><br>(order 36)||1
|(none)
|- align=center
||[(2<sup>+</sup>,4)[3,2,3]]=[2<sup>+</sup>[6,2,6]]<br>{{CDD|node_c1|3|node_c1|2|node_c1|3|node_c1}} = {{CDD|node_c1|6|node|2|node_c1|6|node}}<br>(order 288)||1
| {{CDD|node_1|3|node_1|2|node_1|3|node_1}}
||[(2<sup>+</sup>,4)[3,2,3]<span/>]<sup>+</sup>=[2<sup>+</sup>[6,2,6]<span/>]<sup>+</sup><br>(order 144)||1
| {{CDD|node_h|3|node_h|2|node_h|3|node_h}}
 
|- align=center BGCOLOR="#e0f0e0"
|rowspan=5|[4,2,4]<br>{{CDD|node|4|node|2|node|4|node}}||[4,2,4]<br>{{CDD|node_c1|4|node_c2|2|node_c3|4|node_c4}}<br>(order 64)||0
| (none)
|[(4,2,4]<sup>+</sup><br>(order 32)||0
| (none)
|- align=center BGCOLOR="#e0f0e0"
|[2<sup>+</sup>[4,2,4]]<br>{{CDD|node_c1|4|node_c2|2|node_c2|4|node_c1}}<br>(order 128)||0
| (none)
|[2<sup>+</sup>[(4,2<sup>+</sup>,4,2<sup>+</sup>)]]<br>(order 64)||0
| (none)
|- align=center BGCOLOR="#e0f0e0"
|[(3,3)[4,2<sup>*</sup>,4]]=[4,3,3]<br>{{CDD|node_c1|4|node|2|node|4|node_c1}} = {{CDD|node_c1|4|node|3|node|3|node}}<br>(order 384)||(1)
| {{CDD|node|4|node_1|2|node_1|4|node}}<sub>10</sub>
|[(3,3)[4,2<sup>*</sup>,4]]<sup>+</sup>=[4,3,3]<sup>+</sup><br>(order 192)||(1)
| {{CDD|node|4|node_h|2|node_h|4|node}}<sub>12</sub>
|- align=center BGCOLOR="#e0f0e0"
||[[4],2,4]=[8,2,4]<br>{{CDD|node_c1|4|node_c1|2|node_c2|4|node_c3}} = {{CDD|node_c1|8|node|2|node_c2|4|node_c3}}<br>(order 128)||(1)
| {{CDD|node_1|4|node_1|2|node_1|4|node}}
||[1[4,2,4]]=[[4],2,4]<sup>+</sup>=[8,2,4]<sup>+</sup><br>(order 64)||(1)
| {{CDD|node_h|4|node_h|2|node_h|4|node}}
|- align=center BGCOLOR="#e0f0e0"
||[(2<sup>+</sup>,4)[4,2,4]]=[2<sup>+</sup>[8,2,8]]<br>{{CDD|node_c1|4|node_c1|2|node_c1|4|node_c1}} = {{CDD|node_c1|8|node|2|node_c1|8|node}}<br>(order 512)||(1)
| {{CDD|node_1|4|node_1|2|node_1|4|node_1}}
||[(2<sup>+</sup>,4)[4,2,4]<span/>]<sup>+</sup>=[2<sup>+</sup>[8,2,8]<span/>]<sup>+</sup><br>(order 256)||(1)
| {{CDD|node_h|4|node_h|2|node_h|4|node_h}}
|}
 
== Polychoric symmetry groups ==
{{main|Polychoric symmetry groups}}
There are 5 fundamental mirror symmetry [[point group]] families in 4-dimensions: '''A'''<sub>4</sub>: {{CDD|node|3|node|3|node|3|node}}, '''BC'''<sub>4</sub>: {{CDD|node|4|node|3|node|3|node}}, '''D'''<sub>4</sub>: {{CDD|node|3|node|split1|nodes}}, '''F'''<sub>4</sub>: {{CDD|node|3|node|4|node|3|node}}, '''H'''<sub>4</sub>: {{CDD|node|5|node|3|node|3|node}}, and I<sub>2</sub>(p)×I<sub>2</sub>(q) as {{CDD|node|p|node|2|node|q|node}}. Each group defined by a [[Goursat tetrahedron#3-sphere (finite) solutions|Goursat tetrahedron]] [[fundamental domain]] bounded by mirror planes.
 
== See also==
* [[Regular skew polyhedron#Finite regular skew polyhedra of 4-space]]
* [[Convex uniform honeycomb]] - related infinite 4-polytopes in Euclidean 3-space.
* [[Convex uniform honeycombs in hyperbolic space]] - related infinite 4-polytopes in Hyperbolic 3-space.
 
