Uniform 4-polytope

320px|thumb|[[Schlegel diagram for the truncated 120-cell with tetrahedral cells visible]]

thumb|320px|[[Orthographic projection of the truncated 120-cell, in the Coxeter plane ( symmetry). Only vertices and edges are drawn.]]

In geometry, a uniform 4-polytope (or uniform polychoron) 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 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.
• 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.
• 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.
• 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.
• 1998-2000: 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"). The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t_0,1, cantellation, t_0,2, runcination t_0,3, with single ringed forms called rectified, and bi, tri-prefixes added when the first ring was on the second or third nodes.
• 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.
• 2008: The Symmetries of Things 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.
• 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, 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.
• 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.

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:

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

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

• 6 regular convex 4-polytopes: 5-cell {3,3,3}, 8-cell {4,3,3}, 16-cell {3,3,4}, 24-cell {3,4,3}, 120-cell {5,3,3}, and 600-cell {3,3,5}.
• 10 regular star 4-polytopes: icosahedral 120-cell {3,5,⁵/₂}, small stellated 120-cell {⁵/₂,5,3}, great 120-cell {5,⁵/₂,5}, grand 120-cell {5,3,⁵/₂}, great stellated 120-cell {⁵/₂,3,5}, grand stellated 120-cell {⁵/₂,5,⁵/₂}, great grand 120-cell {5,⁵/₂,3}, great icosahedral 120-cell {3,⁵/₂,5}, grand 600-cell {3,3,⁵/₂}, and great grand stellated 120-cell {⁵/₂,3,3}.

Convex uniform 4-polytopes

Symmetry of uniform 4-polytopes in four dimensions

{| class=wikitable style="float:right; margin-left:10px; width:240px"

|+ Orthogonal subgroups

|-

|The 24 mirrors of F₄ can be decomposed into 2 orthogonal D₄ groups:

1. = (12 mirrors)
2. = (12 mirrors)

|-

|The 10 mirrors of B₃×A₁ can be decomposed into orthogonal groups, 4A₁ and D₃:

1. = (3+1 mirrors)
2. = (6 mirrors)

|}

There are 5 fundamental mirror symmetry point group families in 4-dimensions: A₄ = , B₄ = , D₄ = , F₄ = , H₄ = .

|60px

| align=center|<br />ht_0,1,2,3{3,3,3}

|30px (2)<br />(3.3.3.3.3)

|30px (2)<br />(3.3.3.3)

|30px (4)<br />(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, 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 ⁺, order 120, defining an omnisnub 5-cell which is listed for completeness, but is not uniform.

The B₄ family

This family has diploid hexadecachoric symmetry,<br />(Or omnisnub 16-cell)

|60px

| align=center|<br />ht_0,1,2,3{4,3,3}

|(1)<br />30px<br />(3.3.3.3.4)

|(1)<br />30px<br />(3.3.3.4)

|(1)<br />30px<br />(3.3.3.3)

|(1)<br />30px<br />(3.3.3.3.3)

|(4)<br />30px<br />(3.3.3)

| 272

| 944

| 864

| 192

|}

16-cell truncations

{| class="wikitable"

!rowspan=2|#

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

!rowspan=2| Vertex<br />figure

!rowspan=2|Coxeter diagram<br />and Schläfli<br />symbols

!colspan=5 |Cell counts by location

!colspan=4|Element counts

|-

! Pos. 3<br /><br />(8)

! Pos. 2<br /><br />(24)

! Pos. 1<br /><br />(32)

! Pos. 0<br /><br />(16)

