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Uniform 4-polytope#The H4 family

{{Short description|Class of 4-dimensional polytopes}}

File:Schlegel half-solid truncated 120-cell.png for the truncated 120-cell with tetrahedral cells visible]]

File:120-cell t01 H3.svg 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)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 is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

There are 47 non-prismatic 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 polytopes:
  • 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions.
  • Regular star 4-polytopes (star polyhedron cells and/or vertex figures)
  • 1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}.
  • 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 polytopes: (Various definitions before Coxeter's uniform category)
  • 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) 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.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
  • 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 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.{{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 }}
  • 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.Elte (1912)
  • Convex uniform polytopes:
  • 1940: The search was expanded systematically by 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, 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 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[https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html Uniform Polytopes in Four Dimensions] December 6, 1998 oldest archive-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 roots poly ("many") and choros ("room" or "space").[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 The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t0,1, cantellation, t0,2, runcination t0,3, with single ringed forms called rectified, and bi, tri-prefixes added when the first ring was on the second or third nodes.Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups{{PolyCell | urlname = uniform.html| title = Uniform Polytopes in Four Dimensions}}
  • 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.{{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 }}
  • 2008: The Symmetries of ThingsConway (2008) was published by 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 nonconvex uniform polyhedra)
  • 1966: Johnson describes three nonconvex uniform antiprisms in 4-space in his dissertation.[https://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html Multidimensional Glossary], George Olshevsky
  • 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,https://www.mit.edu/~hlb/Associahedron/program.pdf Convex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract with an additional four discovered in 2006 for a total of 1849. The count includes the 74 prisms of the 75 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.{{Cite web|url=http://www.polytope.net/hedrondude/polychora.htm|title=Uniform Polychora|website=www.polytope.net|access-date=February 20, 2020}}
  • 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.{{Cite web|url=https://polytope.miraheze.org/wiki/Uniform_polytope|title=Uniform polytope|date=6 November 2023|website=Polytope Wiki|accessdate=11 November 2023}}

Regular 4-polytopes

Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements. 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 figures {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:Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135

:\sin \left ( \frac{\pi}{p} \right ) \sin \left(\frac{\pi}{r}\right) > \cos\left(\frac{\pi}{q}\right).

The 16 regular 4-polytopes, with the property that all cells, faces, edges, and vertices are congruent:

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 F4 can be decomposed into 2 orthogonal D4 groups:

  1. {{CDD|node_g|3sg|node_g|4|node_c3|3|node_c4}} = {{CDD|node_c3|branch3_c3|splitsplit2|node_c4}} (12 mirrors)
  2. {{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 B3×A1 can be decomposed into orthogonal groups, 4A1 and D3:
  • {{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: A4 = {{CDD|node|3|node|3|node|3|node}}, B4 = {{CDD|node|4|node|3|node|3|node}}, D4 = {{CDD|node|3|node|split1|nodes}}, F4 = {{CDD|node|3|node|4|node|3|node}}, H4 = {{CDD|node|5|node|3|node|3|node}}. There are also 3 prismatic groups A3A1 = {{CDD|node|3|node|3|node|2|node}}, B3A1 = {{CDD|node|4|node|3|node|2|node}}, H3A1 = {{CDD|node|5|node|3|node|2|node}}, and duoprismatic groups: I2(p)×I2(q) = {{CDD|node|p|node|2|node|q|node}}. Each group defined by a 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 hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,a], have an extended symmetry, a,b,a, 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 operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.

    class=wikitable
    valign=top

    !Weyl
    group

    !Conway
    Quaternion

    !Abstract
    structure

    !Order

    !colspan=2|Coxeter
    diagram

    !Coxeter
    notation

    !Commutator
    subgroup

    !Coxeter
    number
    (h)

    !colspan=4|Mirrors
    m=2h

    colspan=13|Irreducible
    align=center

    !A4

    		+1/60[I×I].21S5120{{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]+510{{CDD|node_c1}}
    align=center

    !D4

    		±1/3[T×T].21/2.2S4192{{CDD|nodes|split2|node|3|node}}{{CDD|nodeab_c1|split2|node_c1|3|node_c1}}[31,1,1]rowspan=2| [31,1,1]+612{{CDD|node_c1}}
    align=center

    !B4

    		±1/6[O×O].22S4 = S2≀S4384{{CDD|node|4|node|3|node|3|node}}{{CDD|node_c2|4|node_c1|3|node_c1|3|node_c1}}[4,3,3]84{{CDD|node_c2}}12{{CDD|node_c1}}
    align=center

    !F4

    		±1/2[O×O].233.2S41152{{CDD|node|3|node|4|node|3|node}}{{CDD|node_c2|3|node_c2|4|node_c1|3|node_c1}}[3,4,3][3+,4,3+]1212{{CDD|node_c2}}12{{CDD|node_c1}}
    align=center

    !H4

    		±[I×I].22.(A5×A5).214400{{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]+3060{{CDD|node_c1}}
    colspan=13|Prismatic groups
    align=center

    !A3A1

    		+1/24[O×O].23S4×D148{{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]+ |
    |6{{CDD|node_c1}}1{{CDD|node_c3}}
    align=center

    !B3A1

    		±1/24[O×O].2S4×D196{{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

    !H3A1

    		±1/60[I×I].2A5×D1240{{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]+-15{{CDD|node_c1}}1{{CDD|node_c3}}
    colspan=13|Duoprismatic groups (Use 2p,2q for even integers)
    align=center

    !I2(p)I2(q)

    		±1/2[D2p×D2q]Dp×Dq4pq{{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+,2,q+]-p {{CDD|node_c1}}q {{CDD|node_c3}}
    align=center

    !I2(2p)I2(q)

    		±1/2[D4p×D2q]D2p×Dq8pq{{CDD|node|2x|p|node|2|node|q|node}}{{CDD|node_c2|2x|p|node_c1|2|node_c3|q|node_c3}}[2p,2,q] = [2p]×[q]-p {{CDD|node_c2}}p {{CDD|node_c1}}q {{CDD|node_c3}}
    align=center

    !I2(2p)I2(2q)

    		±1/2[D4p×D4q]D2p×D2q16pq{{CDD|node|2x|p|node|2|node|2x|q|node}}{{CDD|node_c2|2x|p|node_c1|2|node_c3|2x|q|node_c4}}[2p,2,2q] = [2p]×[2q]-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 duoprisms and the antiprismatic prisms.

    • 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)
    • 13 are polyhedral prisms based on the Archimedean solids
    • 9 are in the self-dual regular A4 [3,3,3] group (5-cell) family.
    • 9 are in the self-dual regular F4 [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
    • 15 are in the regular B4 [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family)
    • 15 are in the regular H4 [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 − 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:

    = The A<sub>4</sub> family =

    {{See|A4 polytope}}

    The 5-cell has diploid pentachoric [3,3,3] symmetry, of 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
    Bowers name (and acronym)

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !colspan=4 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3
    {{CDD|node|3|node|3|node|2|2}}
    (5)

    ! Pos. 2
    {{CDD|node|3|node|2|2|2|node}}
    (10)

    ! Pos. 1
    {{CDD|node|2|2|2|node|3|node}}
    (10)

    ! Pos. 0
    {{CDD|2|2|node|3|node|3|node}}
    (5)

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#f0e0e0" align=center

    !1

    |align=center|5-cell
    Pentachoron (pen)

    |60px

    |align=center|{{CDD|node_1|3|node|3|node|3|node}}
    {3,3,3}

    |(4)
    30px
    (3.3.3)

    |

    |

    |

    | 5

    | 10

    | 10

    | 5

    BGCOLOR="#f0e0e0" align=center

    !2

    |align=center|rectified 5-cell
    Rectified pentachoron (rap)

    |60px

    |align=center|{{CDD|node|3|node_1|3|node|3|node}}
    r{3,3,3}

    |(3)
    30px
    (3.3.3.3)

    |

    |

    |(2)
    30px
    (3.3.3)

    | 10

    | 30

    | 30

    | 10

    BGCOLOR="#f0e0e0" align=center

    !3

    |align=center|truncated 5-cell
    Truncated pentachoron (tip)

    |60px

    |align=center|{{CDD|node_1|3|node_1|3|node|3|node}}
    t{3,3,3}

    |(3)
    30px
    (3.6.6)

    |

    |

    |(1)
    30px
    (3.3.3)

    | 10

    | 30

    | 40

    | 20

    BGCOLOR="#f0e0e0" align=center

    !4

    |align=center|cantellated 5-cell
    Small rhombated pentachoron (srip)

    |60px

    |align=center|{{CDD|node_1|3|node|3|node_1|3|node}}
    rr{3,3,3}

    |(2)
    30px
    (3.4.3.4)

    |

    |(2)
    25px
    (3.4.4)

    |(1)
    30px
    (3.3.3.3)

    | 20

    | 80

    | 90

    | 30

    BGCOLOR="#f0e0e0" align=center

    !7

    |align=center|cantitruncated 5-cell
    Great rhombated pentachoron (grip)

    |60px

    |align=center|{{CDD|node_1|3|node_1|3|node_1|3|node}}
    tr{3,3,3}

    |(2)
    30px
    (4.6.6)

    |

    |(1)
    25px
    (3.4.4)

    |(1)
    30px
    (3.6.6)

    | 20

    | 80

    | 120

    | 60

    BGCOLOR="#f0e0e0" align=center

    !8

    |align=center|runcitruncated 5-cell
    Prismatorhombated pentachoron (prip)

    |60px

    |align=center|{{CDD|node_1|3|node_1|3|node|3|node_1}}
    t0,1,3{3,3,3}

    |(1)
    30px
    (3.6.6)

    |(2)
    30px
    (4.4.6)

    |(1)
    25px
    (3.4.4)

    |(1)
    30px
    (3.4.3.4)

    | 30

    | 120

    | 150

    | 60

    class="wikitable"

    |+ 3,3,3 uniform polytopes

    rowspan=2| #

    !rowspan=2| Name
    Bowers name (and acronym)

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    {{CDD|node_c1|3|node_c2|3|node_c2|3|node_c1}}
    and Schläfli
    symbols

    !colspan=3 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3-0
    {{CDD|node|3|node|3|node|2|2}}
    (10)

    ! Pos. 1-2
    {{CDD|node|3|node|2|2|2|node}}
    (20)

    !Alt

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#e0f0e0" align=center

    !5

    |align=center|*runcinated 5-cell
    Small prismatodecachoron (spid)

    |60px

    |align=center|{{CDD|node_1|3|node|3|node|3|node_1}}
    t0,3{3,3,3}

    |(2)
    30px
    (3.3.3)

    |(6)
    25px
    (3.4.4)

    |

    | 30

    | 70

    | 60

    | 20

    BGCOLOR="#e0f0e0" align=center

    !6

    |align=center|*bitruncated 5-cell
    Decachoron (deca)

    |60px

    |align=center|{{CDD|node|3|node_1|3|node_1|3|node}}
    2t{3,3,3}

    |(4)
    30px
    (3.6.6)

    |

    |

    | 10

    | 40

    | 60

    | 30

    BGCOLOR="#e0f0e0" align=center

    !9

    |align=center|*omnitruncated 5-cell
    Great prismatodecachoron (gippid)

    |60px

    |align=center|{{CDD|node_1|3|node_1|3|node_1|3|node_1}}
    t0,1,2,3{3,3,3}

    |(2)
    30px
    (4.6.6)

    |(2)
    30px
    (4.4.6)

    |

    | 30

    | 150

    | 240

    | 120

    BGCOLOR="#d0f0f0" align=center

    !Nonuniform

    |omnisnub 5-cell
    Snub decachoron (snad)
    Snub pentachoron (snip){{Cite web|url=http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm|title = S3s3s3s}}

    |60px

    | align=center|{{CDD|node_h|3|node_h|3|node_h|3|node_h}}
    ht0,1,2,3{3,3,3}

    |30px (2)
    (3.3.3.3.3)

    |30px (2)
    (3.3.3.3)

    |30px (4)
    (3.3.3)

    | 90

    | 300

    | 270

    | 60

    The three uniform 4-polytopes forms marked with an asterisk, *, have the higher 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]+, order 60, or its doubling {{brackets|3,3,3}}+, 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 diploid hexadecachoric symmetry, [4,3,3], of order 24×16=384: 4!=24 permutations of the four axes, 24=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+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192.

