View source for Uniform 4-polytope

{{Short description|Class of 4-dimensional polytopes}}
[[File:Schlegel half-solid truncated 120-cell.png|320px|thumb|[[Schlegel diagram]] for the [[truncated 120-cell]] with [[Tetrahedron|tetrahedral]] cells visible]]
[[File:120-cell t01 H3.svg|thumb|320px|[[Orthographic projection]] of the truncated 120-cell, in the {{math|H{{sub|3}}}} [[Coxeter plane]] ({{math|D{{sub|10}}}} symmetry). Only vertices and edges are drawn.]]

In [[geometry]], a '''uniform 4-polytope''' (or '''uniform polychoron''')<ref>[[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.1 ''Polytopes and Honeycombs'', p.224</ref> is a 4-dimensional [[polytope]] which is [[vertex-transitive]] and whose cells are [[uniform polyhedron|uniform polyhedra]], and faces are [[regular polygon]]s.

There are 47 non-[[Prism (geometry)|prismatic]] [[Convex polytope|convex]] uniform 4-polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform polyhedra. There are also an unknown number of non-convex star forms.

== History of discovery ==
* Convex '''[[Regular polytope]]s''':
** '''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 4-polytopes]]''' ([[star polyhedron]] cells and/or [[vertex figure]]s)
** '''1852''': [[Ludwig Schläfli]] also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures [[small stellated dodecahedron|{<sup>5</sup>/<sub>2</sub>,5}]] and [[great dodecahedron|{5,<sup>5</sup>/<sub>2</sub>}]].
** '''1883''': [[Edmund Hess]] completed the list of 10 of the nonconvex regular 4-polytopes, 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 Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder, von dr. Edmund Hess. Mit sechzehn lithographierten tafeln.].
* '''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''. In four dimensions, this gives the [[rectified 5-cell]], the [[rectified 600-cell]], and the [[snub 24-cell]].<ref>[[Thorold Gosset|T. Gosset]]: ''[https://www.maths.ed.ac.uk/~v1ranick/papers/gosset.pdf 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 4-polytopes, corresponding to the nonprismatic forms listed below. The snub 24-cell and [[grand antiprism]] were missing from her list.<ref>{{cite web |url=http://dissertations.ub.rug.nl/FILES/faculties/science/2007/i.polo.blanco/c5.pdf |title=Archived copy |access-date=2010-08-13 |url-status=dead |archive-url=https://web.archive.org/web/20091229071352/http://dissertations.ub.rug.nl/FILES/faculties/science/2007/i.polo.blanco/c5.pdf |archive-date=2009-12-29 }}</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>Elte (1912)</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 4-polytopes''':
***'''1965''': The complete list of convex forms was finally enumerated 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 4-polytope, the grand antiprism.
*** '''1966''' [[Norman Johnson (mathematician)|Norman 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.
*** '''1986''' Coxeter published a paper ''Regular and Semi-Regular Polytopes II'' which included analysis of the unique [[snub 24-cell]] structure, and the symmetry of the anomalous grand antiprism.
*** '''1998<ref>[https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html Uniform Polytopes in Four Dimensions] December 6, 1998 oldest archive</ref>-2000''': The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevsky's online indexed enumeration (used as a basis for this listing). Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the [[Greek language|Greek]] roots ''poly'' ("many") and ''choros'' ("room" or "space").<ref>[https://books.google.com/books?id=HrOxRdtYYaMC&q=polychoron&pg=PP1 The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes] By David Darling, (2004) ASIN: B00SB4TU58</ref> The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t<sub>0,1</sub>, cantellation, t<sub>0,2</sub>, runcination t<sub>0,3</sub>, with single ringed forms called rectified, and bi, tri-prefixes added when the first ring was on the second or third nodes.<ref name="polychoric groups">Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 ''full polychoric groups''</ref><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''. Möller reproduced Johnson's naming system in his listing.<ref>{{Cite thesis |url=https://ediss.sub.uni-hamburg.de/volltexte/2004/2196/pdf/Dissertation.pdf |title=Vierdimensionale Archimedische Polytope |last=Möller |first=Marco |date=2004 |publisher=University of Hamburg |type=Doctoral thesis |language=de }}</ref>
*** '''2008''': ''The Symmetries of Things''<ref>Conway (2008)</ref> was published by [[John Horton Conway|John H. Conway]] and contains the first print-published listing of the convex uniform 4-polytopes and higher dimensional polytopes by Coxeter group family, with general [[vertex figure]] diagrams for each ringed [[Coxeter diagram]] permutation—snub, grand antiprism, and duoprisms—which he called proprisms for product prisms. He used his own ''ijk''-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, and all of Johnson's names were included in the book index.
* '''Nonregular uniform star 4-polytopes''': (similar to the [[List of uniform polyhedra#Nonconvex forms with convex faces|nonconvex uniform polyhedra]])
**'''1966''': Johnson describes three nonconvex uniform antiprisms in 4-space in his dissertation.<ref>[https://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html Multidimensional Glossary], George Olshevsky</ref>
**'''1990-2006''': In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes (convex and nonconvex) had been identified by Jonathan Bowers and George Olshevsky,<ref>https://www.mit.edu/~hlb/Associahedron/program.pdf ''Convex and Abstract Polytopes'' workshop (2005), N.Johnson — "Uniform Polychora" abstract</ref> with an additional four discovered in 2006 for a total of 1849. The count includes the 74 prisms of the 75 [[list of uniform polyhedra|non-prismatic uniform polyhedra]] (since that is a finite set – the cubic prism is excluded as it duplicates the tesseract), but not the infinite categories of duoprisms or prisms of antiprisms.<ref name=":0">{{Cite web|url=http://www.polytope.net/hedrondude/polychora.htm|title=Uniform Polychora|website=www.polytope.net|access-date=February 20, 2020}}</ref>
**'''2020-2023''': 342 new polychora were found, bringing up the total number of known uniform 4-polytopes to 2191. The list has not been proven complete.<ref name=":0"/><ref>{{Cite web|url=https://polytope.miraheze.org/wiki/Uniform_polytope|title=Uniform polytope|date=6 November 2023|website=Polytope Wiki|accessdate=11 November 2023}}</ref>

== Regular 4-polytopes ==

Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements. [[List of regular polytopes#Four-dimensional regular polytopes|Regular 4-polytopes]] can be expressed with [[Schläfli symbol]] {''p'',''q'',''r''} have cells of type {''p'',''q''}, faces of type {''p''}, edge figures {''r''}, and [[vertex figure]]s {''q'',''r''}.

The existence of a regular 4-polytope {''p'',''q'',''r''} is constrained by the existence of the regular polyhedra {''p'',''q''} which becomes cells, and {''q'',''r''} which becomes the [[vertex figure]].

Existence as a finite 4-polytope is dependent upon an inequality:<ref>Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135</ref>
:<math>\sin \left ( \frac{\pi}{p} \right ) \sin \left(\frac{\pi}{r}\right) > \cos\left(\frac{\pi}{q}\right).</math>

The 16 [[List of regular polytopes#Four dimensional regular polytopes|regular 4-polytopes]], with the property that all cells, faces, edges, and vertices are congruent:
* 6 [[regular convex 4-polytope]]s: [[5-cell]] {3,3,3}, [[8-cell]] {4,3,3}, [[16-cell]] {3,3,4}, [[24-cell]] {3,4,3}, [[120-cell]] {5,3,3}, and [[600-cell]] {3,3,5}.
* 10 [[regular star 4-polytope]]s: [[icosahedral 120-cell]] {3,5,<sup>5</sup>/<sub>2</sub>}, [[small stellated 120-cell]] {<sup>5</sup>/<sub>2</sub>,5,3}, [[great 120-cell]] {5,<sup>5</sup>/<sub>2</sub>,5}, [[grand 120-cell]] {5,3,<sup>5</sup>/<sub>2</sub>}, [[great stellated 120-cell]] {<sup>5</sup>/<sub>2</sub>,3,5}, [[grand stellated 120-cell]] {<sup>5</sup>/<sub>2</sub>,5,<sup>5</sup>/<sub>2</sub>}, [[great grand 120-cell]] {5,<sup>5</sup>/<sub>2</sub>,3}, [[great icosahedral 120-cell]] {3,<sup>5</sup>/<sub>2</sub>,5}, [[grand 600-cell]] {3,3,<sup>5</sup>/<sub>2</sub>}, and [[great grand stellated 120-cell]] {<sup>5</sup>/<sub>2</sub>,3,3}.

==Convex uniform 4-polytopes==
=== Symmetry of uniform 4-polytopes in four dimensions ===
{{main|Point groups in four dimensions}}
{| class=wikitable align=right width=240
|+ Orthogonal subgroups
|-
|The 24 mirrors of '''F'''<sub>4</sub> can be decomposed into 2 orthogonal '''D'''<sub>4</sub> groups:
#{{CDD|node_g|3sg|node_g|4|node_c3|3|node_c4}} = {{CDD|node_c3|branch3_c3|splitsplit2|node_c4}} (12 mirrors)
#{{CDD|node_c1|3|node_c2|4|node_g|3sg|node_g}} = {{CDD|node_c1|splitsplit1|branch3_c2|node_c2}} (12 mirrors)
|-
|The 10 mirrors of '''B'''<sub>3</sub>×'''A'''<sub>1</sub> can be decomposed into orthogonal groups, 4'''A'''<sub>1</sub> and '''D'''<sub>3</sub>:
# {{CDD|node_c1|4|node_g|3sg|node_g|2|node_c4}} = {{CDD|node_c1|2|nodeab_c1|2|node_c4}} (3+1 mirrors)
# {{CDD|node_h0|4|node_c2|3|node_c3|2|node_h0}} = {{CDD|nodeab_c2|split2|node_c3}} (6 mirrors)
|}
There are 5 fundamental mirror symmetry [[point group]] families in 4-dimensions: '''A'''<sub>4</sub> = {{CDD|node|3|node|3|node|3|node}}, '''B'''<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}}.<ref name="polychoric groups" /> There are also 3 prismatic groups '''A'''<sub>3</sub>'''A'''<sub>1</sub> = {{CDD|node|3|node|3|node|2|node}}, '''B'''<sub>3</sub>'''A'''<sub>1</sub> = {{CDD|node|4|node|3|node|2|node}}, '''H'''<sub>3</sub>'''A'''<sub>1</sub> = {{CDD|node|5|node|3|node|2|node}}, and duoprismatic groups: I<sub>2</sub>(p)×I<sub>2</sub>(q) = {{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.