!Alt

! Cells

! Faces

! Edges

! Vertices

|- BGCOLOR="#e0e0f0" align=center

!12

|align=center|16-cell<br>Hexadecachoron

|60px

|align=center|<br />ht_0,1,2,3{3,4,3}

|(2)<br />30px<br />(3.3.3.3.4)

|(2)<br />30px<br />(3.3.3.3)

|(4)<br />30px<br />(3.3.3)

| 816

| 2832

| 2592

| 576

|}

The H₄ family

This family has diploid hexacosichoric symmetry,<br />(Same as the omnisnub 600-cell)

|60px

| align=center|<br />ht_0,1,2,3{5,3,3}

|30px (1)<br />(3.3.3.3.5)

|30px (1)<br />(3.3.3.5)

|30px (1)<br />(3.3.3.3)

|30px (1)<br />(3.3.3.3.3)

|30px (4)<br />(3.3.3)

| 9840

| 35040

| 32400

| 7200

|}

600-cell truncations

{| class="wikitable"

!rowspan=2|#

!rowspan=2|Name<br />(Bowers style acronym)

!rowspan=2|Vertex<br />figure

!rowspan=2|Coxeter diagram<br />and Schläfli<br />symbols

!rowspan=2|Symmetry

!colspan=4 |Cell counts by location

!colspan=4|Element counts

|-

! Pos. 3<br /><br />(120)

! Pos. 2<br /><br />(720)

! Pos. 1<br /><br />(1200)

! Pos. 0<br /><br />(600)

! Cells

! Faces

! Edges

! Vertices

|- BGCOLOR="#e0e0f0" align=center

!35

|600-cell<br>Hexacosichoron

Polygonal prismatic prisms: [p] × [ ] × [ ]

The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - - 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}.

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

The infinite sets of uniform antiprismatic prisms are constructed from two parallel uniform antiprisms): (p≥2) - - 2 p-gonal antiprisms, connected by 2 p-gonal prisms and 2p triangular 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

320px|thumb|Like the 3-dimensional [[snub cube, , an alternation removes half the vertices, in two chiral sets of vertices from the ringed form , 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 , s{2^1,1,1} which represented an index 24 subgroup (symmetry [2,2,2]⁺, order 8) form of the demitesseract, , h{4,3,3} (symmetry [1⁺,4,3,3] = [3^1,1,1], order 192). The second is , s{3^1,1,1}, which is an index 6 subgroup (symmetry [3^1,1,1]⁺, order 96) form of the snub 24-cell, , s{3,4,3}, (symmetry [3⁺,4,3], order 576).

Other alternations, such as , as an alternation from the omnitruncated tesseract , 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.

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. This category allows a subset of Johnson solids as cells, for example the triangular cupola.

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

The nets and vertex figures of the four convex equilateral cases are given below, along with a list of cells around each vertex.

{| class=wikitable width=400

|+ Four convex vertex-transitive equilateral 4-polytopes with nonuniform cells

!Coxeter<br />diagram

!s₃{2,4,3},

!s₃{3,4,3},

!colspan=2|Others

|-

!Relation

!24 of 48 vertices of <br />rhombicuboctahedral prism

!288 of 576 vertices of<br />runcitruncated 24-cell

!72 of 120 vertices<br />of 600-cell

!600 of 720 vertices<br />of rectified 600-cell

|- valign=top align=center

!valign=center|Projection

|200px

|200px

|200px

|200px<BR>Two rings of pyramids

|- align=center valign=top

!valign=center|Net

|200px<br />runcic snub cubic hosochoron

|200px<br />runcic snub 24-cell

|200px

|200px

|-

!valign=center|Cells

! 40px 40px 40px

! 40px 40px 40px 40px

!40px

!40px 40px 40px

|- valign=top

!valign=center|Vertex<br />figure

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

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

|200px<br />(2) 3.3.3.5: tridiminished icosahedron <br />(4) 3.5.5: tridiminished icosahedron

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

|}

=== Geometric derivations for 46 nonprismatic Wythoffian uniform polychora === <!--This section is linked from Convex uniform honeycomb -->

The 46 Wythoffian 4-polytopes include the six convex regular 4-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<br />Summary chart of truncation operations

|600px<br />Example locations of kaleidoscopic generator point on fundamental domain.

|}

The geometric operations that derive the 40 uniform 4-polytopes from the regular 4-polytopes are truncating operations. A 4-polytope may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below.