    == Tesseract truncations ==

    class="wikitable"

    !rowspan=2|#

    !rowspan=2|Name
    (Bowers name and acronym)

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !colspan=5 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3
    {{CDD|node_n0|4|node_n1|3|node_n2|2|2}}
    (8)

    ! Pos. 2
    {{CDD|node_n0|4|node_n1|2|2|node_n3}}
    (24)

    ! Pos. 1
    {{CDD|node_n0|2|2|node_n2|3|node_n3}}
    (32)

    ! Pos. 0
    {{CDD|2|2|node_n1|3|node_n2|3|node_n3}}
    (16)

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#f0e0e0" align=center

    !10

    |align=center|tesseract or 8-cell
    Tesseract (tes)

    |60px

    | align=center|{{CDD|node_1|4|node|3|node|3|node}}
    {4,3,3}

    |(4)
    30px
    (4.4.4)

    |

    |

    |

    | 8

    | 24

    | 32

    | 16

    BGCOLOR="#f0e0e0" align=center

    !11

    |align=center|Rectified tesseract (rit)

    |60px

    | align=center|{{CDD|node|4|node_1|3|node|3|node}}
    r{4,3,3}

    |(3)
    30px
    (3.4.3.4)

    |

    |

    |(2)
    30px
    (3.3.3)

    | 24

    | 88

    | 96

    | 32

    BGCOLOR="#f0e0e0" align=center

    !13

    |align=center|Truncated tesseract (tat)

    |60px

    | align=center|{{CDD|node_1|4|node_1|3|node|3|node}}
    t{4,3,3}

    |(3)
    30px
    (3.8.8)

    |

    |

    |(1)
    30px
    (3.3.3)

    | 24

    | 88

    | 128

    | 64

    BGCOLOR="#f0e0e0" align=center

    !14

    |align=center|Cantellated tesseract
    Small rhombated tesseract (srit)

    |60px

    | align=center|{{CDD|node_1|4|node|3|node_1|3|node}}
    rr{4,3,3}

    |(2)
    30px
    (3.4.4.4)

    |

    |(2)
    25px
    (3.4.4)

    |(1)
    30px
    (3.3.3.3)

    | 56

    | 248

    | 288

    | 96

    BGCOLOR="#e0f0e0" align=center

    !15

    |align=center|Runcinated tesseract
    (also runcinated 16-cell)
    Small disprismatotesseractihexadecachoron (sidpith)

    |60px

    | align=center|{{CDD|node_1|4|node|3|node|3|node_1}}
    t0,3{4,3,3}

    |(1)
    30px
    (4.4.4)

    |(3)
    30px
    (4.4.4)

    |(3)
    25px
    (3.4.4)

    |(1)
    30px
    (3.3.3)

    | 80

    | 208

    | 192

    | 64

    BGCOLOR="#e0f0e0" align=center

    !16

    |align=center|Bitruncated tesseract
    (also bitruncated 16-cell)
    Tesseractihexadecachoron (tah)

    |60px

    | align=center|{{CDD|node|4|node_1|3|node_1|3|node}}
    2t{4,3,3}

    |(2)
    30px
    (4.6.6)

    |

    |

    |(2)
    30px
    (3.6.6)

    | 24

    | 120

    | 192

    | 96

    BGCOLOR="#f0e0e0" align=center

    !18

    |align=center|Cantitruncated tesseract
    Great rhombated tesseract (grit)

    |60px

    | align=center|{{CDD|node_1|4|node_1|3|node_1|3|node}}
    tr{4,3,3}

    |(2)
    30px
    (4.6.8)

    |

    |(1)
    25px
    (3.4.4)

    |(1)
    30px
    (3.6.6)

    | 56

    | 248

    | 384

    | 192

    BGCOLOR="#f0e0e0" align=center

    !19

    |align=center|Runcitruncated tesseract
    Prismatorhombated hexadecachoron (proh)

    |60px

    | align=center|{{CDD|node_1|4|node_1|3|node|3|node_1}}
    t0,1,3{4,3,3}

    |(1)
    30px
    (3.8.8)

    |(2)
    30px
    (4.4.8)

    |(1)
    25px
    (3.4.4)

    |(1)
    30px
    (3.4.3.4)

    | 80

    | 368

    | 480

    | 192

    BGCOLOR="#e0f0e0" align=center

    !21

    |align=center|Omnitruncated tesseract
    (also omnitruncated 16-cell)
    Great disprismatotesseractihexadecachoron (gidpith)

    |60px

    | align=center|{{CDD|node_1|4|node_1|3|node_1|3|node_1}}
    t0,1,2,3{3,3,4}

    |(1)
    30px
    (4.6.8)

    |(1)
    30px
    (4.4.8)

    |(1)
    30px
    (4.4.6)

    |(1)
    30px
    (4.6.6)

    | 80

    | 464

    | 768

    | 384

    class="wikitable"

    |+ Related half tesseract, [1+,4,3,3] uniform 4-polytopes

    !rowspan=2|#

    !rowspan=2|Name
    (Bowers style acronym)

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !colspan=5 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3
    {{CDD|node_n0|4|node_n1|3|node_n2|2|2}}
    (8)

    ! Pos. 2
    {{CDD|node_n0|4|node_n1|2|2|node_n3}}
    (24)

    ! Pos. 1
    {{CDD|node_n0|2|2|node_n2|3|node_n3}}
    (32)

    ! Pos. 0
    {{CDD|2|2|node_n1|3|node_n2|3|node_n3}}
    (16)

    !Alt

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#d0f0f0" align=center

    ![12]

    |align=center|Half tesseract
    Demitesseract
    = 16-cell (hex)

    |60px

    | align=center|{{CDD|node_h1|4|node|3|node|3|node}} = {{CDD|nodes_10ru|split2|node|3|node}}
    h{4,3,3}={3,3,4}

    |(4)
    30px
    (3.3.3)

    |

    |

    |

    |(4)
    30px
    (3.3.3)

    | 16

    | 32

    | 24

    | 8

    BGCOLOR="#d0f0f0" align=center

    ![17]

    |align=center|Cantic tesseract
    = Truncated 16-cell (thex)

    |60px

    | align=center|{{CDD|node_h1|4|node|3|node_1|3|node}} = {{CDD|nodes_10ru|split2|node_1|3|node}}
    h2{4,3,3}=t{4,3,3}

    |(4)
    30px
    (6.6.3)

    |

    |

    |(1)
    30px
    (3.3.3.3)

    |

    | 24

    | 96

    | 120

    | 48

    BGCOLOR="#d0f0f0" align=center

    ![11]

    |align=center|Runcic tesseract
    = Rectified tesseract (rit)

    |60px

    | align=center|{{CDD|node_h1|4|node|3|node|3|node_1}} = {{CDD|nodes_10ru|split2|node|3|node_1}}
    h3{4,3,3}=r{4,3,3}

    |(3)
    30px
    (3.4.3.4)

    |

    |

    |(2)
    30px
    (3.3.3)

    |

    | 24

    | 88

    | 96

    | 32

    BGCOLOR="#d0f0f0" align=center

    ![16]

    |align=center|Runcicantic tesseract
    = Bitruncated tesseract (tah)

    |60px

    | align=center|{{CDD|node_h1|4|node|3|node_1|3|node_1}} = {{CDD|nodes_10ru|split2|node_1|3|node_1}}
    h2,3{4,3,3}=2t{4,3,3}

    |(2)
    30px
    (3.4.3.4)

    |

    |

    |(2)
    30px
    (3.6.6)

    |

    | 24

    | 120

    | 192

    | 96

    BGCOLOR="#d0f0f0" align=center

    ![11]

    |align=center|= Rectified tesseract (rat)

    |60px

    | align=center|{{CDD|node_h0|4|node_1|3|node|3|node}} = {{CDD|nodes_11|split2|node|3|node}}
    h1{4,3,3}=r{4,3,3}

    |

    |

    |

    |

    |

    | 24

    | 88

    | 96

    | 32

    BGCOLOR="#d0f0f0" align=center

    ![16]

    |align=center|= Bitruncated tesseract (tah)

    |60px

    | align=center|{{CDD|node_h0|4|node_1|3|node_1|3|node}} = {{CDD|nodes_11|split2|node_1|3|node}}
    h1,2{4,3,3}=2t{4,3,3}

    |

    |

    |

    |

    |

    | 24

    | 120

    | 192

    | 96

    BGCOLOR="#d0f0f0" align=center

    ![23]

    |align=center|= Rectified 24-cell (rico)

    |60px

    | align=center|{{CDD|node_h0|4|node_1|3|node|3|node_1}} = {{CDD|nodes_11|split2|node|3|node_1}}
    h1,3{4,3,3}=rr{3,3,4}

    |

    |

    |

    |

    |

    | 48

    | 240

    | 288

    | 96

    BGCOLOR="#d0f0f0" align=center

    ![24]

    |align=center|= Truncated 24-cell (tico)

    |60px

    | align=center|{{CDD|node_h0|4|node_1|3|node_1|3|node_1}} = {{CDD|nodes_11|split2|node_1|3|node_1}}
    h1,2,3{4,3,3}=tr{3,3,4}

    |

    |

    |

    |

    |

    | 48

    | 240

    | 384

    | 192

    class="wikitable"

    !rowspan=2|#

    !rowspan=2|Name
    (Bowers style acronym)

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !colspan=5 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3
    {{CDD|node_n0|4|node_n1|3|node_n2|2|2}}
    (8)

    ! Pos. 2
    {{CDD|node_n0|4|node_n1|2|2|node_n3}}
    (24)

    ! Pos. 1
    {{CDD|node_n0|2|2|node_n2|3|node_n3}}
    (32)

    ! Pos. 0
    {{CDD|2|2|node_n1|3|node_n2|3|node_n3}}
    (16)

    !Alt

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#d0f0f0" align=center

    !Nonuniform

    		omnisnub tesseract
    Snub tesseract (snet){{Cite web|url=http://www.bendwavy.org/klitzing/incmats/s3s3s4s.htm|title = S3s3s4s}}
    (Or omnisnub 16-cell)

    |60px

    | align=center|{{CDD|node_h|4|node_h|3|node_h|3|node_h}}
    ht0,1,2,3{4,3,3}

    |(1)
    30px
    (3.3.3.3.4)

    |(1)
    30px
    (3.3.3.4)

    |(1)
    30px
    (3.3.3.3)

    |(1)
    30px
    (3.3.3.3.3)

    |(4)
    30px
    (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

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !colspan=5 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3
    {{CDD|node_n0|4|node_n1|3|node_n2|2|2}}
    (8)

    ! Pos. 2
    {{CDD|node_n0|4|node_n1|2|2|node_n3}}
    (24)

    ! Pos. 1
    {{CDD|node_n0|2|2|node_n2|3|node_n3}}
    (32)

    ! Pos. 0
    {{CDD|2|2|node_n1|3|node_n2|3|node_n3}}
    (16)