Each reflective uniform 4-polytope can be constructed in one or more reflective point group in 4 dimensions by a [[Wythoff construction]], represented by rings around permutations of nodes in a [[Coxeter diagram]]. Mirror [[hyperplane]]s can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,a], have an extended symmetry, <nowiki>[[a,b,a]]</nowiki>, doubling the symmetry order. This includes [3,3,3], [3,4,3], and [''p'',2,''p'']. Uniform polytopes in these group with symmetric rings contain this extended symmetry.

If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an [[alternation (geometry)|alternation]] operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is [[#Nonuniform alternations|not generally adjustable to create uniform solutions]].
{| class=wikitable
|- valign=top
![[Coxeter group#Weyl groups|Weyl<br />group]]
![[John Horton Conway|Conway]]<br />Quaternion
!Abstract<br />structure
![[Group order|Order]]
!colspan=2|[[Coxeter-Dynkin diagram|Coxeter<br />diagram]]
![[Coxeter notation|Coxeter<br />notation]]
![[Coxeter notation#Commutator subgroups|Commutator<br />subgroup]]
![[Coxeter number|Coxeter<br />number]]<br />(h)
!colspan=4|[[kaleidoscope|Mirrors]]<br />''m''=2''h''
|-
!colspan=13|Irreducible
|- align=center
!A<sub>4</sub>
||+1/60[I×I].21||[[Symmetric group|S<sub>5</sub>]]|| 120|| {{CDD|node|3|node|3|node|3|node}}|| {{CDD|node_c1|3|node_c1|3|node_c1|3|node_c1}}|| [3,3,3] || [3,3,3]<sup>+</sup> ||5||||10{{CDD|node_c1}}|| ||
|- align=center
!D<sub>4</sub>
||±1/3[T×T].2||1/2.<sup>2</sup>S<sub>4</sub>||192|| {{CDD|nodes|split2|node|3|node}}||{{CDD|nodeab_c1|split2|node_c1|3|node_c1}}|| [3<sup>1,1,1</sup>] ||rowspan=2| [3<sup>1,1,1</sup>]<sup>+</sup>||6||||12{{CDD|node_c1}}|| ||
|- align=center
!B<sub>4</sub>
||±1/6[O×O].2||<sup>2</sup>S<sub>4</sub> = S<sub>2</sub>≀S<sub>4</sub>|| 384 || {{CDD|node|4|node|3|node|3|node}}||{{CDD|node_c2|4|node_c1|3|node_c1|3|node_c1}}|| [4,3,3] ||8||4{{CDD|node_c2}}||12{{CDD|node_c1}}|| ||
|- align=center
!F<sub>4</sub>
||±1/2[O×O].2<sub>3</sub>||3.<sup>2</sup>S<sub>4</sub>||1152|| {{CDD|node|3|node|4|node|3|node}}||{{CDD|node_c2|3|node_c2|4|node_c1|3|node_c1}}|| [3,4,3] ||[3<sup>+</sup>,4,3<sup>+</sup>]||12||12{{CDD|node_c2}}||12{{CDD|node_c1}}|| ||
|- align=center
!H<sub>4</sub>
||±[I×I].2||2.(A<sub>5</sub>×A<sub>5</sub>).2||14400|| {{CDD|node|5|node|3|node|3|node}}||{{CDD|node_c1|5|node_c1|3|node_c1|3|node_c1}}|| [5,3,3] || [5,3,3]<sup>+</sup>||30||||60{{CDD|node_c1}}|| ||
|-
!colspan=13|Prismatic groups
|- align=center
!A<sub>3</sub>A<sub>1</sub>
||+1/24[O×O].2<sub>3</sub>||S<sub>4</sub>×D<sub>1</sub>|| 48|| {{CDD|node|3|node|3|node|2|node}}||{{CDD|node_c1|3|node_c1|3|node_c1|2|node_c3}}|| [3,3,2] = [3,3]×[ ] ||rowspan=2|[3,3]<sup>+</sup> ||-  ||||6{{CDD|node_c1}}||1{{CDD|node_c3}} ||
|- align=center
!B<sub>3</sub>A<sub>1</sub>
||±1/24[O×O].2||S<sub>4</sub>×D<sub>1</sub>||96|| {{CDD|node|4|node|3|node|2|node}}||{{CDD|node_c2|4|node_c1|3|node_c1|2|node_c3}}|| [4,3,2] = [4,3]×[ ] || -  ||3{{CDD|node_c2}}||6{{CDD|node_c1}}||1{{CDD|node_c3}} ||
|- align=center
!H<sub>3</sub>A<sub>1</sub>
||±1/60[I×I].2||A<sub>5</sub>×D<sub>1</sub>||240|| {{CDD|node|5|node|3|node|2|node}}|| {{CDD|node_c1|5|node_c1|3|node_c1|2|node_c3}}|| [5,3,2] = [5,3]×[ ] ||[5,3]<sup>+</sup> || -  ||||15{{CDD|node_c1}}||1{{CDD|node_c3}} ||
|-
!colspan=13|Duoprismatic groups (Use 2p,2q for even integers)
|- align=center
!I<sub>2</sub>(''p'')I<sub>2</sub>(''q'')
||±1/2[D<sub>2''p''</sub>×D<sub>2''q''</sub>]||D<sub>''p''</sub>×D<sub>''q''</sub>||4''pq''|| {{CDD|node|p|node|2|node|q|node}}||{{CDD|node_c1|p|node_c1|2|node_c3|q|node_c3}}|| [''p'',2,''q''] = [''p'']×[''q''] ||rowspan=3|[''p''<sup>+</sup>,2,''q''<sup>+</sup>]|| - || ||''p'' {{CDD|node_c1}}|| ''q'' {{CDD|node_c3}} ||

|- align=center
!I<sub>2</sub>(''2p'')I<sub>2</sub>(''q'')
||±1/2[D<sub>4''p''</sub>×D<sub>2''q''</sub>]||D<sub>2''p''</sub>×D<sub>''q''</sub>||8''pq''|| {{CDD|node|2x|p|node|2|node|q|node}}||{{CDD|node_c2|2x|p|node_c1|2|node_c3|q|node_c3}}|| [2''p'',2,''q''] = [2''p'']×[''q''] || - || ''p'' {{CDD|node_c2}}|| ''p'' {{CDD|node_c1}}|| ''q'' {{CDD|node_c3}} ||

|- align=center
!I<sub>2</sub>(''2p'')I<sub>2</sub>(''2q'')
||±1/2[D<sub>4''p''</sub>×D<sub>4''q''</sub>]||D<sub>''2p''</sub>×D<sub>''2q''</sub>||16''pq''|| {{CDD|node|2x|p|node|2|node|2x|q|node}}||{{CDD|node_c2|2x|p|node_c1|2|node_c3|2x|q|node_c4}}|| [2''p'',2,2''q''] = [2''p'']×[2''q''] || - ||''p'' {{CDD|node_c2}}||''p'' {{CDD|node_c1}}|| ''q'' {{CDD|node_c3}}|| ''q'' {{CDD|node_c4}}
|}

=== Enumeration ===

There are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the [[duoprism]]s and the [[Uniform antiprismatic prism|antiprismatic prisms]].
* 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 B<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 4-polytope, the grand antiprism.
* TOTAL: 68 &minus; 4 = 64

These 64 uniform 4-polytopes 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 [[Cartesian product]] of two polygons.

=== The A<sub>4</sub> family ===
{{See|A4 polytope}}
The 5-cell has [[Pentachoric symmetry|''diploid pentachoric'' [3,3,3] symmetry]],<ref name="polychoric groups" /> 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| 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 4-polytopes 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<ref name="polychoric groups" /> (pen)
|[[File:5-cell verf.svg|60px]]
|align=center|{{CDD|node_1|3|node|3|node|3|node}}<br />{3,3,3}
|(4)<br />[[File:Uniform_polyhedron-33-t0.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:Uniform_polyhedron-43-t2.png|30px]]<br />[[octahedron|(3.3.3.3)]]
| 
|
|(2)<br />[[File:Uniform_polyhedron-33-t0.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:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
| 
|
|(1)<br />[[File:Uniform_polyhedron-33-t0.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:Uniform_polyhedron-33-t02.svg|30px]]<br />[[cuboctahedron|(3.4.3.4)]]
|
|(2)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-t1.svg|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:Uniform_polyhedron-33-t012.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
|
|(1)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
| 20
| 80
| 120
| 60
|- BGCOLOR="#f0e0e0" align=center
!8
|align=center|[[runcitruncated 5-cell]] <br />Prismatorhombated 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:Uniform_polyhedron-33-t01.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|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-t02.svg|30px]]<br />[[cuboctahedron|(3.4.3.4)]]
| 30
| 120
| 150
| 60
|}

{| class="wikitable"
|+ <nowiki>[[3,3,3]]</nowiki> uniform polytopes
|-
!rowspan=2| #
!rowspan=2| 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 prismatodecachoron (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:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
|(6)<br />[[File:triangular prism.png|25px]]<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:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
|
|
| 10
| 40
| 60
| 30
|- BGCOLOR="#e0f0e0" align=center
!9
|align=center|*[[omnitruncated 5-cell]] <br />Great prismatodecachoron (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:Uniform_polyhedron-33-t012.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]]
|[[omnisnub 5-cell]]<br/>Snub decachoron (snad)<br>Snub pentachoron (snip)<ref>{{Cite web|url=http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm|title = S3s3s3s}}</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>{3,3,3}
|[[File:Uniform_polyhedron-33-s012.png|30px]] (2)<br />[[Snub tetrahedron|(3.3.3.3.3)]]
|[[File:Trigonal_antiprism.png|30px]] (2)<br />[[triangular antiprism|(3.3.3.3)]]
|[[File:Uniform_polyhedron-33-t0.png|30px]] (4)<br />[[tetrahedron|(3.3.3)]]
| 90
| 300
| 270
| 60
|}

The three uniform 4-polytopes forms marked with an [[asterisk]], '''*''', have the higher [[Pentachoric symmetry|extended pentachoric symmetry]], of order 240, {{brackets|3,3,3}} 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 {{brackets|3,3,3}}<sup>+</sup>, order 120, defining an [[omnisnub 5-cell]] which is listed for completeness, but is not uniform.