The Coxeter-Dynkin diagram shows the four mirrors of the Wythoffian kaleidoscope as nodes, and the edges between the nodes are labeled by an integer showing the angle between the mirrors (&pi;/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"

!Row

!Operation

!Schläfli symbol

!Symmetry

!Coxeter diagram

!Description

|- align=center

!1

!Parent

|t₀{p,q,r}

|rowspan=15|[p,q,r]

|

|Original regular form {p,q,r}

|- align=center

!2

!Rectification

|t₁{p,q,r}

|

|Truncation operation applied until the original edges are degenerated into points.

|- align=center

!3

!Birectification<br />(Rectified dual)

|t₂{p,q,r}

|

|Face are fully truncated to points. Same as rectified dual.

|- align=center

!4

!Trirectification<br />(dual)

|t₃{p,q,r}

|

|Cells are truncated to points. Regular dual {r,q,p}

|- align=center

!5

!Truncation

|t_0,1{p,q,r}

|

|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

!6

!Cantellation

|t_0,2{p,q,r}

|

| A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.

|- align=center

!7

!Runcination<br />(or expansion)

|t_0,3{p,q,r}

|

|A truncation applied to the cells, faces and edges; defines a progression between a regular form and the dual.

|- align=center

!8

!Bitruncation

|t_1,2{p,q,r}

|

| A truncation between a rectified form and the dual rectified form.

|- align=center

!9

!Bicantellation

|t_1,3{p,q,r}

|

|Cantellated dual {r,q,p}.

|- align=center

!10

!Tritruncation

|t_2,3{p,q,r}

|

|Truncated dual {r,q,p}.

|- align=center

!11

!Cantitruncation

|t_0,1,2{p,q,r}

|

|Both the cantellation and truncation operations applied together.

|- align=center

!12

!Runcitruncation

|t_0,1,3{p,q,r}

|

|Both the runcination and truncation operations applied together.

|- align=center

!13

!Runcicantellation

|t_0,2,3{p,q,r}

|

|Runcitruncated dual {r,q,p}.

|- align=center

!14

!Bicantitruncation

|t_1,2,3{p,q,r}

|

|Cantitruncated dual {r,q,p}.

|- align=center

!15

!Omnitruncation<br />(runcicantitruncation)

|t_0,1,2,3{p,q,r}

|

| Application of all three operators.

|- align=center

!16

!Snub

|s{p,2q,r}

|rowspan=4|[p⁺,2q,r]

|

|Alternated truncation

|- align=center

!17

!Cantic snub

|s₂{p,2q,r}

|

|Cantellated alternated truncation

|- align=center

!18

!Runcic snub

|s₃{p,2q,r}

|

|Runcinated alternated truncation

|- align=center

!19

!Runcicantic snub

|s_2,3{p,2q,r}

|

|Runcicantellated alternated truncation

|- align=center

!20

!Snub rectified

|sr{p,q,2r}

|[(p,q)⁺,2r]

|

|Alternated truncated rectification

|- align=center

!21

!

|ht_0,3{2p,q,2r}

|[(2p,q,2r,2⁺)]

|

|Alternated runcination

|- align=center

!22

!Bisnub

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

|[2p,q⁺,2r]

|

|Alternated bitruncation

|- align=center

!23

!Omnisnub

|ht_0,1,2,3{p,q,r}

|[p,q,r]⁺

|

|Alternated omnitruncation

|- align=center

!24

!Half

|h{2p,3,q}

|rowspan=4|[1⁺,2p,3,q]<br />=[(3,p,3),q]

|

|Alternation of , same as

|- align=center

!25

!Cantic

|h₂{2p,3,q}

|

|Same as

|- align=center

!26

!Runcic

|h₃{2p,3,q}

|

|Same as

|- align=center

!27

!Runcicantic

|h_2,3{2p,3,q}

|

|Same as

|- align=center

!28

!Quarter

|q{2p,3,2q}

|[1⁺,2p,3,2q,1⁺]