    !Alt

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#e0e0f0" align=center

    !12

    |align=center|16-cell
    Hexadecachoron (hex)

    |60px

    | align=center|{{CDD|node|4|node|3|node|3|node_1}}
    {3,3,4}

    |

    |

    |

    |(8)
    30px
    (3.3.3)

    |

    | 16

    | 32

    | 24

    | 8

    BGCOLOR="#e0e0f0" align=center

    ![22]

    |align=center|*Rectified 16-cell
    (Same as 24-cell) (ico)

    |60px

    | align=center|{{CDD|node|4|node|3|node_1|3|node}} = {{CDD|node_1|3|node|4|node|3|node}}
    r{3,3,4}

    |(2)
    30px
    (3.3.3.3)

    |

    |

    |(4)
    30px
    (3.3.3.3)

    |

    | 24

    | 96

    | 96

    | 24

    BGCOLOR="#e0e0f0" align=center

    !17

    |align=center|Truncated 16-cell
    Truncated hexadecachoron (thex)

    |60px

    | align=center|{{CDD|node|4|node|3|node_1|3|node_1}}
    t{3,3,4}

    |(1)
    30px
    (3.3.3.3)

    |

    |

    |(4)
    30px
    (3.6.6)

    |

    | 24

    | 96

    | 120

    | 48

    BGCOLOR="#e0e0f0" align=center

    ![23]

    |align=center|*Cantellated 16-cell
    (Same as rectified 24-cell) (rico)

    |60px

    | align=center|{{CDD|node|4|node_1|3|node|3|node_1}} = {{CDD|node|3|node_1|4|node|3|node}}
    rr{3,3,4}

    |(1)
    30px
    (3.4.3.4)

    |(2)
    30px
    (4.4.4)

    |

    |(2)
    30px
    (3.4.3.4)

    |

    | 48

    | 240

    | 288

    | 96

    BGCOLOR="#e0f0e0" align=center

    ![15]

    |align=center|Runcinated 16-cell
    (also runcinated tesseract) (sidpith)

    |60px

    | align=center|{{CDD|node_1|4|node|3|node|3|node_1}}
    t0,3{3,3,4}

    |(1)
    30px
    (4.4.4)

    |(3)
    30px
    (4.4.4)

    |(3)
    25px
    (3.4.4)

    |(1)
    30px
    (3.3.3)

    |

    | 80

    | 208

    | 192

    | 64

    BGCOLOR="#e0f0e0" align=center

    ![16]

    |align=center|Bitruncated 16-cell
    (also bitruncated tesseract) (tah)

    |60px

    | align=center|{{CDD|node|4|node_1|3|node_1|3|node}}
    2t{3,3,4}

    |(2)
    30px
    (4.6.6)

    |

    |

    |(2)
    30px
    (3.6.6)

    |

    | 24

    | 120

    | 192

    | 96

    BGCOLOR="#e0e0f0" align=center

    ![24]

    |align=center|*Cantitruncated 16-cell
    (Same as truncated 24-cell) (tico)

    |60px

    | align=center|{{CDD|node|4|node_1|3|node_1|3|node_1}} = {{CDD|node_1|3|node_1|4|node|3|node}}
    tr{3,3,4}

    |(1)
    30px
    (4.6.6)

    |(1)
    30px
    (4.4.4)

    |

    |(2)
    30px
    (4.6.6)

    |

    | 48

    | 240

    | 384

    | 192

    BGCOLOR="#e0e0f0" align=center

    !20

    |align=center|Runcitruncated 16-cell
    Prismatorhombated tesseract (prit)

    |60px

    | align=center|{{CDD|node_1|4|node|3|node_1|3|node_1}}
    t0,1,3{3,3,4}

    |(1)
    30px
    (3.4.4.4)

    |(1)
    30px
    (4.4.4)

    |(2)
    30px
    (4.4.6)

    |(1)
    30px
    (3.6.6)

    |

    | 80

    | 368

    | 480

    | 192

    BGCOLOR="#e0f0e0" align=center

    ![21]

    |align=center|Omnitruncated 16-cell
    (also omnitruncated tesseract) (gidpith)

    |60px

    | align=center|{{CDD|node_1|4|node_1|3|node_1|3|node_1}}
    t0,1,2,3{3,3,4}

    |(1)
    30px
    (4.6.8)

    |(1)
    30px
    (4.4.8)

    |(1)
    30px
    (4.4.6)

    |(1)
    30px
    (4.6.6)

    |

    | 80

    | 464

    | 768

    | 384

    BGCOLOR="#d0f0f0" align=center

    ![31]

    |align=center|alternated cantitruncated 16-cell
    (Same as the snub 24-cell) (sadi)

    |60px

    | align=center|{{CDD|node|4|node_h|3|node_h|3|node_h}}
    sr{3,3,4}

    |(1)
    30px
    (3.3.3.3.3)

    |

    |(1)
    30px
    (3.3.3)

    |(2)
    30px
    (3.3.3.3.3)

    |(4)
    30px
    (3.3.3)

    | 144

    | 480

    | 432

    | 96

    BGCOLOR="#d0f0f0" align=center

    !Nonuniform

    |Runcic snub rectified 16-cell
    Pyritosnub tesseract (pysnet)

    |60px

    | align=center|{{CDD|node_1|4|node_h|3|node_h|3|node_h}}
    sr3{3,3,4}

    |(1)
    30px
    (3.4.4.4)

    |(2)
    25px
    (3.4.4)

    |(1)
    30px
    (4.4.4)

    |(1)
    30px
    (3.3.3.3.3)

    |(2)
    25px
    (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)+,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 diploid icositetrachoric symmetry, [3,4,3], of order 24×48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There are 3 small index subgroups, with the first two isomorphic pairs generating uniform 4-polytopes which are also repeated in other families, [3+,4,3], [3,4,3+], and [3,4,3]+, all order 576.

    class="wikitable"

    |+ [3,4,3] uniform 4-polytopes

    rowspan=2|#

    !rowspan=2|Name

    !rowspan=2|Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !colspan=4 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3
    {{CDD|node_n0|3|node_n1|4|node_n2|2|2}}
    (24)

    ! Pos. 2
    {{CDD|node_n0|3|node_n1|2|2|node_n3}}
    (96)

    ! Pos. 1
    {{CDD|node_n0|2|2|2|node_n2|3|node_n3}}
    (96)

    ! Pos. 0
    {{CDD|2|2|node_n1|4|node_n2|3|node_n3}}
    (24)

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#f0e0e0" align=center

    !22

    |align=center|24-cell
    (Same as rectified 16-cell)
    Icositetrachoron (ico)

    |60px

    |align=center|{{CDD|node_1|3|node|4|node|3|node}}
    {3,4,3}

    |(6)
    30px
    (3.3.3.3)

    |

    |

    |

    | 24

    | 96

    | 96

    | 24

    BGCOLOR="#f0e0e0" align=center

    !23

    |align=center|rectified 24-cell
    (Same as cantellated 16-cell)
    Rectified icositetrachoron (rico)

    |60px

    |align=center|{{CDD|node|3|node_1|4|node|3|node}}
    r{3,4,3}

    |(3)
    30px
    (3.4.3.4)

    |

    |

    |(2)
    30px
    (4.4.4)

    | 48

    | 240

    | 288

    | 96

    BGCOLOR="#f0e0e0" align=center

    !24

    |align=center|truncated 24-cell
    (Same as cantitruncated 16-cell)
    Truncated icositetrachoron (tico)

    |60px

    |align=center|{{CDD|node_1|3|node_1|4|node|3|node}}
    t{3,4,3}

    |(3)
    30px
    (4.6.6)

    |

    |

    |(1)
    30px
    (4.4.4)

    | 48

    | 240

    | 384

    | 192

    BGCOLOR="#f0e0e0" align=center

    !25

    |align=center|cantellated 24-cell
    Small rhombated icositetrachoron (srico)

    |60px

    |align=center|{{CDD|node_1|3|node|4|node_1|3|node}}
    rr{3,4,3}

    |(2)
    30px
    (3.4.4.4)

    |

    |(2)
    25px
    (3.4.4)

    |(1)
    30px
    (3.4.3.4)

    | 144

    | 720

    | 864

    | 288

    BGCOLOR="#f0e0e0" align=center

    !28

    |align=center|cantitruncated 24-cell
    Great rhombated icositetrachoron (grico)

    |60px

    |align=center|{{CDD|node_1|3|node_1|4|node_1|3|node}}
    tr{3,4,3}

    |(2)
    30px
    (4.6.8)

    |

    |(1)
    25px
    (3.4.4)

    |(1)
    30px
    (3.8.8)

    | 144

    | 720

    | 1152

    | 576

    BGCOLOR="#f0e0e0" align=center

    !29

    |align=center|runcitruncated 24-cell
    Prismatorhombated icositetrachoron (prico)

    |60px

    |align=center|{{CDD|node_1|3|node_1|4|node|3|node_1}}
    t0,1,3{3,4,3}

    |(1)
    30px
    (4.6.6)

    |(2)
    30px
    (4.4.6)

    |(1)
    25px
    (3.4.4)

    |(1)
    30px
    (3.4.4.4)

    | 240

    | 1104

    | 1440

    | 576

    class="wikitable"

    |+ [3+,4,3] uniform 4-polytopes

    rowspan=2|#

    !rowspan=2|Name

    !rowspan=2|Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !colspan=5 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3
    {{CDD|node_n0|3|node_n1|4|node_n2|2|2}}
    (24)

    ! Pos. 2
    {{CDD|node_n0|3|node_n1|2|2|node_n3}}
    (96)

    ! Pos. 1
    {{CDD|node_n0|2|2|2|node_n2|3|node_n3}}
    (96)

    ! Pos. 0
    {{CDD|2|2|node_n1|4|node_n2|3|node_n3}}
    (24)

    !Alt

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#d0f0f0" align=center

    !31

    |align=center|†snub 24-cell
    Snub disicositetrachoron (sadi)

    |60px

    |align=center|{{CDD|node_h|3|node_h|4|node|3|node}}
    s{3,4,3}

    |(3)
    30px
    (3.3.3.3.3)

    |

    |

    |(1)
    30px
    (3.3.3)

    |(4)
    30px
    (3.3.3)

    | 144

    | 480

    | 432

    | 96

    BGCOLOR="#d0f0f0" align=center

    !Nonuniform

    |runcic snub 24-cell
    Prismatorhombisnub icositetrachoron (prissi)

    |60px

    |align=center|{{CDD|node_h|3|node_h|4|node|3|node_1}}
    s3{3,4,3}

    |(1)
    30px
    (3.3.3.3.3)

    |(2)
    25px
    (3.4.4)

    |

    |(1)
    30px
    (3.6.6)

    |(3)
    30px
    Tricup

    |240

    |960

    |1008

    |288

    BGCOLOR="#d0f0f0" align=center

    ![25]

    |cantic snub 24-cell
    (Same as cantellated 24-cell) (srico)

    |60px

    |{{CDD|node_h|3|node_h|4|node_1|3|node}}
    s2{3,4,3}

    |(2)
    30px
    (3.4.4.4)

    |

    |

    |(1)
    30px
    (3.4.3.4)

    |(2)
    25px
    (3.4.4)

    | 144

    | 720

    | 864

    | 288

    BGCOLOR="#d0f0f0" align=center

    ![29]

    |runcicantic snub 24-cell
    (Same as runcitruncated 24-cell) (prico)

    |60px

    |{{CDD|node_h|3|node_h|4|node_1|3|node_1}}
    s2,3{3,4,3}

    |(1)
    30px
    (4.6.6)

    |

    |(1)
    25px
    (3.4.4)

    |(1)
    30px
    (3.4.4.4)

    |(2)
    30px
    (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 of the truncated 24-cell. Its symmetry number is only 576, (the ionic diminished icositetrachoric group, [3+,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 (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

    !rowspan=2|Coxeter diagram
    {{CDD|node_c1|3|node_c2|4|node_c2|3|node_c1}}
    and Schläfli
    symbols

    !colspan=2|Cell counts by location

    !colspan=4|Element counts

    Pos. 3-0
    {{CDD|node|3|node|4|node|2|2}}
    {{CDD|2|2|node|4|node|3|node}}
    (48)

    ! Pos. 2-1
    {{CDD|node|3|node|2|2|node}}
    {{CDD|node|2|2|node|3|node}}
    (192)

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#e0f0e0" align=center

    !26

    |align=center|runcinated 24-cell
    Small prismatotetracontoctachoron (spic)

    |60px

    |align=center|{{CDD|node_1|3|node|4|node|3|node_1}}
    t0,3{3,4,3}

    |(2)
    30px
    (3.3.3.3)

    |(6)
    25px
    (3.4.4)

    | 240

    | 672

    | 576

    | 144

    BGCOLOR="#e0f0e0" align=center

    !27

    |align=center|bitruncated 24-cell
    Tetracontoctachoron (cont)

    |60px

    |align=center|{{CDD|node|3|node_1|4|node_1|3|node}}
    2t{3,4,3}

    |(4)
    30px
    (3.8.8)

    |

    | 48

    | 336

    | 576

    | 288

    BGCOLOR="#e0f0e0" align=center

    !30

    |align=center|omnitruncated 24-cell
    Great prismatotetracontoctachoron (gippic)

    |60px

    |align=center|{{CDD|node_1|3|node_1|4|node_1|3|node_1}}
    t0,1,2,3{3,4,3}

    |(2)
    30px
    (4.6.8)

    |(2)
    30px
    (4.4.6)

    | 240

    | 1392

    | 2304

    | 1152

    class="wikitable"

    |+ 3,4,3+ isogonal 4-polytope

    rowspan=2|#

    !rowspan=2|Name

    !rowspan=2|Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !colspan=3 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3-0
    {{CDD|node|3|node|4|node|2|2}}
    {{CDD|2|2|node|4|node|3|node}}
    (48)

    ! Pos. 2-1
    {{CDD|node|3|node|2|2|node}}
    {{CDD|node|2|2|node|3|node}}
    (192)

    !Alt

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#d0f0f0" align=center

    !Nonuniform

    |align=center|omnisnub 24-cell
    Snub tetracontoctachoron (snoc)
    Snub icositetrachoron (sni){{Cite web|url=http://www.bendwavy.org/klitzing/incmats/s3s4s3s.htm|title = S3s4s3s}}

    |60px

    |align=center|{{CDD|node_h|3|node_h|4|node_h|3|node_h}}
    ht0,1,2,3{3,4,3}

    |(2)
    30px
    (3.3.3.3.4)

    |(2)
    30px
    (3.3.3.3)

    |(4)
    30px
    (3.3.3)

    | 816

    | 2832

    | 2592

    | 576

    =The H<sub>4</sub> family=

    {{See|H4 polytope}}

    This family has diploid hexacosichoric symmetry, [5,3,3], of order 120×120=24×600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra. There is one small index subgroups [5,3,3]+, all order 7200.