===The B<sub>4</sub> family===
{{See|B4 polytope}}

This family has [[Hexadecachoric symmetry|''diploid hexadecachoric'' symmetry]],<ref name="polychoric groups" /> [4,3,3], of [[Symmetry order|order]] 24&times;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 4-polytopes 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|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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br />{{CDD|node_n0|4|node_n1|3|node_n2|2|2}}<br />(8)
! Pos. 2<br />{{CDD|node_n0|4|node_n1|2|2|node_n3}}<br />(24)
! Pos. 1<br />{{CDD|node_n0|2|2|node_n2|3|node_n3}}<br />(32)
! Pos. 0<br />{{CDD|2|2|node_n1|3|node_n2|3|node_n3}}<br />(16)
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#f0e0e0" align=center
!10
|align=center|[[tesseract]] or 8-cell<br />Tesseract (tes)
|[[File:8-cell verf.svg|60px]]
| align=center|{{CDD|node_1|4|node|3|node|3|node}}<br />{4,3,3}
|(4)<br />[[File:Uniform_polyhedron-43-t0.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:Uniform_polyhedron-43-t1.svg|30px]]<br />[[cuboctahedron|(3.4.3.4)]]
| 
| 
|(2)<br />[[File:Uniform_polyhedron-33-t0.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:Uniform_polyhedron-43-t01.png|30px]]<br />[[truncated cube|(3.8.8)]]
|
|
|(1)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 24
| 88
| 128
| 64
|- BGCOLOR="#f0e0e0" align=center
!14
|align=center|[[Cantellated tesseract]]<br>Small rhombated 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}
|(2)<br />[[File:Uniform_polyhedron-43-t02.png|30px]]<br />[[small rhombicuboctahedron|(3.4.4.4)]]
|
|(2)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-43-t2.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'')<br>Small disprismatotesseractihexadecachoron (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:Uniform_polyhedron-43-t0.png|30px]]<br />[[cube|(4.4.4)]]
|(3)<br />[[File:Uniform_polyhedron-43-t0.png|30px]]<br />[[cube|(4.4.4)]]
|(3)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-t0.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'')<br>Tesseractihexadecachoron (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:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
|
|
|(2)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
| 24
| 120
| 192
| 96
|- BGCOLOR="#f0e0e0" align=center
!18
|align=center|[[Cantitruncated tesseract]]<br>Great rhombated 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:Uniform_polyhedron-43-t012.png|30px]]<br />[[truncated cuboctahedron|(4.6.8)]]
|
|(1)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
| 56
| 248
| 384
| 192
|- BGCOLOR="#f0e0e0" align=center
!19
|align=center|[[Runcitruncated tesseract]]<br>Prismatorhombated hexadecachoron (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:Uniform_polyhedron-43-t01.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|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform polyhedron-43-t1.svg|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'')<br>Great disprismatotesseractihexadecachoron (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:Uniform_polyhedron-43-t012.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:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
| 80
| 464
| 768
| 384
|}

{| class="wikitable"
|+ Related half tesseract, [1<sup>+</sup>,4,3,3] uniform 4-polytopes
!rowspan=2|#
!rowspan=2|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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br />{{CDD|node_n0|4|node_n1|3|node_n2|2|2}}<br />(8)
! Pos. 2<br />{{CDD|node_n0|4|node_n1|2|2|node_n3}}<br />(24)
! Pos. 1<br />{{CDD|node_n0|2|2|node_n2|3|node_n3}}<br />(32)
! Pos. 0<br />{{CDD|2|2|node_n1|3|node_n2|3|node_n3}}<br />(16)
!Alt
! Cells
! Faces
! Edges
! Vertices
|-  BGCOLOR="#d0f0f0" align=center
![12]
|align=center|Half tesseract<br />Demitesseract<br />= [[16-cell]] (hex)
|[[File:16-cell verf.svg|60px]]
| align=center|{{CDD|node_h1|4|node|3|node|3|node}} = {{CDD|nodes_10ru|split2|node|3|node}}<br />h{4,3,3}={3,3,4}
|(4)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 
| 
|
|(4)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 16
| 32
| 24
| 8

|-  BGCOLOR="#d0f0f0" align=center
![17]
|align=center|[[Cantic tesseract]]<br />= [[Truncated 16-cell]] (thex)
|[[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}=t{4,3,3}
|(4)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(6.6.3)]]
| 
| 
|(1)<br />[[File:Uniform_polyhedron-43-t2.png|30px]]<br />[[Octahedron|(3.3.3.3)]]
|
| 24  
| 96
| 120
| 48

|-  BGCOLOR="#d0f0f0" align=center
![11]
|align=center|[[Runcic tesseract]]<br />= [[Rectified tesseract]] (rit)
|[[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}=r{4,3,3}
|(3)<br />[[File:Uniform_polyhedron-43-t1.svg|30px]]<br />[[Cuboctahedron|(3.4.3.4)]]
| 
| 
|(2)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[Tetrahedron|(3.3.3)]]
|
| 24
| 88
| 96
| 32

|-  BGCOLOR="#d0f0f0" align=center
![16]
|align=center|[[Runcicantic tesseract]]<br />= [[Bitruncated tesseract]] (tah)
|[[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}=2t{4,3,3}
|(2)<br />[[File:Uniform_polyhedron-43-t12.png|30px]]<br />[[Truncated octahedron|(3.4.3.4)]]
| 
| 
|(2)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[Truncated tetrahedron|(3.6.6)]]
|
| 24
| 120
| 192
| 96
|-  BGCOLOR="#d0f0f0" align=center
![11]
|align=center|= [[Rectified tesseract]] (rat)
|[[File:Cantellated demitesseract verf.png|60px]]
| align=center|{{CDD|node_h0|4|node_1|3|node|3|node}} = {{CDD|nodes_11|split2|node|3|node}}<br />h<sub>1</sub>{4,3,3}=r{4,3,3}
|
| 
|
|
|
| 24
| 88
| 96
| 32
|-  BGCOLOR="#d0f0f0" align=center
![16]
|align=center|= [[Bitruncated tesseract]] (tah)
|[[File:Cantitruncated demitesseract verf.png|60px]]
| align=center|{{CDD|node_h0|4|node_1|3|node_1|3|node}} = {{CDD|nodes_11|split2|node_1|3|node}}<br />h<sub>1,2</sub>{4,3,3}=2t{4,3,3}
|
| 
| 
|
|
| 24
| 120
| 192
| 96

|-  BGCOLOR="#d0f0f0" align=center
![23]
|align=center|= [[Rectified 24-cell]] (rico)
|[[File:Runcicantellated demitesseract verf.png|60px]]
| align=center|{{CDD|node_h0|4|node_1|3|node|3|node_1}} = {{CDD|nodes_11|split2|node|3|node_1}}<br />h<sub>1,3</sub>{4,3,3}=rr{3,3,4}
|
| 
| 
|
|
| 48
| 240
| 288
| 96
|-  BGCOLOR="#d0f0f0" align=center
![24]
|align=center|= [[Truncated 24-cell]] (tico)
|[[File:Omnitruncated demitesseract verf.png|60px]]
| align=center|{{CDD|node_h0|4|node_1|3|node_1|3|node_1}} = {{CDD|nodes_11|split2|node_1|3|node_1}}<br />h<sub>1,2,3</sub>{4,3,3}=tr{3,3,4}
|
| 
| 
|
|
| 48
| 240
| 384
| 192
|}

{| class="wikitable"
!rowspan=2|#
!rowspan=2|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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br />{{CDD|node_n0|4|node_n1|3|node_n2|2|2}}<br />(8)
! Pos. 2<br />{{CDD|node_n0|4|node_n1|2|2|node_n3}}<br />(24)
! Pos. 1<br />{{CDD|node_n0|2|2|node_n2|3|node_n3}}<br />(32)
! Pos. 0<br />{{CDD|2|2|node_n1|3|node_n2|3|node_n3}}<br />(16)
!Alt
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#d0f0f0"  align=center
![[#Nonuniform alternations|Nonuniform]]
||[[omnisnub tesseract]]<br>Snub tesseract (snet)<ref>{{Cite web|url=http://www.bendwavy.org/klitzing/incmats/s3s3s4s.htm|title = S3s3s4s}}</ref><br />(Or ''omnisnub 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:Uniform_polyhedron-43-s012.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:Trigonal_antiprism.png|30px]]<br />[[triangular antiprism|(3.3.3.3)]]
|(1)<br />[[File:Uniform_polyhedron-33-s012.png|30px]]<br />[[Snub tetrahedron|(3.3.3.3.3)]]
|(4)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 272
| 944
| 864
| 192
|}

==== 16-cell truncations ====

{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name (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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br />{{CDD|node_n0|4|node_n1|3|node_n2|2|2}}<br />(8)
! Pos. 2<br />{{CDD|node_n0|4|node_n1|2|2|node_n3}}<br />(24)
! Pos. 1<br />{{CDD|node_n0|2|2|node_n2|3|node_n3}}<br />(32)
! Pos. 0<br />{{CDD|2|2|node_n1|3|node_n2|3|node_n3}}<br />(16)
!Alt
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#e0e0f0" align=center
!12
|align=center|[[16-cell]]<br>Hexadecachoron<ref name="polychoric groups"/> (hex)
|[[File:16-cell verf.svg|60px]]
| align=center|{{CDD|node|4|node|3|node|3|node_1}}<br />{3,3,4}
| 
| 
| 
|(8)<br />[[File:Uniform_polyhedron-33-t0.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}} = {{CDD|node_1|3|node|4|node|3|node}}<br />r{3,3,4}
|(2)<br />[[File:Uniform_polyhedron-43-t2.png|30px]]<br />[[octahedron|(3.3.3.3)]]
| 
| 
|(4)<br />[[File:Uniform_polyhedron-43-t2.png|30px]]<br />[[octahedron|(3.3.3.3)]]
| 
| 24
| 96
| 96
| 24
|- BGCOLOR="#e0e0f0" align=center
!17
|align=center|[[Truncated 16-cell]]<br>Truncated hexadecachoron (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:Uniform_polyhedron-43-t2.png|30px]]<br />[[octahedron|(3.3.3.3)]]
| 
|
|(4)<br />[[File:Uniform_polyhedron-33-t01.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}} = {{CDD|node|3|node_1|4|node|3|node}}<br />rr{3,3,4}
|(1)<br />[[File:Uniform_polyhedron-43-t1.svg|30px]]<br />[[cuboctahedron|(3.4.3.4)]]
|(2)<br />[[File:Tetragonal prism.png|30px]]<br />[[cube|(4.4.4)]]
|
|(2)<br />[[File:Uniform_polyhedron-43-t1.svg|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 tesseract'') (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:Uniform_polyhedron-43-t0.png|30px]]<br />[[cube|(4.4.4)]]
|(3)<br />[[File:Tetragonal prism.png|30px]]<br />[[cube|(4.4.4)]]
|(3)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-t0.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 tesseract'') (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:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
|
|
|(2)<br />[[File:Uniform_polyhedron-33-t01.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}} = {{CDD|node_1|3|node_1|4|node|3|node}}<br />tr{3,3,4}
|(1)<br />[[File:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
|(1)<br />[[File:Tetragonal prism.png|30px]]<br />[[cube|(4.4.4)]]
|
|(2)<br />[[File:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
| 
| 48
| 240
| 384
| 192
|- BGCOLOR="#e0e0f0" align=center
!20
|align=center|[[Runcitruncated 16-cell]]<br>Prismatorhombated tesseract (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:Uniform_polyhedron-43-t02.png|30px]]<br />[[rhombicuboctahedron|(3.4.4.4)]]
|(1)<br />[[File:Tetragonal prism.png|30px]]<br />[[cube|(4.4.4)]]
|(2)<br />[[File:hexagonal prism.png|30px]]<br />[[hexagonal prism|(4.4.6)]]
|(1)<br />[[File:Uniform_polyhedron-33-t01.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 tesseract'') (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:Uniform_polyhedron-43-t012.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:Uniform_polyhedron-43-t12.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:Uniform_polyhedron-43-h01.svg|30px]]<br />[[regular icosahedron|(3.3.3.3.3)]]
|
|(1)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
|(2)<br />[[File:Uniform_polyhedron-33-s012.png|30px]]<br />[[regular icosahedron|(3.3.3.3.3)]]
|(4)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 144
| 480
| 432
| 96
|- BGCOLOR="#d0f0f0" align=center
!Nonuniform
|Runcic snub rectified 16-cell<br>Pyritosnub tesseract (pysnet)
|[[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:Rhombicuboctahedron uniform edge coloring.png|30px]]<br />[[rhombicuboctahedron|(3.4.4.4)]]
|(2)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Tetragonal prism.png|30px]]<br />[[cube|(4.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-s012.png|30px]]<br />[[regular icosahedron|(3.3.3.3.3)]]
|(2)<br />[[File:triangular prism.png|25px]]<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.