==Notes==
{{reflist}}
 
== References ==
* [[Alicia Boole Stott|A. Boole Stott]]: ''Geometrical deduction of semiregular from regular polytopes and space fillings'', Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: ''Uniform Polyhedra'', Philosophical Transactions of the Royal Society of London, Londne, 1954
* {{citation|first=Pieter Hendrik|last=Schoute|authorlink=Pieter Hendrik Schoute|title=Analytic treatment of the polytopes regularly derived from the regular polytopes|journal=Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam|volume=11|issue=3|year=1911|pages=87 pp.}} [http://books.google.com/books?id=qC5LAAAAYAAJ&pg=PA357&lpg=PA357&dq=Analytic+treatment+of+the+polytopes+regularly+derived+from+the+regular+polytopes&source=bl&ots=SSICDM5u2d&sig=YyC2qp3xlErwN8b2_slTPfFEle4&hl=en&ei=tdBQTN6kFIeonQf_5dyPBw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CCMQ6AEwBA#v=onepage&q&f=false Googlebook, 370-381]
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
* ''[http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Kaleidoscopes: Selected Writings of H.S.M. Coxeter]'', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* H.S.M. Coxeter and W. O. J. Moser. ''Generators and Relations for Discrete Groups'' 4th ed, Springer-Verlag. New York. 1980 p92, p122.
* [[John Horton Conway|J.H. Conway]] and [[Michael Guy (computer scientist)|M.J.T. Guy]]: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
* [[Branko Grünbaum|B. Grünbaum]] ''Convex polytopes'',  New York ; London : Springer, c2003. ISBN 0-387-00424-6. <br>Second edition prepared by Volker Kaibel, [[Victor Klee]], and Günter M. Ziegler.
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26)
* Richard Klitzing, ''Snubs, alternated facetings, and Stott-Coxeter-Dynkin diagrams'', Symmetry: Culture and Science, Vol. 21, No.4, 329-344, (2010) [http://bendwavy.org/klitzing/pdf/Stott_v8.pdf]
 
==External links==
* {{mathworld | urlname = UniformPolychoron  | title = Uniform polychoron }}
* Convex uniform polychora
** [http://www.polytope.de Uniform, convex polytopes in four dimensions:], Marco Möller {{de icon}}
*** [http://ediss.sub.uni-hamburg.de/volltexte/2004/2196/pdf/Dissertation.pdf 2004 Dissertation] Four-dimensional Archimedean polytopes {{de icon}}
**{{PolyCell | urlname = uniform.html| title = Uniform Polytopes in Four Dimensions}}
**# {{PolyCell | urlname = section1.html| title = Convex uniform polychora based on the pentachoron}}
**# {{PolyCell | urlname = section2.html| title = Convex uniform polychora based on the tesserract/16-cell}}
**#{{PolyCell | urlname = section3.html| title = Convex uniform polychora based on the 24-cell}}
**#{{PolyCell | urlname = section4.html| title = Convex uniform polychora based on the 120-cell/600-cell}}
**#{{PolyCell | urlname = section5.html| title = Anomalous convex uniform polychoron: (grand antiprism)}}
**#{{PolyCell | urlname = section6.html| title = Convex uniform prismatic polychora}}
**#{{PolyCell | urlname = section7.html| title = Uniform polychora derived from glomeric tetrahedron B4}}
**[http://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.html Regular and semi-regular convex polytopes a short historical overview]
**[http://public.beuth-hochschule.de/~meiko/pentatope.html Java3D Applets with sources]
* Nonconvex uniform polychora
**[http://www.polytope.net/hedrondude/polychora.htm Uniform polychora] by Jonathan Bowers
** [http://www.software3d.com/Stella.php#stella4D Stella4D] [[Stella (software)]] produces interactive views of all 1849 known uniform polychora including the 64 convex forms and the infinite prismatic families. Was used to create most images on this page.
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes}}
* [http://arxiv-web3.library.cornell.edu/abs/1102.1132 4D-Polytopes and Their Dual Polytopes of the Coxeter Group W(A4) Represented by Quaternions] International Journal of Geometric Methods in Modern Physics,Vol. 9, No. 4 (2012) Mehmet Koca, Nazife Ozdes Koca, Mudhahir Al-Ajmi (2012) [http://arxiv.org/ftp/arxiv/papers/1102/1102.1132.pdf]
 
{{Polytopes}}
 
[[Category:Four-dimensional geometry]]
[[Category:Polychora]]

Revision as of 10:48, 14 February 2014

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