    == 120-cell truncations ==

    class="wikitable"

    !rowspan=2|#

    !rowspan=2|Name
    (Bowers name and acronym)

    !rowspan=2|Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !colspan=5 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3
    {{CDD|node_n0|5|node_n1|3|node_n2|2}}
    (120)

    ! Pos. 2
    {{CDD|node_n0|5|node_n1|2|2|node_n3}}
    (720)

    ! Pos. 1
    {{CDD|node_n0|2|2|node_n2|3|node_n3}}
    (1200)

    ! Pos. 0
    {{CDD|2|node_n1|3|node_n2|3|node_n3}}
    (600)

    !Alt

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#f0e0e0" align=center

    !32

    |120-cell
    (hecatonicosachoron or dodecacontachoron)
    Hecatonicosachoron (hi)

    |60px

    | align=center|{{CDD|node_1|5|node|3|node|3|node}}
    {5,3,3}

    |(4)
    30px
    (5.5.5)

    |

    |

    |

    |

    | 120

    | 720

    | 1200

    | 600

    BGCOLOR="#f0e0e0" align=center

    !33

    |rectified 120-cell
    Rectified hecatonicosachoron (rahi)

    |60px

    | align=center|{{CDD|node|5|node_1|3|node|3|node}}
    r{5,3,3}

    |(3)
    30px
    (3.5.3.5)

    |

    |

    |(2)
    30px
    (3.3.3)

    |

    | 720

    | 3120

    | 3600

    | 1200

    BGCOLOR="#f0e0e0" align=center

    !36

    |truncated 120-cell
    Truncated hecatonicosachoron (thi)

    |60px

    | align=center|{{CDD|node_1|5|node_1|3|node|3|node}}
    t{5,3,3}

    |(3)
    30px
    (3.10.10)

    |

    |

    |(1)
    30px
    (3.3.3)

    |

    | 720

    | 3120

    | 4800

    | 2400

    BGCOLOR="#f0e0e0" align=center

    !37

    |cantellated 120-cell
    Small rhombated hecatonicosachoron (srahi)

    |60px

    | align=center|{{CDD|node_1|5|node|3|node_1|3|node}}
    rr{5,3,3}

    |(2)
    30px
    (3.4.5.4)

    |

    |(2)
    25px
    (3.4.4)

    |(1)
    30px
    (3.3.3.3)

    |

    | 1920

    | 9120

    | 10800

    | 3600

    BGCOLOR="#e0f0e0" align=center

    !38

    |align=center|runcinated 120-cell
    (also runcinated 600-cell)
    Small disprismatohexacosihecatonicosachoron (sidpixhi)

    |60px

    | align=center|{{CDD|node_1|5|node|3|node|3|node_1}}
    t0,3{5,3,3}

    |(1)
    30px
    (5.5.5)

    |(3)
    30px
    (4.4.5)

    |(3)
    25px
    (3.4.4)

    |(1)
    30px
    (3.3.3)

    |

    | 2640

    | 7440

    | 7200

    | 2400

    BGCOLOR="#e0f0e0" align=center

    !39

    |align=center|bitruncated 120-cell
    (also bitruncated 600-cell)
    Hexacosihecatonicosachoron (xhi)

    |60px

    | align=center|{{CDD|node|5|node_1|3|node_1|3|node}}
    2t{5,3,3}

    |(2)
    30px
    (5.6.6)

    |

    |

    |(2)
    30px
    (3.6.6)

    |

    | 720

    | 4320

    | 7200

    | 3600

    BGCOLOR="#f0e0e0" align=center

    !42

    |cantitruncated 120-cell
    Great rhombated hecatonicosachoron (grahi)

    |60px

    | align=center|{{CDD|node_1|5|node_1|3|node_1|3|node}}
    tr{5,3,3}

    |(2)
    30px
    (4.6.10)

    |

    |(1)
    25px
    (3.4.4)

    |(1)
    30px
    (3.6.6)

    |

    | 1920

    | 9120

    | 14400

    | 7200

    BGCOLOR="#f0e0e0" align=center

    !43

    |runcitruncated 120-cell
    Prismatorhombated hexacosichoron (prix)

    |60px

    | align=center|{{CDD|node_1|5|node_1|3|node|3|node_1}}
    t0,1,3{5,3,3}

    |(1)
    30px
    (3.10.10)

    |(2)
    30px
    (4.4.10)

    |(1)
    25px
    (3.4.4)

    |(1)
    30px
    (3.4.3.4)

    |

    | 2640

    | 13440

    | 18000

    | 7200

    BGCOLOR="#e0f0e0" align=center

    !46

    |align=center|omnitruncated 120-cell
    (also omnitruncated 600-cell)
    Great disprismatohexacosihecatonicosachoron (gidpixhi)

    |60px

    | align=center|{{CDD|node_1|5|node_1|3|node_1|3|node_1}}
    t0,1,2,3{5,3,3}

    |(1)
    30px
    (4.6.10)

    |(1)
    30px
    (4.4.10)

    |(1)
    30px
    (4.4.6)

    |(1)
    30px
    (4.6.6)

    |

    | 2640

    | 17040

    | 28800

    | 14400

    BGCOLOR="#d0f0f0" align=center

    !Nonuniform

    		omnisnub 120-cell
    Snub hecatonicosachoron (snixhi){{Cite web|url=http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm|title = S3s3s5s}}
    (Same as the omnisnub 600-cell)

    |60px

    | align=center|{{CDD|node_h|5|node_h|3|node_h|3|node_h}}
    ht0,1,2,3{5,3,3}

    |30px (1)
    (3.3.3.3.5)

    |30px (1)
    (3.3.3.5)

    |30px (1)
    (3.3.3.3)

    |30px (1)
    (3.3.3.3.3)

    |30px (4)
    (3.3.3)

    | 9840

    | 35040

    | 32400

    | 7200

    == 600-cell truncations ==

    class="wikitable"

    !rowspan=2|#

    !rowspan=2|Name
    (Bowers style acronym)

    !rowspan=2|Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !rowspan=2|Symmetry

    !colspan=4 |Cell counts by location

    !colspan=4|Element counts

    Pos. 3
    {{CDD|node|5|node|3|node}}
    (120)

    ! Pos. 2
    {{CDD|node|5|node|2|node}}
    (720)

    ! Pos. 1
    {{CDD|node|2|node|3|node}}
    (1200)

    ! Pos. 0
    {{CDD|node|3|node|3|node}}
    (600)

    ! Cells

    ! Faces

    ! Edges

    ! Vertices

    BGCOLOR="#e0e0f0" align=center

    !35

    |600-cell
    Hexacosichoron (ex)

    |60px

    | align=center|{{CDD|node|5|node|3|node|3|node_1}}
    {3,3,5}

    |[5,3,3]
    order 14400

    |

    |

    |

    |(20)
    30px
    (3.3.3)

    | 600

    | 1200

    | 720

    | 120

    BGCOLOR="#f0e0f0" align=center

    ![47]

    |20-diminished 600-cell
    = Grand antiprism (gap)

    |60px

    |Nonwythoffian
    construction

    | 10,2+,10
    order 400
    Index 36

    |(2)
    30px
    (3.3.3.5)

    |

    |

    |(12)
    30px
    (3.3.3)

    |320

    |720

    |500

    |100

    BGCOLOR="#f0e0f0" align=center

    ![31]

    |24-diminished 600-cell
    = Snub 24-cell (sadi)

    |60px

    |Nonwythoffian
    construction

    | [3+,4,3]
    order 576
    index 25

    |(3)
    30px
    (3.3.3.3.3)

    |

    |

    |(5)
    30px
    (3.3.3)

    | 144

    | 480

    | 432

    | 96

    BGCOLOR="#f0e0f0" align=center

    !Nonuniform

    |bi-24-diminished 600-cell
    Bi-icositetradiminished hexacosichoron (bidex)

    |60px

    |Nonwythoffian
    construction

    | order 144
    index 100

    |(6)
    30px
    tdi

    |

    |

    |

    | 48

    | 192

    | 216

    | 72

    BGCOLOR="#e0e0f0" align=center

    !34

    |rectified 600-cell
    Rectified hexacosichoron (rox)

    |60px

    | align=center|{{CDD|node|5|node|3|node_1|3|node}}
    r{3,3,5}

    | [5,3,3]

    |(2)
    30px
    (3.3.3.3.3)

    |

    |

    |(5)
    30px
    (3.3.3.3)

    | 720

    | 3600

    | 3600

    | 720

    BGCOLOR="#f0e0f0" align=center

    !Nonuniform

    |120-diminished rectified 600-cell
    Swirlprismatodiminished rectified hexacosichoron (spidrox)

    |60px

    |Nonwythoffian
    construction

    | order 1200
    index 12

    |(2)
    30px
    3.3.3.5

    |(2)
    30px
    4.4.5

    |

    |(5)
    30px
    P4

    | 840

    | 2640

    | 2400

    | 600

    BGCOLOR="#e0e0f0" align=center

    !41

    |truncated 600-cell
    Truncated hexacosichoron (tex)

    |60px

    | align=center|{{CDD|node|5|node|3|node_1|3|node_1}}
    t{3,3,5}

    | [5,3,3]

    |(1)
    30px
    (3.3.3.3.3)

    |

    |

    |(5)
    30px
    (3.6.6)

    | 720

    | 3600

    | 4320

    | 1440

    BGCOLOR="#e0e0f0" align=center

    !40

    |cantellated 600-cell
    Small rhombated hexacosichoron (srix)

    |60px

    | align=center|{{CDD|node|5|node_1|3|node|3|node_1}}
    rr{3,3,5}

    | [5,3,3]

    |(1)
    30px
    (3.5.3.5)

    |(2)
    30px
    (4.4.5)

    |

    |(1)
    30px
    (3.4.3.4)

    | 1440

    | 8640

    | 10800

    | 3600

    BGCOLOR="#e0f0e0" align=center

    ![38]

    |align=center|runcinated 600-cell
    (also runcinated 120-cell) (sidpixhi)

    |60px

    | align=center|{{CDD|node_1|5|node|3|node|3|node_1}}
    t0,3{3,3,5}

    | [5,3,3]

    |(1)
    30px
    (5.5.5)

    |(3)
    30px
    (4.4.5)

    |(3)
    25px
    (3.4.4)

    |(1)
    30px
    (3.3.3)

    | 2640

    | 7440

    | 7200

    | 2400

    BGCOLOR="#e0f0e0" align=center

    ![39]

    |align=center|bitruncated 600-cell
    (also bitruncated 120-cell) (xhi)

    |60px

    | align=center|{{CDD|node|5|node_1|3|node_1|3|node}}
    2t{3,3,5}

    | [5,3,3]

    |(2)
    30px
    (5.6.6)

    |

    |

    |(2)
    30px
    (3.6.6)