===The F<sub>4</sub> family===
{{See|F4 polytope}}

This family has [[Icositetrachoric symmetry|''diploid icositetrachoric'' symmetry]],<ref name="polychoric groups" /> [3,4,3], of [[Symmetry order|order]] 24&times;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 4-polytopes 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 4-polytopes
|-
!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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!colspan=4 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br />{{CDD|node_n0|3|node_n1|4|node_n2|2|2}}<br />(24)
! Pos. 2<br />{{CDD|node_n0|3|node_n1|2|2|node_n3}}<br />(96)
! Pos. 1<br />{{CDD|node_n0|2|2|2|node_n2|3|node_n3}}<br />(96)
! Pos. 0<br />{{CDD|2|2|node_n1|4|node_n2|3|node_n3}}<br />(24)
! Cells
! Faces
! Edges
! Vertices
|-  BGCOLOR="#f0e0e0" align=center 
!22
|align=center|[[24-cell]]<br />(Same as ''rectified 16-cell'')<br>Icositetrachoron<ref name="polychoric groups"/> (ico)
|[[File:24 cell verf.svg|60px]]
|align=center|{{CDD|node_1|3|node|4|node|3|node}}<br />{3,4,3}
|(6)<br />[[File:Uniform_polyhedron-43-t2.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'')<Br>Rectified icositetrachoron (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:Uniform_polyhedron-43-t1.svg|30px]]<br />[[cuboctahedron|(3.4.3.4)]]
| 
| 
|(2)<br />[[File:Uniform_polyhedron-43-t0.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'')<br>Truncated icositetrachoron (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:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
| 
|
|(1)<br />[[File:Uniform_polyhedron-43-t0.png|30px]]<br />[[cube|(4.4.4)]]
| 48
| 240
| 384
| 192
|- BGCOLOR="#f0e0e0" align=center 
!25
|align=center|[[cantellated 24-cell]]<br>Small rhombated icositetrachoron (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:Uniform_polyhedron-43-t02.png|30px]]<br />[[small rhombicuboctahedron|(3.4.4.4)]]
|
|(2)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-43-t1.svg|30px]]<br />[[cuboctahedron|(3.4.3.4)]]
| 144
| 720
| 864
| 288
|- BGCOLOR="#f0e0e0" align=center
!28
|align=center|[[cantitruncated 24-cell]]<br>Great rhombated icositetrachoron (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:Uniform_polyhedron-43-t012.png|30px]]<br />[[truncated cuboctahedron|(4.6.8)]]
|
|(1)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-43-t01.png|30px]]<br />[[truncated cube|(3.8.8)]]
| 144
| 720
| 1152
| 576
|- BGCOLOR="#f0e0e0" align=center 
!29
|align=center|[[runcitruncated 24-cell]]<br>Prismatorhombated icositetrachoron (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:Uniform_polyhedron-43-t12.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|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-43-t02.png|30px]]<br />[[small rhombicuboctahedron|(3.4.4.4)]]
| 240
| 1104
| 1440
| 576
|}

{| class="wikitable"
|+ [3<sup>+</sup>,4,3] uniform 4-polytopes
|-
!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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!colspan=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br />{{CDD|node_n0|3|node_n1|4|node_n2|2|2}}<br />(24)
! Pos. 2<br />{{CDD|node_n0|3|node_n1|2|2|node_n3}}<br />(96)
! Pos. 1<br />{{CDD|node_n0|2|2|2|node_n2|3|node_n3}}<br />(96)
! Pos. 0<br />{{CDD|2|2|node_n1|4|node_n2|3|node_n3}}<br />(24)
!Alt
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#d0f0f0" align=center 
!31
|align=center|†[[snub 24-cell]]<br>Snub disicositetrachoron (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:Uniform polyhedron-43-h01.svg|30px]]<br />[[regular icosahedron|(3.3.3.3.3)]]
|
|
|(1)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
|(4)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]] 
| 144
| 480
| 432
| 96
|- BGCOLOR="#d0f0f0" align=center 
![[#Nonuniform alternations|Nonuniform]]
|[[runcic snub 24-cell]]<br>Prismatorhombisnub icositetrachoron (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:Uniform polyhedron-43-h01.svg|30px]]<br />[[regular icosahedron|(3.3.3.3.3)]]
|(2)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|
|(1)<br />[[File:Uniform_polyhedron-33-t01.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 snub 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:Rhombicuboctahedron uniform edge coloring.png|30px]]<br />[[small rhombicuboctahedron|(3.4.4.4)]]
|
|
|(1)<br />[[File:Uniform_polyhedron-43-t1.svg|30px]]<br />[[cuboctahedron|(3.4.3.4)]]
|(2)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
| 144
| 720
| 864
| 288
|- BGCOLOR="#d0f0f0" align=center
![29]
|runcicantic snub 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:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
|
|(1)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Rhombicuboctahedron uniform edge coloring.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, it 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]).

Like the 5-cell, the 24-cell is self-dual, and so the following three forms have twice as many symmetries, bringing their total to 2304 ([[Icositetrachoric symmetry|extended icositetrachoric symmetry]] {{brackets|3,4,3}}).

{| class="wikitable"
|+ {{brackets|3,4,3}} uniform 4-polytopes
|-
!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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!colspan=2|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)
! Cells
! Faces
! Edges
! Vertices
|- BGCOLOR="#e0f0e0" align=center 
!26
|align=center|[[runcinated 24-cell]]<br>Small prismatotetracontoctachoron (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:Uniform_polyhedron-43-t2.png|30px]]<br />[[octahedron|(3.3.3.3)]]
|(6)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
| 240
| 672
| 576
| 144
|- BGCOLOR="#e0f0e0" align=center 
!27
|align=center|[[bitruncated 24-cell]]<br />Tetracontoctachoron (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:Uniform_polyhedron-43-t01.png|30px]]<br />[[truncated cube|(3.8.8)]]
|
| 48
| 336
| 576
| 288
|- BGCOLOR="#e0f0e0" align=center 
!30
|align=center|[[omnitruncated 24-cell]]<br>Great prismatotetracontoctachoron (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:Uniform_polyhedron-43-t012.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
|}

{| class="wikitable"
|+ <nowiki>[[3,4,3]]</nowiki><sup>+</sup> isogonal 4-polytope
|-
!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 4-polytopes 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="#d0f0f0" align=center 
![[#Nonuniform alternations|Nonuniform]]
|align=center|[[omnisnub 24-cell]]<br>Snub tetracontoctachoron (snoc)<br>Snub icositetrachoron (sni)<ref>{{Cite web|url=http://www.bendwavy.org/klitzing/incmats/s3s4s3s.htm|title = S3s4s3s}}</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:Uniform_polyhedron-43-s012.png|30px]]<br />[[snub cube|(3.3.3.3.4)]]
|(2)<br />[[File:Trigonal antiprism.png|30px]]<br />[[octahedron|(3.3.3.3)]]
|(4)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 816
| 2832
| 2592
| 576
|}

===The H<sub>4</sub> family===
{{See|H4 polytope}}

This family has [[Hexacosichoric symmetry|''diploid hexacosichoric'' symmetry]],<ref name="polychoric groups" /> [5,3,3], of [[Symmetry order|order]] 120&times;120=24&times;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|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=5 |Cell counts by location
!colspan=4|Element counts
|-
! Pos. 3<br />{{CDD|node_n0|5|node_n1|3|node_n2|2}}<br />(120)
! Pos. 2<br />{{CDD|node_n0|5|node_n1|2|2|node_n3}}<br />(720)
! Pos. 1<br />{{CDD|node_n0|2|2|node_n2|3|node_n3}}<br />(1200)
! Pos. 0<br />{{CDD|2|node_n1|3|node_n2|3|node_n3}}<br />(600)
!Alt
! Cells
! Faces
! Edges
! Vertices
|-  BGCOLOR="#f0e0e0" align=center
!32
|[[120-cell]]<br />(hecatonicosachoron or dodecacontachoron)<ref name="polychoric groups" /><br>Hecatonicosachoron (hi)
|[[File:120-cell verf.svg|60px]]
| align=center|{{CDD|node_1|5|node|3|node|3|node}}<br />{5,3,3}
|(4)<br />[[File:Uniform_polyhedron-53-t0.png|30px]]<br />[[dodecahedron|(5.5.5)]]
| 
| 
| 
| 
| 120
| 720
| 1200
| 600
|- BGCOLOR="#f0e0e0"  align=center
!33
|[[rectified 120-cell]]<br>Rectified hecatonicosachoron (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:Uniform_polyhedron-53-t1.png|30px]]<br />[[icosidodecahedron|(3.5.3.5)]]
|
|
|(2)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 
| 720
| 3120
| 3600
| 1200
|- BGCOLOR="#f0e0e0"  align=center
!36
|[[truncated 120-cell]]<br>Truncated hecatonicosachoron (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:Uniform_polyhedron-53-t01.png|30px]]<br />[[truncated dodecahedron|(3.10.10)]]
|
|
|(1)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 
| 720
| 3120
| 4800
| 2400
|- BGCOLOR="#f0e0e0"  align=center
!37
|[[cantellated 120-cell]]<br>Small rhombated hecatonicosachoron (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}
|(2)<br />[[File:Uniform_polyhedron-53-t02.png|30px]]<br />[[small rhombicosidodecahedron|(3.4.5.4)]]
|
|(2)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-43-t2.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'')<br>Small disprismatohexacosihecatonicosachoron (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:Uniform_polyhedron-53-t0.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|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-t0.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'')<br>Hexacosihecatonicosachoron (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:Uniform_polyhedron-53-t12.png|30px]]<br />[[truncated icosahedron|(5.6.6)]]
|
|
|(2)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
| 
| 720
| 4320
| 7200
| 3600
|- BGCOLOR="#f0e0e0"  align=center
!42
|[[cantitruncated 120-cell]]<br>Great rhombated hecatonicosachoron (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:Uniform_polyhedron-53-t012.png|30px]]<br />[[truncated icosidodecahedron|(4.6.10)]]
|
|(1)<br />[[File:triangular prism.png|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
| 
| 1920
| 9120
| 14400
| 7200
|- BGCOLOR="#f0e0e0" align=center
!43
|[[runcitruncated 120-cell]]<br>Prismatorhombated hexacosichoron (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:Uniform_polyhedron-53-t01.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|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-43-t1.svg|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'')<br>Great disprismatohexacosihecatonicosachoron (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:Uniform_polyhedron-53-t012.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:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
| 
| 2640
| 17040
| 28800
| 14400
|- BGCOLOR="#d0f0f0" align=center
![[#Nonuniform alternations|Nonuniform]]
||[[omnisnub 120-cell]]<br>Snub hecatonicosachoron (snixhi)<ref>{{Cite web|url=http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm|title = S3s3s5s}}</ref><br />(Same as the ''omnisnub 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:Uniform_polyhedron-53-s012.png|30px]] (1)<br />[[Snub dodecahedron|(3.3.3.3.5)]]
|[[File:pentagonal antiprism.png|30px]] (1)<br />[[pentagonal antiprism|(3.3.3.5)]]
|[[File:Trigonal antiprism.png|30px]] (1)<br />[[triangular antiprism|(3.3.3.3)]]
|[[File:Uniform_polyhedron-33-s012.png|30px]] (1)<br />[[Snub tetrahedron|(3.3.3.3.3)]]
|[[File:Uniform_polyhedron-33-t0.png|30px]] (4)<br />[[tetrahedron|(3.3.3)]]
| 9840
| 35040
| 32400
| 7200
|}

==== 600-cell truncations ====

{| class="wikitable"
!rowspan=2|#
!rowspan=2|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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!rowspan=2|Symmetry
!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]]<br>Hexacosichoron<ref name="polychoric groups"/> (ex)
|[[File:600-cell verf.svg|60px]]
| align=center|{{CDD|node|5|node|3|node|3|node_1}}<br />{3,3,5}
|[5,3,3]<br />order 14400
| 
| 
| 
|(20)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 600
| 1200
| 720
| 120
|- BGCOLOR="#f0e0f0" align=center
![47]
|[[600-cell#Diminished 600-cells|20-diminished 600-cell]]<br />= [[Grand antiprism]] (gap)
|[[File:Grand antiprism verf.png|60px]]
|Nonwythoffian<br />construction
| [[10,2<sup>+</sup>,10]]<br />order 400<br />Index 36
|(2)<br />[[File:pentagonal antiprism.png|30px]]<br />[[pentagonal antiprism|(3.3.3.5)]]
|
|
|(12)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
|320 
|720
|500 
|100 
|- BGCOLOR="#f0e0f0" align=center 
![31]
|[[600-cell#Diminished 600-cells|24-diminished 600-cell]]<br />= [[Snub 24-cell]] (sadi)
|[[File:Snub 24-cell verf.png|60px]]
|Nonwythoffian<br />construction
| [3<sup>+</sup>,4,3]<br />order 576<br />index 25
|(3)<br />[[File:Uniform_polyhedron-53-t2.png|30px]]<br />[[regular icosahedron|(3.3.3.3.3)]]
|
|
|(5)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]] 
| 144
| 480
| 432
| 96

|- BGCOLOR="#f0e0f0" align=center 
!Nonuniform
|[[600-cell#Diminished 600-cells|bi-24-diminished 600-cell]]<br>Bi-icositetradiminished hexacosichoron (bidex)
|[[File:Biicositetradiminished 600-cell vertex figure.png|60px]]
|Nonwythoffian<br />construction
| order 144<br />index 100
|(6)<br />[[File:tridiminished icosahedron.png|30px]]<br />[[tridiminished icosahedron|tdi]]
|
|
|
| 48
| 192
| 216
| 72

|- BGCOLOR="#e0e0f0" align=center
!34
|[[rectified 600-cell]]<br>Rectified hexacosichoron (rox)
|[[File:Rectified 600-cell verf.png|60px]]
| align=center|{{CDD|node|5|node|3|node_1|3|node}}<br />r{3,3,5}
| [5,3,3]
|(2)<br />[[File:Uniform_polyhedron-53-t2.png|30px]]<br />[[regular icosahedron|(3.3.3.3.3)]]
| 
|
|(5)<br />[[File:Uniform_polyhedron-43-t2.png|30px]]<br />[[octahedron|(3.3.3.3)]]
| 720
| 3600
| 3600
| 720

|- BGCOLOR="#f0e0f0" align=center 
!Nonuniform
|[[Rectified 600-cell#Diminished rectified 600-cell|120-diminished rectified 600-cell]]<br>Swirlprismatodiminished rectified hexacosichoron (spidrox)
|[[File:Spidrox-vertex figure.png|60px]]
|Nonwythoffian<br />construction
| order 1200<br />index 12
|(2)<br />[[File:pentagonal antiprism.png|30px]]<br />[[pentagonal antiprism|3.3.3.5]]
|(2)<br />[[File:pentagonal prism.png|30px]]<br />[[pentagonal prism|4.4.5]]
|
|(5)<br />[[File:Square pyramid.png|30px]]<br />[[Square pyramid|P4]]
| 840
| 2640 
| 2400
| 600

|- BGCOLOR="#e0e0f0" align=center
!41
|[[truncated 600-cell]]<br>Truncated hexacosichoron (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}
| [5,3,3]
|(1)<br />[[File:Uniform_polyhedron-53-t2.png|30px]]<br />[[regular icosahedron|(3.3.3.3.3)]]
| 
|
|(5)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
| 720
| 3600
| 4320
| 1440
|-  BGCOLOR="#e0e0f0" align=center
!40
|[[cantellated 600-cell]]<br>Small rhombated hexacosichoron (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}
| [5,3,3]
|(1)<br />[[File:Uniform_polyhedron-53-t1.png|30px]]<br />[[icosidodecahedron|(3.5.3.5)]]
|(2)<br />[[File:pentagonal prism.png|30px]]<br />[[pentagonal prism|(4.4.5)]]
|
|(1)<br />[[File:Uniform_polyhedron-43-t1.svg|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}
| [5,3,3]
|(1)<br />[[File:Uniform_polyhedron-53-t0.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|25px]]<br />[[triangular prism|(3.4.4)]]
|(1)<br />[[File:Uniform_polyhedron-33-t0.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}
| [5,3,3]
|(2)<br />[[File:Uniform_polyhedron-53-t12.png|30px]]<br />[[truncated icosahedron|(5.6.6)]]
|
|
|(2)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
| 720
| 4320
| 7200
| 3600
|- BGCOLOR="#e0e0f0" align=center
!45
|[[cantitruncated 600-cell]]<br>Great rhombated hexacosichoron (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}
| [5,3,3]
|(1)<br />[[File:Uniform_polyhedron-53-t12.png|30px]]<br />[[truncated icosahedron|(5.6.6)]]
|(1)<br />[[File:pentagonal prism.png|30px]]<br />[[pentagonal prism|(4.4.5)]]
|
|(2)<br />[[File:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
| 1440
| 8640
| 14400
| 7200
|- BGCOLOR="#e0e0f0" align=center
!44
|[[runcitruncated 600-cell]]<br>Prismatorhombated hecatonicosachoron (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}
| [5,3,3]
|(1)<br />[[File:Uniform_polyhedron-53-t02.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:Uniform_polyhedron-33-t01.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}
| [5,3,3]
|(1)<br />[[File:Uniform_polyhedron-53-t012.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:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
| 2640
| 17040
| 28800
| 14400
|}

=== The D<sub>4</sub> family ===
{{See|D4 polytope}}

This [[Demitesseractic symmetry|demitesseract family]], [3<sup>1,1,1</sup>], introduces no new uniform 4-polytopes, but it is worthy to repeat these alternative constructions. This family has [[Symmetry order|order]] 12&times;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 4-polytopes, [3<sup>1,1,1</sup>]<sup>+</sup>, order 96.

{| class="wikitable"
|+ [3<sup>1,1,1</sup>] uniform 4-polytopes
|-
!rowspan=2|#
!rowspan=2|Name (Bowers style acronym)
!rowspan=2| [[Vertex figure|Vertex<br />figure]]
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]<br />[[File:CD B4 nodes.png]]<br />{{CDD|nodes_10ru|split2|node_n2|3|node_n3}} = {{CDD|node_h1|4|node|3|node_n2|3|node_n3}}<br />{{CDD|nodes_10ru|split2|node_c1|3|node_c2}} = {{CDD|node_h1|4|node|3|node_c1|3|node_c2}}
!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.svg|60px]]
| align=center|{{CDD|nodes_10ru|split2|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node}}<br />h{4,3,3}
| 
| 
|(4)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
|(4)<br />[[File:Uniform_polyhedron-33-t0.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:Uniform_polyhedron-43-t2.png|30px]]<br />[[octahedron|(3.3.3.3)]]
| 
|(2)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
|(2)<br />[[File:Uniform_polyhedron-33-t01.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:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
| 
|(1)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
|(3)<br />[[File:Uniform_polyhedron-43-t1.svg|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:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
|
|(1)<br />[[File:Uniform_polyhedron-33-t01.png|30px]]<br />[[truncated tetrahedron|(3.6.6)]]
|(2)<br />[[File:Uniform_polyhedron-43-t12.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 4-polytopes
|-
!rowspan=2|#
!rowspan=2|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:Uniform_polyhedron-43-t2.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:Uniform_polyhedron-43-t1.svg|30px]]<br />[[cuboctahedron|(3.4.3.4)]]
|(2)<br />[[File:Uniform_polyhedron-43-t0.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:Uniform_polyhedron-43-t12.png|30px]]<br />[[truncated octahedron|(4.6.6)]]
|(1)<br />[[File:Uniform_polyhedron-43-t0.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:Uniform_polyhedron-33-s012.png|30px]]<br />[[regular icosahedron|(3.3.3.3.3)]]
|(1)<br />[[File:Uniform_polyhedron-33-t0.png|30px]]<br />[[tetrahedron|(3.3.3)]]
|(4)<br />[[File:Uniform_polyhedron-33-t0.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 4-polytope, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the [[snub cube]] and the [[snub dodecahedron]].