    | 720

    | 4320

    | 7200

    | 3600

    BGCOLOR="#e0e0f0" align=center

    !45

    |cantitruncated 600-cell
    Great rhombated hexacosichoron (grix)

    |60px

    | align=center|{{CDD|node|5|node_1|3|node_1|3|node_1}}
    tr{3,3,5}

    | [5,3,3]

    |(1)
    30px
    (5.6.6)

    |(1)
    30px
    (4.4.5)

    |

    |(2)
    30px
    (4.6.6)

    | 1440

    | 8640

    | 14400

    | 7200

    BGCOLOR="#e0e0f0" align=center

    !44

    |runcitruncated 600-cell
    Prismatorhombated hecatonicosachoron (prahi)

    |60px

    | align=center|{{CDD|node_1|5|node|3|node_1|3|node_1}}
    t0,1,3{3,3,5}

    | [5,3,3]

    |(1)
    30px
    (3.4.5.4)

    |(1)
    30px
    (4.4.5)

    |(2)
    30px
    (4.4.6)

    |(1)
    30px
    (3.6.6)

    | 2640

    | 13440

    | 18000

    | 7200

    BGCOLOR="#e0f0e0" align=center

    ![46]

    |align=center|omnitruncated 600-cell
    (also omnitruncated 120-cell) (gidpixhi)

    |60px

    | align=center|{{CDD|node_1|5|node_1|3|node_1|3|node_1}}
    t0,1,2,3{3,3,5}

    | [5,3,3]

    |(1)
    30px
    (4.6.10)

    |(1)
    30px
    (4.4.10)

    |(1)
    30px
    (4.4.6)

    |(1)
    30px
    (4.6.6)

    | 2640

    | 17040

    | 28800

    | 14400

    = The D<sub>4</sub> family =

    {{See|D4 polytope}}

    This demitesseract family, [31,1,1], introduces no new uniform 4-polytopes, but it is worthy to repeat these alternative constructions. This family has order 12×16=192: 4!/2=12 permutations of the four axes, half as alternated, 24=16 for reflection in each axis. There is one small index subgroups that generating uniform 4-polytopes, [31,1,1]+, order 96.

    class="wikitable"

    |+ [31,1,1] uniform 4-polytopes

    rowspan=2|#

    !rowspan=2|Name (Bowers style acronym)

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    File:CD B4 nodes.png
    {{CDD|nodes_10ru|split2|node_n2|3|node_n3}} = {{CDD|node_h1|4|node|3|node_n2|3|node_n3}}
    {{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
    {{CDD|node|3|node|3|node}}
    (8)

    ! Pos. 2
    {{CDD|nodes|2|node}}
    (24)

    ! Pos. 1
    {{CDD|nodes|split2|node}}
    (8)

    ! Pos. 3
    {{CDD|node|3|node|3|node}}
    (8)

    ! Pos. Alt
    (96)

    !3

    !2

    !1

    !0

    align=center

    ![12]

    | align=center|demitesseract
    half tesseract
    (Same as 16-cell) (hex)

    |60px

    | align=center|{{CDD|nodes_10ru|split2|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node}}
    h{4,3,3}

    |

    |

    |(4)
    30px
    (3.3.3)

    |(4)
    30px
    (3.3.3)

    |

    | 16

    | 32

    | 24

    | 8

    align=center

    ![17]

    | align=center|cantic tesseract
    (Same as truncated 16-cell) (thex)

    |60px

    | align=center|{{CDD|nodes_10ru|split2|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node}}
    h2{4,3,3}

    |(1)
    30px
    (3.3.3.3)

    |

    |(2)
    30px
    (3.6.6)

    |(2)
    30px
    (3.6.6)

    |

    | 24

    | 96

    | 120

    | 48

    align=center

    ![11]

    | align=center|runcic tesseract
    (Same as rectified tesseract) (rit)

    |60px

    | align=center|{{CDD|nodes_10ru|split2|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1}}
    h3{4,3,3}

    |(1)
    30px
    (3.3.3)

    |

    |(1)
    30px
    (3.3.3)

    |(3)
    30px
    (3.4.3.4)

    |

    | 24

    | 88

    | 96

    | 32

    align=center

    ![16]

    | align=center|runcicantic tesseract
    (Same as bitruncated tesseract) (tah)

    |60px

    | align=center|{{CDD|nodes_10ru|split2|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1}}
    h2,3{4,3,3}

    |(1)
    30px
    (3.6.6)

    |

    |(1)
    30px
    (3.6.6)

    |(2)
    30px
    (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[31,1,1]] = [3,4,3], and thus these polytopes are repeated from the 24-cell family.

    class="wikitable"

    |+ [3[31,1,1]] uniform 4-polytopes

    rowspan=2|#

    !rowspan=2|Name (Bowers style acronym)

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    {{CDD|nodeab_c1|split2|node_c2|3|node_c1}} = {{CDD|node|4|node_c1|3|node_c2|3|node_c1}}
    {{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
    {{CDD|node|3|node|3|node}}
    (24)

    ! Pos. 2
    {{CDD|nodes|2|node}}
    (24)

    ! Pos. Alt
    (96)

    !3

    !2

    !1

    !0

    align=center

    ![22]

    |align=center|rectified 16-cell
    (Same as 24-cell) (ico)

    |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}}
    {31,1,1} = r{3,3,4} = {3,4,3}

    |(6)
    30px
    (3.3.3.3)

    |

    |

    | 48

    | 240

    | 288

    | 96

    align=center

    ![23]

    | align=center|cantellated 16-cell
    (Same as rectified 24-cell) (rico)

    |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}}
    r{31,1,1} = rr{3,3,4} = r{3,4,3}

    |(3)
    30px
    (3.4.3.4)

    |(2)
    30px
    (4.4.4)

    |

    | 24

    | 120

    | 192

    | 96

    align=center

    ![24]

    | align=center|cantitruncated 16-cell
    (Same as truncated 24-cell) (tico)

    |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}}
    t{31,1,1} = tr{3,3,4} = t{3,4,3}

    |(3)
    30px
    (4.6.6)

    |(1)
    30px
    (4.4.4)

    |

    | 48

    | 240

    | 384

    | 192

    BGCOLOR="#d0f0f0" align=center

    ![31]

    |align=center|snub 24-cell (sadi)

    |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}}
    s{31,1,1} = sr{3,3,4} = s{3,4,3}

    |(3)
    30px
    (3.3.3.3.3)

    |(1)
    30px
    (3.3.3)

    |(4)
    30px
    (3.3.3)

    | 144

    | 480

    | 432

    | 96

    Here again the snub 24-cell, with the symmetry group [31,1,1]+ 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 antiprisms forming two perpendicular rings joined by 300 tetrahedra. It is loosely analogous to the three-dimensional antiprisms, which consist of two parallel polygons joined by a band of triangles. 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+,10, order 400.

    class="wikitable"

    !rowspan=2|#

    !rowspan=2| Name (Bowers style acronym)

    !rowspan=2| Picture

    !rowspan=2|Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    symbols

    !rowspan=2 colspan=2|Cells by type

    !colspan=4|Element counts

    !rowspan=2|Net

    Cells

    ! Faces

    ! Edges

    ! Vertices

    47

    !grand antiprism (gap)

    |50px

    |50px

    |No symbol

    |300 20px
    (3.3.3)

    |20 20px
    (3.3.3.5)

    |320

    |20 {5}
    700 {3}

    |500

    |100

    |50px

    =Prismatic uniform 4-polytopes=

    A prismatic polytope is a Cartesian product of two polytopes of lower dimension; familiar examples are the 3-dimensional 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 antiprisms.
    • 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 hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic 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 solids and 13 Archimedean solids as well as for the infinite families of three-dimensional prisms and antiprisms.{{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 prismatic tetrahedral symmetry is [3,3,2], order 48. There are two index 2 subgroups, [(3,3)+,2] and [3,3,2]+, 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| Picture

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    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)

    |75px

    |75px

    |align=center|{{CDD|node_1|3|node|3|node|2|node_1}}
    {3,3}×{ }
    t0,3{3,3,2}

    |2 20px
    3.3.3

    |4 20px
    3.4.4

    |

    |6

    |8 {3}
    6 {4}

    |16

    |8

    |75px

    align=center

    !49

    !Truncated tetrahedral prism (tuttip)

    |75px

    |75px

    |align=center|{{CDD|node_1|3|node_1|3|node|2|node_1}}
    t{3,3}×{ }
    t0,1,3{3,3,2}

    |2 20px
    3.6.6

    |4 20px
    3.4.4

    |4 20px
    4.4.6

    |10

    |8 {3}
    18 {4}
    8 {6}

    |48

    |24

    |75px

    class="wikitable"

    |+ [[3,3],2] uniform 4-polytopes

    !rowspan=2|#

    !rowspan=2| Name (Bowers style acronym)

    !rowspan=2| Picture

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    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
    (Same as octahedral prism) (ope)

    |75px

    |75px

    |align=center|{{CDD|node|3|node_1|3|node|2|node_1}}
    r{3,3}×{ }
    t1,3{3,3,2}

    |2 20px
    3.3.3.3

    |4 20px
    3.4.4

    |

    |6

    |16 {3}
    12 {4}

    |30

    |12

    |75px

    align=center

    ![50]

    |align=center|Cantellated tetrahedral prism
    (Same as cuboctahedral prism) (cope)

    | 75px

    |75px

    |align=center|{{CDD|node_1|3|node|3|node_1|2|node_1}}
    rr{3,3}×{ }
    t0,2,3{3,3,2}

    |2 20px
    3.4.3.4

    |8 20px
    3.4.4

    |6 20px
    4.4.4

    |16

    |16 {3}
    36 {4}

    |60

    |24

    |75px

    align=center

    ![54]

    |align=center|Cantitruncated tetrahedral prism
    (Same as truncated octahedral prism) (tope)

    |75px

    |75px

    |align=center|{{CDD|node_1|3|node_1|3|node_1|2|node_1}}
    tr{3,3}×{ }
    t0,1,2,3{3,3,2}

    |2 20px
    4.6.6

    |8 20px
    6.4.4

    |6 20px
    4.4.4

    |16

    |48 {4}
    16 {6}

    |96

    |48

    |75px

    align=center BGCOLOR="#d0f0f0"

    ![59]

    |align=center|Snub tetrahedral prism
    (Same as icosahedral prism) (ipe)

    |75px

    |75px

    |align=center|{{CDD|node_h|3|node_h|3|node_h|2|node_1}}
    sr{3,3}×{ }

    |2 20px
    3.3.3.3.3

    |20 20px
    3.4.4

    |

    |22

    |40 {3}
    30 {4}

    |72

    |24

    |75px

    align=center BGCOLOR="#d0f0f0"

    !Nonuniform

    |omnisnub tetrahedral antiprism
    Pyritohedral icosahedral antiprism (pikap)

    |

    |75px

    |align=center|{{CDD|node_h|2x|node_h|3|node_h|3|node_h}}
    s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}

    |2 20px
    3.3.3.3.3

    |8 20px
    3.3.3.3

    |6+24 20px
    3.3.3

    |40

    |16+96 {3}

    |96

    |24

    |

    == Octahedral prisms: B<sub>3</sub> × A<sub>1</sub> ==

    This 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. Symmetries are [(4,3)+,2], [1+,4,3,2], [4,3,2+], [4,3+,2], [4,(3,2)+], and [4,3,2]+.

    class="wikitable"

    !rowspan=2| #

    !rowspan=2| Name (Bowers style acronym)