===The grand antiprism===
There is one non-Wythoffian uniform convex 4-polytope, 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| 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
!rowspan=2|Net
|-
! Cells
! Faces
! Edges
! Vertices
|-
!47
![[grand antiprism]] (gap)
|[[File:Grand antiprism.png|50px]]
|[[File:Grand antiprism verf.png|50px]]
|No symbol
|300 [[File:Uniform_polyhedron-33-t0.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
|[[File:Pentagonal double antiprismoid net.png|50px]]
|}

===Prismatic uniform 4-polytopes===

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 4-polytopes 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 4-polytopes is the ''polyhedral prisms,'' i.e. products of a polyhedron with a [[line segment]].  The cells of such a 4-polytopes 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 4-polytope.

{| class="wikitable"
|+ [3,3,2] uniform 4-polytopes
!rowspan=2|#
!rowspan=2| 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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!rowspan=2 colspan=3|Cells by type
!colspan=4|Element counts
!rowspan=2|Net
|-
! 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:Uniform_polyhedron-33-t0.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
|[[File:Tetrahedron prism net.png|75px]]
|- 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:Uniform_polyhedron-33-t01.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
|[[File:Truncated tetrahedral prism net.png|75px]]
|}

{| class="wikitable"
|+ <nowiki>[[</nowiki>3,3],2] uniform 4-polytopes
!rowspan=2|#
!rowspan=2| 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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!rowspan=2 colspan=3|Cells by type
!colspan=4|Element counts
!rowspan=2|Net
|-
! 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:Uniform_polyhedron-43-t2.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
|[[File:Octahedron prism net.png|75px]]
|- 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:Uniform_polyhedron-43-t1.svg|20px]]<br />[[Cuboctahedron|3.4.3.4]]
|8 [[File:Triangular prism.png|20px]]<br />[[Triangular prism|3.4.4]]
|6 [[File:Uniform_polyhedron-43-t0.png|20px]]<br />[[Cube|4.4.4]]
|16
|16 {3}<br />36 {4}
|60
|24
|[[File:Cuboctahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-43-t12.png|20px]]<br />[[Truncated octahedron|4.6.6]] 
|8 [[File:Hexagonal prism.png|20px]]<br />[[Hexagonal prism|6.4.4]]
|6 [[File:Uniform_polyhedron-43-t0.png|20px]]<br />[[Cube|4.4.4]]
|16
|48 {4}<br />16 {6}
|96
|48
|[[File:Truncated octahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0" 
![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:Uniform_polyhedron-53-t2.png|20px]]<br />[[regular 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
|[[File:Icosahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0" 
![[#Nonuniform alternations|Nonuniform]]
|[[omnisnub tetrahedral antiprism]]<br>Pyritohedral icosahedral antiprism (pikap)
|
|[[File:Snub 332 verf.png|75px]]
|align=center|{{CDD|node_h|2x|node_h|3|node_h|3|node_h}}<br /><math>s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}</math>
|2 [[File:Uniform polyhedron-33-s012.png|20px]]<br />[[snub tetrahedron|3.3.3.3.3]]
|8 [[File:Trigonal antiprism.png|20px]]<br />[[triangular antiprism|3.3.3.3]]
|6+24 [[File:Uniform_polyhedron-33-t0.png|20px]]<br />[[tetrahedron|3.3.3]]
|40
|16+96 {3}
|96
|24
|
|}

==== Octahedral prisms: B<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 alternated 4-polytopes below. [[Coxeter notation|Symmetries]] 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| 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 4-polytopes and 3-space honeycombs|Schläfli]]<br />symbols
!rowspan=2 colspan=4|Cells by type
!colspan=4|Element counts
!rowspan=2|Net
|-
! 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:Uniform_polyhedron-43-t0.png|20px]]<br />[[cube|4.4.4]]
|6 [[File:Uniform_polyhedron-43-t0.png|20px]]<br />[[cube|4.4.4]]
|
|
|8||24 {4}||32||16
|[[File:8-cell net.png|75px]]
|- 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:Uniform_polyhedron-43-t1.svg|20px]]<br />[[Cuboctahedron|3.4.3.4]] 
| 8 [[File:Triangular prism.png|20px]]<br />[[Triangular prism|3.4.4]]
| 6 [[File:Uniform_polyhedron-43-t0.png|20px]]<br />[[Cube|4.4.4]]
|
| 16|| 16 {3}<br />36 {4}|| 60|| 24
|[[File:Cuboctahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-43-t2.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
|[[File:Octahedron prism net.png|75px]]
|- 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:Uniform_polyhedron-43-t02.png|20px]]<br />[[Rhombicuboctahedron|3.4.4.4]] 
| 8 [[File:Triangular prism.png|20px]]<br />[[Triangular prism|3.4.4]]
| 18 [[File:Uniform_polyhedron-43-t0.png|20px]]<br />[[Cube|4.4.4]]
|
| 28|| 16 {3}<br />84 {4}|| 120|| 48
|[[File:Small rhombicuboctahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-43-t01.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
|[[File:Truncated cubic prism net.png|75px]]
|- 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:Uniform_polyhedron-43-t12.png|20px]]<br />[[Truncated octahedron|4.6.6]]
| 6 [[File:Uniform_polyhedron-43-t0.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
|[[File:Truncated octahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-43-t012.png|20px]]<br />[[Truncated cuboctahedron|4.6.8]]
| 12 [[File:Uniform_polyhedron-43-t0.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
|[[File:Great rhombicuboctahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0"
!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:Uniform_polyhedron-43-t0.png|20px]]<br />[[Cube|4.4.4]]
|
| 40|| 64 {3}<br />72 {4}|| 144|| 48
|[[File:Snub cuboctahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0"
![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:Uniform_polyhedron-33-t0.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
|[[File:Tetrahedron prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0"
![49]
|[[Truncated tetrahedral prism]] (tuttip)
|[[File:Truncated tetrahedral prism.png|75px]]
|[[File:Truncated tetrahedral prism verf.png|75px]]
|align=center|{{CDD|node_h1|4|node|3|node_1|2|node_1}}<br />h<sub>2</sub>{4,3}×{&nbsp;}
|2 [[File:Uniform_polyhedron-33-t01.png|20px]]<br />[[Truncated tetrahedron|3.3.6]]
|4 [[File:Triangular prism.png|20px]]<br />[[Triangular prism|3.4.4]]
|4 [[File:Hexagonal prism.png|20px]]<br />[[Hexagonal prism|4.4.6]]
|
|6||8 {3}<br />6 {4}||16||8
|[[File:Truncated tetrahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0"
![50]
|align=center|[[Cuboctahedral prism]] (cope)
|[[File:Cuboctahedral prism.png|75px]]
|[[File:Cuboctahedral prism verf.png|75px]]
|align=center|{{CDD|node_h0|4|node_1|3|node|2|node_1}}<br />rr{3,3}×{&nbsp;}
| 2 [[File:Uniform_polyhedron-43-t1.svg|20px]]<br />[[Cuboctahedron|3.4.3.4]] 
| 8 [[File:Triangular prism.png|20px]]<br />[[Triangular prism|3.4.4]]
| 6 [[File:Uniform_polyhedron-43-t0.png|20px]]<br />[[Cube|4.4.4]]
|
| 16|| 16 {3}<br />36 {4}|| 60|| 24
|[[File:Cuboctahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0"
![52]
|[[Rhombicuboctahedral prism]] (sircope)
|[[File:Rhombicuboctahedral prism.png|75px]]
|[[File:Rhombicuboctahedron prism verf.png|75px]]
|align=center|{{CDD|node_1|4|node_h|3|node_h|2|node_1}}<br />s<sub>2</sub>{3,4}×{&nbsp;}
| 2 [[File:Rhombicuboctahedron uniform edge coloring.png|20px]]<br />[[Rhombicuboctahedron|3.4.4.4]] 
| 8 [[File:Triangular prism.png|20px]]<br />[[Triangular prism|3.4.4]]
| 18 [[File:Uniform_polyhedron-43-t0.png|20px]]<br />[[Cube|4.4.4]]
|
| 28|| 16 {3}<br />84 {4}|| 120|| 48
|[[File:Small rhombicuboctahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0"
![54]
|align=center|[[Truncated octahedral prism]] (tope)
|[[File:Truncated octahedral prism.png|75px]]
|[[File:Truncated octahedral prism verf.png|75px]]
|align=center|{{CDD|node_h0|4|node_1|3|node_1|2|node_1}}<br />tr{3,3}×{&nbsp;}
| 2 [[File:Uniform_polyhedron-43-t12.png|20px]]<br />[[Truncated octahedron|4.6.6]]
| 6 [[File:Uniform_polyhedron-43-t0.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
|[[File:Truncated octahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0" 
![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:Uniform_polyhedron-53-t2.png|20px]]<br />[[regular 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
|[[File:Icosahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0" 
![12]
|align=center|[[16-cell]] (hex)
|[[File:Schlegel wireframe 16-cell.png|75px]]
|[[File:16-cell verf.svg|75px]]
|align=center|{{CDD|node_h|2x|node_h|4|node|3|node}}<br />s{2,4,3}
|2+6+8 [[File:Uniform_polyhedron-33-t0.png|20px]]<br />[[tetrahedron|3.3.3.3]]
|
|
|
|16||32 {3}||24||8
|[[File:16-cell net.png|75px]]
|- align=center BGCOLOR="#d0f0f0" 
![[#Nonuniform alternations|Nonuniform]]
|[[Omnisnub tetrahedral antiprism]]<br>= Pyritohedral icosahedral antiprism (pikap)
|
|[[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:Uniform_polyhedron-53-t2.png|20px]]<br />[[regular icosahedron|3.3.3.3.3]]
|8 [[File:Trigonal_antiprism.png|20px]]<br />[[octahedron|3.3.3.3]]
|6+24 [[File:Uniform_polyhedron-33-t0.png|20px]]<br />[[tetrahedron|3.3.3]]
|
|40||16+96 {3}||96||24
|
|- align=center BGCOLOR="#d0f0f0" 
![[#Nonuniform alternations|Nonuniform]]
|[[Truncated cuboctahedral prism#Related polytopes|Edge-snub octahedral hosochoron]]<br>Pyritosnub alterprism (pysna)
|
|[[File:Bialternatosnub octahedral hosochoron vertex figure.png|75px]]
|align=center|{{CDD|node_1|4|node_h|3|node_h|2x|node_h}}<br />sr<sub>3</sub>{2,3,4}
|2 [[File:Rhombicuboctahedron uniform edge coloring.png|20px]]<br />[[rhombicuboctahedron|3.4.4.4]]
|6 [[File:Cube rotorotational symmetry.png|20px]]<br />[[cube|4.4.4]]
|8 [[File:Trigonal antiprism.png|20px]]<br />[[octahedron|3.3.3.3]]
|24 [[File:Triangular prism.png|20px]]<br />[[triangular prism|3.4.4]]
|40||16+48 {3}<br>12+12+24+24 {4} ||144||48
|
|- align=center BGCOLOR="#d0f0f0" 
![[#Nonuniform alternations|Nonuniform]]
|[[Omnisnub cubic antiprism]]<br>Snub cubic antiprism (sniccap)
|
|[[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:Uniform_polyhedron-33-t0.png|20px]]<br />[[tetrahedron|3.3.3]]
| 8 [[File:Trigonal_antiprism.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
|
|- align=center BGCOLOR="#d0f0f0" 
![[#Nonuniform alternations|Nonuniform]]
|[[Runcic snub cubic hosochoron]]<br />Truncated tetrahedral alterprism (tuta)
|[[File:Runcic snub cubic hosochoron.png|75px]]
| [[File:Runcic snub 243 verf.png|75px]]
|{{CDD|node_h|2x|node_h|4|node|3|node_1}}<br />s<sub>3</sub>{2,4,3}
| 2 [[File:Uniform_polyhedron-33-t01.png|20px]]<br />[[truncated tetrahedron|3.6.6]]
| 6 [[File:Uniform_polyhedron-33-t0.png|20px]]<br />[[tetrahedron|3.3.3]]
| 8 [[File:Triangular cupola.png|20px]]<br />[[triangular cupola]]
|
|16 ||52 ||60 ||24
|[[File:Truncated tetrahedral cupoliprism net.png|75px]]
|}

==== 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| Name (Bowers name and 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
!rowspan=2|Net
|-
! 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:Uniform_polyhedron-53-t0.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
|[[File:Dodecahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-53-t1.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
|[[File:Icosidodecahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-53-t2.png|20px]]<br />[[regular 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
|[[File:Icosahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-53-t01.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|4.4.10]]
| 
| 34
| 40 {3}<br />90 {4}<br />24 {10}
| 240
| 120
|[[File:Truncated dodecahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-53-t02.png|20px]]<br />[[rhombicosidodecahedron|3.4.5.4]]
|20 [[File:Triangular prism.png|20px]]<br />[[Triangular prism|3.4.4]]
|30 [[File:Uniform_polyhedron-43-t0.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
|[[File:Small rhombicosidodecahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-53-t12.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
|[[File:Truncated icosahedral prism net.png|75px]]
|- 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:Uniform_polyhedron-53-t012.png|20px]]<br />[[Truncated icosidodecahedron|4.6.10]] 
|30 [[File:Uniform_polyhedron-43-t0.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
|[[File:Great rhombicosidodecahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0"
!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
| 160 {3}<br />150 {4}<br />24 {5}
| 360
| 120
|[[File:Snub icosidodecahedral prism net.png|75px]]
|- align=center BGCOLOR="#d0f0f0"
![[#Nonuniform alternations|Nonuniform]]
|[[Omnisnub dodecahedral antiprism]]<br>Snub dodecahedral antiprism (sniddap)
|
|[[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 ccw.png|20px]]<br />[[snub dodecahedron|3.3.3.3.5]] 
|30+120 [[File:Uniform_polyhedron-33-t0.png|20px]]<br />[[tetrahedron|3.3.3]]
|20 [[File:Uniform_polyhedron-43-t2.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 a [[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 extended [[f-vector]] of {''p''}×{''q''} is (''p'',''p'',1)*(''q'',''q'',1) = (''pq'',2''pq'',''pq''+''p''+''q'',''p''+''q'').
* 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
!Net
|- 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]]||[[File:3-3 duoprism net.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]]||[[File:4-3 duoprism net.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]]||[[File:8-cell net.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]]||[[File:5-3 duoprism net.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:4-5 duoprism.png|75px]] [[File:5-4 duoprism.png|75px]]||[[File:5-4 duoprism net.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]]||[[File:5-5 duoprism net.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]]||[[File:6-3 duoprism net.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]]||[[File:6-4 duoprism net.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]]||[[File:6-5 duoprism net.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]]||[[File:6-6 duoprism net.png|75px]]
|}

{| class="wikitable" width=600px
|- align=center
|[[File:3-3 duoprism.png|75px]]<br />[[3-3 duoprism|3-3]]
|[[File:3-4 duoprism.png|75px]]<br />[[3-4 duoprism|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 />[[3-4 duoprism|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
|}

Alternations are possible. {{CDD|node_h|p|node_h|2x|node_h|q|node_h}} = {{CDD|node_h|2x|p|node|2x|node_h|2x|q|node}} gives the family of ''duoantiprisms'', but they generally cannot be made uniform. p=q=2 is the only ''convex'' case that can be made uniform, giving the regular 16-cell. p=5, q=5/3 is the only nonconvex case that can be made uniform, giving the so-called [[great duoantiprism]]. {{CDD|node_h|p|node_h|2x|node_h|2x|q|node_1}} gives the p-2q-gonal ''prismantiprismoid'' (an edge-alternation of the 2p-4q duoprism), but this cannot be made uniform in any cases.<ref>[https://bendwavy.org/klitzing/incmats/sns2s2mx.htm sns2s2mx], Richard Klitzing</ref>

==== 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''') The second polytope in the series is a lower symmetry of the regular [[tesseract]], {4}×{4}.

{{Convex prismatic prisms}}

==== Polygonal antiprismatic prisms: [p] × [ ] × [ ] ====

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 ===
[[File:Snubcubes in grCO.svg|320px|thumb|Like the 3-dimensional [[snub cube]], {{CDD|node_h|4|node_h|3|node_h}}, an [[alternation (geometry)|alternation]] removes half the vertices, in two chiral sets of vertices from the ringed form {{CDD|node_1|4|node_1|3|node_1}}, however the ''uniform'' solution requires the vertex positions be adjusted for equal lengths. In four dimensions, this adjustment is only possible for 2 alternated figures, while the rest only exist as nonequilateral alternated figures.]]
Coxeter showed only two uniform solutions for rank 4 Coxeter groups with all rings [[alternation (geometry)|alternated]] (shown with empty circle nodes). The first is {{CDD|node_h|2x|node_h|2x|node_h|2x|node_h}}, s{2<sup>1,1,1</sup>} which represented an index 24 subgroup ([[Coxeter notation|symmetry]] [2,2,2]<sup>+</sup>, order 8) form of the [[demitesseract]], {{CDD|node_h|4|node|3|node|3|node}}, h{4,3,3} (symmetry [1<sup>+</sup>,4,3,3] = [3<sup>1,1,1</sup>], order 192). The second is {{CDD|node_h|3|node_h|split1|nodes_hh}}, s{3<sup>1,1,1</sup>}, which is an index 6 subgroup (symmetry [3<sup>1,1,1</sup>]<sup>+</sup>, order 96) form of the [[snub 24-cell]], {{CDD|node_h|3|node_h|4|node|3|node}}, s{3,4,3}, (symmetry [3<sup>+</sup>,4,3], order 576).

Other alternations, such as {{CDD|node_h|4|node_h|3|node_h|3|node_h}}, as an alternation from the [[omnitruncated tesseract]] {{CDD|node_1|4|node_1|3|node_1|3|node_1}}, can not be made uniform as solving for equal edge lengths are in general [[overdetermined system|overdetermined]] (there are six equations but only four variables). Such nonuniform alternated figures can be constructed as [[vertex-transitive]] 4-polytopes by the removal of one of two half sets of the vertices of the full ringed figure, but will have unequal edge lengths. Just like uniform alternations, they will have half of the symmetry of uniform figure, like [4,3,3]<sup>+</sup>, order 192, is the symmetry of the ''alternated omnitruncated tesseract''.<ref>H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) p. 582-588 2.7 ''The four-dimensional analogues of the snub cube''</ref>

Wythoff constructions with alternations produce [[vertex-transitive]] figures that can be made equilateral, but not uniform because the alternated gaps (around the removed vertices) create cells that are not regular or semiregular. A proposed name for such figures is '''scaliform polytopes'''.<ref>{{Cite web|url=http://bendwavy.org/klitzing/explain/polytope-tree.htm#scaliform|title = Polytope-tree}}</ref> This category allows a subset of [[Johnson solid]]s as cells, for example [[triangular cupola]].

Each [[vertex configuration]] within a Johnson solid must exist within the vertex figure. For example, a square pyramid has two vertex configurations: 3.3.4 around the base, and 3.3.3.3 at the apex.