    !rowspan=2| Picture

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    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
    (Same as tesseract)
    (Same as 4-4 duoprism) (tes)

    |75px

    |75px

    |align=center|{{CDD|node_1|4|node|3|node|2|node_1}}
    {4,3}×{ }
    t0,3{4,3,2}

    |2 20px
    4.4.4

    |6 20px
    4.4.4

    |

    |

    |8

    		24 {4}3216

    |75px

    align=center

    !50

    |align=center|Cuboctahedral prism
    (Same as cantellated tetrahedral prism) (cope)

    |75px

    |75px

    |align=center|{{CDD|node|4|node_1|3|node|2|node_1}}
    r{4,3}×{ }
    t1,3{4,3,2}

    | 2 20px
    3.4.3.4

    | 8 20px
    3.4.4

    | 6 20px
    4.4.4

    |

    | 16

    		16 {3}
    36 {4}    		6024

    |75px

    align=center

    !51

    |align=center|Octahedral prism
    (Same as rectified tetrahedral prism)
    (Same as triangular antiprismatic prism) (ope)

    |75px

    |75px

    |align=center|{{CDD|node|4|node|3|node_1|2|node_1}}
    {3,4}×{ }
    t2,3{4,3,2}

    |2 20px
    3.3.3.3

    |8 20px
    3.4.4

    |

    |

    |10

    		16 {3}
    12 {4}    		3012

    |75px

    align=center

    !52

    |Rhombicuboctahedral prism (sircope)

    |75px

    |75px

    |align=center|{{CDD|node_1|4|node|3|node_1|2|node_1}}
    rr{4,3}×{ }
    t0,2,3{4,3,2}

    | 2 20px
    3.4.4.4

    | 8 20px
    3.4.4

    | 18 20px
    4.4.4

    |

    | 28

    		16 {3}
    84 {4}    		12048

    |75px

    align=center

    !53

    |Truncated cubic prism (ticcup)

    |75px

    |75px

    |align=center|{{CDD|node_1|4|node_1|3|node|2|node_1}}
    t{4,3}×{ }
    t0,1,3{4,3,2}

    | 2 20px
    3.8.8

    | 8 20px
    3.4.4

    | 6 20px
    4.4.8

    |

    | 16

    		16 {3}
    36 {4}
    12 {8}    		9648

    |75px

    align=center

    !54

    |align=center|Truncated octahedral prism
    (Same as cantitruncated tetrahedral prism) (tope)

    |75px

    |75px

    |align=center|{{CDD|node|4|node_1|3|node_1|2|node_1}}
    t{3,4}×{ }
    t1,2,3{4,3,2}

    | 2 20px
    4.6.6

    | 6 20px
    4.4.4

    | 8 20px
    4.4.6

    |

    | 16

    		48 {4}
    16 {6}    		9648

    |75px

    align=center

    !55

    |Truncated cuboctahedral prism (gircope)

    |75px

    |75px

    |align=center|{{CDD|node_1|4|node_1|3|node_1|2|node_1}}
    tr{4,3}×{ }
    t0,1,2,3{4,3,2}

    | 2 20px
    4.6.8

    | 12 20px
    4.4.4

    | 8 20px
    4.4.6

    | 6 20px
    4.4.8

    | 28

    		96 {4}
    16 {6}
    12 {8}    		19296

    |75px

    align=center BGCOLOR="#d0f0f0"

    !56

    |Snub cubic prism (sniccup)

    |75px

    |75px

    |align=center|{{CDD|node_h|4|node_h|3|node_h|2|node_1}}
    sr{4,3}×{ }

    | 2 20px
    3.3.3.3.4

    | 32 20px
    3.4.4

    | 6 20px
    4.4.4

    |

    | 40

    		64 {3}
    72 {4}    		14448

    |75px

    align=center BGCOLOR="#d0f0f0"

    ![48]

    |Tetrahedral prism (tepe)

    |75px

    |75px

    |align=center|{{CDD|node_h1|4|node|3|node|2|node_1}}
    h{4,3}×{ }

    |2 20px
    3.3.3

    |4 20px
    3.4.4

    |

    |

    |6

    		8 {3}
    6 {4}    		168

    |75px

    align=center BGCOLOR="#d0f0f0"

    ![49]

    |Truncated tetrahedral prism (tuttip)

    |75px

    |75px

    |align=center|{{CDD|node_h1|4|node|3|node_1|2|node_1}}
    h2{4,3}×{ }

    |2 20px
    3.3.6

    |4 20px
    3.4.4

    |4 20px
    4.4.6

    |

    |6

    		8 {3}
    6 {4}    		168

    |75px

    align=center BGCOLOR="#d0f0f0"

    ![50]

    |align=center|Cuboctahedral prism (cope)

    |75px

    |75px

    |align=center|{{CDD|node_h0|4|node_1|3|node|2|node_1}}
    rr{3,3}×{ }

    | 2 20px
    3.4.3.4

    | 8 20px
    3.4.4

    | 6 20px
    4.4.4

    |

    | 16

    		16 {3}
    36 {4}    		6024

    |75px

    align=center BGCOLOR="#d0f0f0"

    ![52]

    |Rhombicuboctahedral prism (sircope)

    |75px

    |75px

    |align=center|{{CDD|node_1|4|node_h|3|node_h|2|node_1}}
    s2{3,4}×{ }

    | 2 20px
    3.4.4.4

    | 8 20px
    3.4.4

    | 18 20px
    4.4.4

    |

    | 28

    		16 {3}
    84 {4}    		12048

    |75px

    align=center BGCOLOR="#d0f0f0"

    ![54]

    |align=center|Truncated octahedral prism (tope)

    |75px

    |75px

    |align=center|{{CDD|node_h0|4|node_1|3|node_1|2|node_1}}
    tr{3,3}×{ }

    | 2 20px
    4.6.6

    | 6 20px
    4.4.4

    | 8 20px
    4.4.6

    |

    | 16

    		48 {4}
    16 {6}    		9648

    |75px

    align=center BGCOLOR="#d0f0f0"

    ![59]

    |align=center|Icosahedral prism (ipe)

    |75px

    |75px

    |align=center|{{CDD|node|4|node_h|3|node_h|2|node_1}}
    s{3,4}×{ }

    |2 20px
    3.3.3.3.3

    |20 20px
    3.4.4

    |

    |

    | 22

    		40 {3}
    30 {4}    		7224

    |75px

    align=center BGCOLOR="#d0f0f0"

    ![12]

    |align=center|16-cell (hex)

    |75px

    |75px

    |align=center|{{CDD|node_h|2x|node_h|4|node|3|node}}
    s{2,4,3}

    |2+6+8 20px
    3.3.3.3

    |

    |

    |

    |16

    		32 {3}248

    |75px

    align=center BGCOLOR="#d0f0f0"

    !Nonuniform

    |Omnisnub tetrahedral antiprism
    = Pyritohedral icosahedral antiprism (pikap)

    |

    |75px

    |align=center|{{CDD|node|4|node_h|3|node_h|2x|node_h}}
    sr{2,3,4}

    |2 20px
    3.3.3.3.3

    |8 20px
    3.3.3.3

    |6+24 20px
    3.3.3

    |

    |40

    		16+96 {3}9624

    |

    align=center BGCOLOR="#d0f0f0"

    !Nonuniform

    |Edge-snub octahedral hosochoron
    Pyritosnub alterprism (pysna)

    |

    |75px

    |align=center|{{CDD|node_1|4|node_h|3|node_h|2x|node_h}}
    sr3{2,3,4}

    |2 20px
    3.4.4.4

    |6 20px
    4.4.4

    |8 20px
    3.3.3.3

    |24 20px
    3.4.4

    |40

    		16+48 {3}
    12+12+24+24 {4}    		14448

    |

    align=center BGCOLOR="#d0f0f0"

    !Nonuniform

    |Omnisnub cubic antiprism
    Snub cubic antiprism (sniccap)

    |

    |75px

    |align=center|{{CDD|node_h|4|node_h|3|node_h|2x|node_h}}
    s\left\{\begin{array}{l}4\\3\\2\end{array}\right\}

    | 2 20px
    3.3.3.3.4

    | 12+48 20px
    3.3.3

    | 8 20px
    3.3.3.3

    | 6 20px
    3.3.3.4

    |76

    		16+192 {3}
    12 {4}    		19248

    |

    align=center BGCOLOR="#d0f0f0"

    !Nonuniform

    |Runcic snub cubic hosochoron
    Truncated tetrahedral alterprism (tuta)

    |75px

    | 75px

    |{{CDD|node_h|2x|node_h|4|node|3|node_1}}
    s3{2,4,3}

    | 2 20px
    3.6.6

    | 6 20px
    3.3.3

    | 8 20px
    triangular cupola

    |

    |16

    		526024

    |75px

    == Icosahedral prisms: H<sub>3</sub> × A<sub>1</sub> ==

    This prismatic icosahedral symmetry is [5,3,2], order 240. There are two index 2 subgroups, [(5,3)+,2] and [5,3,2]+, but the second doesn't generate a uniform polychoron.

    class="wikitable"

    !rowspan=2|#

    !rowspan=2| Name (Bowers name and acronym)

    !rowspan=2| Picture

    !rowspan=2| Vertex
    figure

    !rowspan=2|Coxeter diagram
    and Schläfli
    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)

    |75px

    |75px

    |align=center|{{CDD|node_1|5|node|3|node|2|node_1}}
    {5,3}×{ }
    t0,3{5,3,2}

    | 2 20px
    5.5.5

    |12 20px
    4.4.5

    |

    |

    | 14

    | 30 {4}
    24 {5}

    | 80

    | 40

    |75px

    align=center

    !58

    |Icosidodecahedral prism (iddip)

    |75px

    |75px

    |align=center|{{CDD|node|5|node_1|3|node|2|node_1}}
    r{5,3}×{ }
    t1,3{5,3,2}

    |2 20px
    3.5.3.5

    |20 20px
    3.4.4

    |12 20px
    4.4.5

    |

    | 34

    | 40 {3}
    60 {4}
    24 {5}

    | 150

    | 60

    |75px

    align=center

    !59

    |align=center|Icosahedral prism
    (same as snub tetrahedral prism) (ipe)

    |75px

    |75px

    |align=center|{{CDD|node|5|node|3|node_1|2|node_1}}
    {3,5}×{ }
    t2,3{5,3,2}

    |2 20px
    3.3.3.3.3

    |20 20px
    3.4.4

    |

    |

    | 22

    | 40 {3}
    30 {4}

    | 72

    | 24

    |75px

    align=center

    !60

    |Truncated dodecahedral prism (tiddip)

    |75px

    |75px

    |align=center|{{CDD|node_1|5|node_1|3|node|2|node_1}}
    t{5,3}×{ }
    t0,1,3{5,3,2}

    |2 20px
    3.10.10

    |20 20px
    3.4.4

    |12 20px
    4.4.10

    |

    | 34

    | 40 {3}
    90 {4}
    24 {10}

    | 240

    | 120

    |75px

    align=center

    !61

    |Rhombicosidodecahedral prism (sriddip)

    |75px

    |75px

    |align=center|{{CDD|node_1|5|node|3|node_1|2|node_1}}
    rr{5,3}×{ }
    t0,2,3{5,3,2}

    |2 20px
    3.4.5.4

    |20 20px
    3.4.4

    |30 20px
    4.4.4

    |12 20px
    4.4.5

    | 64

    | 40 {3}
    180 {4}
    24 {5}

    | 300

    | 120

    |75px

    align=center

    !62

    |Truncated icosahedral prism (tipe)

    |75px

    |75px

    |align=center|{{CDD|node|5|node_1|3|node_1|2|node_1}}
    t{3,5}×{ }
    t1,2,3{5,3,2}

    |2 20px
    5.6.6

    |12 20px
    4.4.5

    |20 20px
    4.4.6

    |

    | 34

    | 90 {4}
    24 {5}
    40 {6}

    | 240

    | 120

    |75px

    align=center

    !63

    |Truncated icosidodecahedral prism (griddip)

    |75px

    |75px

    |align=center|{{CDD|node_1|5|node_1|3|node_1|2|node_1}}
    tr{5,3}×{ }
    t0,1,2,3{5,3,2}

    |2 20px
    4.6.10

    |30 20px
    4.4.4

    |20 20px
    4.4.6

    |12 20px
    4.4.10

    | 64

    | 240 {4}
    40 {6}
    24 {10}

    | 480

    | 240

    |75px

    align=center BGCOLOR="#d0f0f0"

    !64

    |Snub dodecahedral prism (sniddip)

    |75px

    |75px

    |align=center|{{CDD|node_h|5|node_h|3|node_h|2|node_1}}
    sr{5,3}×{ }

    |2 20px
    3.3.3.3.5

    |80 20px
    3.4.4

    |12 20px
    4.4.5

    |

    | 94

    | 160 {3}
    150 {4}
    24 {5}

    | 360

    | 120

    |75px

    align=center BGCOLOR="#d0f0f0"

    !Nonuniform

    |Omnisnub dodecahedral antiprism
    Snub dodecahedral antiprism (sniddap)

    |

    |75px

    |align=center|{{CDD|node_h|5|node_h|3|node_h|2x|node_h}}
    s\left\{\begin{array}{l}5\\3\\2\end{array}\right\}

    |2 20px
    3.3.3.3.5

    |30+120 20px
    3.3.3

    |20 20px
    3.3.3.3

    |12 20px
    3.3.3.5

    |184

    		20+240 {3}
    24 {5}    		220120

    |

    == Duoprisms: [p] × [q] ==

    File:3-3 duoprism.png, one of 6 triangular prism cells shown.]]

    The second is the infinite family of uniform duoprisms, products of two regular polygons. A duoprism's Coxeter-Dynkin diagram is {{CDD|node_1|p|node|2|node_1|q|node}}. Its vertex figure is a disphenoid tetrahedron, 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 4pq if p≠q; if the factors are both p-gons, the symmetry number is 8p2. 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,2pq,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 antiprisms.

    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 prisms75px75px
    align=center

    ! 3-4 duoprism (tisdip)

    		{{CDD|node_1|3|node|2|node_1|4|node}}3 cubes
    4 triangular prisms    		75px 75px75px
    align=center

    ! 4-4 duoprism (tes)
    (same as tesseract)

    		{{CDD|node_1|4|node|2|node_1|4|node}}4+4 cubes75px75px
    align=center

    ! 3-5 duoprism (trapedip)

    		{{CDD|node_1|3|node|2|node_1|5|node}}3 pentagonal prisms
    5 triangular prisms    		75px 75px75px
    align=center

    ! 4-5 duoprism (squipdip)

    		{{CDD|node_1|4|node|2|node_1|5|node}}4 pentagonal prisms
    5 cubes    		75px 75px75px
    align=center

    ! 5-5 duoprism (pedip)

    		{{CDD|node_1|5|node|2|node_1|5|node}}5+5 pentagonal prisms75px75px
    align=center

    ! 3-6 duoprism (thiddip)

    		{{CDD|node_1|3|node|2|node_1|6|node}}3 hexagonal prisms
    6 triangular prisms    		75px 75px75px
    align=center

    ! 4-6 duoprism (shiddip)

    		{{CDD|node_1|4|node|2|node_1|6|node}}4 hexagonal prisms
    6 cubes    		75px 75px75px
    align=center

    ! 5-6 duoprism (phiddip)

    		{{CDD|node_1|5|node|2|node_1|6|node}}5 hexagonal prisms
    6 pentagonal prisms    		75px 75px75px
    align=center

    ! 6-6 duoprism (hiddip)

    		{{CDD|node_1|6|node|2|node_1|6|node}}6+6 hexagonal prisms75px75px

    class="wikitable" width=600px
    align=center

    |75px
    3-3

    |75px
    3-4

    |75px
    3-5

    |75px
    3-6

    |75px
    3-7

    |75px
    3-8

    align=center

    |75px
    4-3

    |75px
    4-4

    |75px
    4-5

    |75px
    4-6

    |75px
    4-7

    |75px
    4-8

    align=center

    |75px
    5-3

    |75px
    5-4

    |75px
    5-5

    |75px
    5-6

    |75px
    5-7

    |75px
    5-8

    align=center

    |75px
    6-3

    |75px
    6-4

    |75px
    6-5

    |75px
    6-6

    |75px
    6-7

    |75px
    6-8

    align=center

    |75px
    7-3

    |75px
    7-4

    |75px
    7-5

    |75px
    7-6

    |75px
    7-7

    |75px
    7-8

    align=center

    |75px
    8-3

    |75px
    8-4

    |75px
    8-5

    |75px
    8-6

    |75px
    8-7

    |75px
    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.[https://bendwavy.org/klitzing/incmats/sns2s2mx.htm sns2s2mx], Richard Klitzing

    == 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 prisms are constructed from two parallel uniform antiprisms): (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, {{CDD|node_h|4|node_h|3|node_h}}, an 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 alternated (shown with empty circle nodes). The first is {{CDD|node_h|2x|node_h|2x|node_h|2x|node_h}}, s{21,1,1} which represented an index 24 subgroup (symmetry [2,2,2]+, order 8) form of the demitesseract, {{CDD|node_h|4|node|3|node|3|node}}, h{4,3,3} (symmetry [1+,4,3,3] = [31,1,1], order 192). The second is {{CDD|node_h|3|node_h|split1|nodes_hh}}, s{31,1,1}, which is an index 6 subgroup (symmetry [31,1,1]+, order 96) form of the snub 24-cell, {{CDD|node_h|3|node_h|4|node|3|node}}, s{3,4,3}, (symmetry [3+,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 (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]+, order 192, is the symmetry of the alternated omnitruncated tesseract.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

    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.{{Cite web|url=http://bendwavy.org/klitzing/explain/polytope-tree.htm#scaliform|title = Polytope-tree}} This category allows a subset of Johnson solids 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
    diagram

    !s3{2,4,3}, {{CDD|node_h|2x|node_h|4|node|3|node_1}}

    !s3{3,4,3}, {{CDD|node_h|3|node_h|4|node|3|node_1}}

    !colspan=2|Others

    Relation

    !24 of 48 vertices of
    rhombicuboctahedral prism

    !288 of 576 vertices of
    runcitruncated 24-cell

    !72 of 120 vertices
    of 600-cell

    !600 of 720 vertices
    of rectified 600-cell

    valign=top align=center

    !valign=center|Projection

    |200px

    |200px

    |200px

    |200px
    Two rings of pyramids

    align=center valign=top

    !valign=center|Net

    |200px
    runcic snub cubic hosochoron{{cite web |url=http://bendwavy.org/klitzing/incmats/tut=invtut.htm |title = tuta}}[http://www.polytope.net/hedrondude/simplesc.htm Category S1: Simple Scaliforms] tutcup

    |200px
    runcic snub 24-cell{{Cite web|url=http://bendwavy.org/klitzing/incmats/prissi.htm|title = Prissi}}[http://www.polytope.net/hedrondude/special.htm Category S3: Special Scaliforms] prissi

    |200px{{Cite web|url=https://bendwavy.org/klitzing/incmats/bidex.htm|title=bidex|website=bendwavy.org|accessdate=11 November 2023}}[http://www.polytope.net/hedrondude/special.htm Category S3: Special Scaliforms] bidex[http://eusebeia.dyndns.org/4d/bi24dim600cell The Bi-icositetradiminished 600-cell]

    |200px{{Cite web|url=https://bendwavy.org/klitzing/incmats/spidrox.htm|title=spidrox|website=bendwavy.org|accessdate=11 November 2023}}[http://www.polytope.net/hedrondude/scaleswirl.htm Category S4: Scaliform Swirlprisms] spidrox

    valign=center|Cells

    ! 40px 40px 40px

    ! 40px 40px 40px 40px

    !40px

    !40px 40px 40px

    valign=top

    !valign=center|Vertex
    figure

    |200px
    (1) 3.4.3.4: triangular cupola
    (2) 3.4.6: triangular cupola
    (1) 3.3.3: tetrahedron
    (1) 3.6.6: truncated tetrahedron

    |150px
    (1) 3.4.3.4: triangular cupola
    (2) 3.4.6: triangular cupola
    (2) 3.4.4: triangular prism
    (1) 3.6.6: truncated tetrahedron
    (1) 3.3.3.3.3: icosahedron

    |200px
    (2) 3.3.3.5: tridiminished icosahedron
    (4) 3.5.5: tridiminished icosahedron

    |150px
    (1) 3.3.3.3: square pyramid
    (4) 3.3.4: square pyramid
    (2) 4.4.5: pentagonal prism
    (2) 3.3.3.5 pentagonal antiprism

    === Geometric derivations for 46 nonprismatic Wythoffian uniform polychora ===

    The 46 Wythoffian 4-polytopes include the six convex regular 4-polytopes. The other forty can be derived from the regular polychora by geometric operations which preserve most or all of their symmetries, and therefore may be classified by the symmetry groups that they have in common.

    class=wikitable width=1000

    |400px
    Summary chart of truncation operations

    |600px
    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 (π/n radians 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

    !Symmetry

    !Coxeter diagram

    !Description

    align=center

    !Parent

    |t0{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

    |t1{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
    (Rectified dual)

    |t2{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
    (dual)

    |t3{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

    |t0,1{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

    |t1,2{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

    |t2,3{p,q,r}

    |{{CDD|node|p|node|q|node_1|r|node_1}}

    |Truncated dual {r,q,p}.

    align=center

    !Cantellation

    |t0,2{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

    |t1,3{p,q,r}

    |{{CDD|node|p|node_1|q|node|r|node_1}}

    |Cantellated dual {r,q,p}.

    align=center

    !Runcination
    (or expansion)

    |t0,3{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

    |t0,1,2{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

    |t1,2,3{p,q,r}

    |{{CDD|node|p|node_1|q|node_1|r|node_1}}

    |Cantitruncated dual {r,q,p}.

    align=center

    !Runcitruncation

    |t0,1,3{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

    |t0,2,3{p,q,r}

    |{{CDD|node_1|p|node|q|node_1|r|node_1}}

    |Runcitruncated dual {r,q,p}.

    align=center

    !Omnitruncation
    (runcicantitruncation)

    |t0,1,2,3{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+,2p,3,q]
    =[(3,p,3),q]

    |{{CDD|node_h1|2x|p|node|3|node|q|node}}

    |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

    |h2{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

    |h3{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

    |h2,3{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+,2p,3,2q,1+]

    |{{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+,2q,r]

    |{{CDD|node_h|p|node_h|2x|q|node|r|node}}

    |Alternated truncation

    align=center

    !Cantic snub

    |s2{p,2q,r}

    |{{CDD|node_h|p|node_h|2x|q|node_1|r|node}}

    |Cantellated alternated truncation

    align=center

    !Runcic snub

    |s3{p,2q,r}

    |{{CDD|node_h|p|node_h|2x|q|node|r|node_1}}

    |Runcinated alternated truncation

    align=center

    !Runcicantic snub

    |s2,3{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)+,2r]

    |{{CDD|node_h|p|node_h|q|node_h|2x|r|node}}

    |Alternated truncated rectification

    align=center

    !

    |ht0,3{2p,q,2r}

    |[(2p,q,2r,2+)]

    |{{CDD|node_h|2x|p|node|q|node|2x|r|node_h}}

    |Alternated runcination

    align=center

    !Bisnub

    |2s{2p,q,2r}

    |[2p,q+,2r]

    |{{CDD|node|2x|p|node_h|q|node_h|2x|r|node}}

    |Alternated bitruncation

    align=center

    !Omnisnub

    |ht0,1,2,3{p,q,r}

    |[p,q,r]+

    |{{CDD|node_h|p|node_h|q|node_h|r|node_h}}

    |Alternated omnitruncation

    See also convex uniform honeycombs, some of which illustrate these operations as applied to the regular cubic honeycomb.