The nets and vertex figures of the four convex equilateral cases are given below, along with a list of cells around each vertex.
{| class=wikitable width=400
|+ Four convex vertex-transitive equilateral 4-polytopes with nonuniform cells
!Coxeter<br />diagram
!s<sub>3</sub>{2,4,3}, {{CDD|node_h|2x|node_h|4|node|3|node_1}}
!s<sub>3</sub>{3,4,3}, {{CDD|node_h|3|node_h|4|node|3|node_1}}
!colspan=2|Others
|-
!Relation
!24 of 48 vertices of <br />[[rhombicuboctahedral prism]]
!288 of 576 vertices of<br />[[runcitruncated 24-cell]]
!72 of 120 vertices<br />of [[600-cell]]
!600 of 720 vertices<br />of [[rectified 600-cell]]
|- valign=top align=center
!valign=center|Projection
|[[File:Tutcup-ortho.png|200px]]
|[[File:24-cell s3 B3.png|200px]]
|[[File:Bidex ortho 12-gon.png|200px]]
|[[File:Spidrox-square pyramid ring.png|200px]]<BR>Two rings of pyramids
|- align=center valign=top
!valign=center|Net
|[[File:Truncated tetrahedral cupoliprism net.png|200px]]<br />[[runcic snub cubic hosochoron]]<ref>{{cite web |url=http://bendwavy.org/klitzing/incmats/tut=invtut.htm |title = tuta}}</ref><ref>[http://www.polytope.net/hedrondude/simplesc.htm Category S1: Simple Scaliforms] tutcup</ref>
|[[File:Prismatorhombisnub icositetrachoron net.png|200px]]<br />[[runcic snub 24-cell]]<ref>{{Cite web|url=http://bendwavy.org/klitzing/incmats/prissi.htm|title = Prissi}}</ref><ref>[http://www.polytope.net/hedrondude/special.htm Category S3: Special Scaliforms] prissi</ref>
|[[File:Biicositetradiminished hexacosichoron net.png|200px]]<ref>{{Cite web|url=https://bendwavy.org/klitzing/incmats/bidex.htm|title=bidex|website=bendwavy.org|accessdate=11 November 2023}}</ref><ref>[http://www.polytope.net/hedrondude/special.htm Category S3: Special Scaliforms] bidex</ref><ref>[http://eusebeia.dyndns.org/4d/bi24dim600cell The Bi-icositetradiminished 600-cell]</ref>
|[[File:Swirlprismatodiminished rectified hexacosichoron net.png|200px]]<ref>{{Cite web|url=https://bendwavy.org/klitzing/incmats/spidrox.htm|title=spidrox|website=bendwavy.org|accessdate=11 November 2023}}</ref><ref>[http://www.polytope.net/hedrondude/scaleswirl.htm Category S4: Scaliform Swirlprisms] spidrox</ref> 
|-
!valign=center|Cells
! [[File:Triangular cupola.png|40px]] [[File:Tetrahedron.png|40px]] [[File:Truncated tetrahedron.png|40px]]
! [[File:Triangular cupola.png|40px]] [[File:Truncated tetrahedron.png|40px]] [[File:Icosahedron.png|40px]] [[File:Triangular prism.png|40px]] 
![[File:tridiminished icosahedron.png|40px]]
![[File:Square pyramid.png|40px]] [[File:Pentagonal prism.png|40px]] [[File:Pentagonal antiprism.png|40px]]
|- valign=top
!valign=center|Vertex<br />figure
|[[File:Runcic snub 243 verf.png|200px]]<br />(1) 3.4.3.4: [[triangular cupola]]<br />(2) 3.4.6: triangular cupola<br />(1) 3.3.3: [[tetrahedron]]<br />(1) 3.6.6: [[truncated tetrahedron]]
|[[File:Runcic snub 24-cell verf.png|150px]]<br />(1) 3.4.3.4: triangular cupola<br />(2) 3.4.6: triangular cupola<br />(2) 3.4.4: [[triangular prism]]<br />(1) 3.6.6: [[truncated tetrahedron]]<br />(1) 3.3.3.3.3: [[icosahedron]]
|[[File:Biicositetradiminished_600-cell_vertex_figure.png|200px]]<br />(2) 3.3.3.5: [[tridiminished icosahedron]] <br />(4) 3.5.5: tridiminished icosahedron
|[[File:Spidrox-vertex figure.png|150px]]<br />(1) 3.3.3.3: [[square pyramid]]<br />(4) 3.3.4: square pyramid<br />(2) 4.4.5: [[pentagonal prism]]<br />(2) 3.3.3.5 [[pentagonal antiprism]]
|}

=== Geometric derivations for 46 nonprismatic Wythoffian uniform polychora === <!--This section is linked from [[Convex uniform honeycomb]] -->
The 46 Wythoffian 4-polytopes include the six [[convex regular 4-polytope]]s. 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 width=1000
|[[File:Polychoron truncation chart.png|400px]]<br />Summary chart of truncation operations
|[[File:Uniform honeycomb truncations.png|600px]]<br />Example locations of kaleidoscopic generator point on fundamental domain.
|}

The geometric operations that derive the 40 uniform 4-polytopes from the regular 4-polytopes are ''truncating'' operations. A 4-polytope 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>0,2,3</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]<br />=[(3,p,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_10ru|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_10ru|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_10ru|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_10ru|split2|node_1|q|node_1}}
|- align=center
!Quarter
|q{2p,3,2q}
|[1<sup>+</sup>,2p,3,2q,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
!Bisnub
|2s{2p,q,2r}
|[2p,q<sup>+</sup>,2r]
|{{CDD|node|2x|p|node_h|q|node_h|2x|r|node}}
|Alternated bitruncation
|- align=center
!Omnisnub
|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>, B<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. The D<sub>4</sub> symmetry is also included, though it only creates duplicates. Alternations are grouped by their chiral symmetry. All alternations are given, although the [[snub 24-cell]], with its 3 constructions from different families is the only one that is uniform. Counts in parentheses 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 B<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>
|BGCOLOR="#e0f0e0"|[2<sup>+</sup>[3,3,3]]<sup>+</sup><br />(order 120)||BGCOLOR="#e0f0e0"|(1)
|BGCOLOR="#e0f0e0"|{{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|4|node|3|node}})<br />{{CDD|node_h|3|node_h|4|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>
|BGCOLOR="#e0f0e0"|[1<sup>+</sup>,4,3,3]<sup>+</sup><br />(order 96)||BGCOLOR="#e0f0e0"|(2)
|BGCOLOR="#e0f0e0"|{{CDD|node_h1|4|node|3|node|3|node}}<sub>(12)</sub> (= {{CDD|nodes_10ru|split2|node|3|node}})<br />{{CDD|node|4|node_h|3|node_h|3|node_h}}<sub>(31)</sub><br />{{CDD|node_1|4|node_h|3|node_h|3|node_h}}<sub>(−)</sub>
|- align=center
|BGCOLOR="#e0f0e0"|[4,3,3]<sup>+</sup><br />(order 192)||BGCOLOR="#e0f0e0"|(1)
|BGCOLOR="#e0f0e0"|{{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>
|BGCOLOR="#e0f0e0"|[2<sup>+</sup>[3,4,3]]<sup>+</sup><br />(order 1152)||BGCOLOR="#e0f0e0"|(1)
|BGCOLOR="#e0f0e0"|{{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>
|BGCOLOR="#e0f0e0"|[5,3,3]<sup>+</sup><br />(order 7200)||BGCOLOR="#e0f0e0"|(1)
|BGCOLOR="#e0f0e0"|{{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}}
|BGCOLOR="#e0f0e0"|[1[3,2,3]]=[[3],2,3]<sup>+</sup>=[6,2,3]<sup>+</sup><br />(order 36)||BGCOLOR="#e0f0e0"|(1)
|BGCOLOR="#e0f0e0"|{{CDD|node_h|3|node_h|2|node_1|3|node}}
|- 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}}
|BGCOLOR="#e0f0e0"|[(2<sup>+</sup>,4)[3,2,3]]<sup>+</sup>=[2<sup>+</sup>[6,2,6]]<sup>+</sup><br />(order 144)||BGCOLOR="#e0f0e0"|(1)
|BGCOLOR="#e0f0e0"|{{CDD|node_h|3|node_h|2x|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*,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>
|BGCOLOR="#e0f0e0"|[(3,3)[4,2*,4]]<sup>+</sup>=[4,3,3]<sup>+</sup><br />(order 192)||BGCOLOR="#e0f0e0"|(1)
|BGCOLOR="#e0f0e0"| {{CDD|node|4|node_h|2x|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}}
|BGCOLOR="#e0f0e0"|[1[4,2,4]]=[[4],2,4]<sup>+</sup>=[8,2,4]<sup>+</sup><br />(order 64)||BGCOLOR="#e0f0e0"|(1)
| {{CDD|node_h|4|node_h|2x|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}}
|BGCOLOR="#e0f0e0"|[(2<sup>+</sup>,4)[4,2,4]]<sup>+</sup>=[2<sup>+</sup>[8,2,8]]<sup>+</sup><br />(order 256)||BGCOLOR="#e0f0e0"|(1)
|BGCOLOR="#e0f0e0"|{{CDD|node_h|4|node_h|2x|node_h|4|node_h}}
|}

== Uniform star polychora ==
Other than the aforementioned infinite duoprism and antiprism prism families, which have infinitely many nonconvex members, many uniform star polychora have been discovered. In 1852, Ludwig Schläfli discovered four ''regular'' star polychora: {5,3,5/2}, {5/2,3,5}, {3,3,5/2}, and {5/2,3,3}. In 1883, Edmund Hess found the other six: {3,5,5/2}, {5/2,5,3}, {5,5/2,5}, {5/2,5,5/2}, {5,5/2,3}, and {3,5/2,5}. Norman Johnson described three uniform antiprism-like star polychora in his doctoral dissertation of 1966: they are based on the three [[ditrigonal polyhedron|ditrigonal polyhedra]] sharing the edges and vertices of the regular dodecahedron. Many more have been found since then by other researchers, including Jonathan Bowers and George Olshevsky, creating a total count of 2127<!--2191 – 64 = 2127--> known uniform star polychora at present (not counting the infinite set of duoprisms based on star polygons). There is currently no proof of the set's completeness.

== See also==
* [[Regular skew polyhedron#Finite regular skew polyhedra of 4-space|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.
* [[Paracompact uniform honeycombs]]

==References==
{{reflist}}

*[[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
*[[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.
*{{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}} [https://www.amazon.com/Semiregular-Polytopes-Hyperspaces-Emanuel-Lodewijk/dp/141817968X The semiregular polytopes of the hyperspaces.] [http://hdl.handle.net/2027/miun.abr2632.0001.001 The semiregular polytopes of the hyperspaces.]
*[[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, Londen, 1954
**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 p.&nbsp;92, p.&nbsp;122.
*[[John Horton Conway|John H. Conway]], Heidi Burgiel, [[Chaim Goodman-Strauss]], ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 26)
*[[John Horton Conway|John 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
* N.W. Johnson: ''Geometries and Transformations'', (2015) Chapter 11: Finite symmetry groups
* 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]
*{{citation|first=Pieter Hendrik|last=Schoute|author-link=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}} [https://books.google.com/books?id=qC5LAAAAYAAJ&dq=Analytic+treatment+of+the+polytopes+regularly+derived+from+the+regular+polytopes&pg=PA357 Googlebook, 370-381]

==External links==
<!--* {{mathworld | urlname = UniformPolychoron  | title = Uniform polychoron }} unhelpful-->
* Convex uniform 4-polytopes
**[http://www.polytope.de Uniform, convex polytopes in four dimensions], Marco Möller {{in lang|de}}. Includes alternative names for these figures, including those from Jonathan Bowers, George Olshevsky, and Norman Johnson.
**[https://web.archive.org/web/20030818121038/http://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.html Regular and semi-regular convex polytopes a short historical overview]
**[https://web.archive.org/web/20110718202453/http://public.beuth-hochschule.de/~meiko/pentatope.html Java3D Applets with sources]
* Nonconvex uniform 4-polytopes
**[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 known uniform polychora including the 64 convex forms and the infinite prismatic families. 
* {{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) [https://arxiv.org/ftp/arxiv/papers/1102/1102.1132.pdf]

{{Polytopes}}

[[Category:Uniform 4-polytopes| ]]