    If two polytopes are duals 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 A4, B4, F4, H4 symmetry are given in this table by their full extended symmetry and Coxeter diagrams. The D4 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 B4 family.

    class=wikitable

    !Coxeter group

    !Extended
    symmetry

    !colspan=2|Polychora

    !Chiral
    extended
    symmetry

    !colspan=2|Alternation honeycombs

    align=center

    |rowspan=2|[3,3,3]
    {{CDD|node|3|node|3|node|3|node}}

    		[3,3,3]
    {{CDD|node_c1|3|node_c2|3|node_c3|3|node_c4}}
    (order 120)    		6

    | {{CDD|node_1|3|node|3|node|3|node}}(1) | {{CDD|node|3|node_1|3|node|3|node}}(2) | {{CDD|node_1|3|node_1|3|node|3|node}}(3)
    {{CDD|node_1|3|node|3|node_1|3|node}}(4) | {{CDD|node_1|3|node_1|3|node_1|3|node}}(7) | {{CDD|node_1|3|node_1|3|node|3|node_1}}(8)

    |colspan=3|

    align=center

    |[2+[3,3,3]]
    {{CDD|node_c1|3|node_c2|3|node_c2|3|node_c1}}
    (order 240)

    		3

    |{{CDD|node_1|3|node|3|node|3|node_1}}(5)| {{CDD|node|3|node_1|3|node_1|3|node}}(6) | {{CDD|node_1|3|node_1|3|node_1|3|node_1}}(9)

    |BGCOLOR="#e0f0e0"|[2+[3,3,3]]+
    (order 120)

    		BGCOLOR="#e0f0e0"|(1)

    |BGCOLOR="#e0f0e0"|{{CDD|node_h|3|node_h|3|node_h|3|node_h}}(−)

    align=center

    |rowspan=3|[3,31,1]
    {{CDD|node|3|node|split1|nodes}}

    		[3,31,1]
    {{CDD|node_c3|3|node_c4|split1|nodeab_c1-2}}
    (order 192)    		0

    | (none)

    |colspan=3|

    BGCOLOR="#e0f0e0" align=center

    |[1[3,31,1]]=[4,3,3]
    {{CDD|node_c1|3|node_c2|split1|nodeab_c3}} = {{CDD|node_c1|3|node_c2|3|node_c3|4|node}}
    (order 384)

    		(4)

    |{{CDD|node_1|3|node|split1|nodes}}(12) | {{CDD|node_1|3|node_1|split1|nodes}}(17) | {{CDD|node|3|node|split1|nodes_11}}(11) | {{CDD|node|3|node_1|split1|nodes_11}}(16)

    |colspan=3|

    BGCOLOR="#e0f0e0" align=center

    |[3[31,1,1]]=[3,4,3]
    {{CDD|node_c1|3|node_c2|split1|nodeab_c1}} = {{CDD|node_c1|3|node_c2|4|node|3|node}}
    (order 1152)

    		(3)

    |{{CDD|node|3|node_1|split1|nodes}}(22) | {{CDD|node_1|3|node|split1|nodes_11}}(23) | {{CDD|node_1|3|node_1|split1|nodes_11}}(24)

    |[3[3,31,1]]+
    =[3,4,3]+
    (order 576)

    		(1)

    |{{CDD|node_h|3|node_h|split1|nodes_hh}}(31) (= {{CDD|node_h|3|node_h|4|node|3|node}})
    {{CDD|node_h|3|node_h|4|node|3|node_1}}(−)

    align=center

    |rowspan=3|[4,3,3]
    {{CDD|node|4|node|3|node|3|node}}

    |BGCOLOR="#e0f0e0"|[3[1+,4,3,3]]=[3,4,3]
    {{CDD|node|4|node_c1|3|node_c2|3|node_c1}} = {{CDD|node_c2|3|node_c1|4|node|3|node}}
    (order 1152)

    		BGCOLOR="#e0f0e0"|(3)BGCOLOR="#e0f0e0"| {{CDD|node|4|node|3|node_1|3|node}}(22) | {{CDD|node|4|node_1|3|node|3|node_1}}(23) | {{CDD|node|4|node_1|3|node_1|3|node_1}}(24)

    |BGCOLOR="#e0f0e0" colspan=3|

    align=center

    |rowspan=2|[4,3,3]
    {{CDD|node_c1|4|node_c2|3|node_c3|3|node_c4}}
    (order 384)

    		rowspan=2|12

    |rowspan=2|{{CDD|node_1|4|node|3|node|3|node}}(10) | {{CDD|node|4|node_1|3|node|3|node}}(11) | {{CDD|node|4|node|3|node|3|node_1}}(12) | {{CDD|node_1|4|node_1|3|node|3|node}}(13) | {{CDD|node_1|4|node|3|node_1|3|node}}(14)
    {{CDD|node_1|4|node|3|node|3|node_1}}(15) | {{CDD|node|4|node_1|3|node_1|3|node}}(16) | {{CDD|node|4|node|3|node_1|3|node_1}}(17) | {{CDD|node_1|4|node_1|3|node_1|3|node}}(18) | {{CDD|node_1|4|node_1|3|node|3|node_1}}(19)
    {{CDD|node_1|4|node|3|node_1|3|node_1}}(20) | {{CDD|node_1|4|node_1|3|node_1|3|node_1}}(21)

    |BGCOLOR="#e0f0e0"|[1+,4,3,3]+
    (order 96)

    		BGCOLOR="#e0f0e0"|(2)

    |BGCOLOR="#e0f0e0"|{{CDD|node_h1|4|node|3|node|3|node}}(12) (= {{CDD|nodes_10ru|split2|node|3|node}})
    {{CDD|node|4|node_h|3|node_h|3|node_h}}(31)
    {{CDD|node_1|4|node_h|3|node_h|3|node_h}}(−)

    align=center

    |BGCOLOR="#e0f0e0"|[4,3,3]+
    (order 192)

    		BGCOLOR="#e0f0e0"|(1)

    |BGCOLOR="#e0f0e0"|{{CDD|node_h|4|node_h|3|node_h|3|node_h}}(−)

    align=center

    |rowspan=2|[3,4,3]
    {{CDD|node|3|node|4|node|3|node}}

    		[3,4,3]
    {{CDD|node_c1|3|node_c2|4|node_c3|3|node_c4}}
    (order 1152)    		6

    |{{CDD|node_1|3|node|4|node|3|node}}(22) | {{CDD|node|3|node_1|4|node|3|node}}(23) | {{CDD|node_1|3|node_1|4|node|3|node}}(24)
    {{CDD|node_1|3|node|4|node_1|3|node}}(25) | {{CDD|node_1|3|node_1|4|node_1|3|node}}(28) | {{CDD|node_1|3|node_1|4|node|3|node_1}}(29)

    |[2+[3+,4,3+]]
    (order 576)

    		1

    |{{CDD|node_h|3|node_h|4|node|3|node}}(31)

    align=center

    |[2+[3,4,3]]
    {{CDD|node_c1|3|node_c2|4|node_c2|3|node_c1}}
    (order 2304)

    		3

    |{{CDD|node_1|3|node|4|node|3|node_1}}(26) | {{CDD|node|3|node_1|4|node_1|3|node}}(27) | {{CDD|node_1|3|node_1|4|node_1|3|node_1}}(30)

    |BGCOLOR="#e0f0e0"|[2+[3,4,3]]+
    (order 1152)

    		BGCOLOR="#e0f0e0"|(1)

    |BGCOLOR="#e0f0e0"|{{CDD|node_h|3|node_h|4|node_h|3|node_h}}(−)

    align=center

    |[5,3,3]
    {{CDD|node|5|node|3|node|3|node}}

    		[5,3,3]
    {{CDD|node_c1|5|node_c2|3|node_c3|3|node_c4}}
    (order 14400)    		15

    | {{CDD|node_1|5|node|3|node|3|node}}(32) | {{CDD|node|5|node_1|3|node|3|node}}(33) | {{CDD|node|5|node|3|node_1|3|node}}(34) | {{CDD|node|5|node|3|node|3|node_1}}(35) | {{CDD|node_1|5|node_1|3|node|3|node}}(36)
    {{CDD|node_1|5|node|3|node_1|3|node}}(37) | {{CDD|node_1|5|node|3|node|3|node_1}}(38) | {{CDD|node|5|node_1|3|node_1|3|node}}(39) | {{CDD|node|5|node_1|3|node|3|node_1}}(40) | {{CDD|node|5|node|3|node_1|3|node_1}}(41)
    {{CDD|node_1|5|node_1|3|node_1|3|node}}(42) | {{CDD|node_1|5|node_1|3|node|3|node_1}}(43) | {{CDD|node_1|5|node|3|node_1|3|node_1}}(44) | {{CDD|node|5|node_1|3|node_1|3|node_1}}(45) | {{CDD|node_1|5|node_1|3|node_1|3|node_1}}(46)

    |BGCOLOR="#e0f0e0"|[5,3,3]+
    (order 7200)

    		BGCOLOR="#e0f0e0"|(1)

    |BGCOLOR="#e0f0e0"|{{CDD|node_h|5|node_h|3|node_h|3|node_h}}(−)

    align=center

    |rowspan=4|[3,2,3]
    {{CDD|node|3|node|2|node|3|node}}

    		[3,2,3]
    {{CDD|node_c1|3|node_c2|2|node_c3|3|node_c3}}
    (order 36)    		0

    | (none)

    |[3,2,3]+
    (order 18)

    		0

    | (none)

    align=center

    |[2+[3,2,3]]
    {{CDD|node_c1|3|node_c2|2|node_c2|3|node_c1}}
    (order 72)

    		0

    | {{CDD|node_1|3|node|2|node|3|node_1}}

    | [2+[3,2,3]]+
    (order 36)

    		0

    | (none)

    align=center[[3],2,3]=[6,2,3]
    {{CDD|node_c1|3|node_c1|2|node_c2|3|node_c3}} = {{CDD|node_c1|6|node|2|node_c2|3|node_c3}}
    (order 72)    		1

    | {{CDD|node_1|3|node_1|2|node_1|3|node}}

    |BGCOLOR="#e0f0e0"|[1[3,2,3]]=[[3],2,3]+=[6,2,3]+
    (order 36)

    		BGCOLOR="#e0f0e0"|(1)

    |BGCOLOR="#e0f0e0"|{{CDD|node_h|3|node_h|2|node_1|3|node}}

    align=center[(2+,4)[3,2,3]]=[2+[6,2,6]]
    {{CDD|node_c1|3|node_c1|2|node_c1|3|node_c1}} = {{CDD|node_c1|6|node|2|node_c1|6|node}}
    (order 288)    		1

    | {{CDD|node_1|3|node_1|2|node_1|3|node_1}}

    |BGCOLOR="#e0f0e0"|[(2+,4)[3,2,3]]+=[2+[6,2,6]]+
    (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]
    {{CDD|node|4|node|2|node|4|node}}

    		[4,2,4]
    {{CDD|node_c1|4|node_c2|2|node_c3|4|node_c4}}
    (order 64)    		0

    | (none)

    |[4,2,4]+
    (order 32)

    		0

    | (none)

    align=center BGCOLOR="#e0f0e0"

    |[2+[4,2,4]]
    {{CDD|node_c1|4|node_c2|2|node_c2|4|node_c1}}
    (order 128)

    		0

    | (none)

    |[2+[(4,2+,4,2+)]]
    (order 64)

    		0

    | (none)

    align=center BGCOLOR="#e0f0e0"

    |[(3,3)[4,2*,4]]=[4,3,3]
    {{CDD|node_c1|4|node|2|node|4|node_c1}} = {{CDD|node_c1|4|node|3|node|3|node}}
    (order 384)

    		(1)

    | {{CDD|node|4|node_1|2|node_1|4|node}}(10)

    |BGCOLOR="#e0f0e0"|[(3,3)[4,2*,4]]+=[4,3,3]+
    (order 192)

    		BGCOLOR="#e0f0e0"|(1)

    |BGCOLOR="#e0f0e0"| {{CDD|node|4|node_h|2x|node_h|4|node}}(12)

    align=center BGCOLOR="#e0f0e0"[[4],2,4]=[8,2,4]
    {{CDD|node_c1|4|node_c1|2|node_c2|4|node_c3}} = {{CDD|node_c1|8|node|2|node_c2|4|node_c3}}
    (order 128)    		(1)

    | {{CDD|node_1|4|node_1|2|node_1|4|node}}

    |BGCOLOR="#e0f0e0"|[1[4,2,4]]=[[4],2,4]+=[8,2,4]+
    (order 64)

    		BGCOLOR="#e0f0e0"|(1)

    | {{CDD|node_h|4|node_h|2x|node_h|4|node}}

    align=center BGCOLOR="#e0f0e0"[(2+,4)[4,2,4]]=[2+[8,2,8]]
    {{CDD|node_c1|4|node_c1|2|node_c1|4|node_c1}} = {{CDD|node_c1|8|node|2|node_c1|8|node}}
    (order 512)    		(1)

    | {{CDD|node_1|4|node_1|2|node_1|4|node_1}}

    |BGCOLOR="#e0f0e0"|[(2+,4)[4,2,4]]+=[2+[8,2,8]]+
    (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 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 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

    References

    {{reflist}}

    • 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
    • B. Grünbaum Convex Polytopes, New York; London : Springer, c2003. {{isbn|0-387-00424-6}}.
      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.]
    • 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. 92, p. 122.
    • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} (Chapter 26)
    • John H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
    • 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

    • 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}}