|

|Same as

|}

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<br />symmetry

!colspan=2|Polychora

!Chiral<br />extended<br />symmetry

!colspan=2|Alternation honeycombs

|- align=center

|rowspan=2|[3,3,3]<br />||[3,3,3]<br /><br />(order 120)||6

| ₍₁₎ | ₍₂₎ | ₍₃₎<br />₍₄₎ | ₍₇₎ | ₍₈₎

|colspan=3|

|- align=center

|[2⁺[3,3,3<nowiki>]]</nowiki><br /><br />(order 240)||3

|₍₅₎| ₍₆₎ | ₍₉₎

|BGCOLOR="#e0f0e0"|[2⁺[3,3,3<nowiki>]]</nowiki>⁺<br />(order 120)||BGCOLOR="#e0f0e0"|(1)

|BGCOLOR="#e0f0e0"|₍₋₎

|- align=center

|rowspan=3|[3,3^1,1]<br /> ||[3,3^1,1]<br /><br />(order 192)||0

| (none)

|colspan=3|

|- BGCOLOR="#e0f0e0" align=center

|[1[3,3^1,1<nowiki>]]</nowiki>=[4,3,3]<br /> = <br />(order 384)||(4)

|₍₁₂₎ | ₍₁₇₎ |₍₁₁₎ |₍₁₆₎

|colspan=3|

|- BGCOLOR="#e0f0e0" align=center

|[3[3^1,1,1<nowiki>]]</nowiki>=[3,4,3]<br /> = <br />(order 1152)||(3)

|₍₂₂₎ | ₍₂₃₎ | ₍₂₄₎

|[3[3,3^1,1<nowiki>]]</nowiki>⁺<br />=[3,4,3]⁺<br />(order 576)||(1)

|₍₃₁₎ (= )<br />₍₋₎

|- align=center

|rowspan=3|[4,3,3]<br />

|BGCOLOR="#e0f0e0"|[3[1⁺,4,3,3<nowiki>]]</nowiki>=[3,4,3]<br /> = <br />(order 1152)||BGCOLOR="#e0f0e0"|(3)||BGCOLOR="#e0f0e0"| ₍₂₂₎ | ₍₂₃₎ | ₍₂₄₎

|BGCOLOR="#e0f0e0" colspan=3|

|- align=center

|rowspan=2|[4,3,3]<br /><br />(order 384)||rowspan=2|12

|rowspan=2|₍₁₀₎ | ₍₁₁₎ | ₍₁₂₎ | ₍₁₃₎ | ₍₁₄₎<br />₍₁₅₎ | ₍₁₆₎ | ₍₁₇₎ | ₍₁₈₎ | ₍₁₉₎<br />₍₂₀₎ | ₍₂₁₎

|BGCOLOR="#e0f0e0"|[1⁺,4,3,3]⁺<br />(order 96)||BGCOLOR="#e0f0e0"|(2)

|BGCOLOR="#e0f0e0"|₍₁₂₎ (= )<br />₍₃₁₎<br />₍₋₎

|- align=center

|BGCOLOR="#e0f0e0"|[4,3,3]⁺<br />(order 192)||BGCOLOR="#e0f0e0"|(1)

|BGCOLOR="#e0f0e0"|₍₋₎

|- align=center

|rowspan=2|[3,4,3]<br />||[3,4,3]<br /><br />(order 1152)||6

|₍₂₂₎ | ₍₂₃₎ | ₍₂₄₎<br />₍₂₅₎ | ₍₂₈₎ | ₍₂₉₎

|[2⁺[3⁺,4,3⁺<nowiki>]]</nowiki><br />(order 576)||1

|₍₃₁₎

|- align=center

|[2⁺[3,4,3<nowiki>]]</nowiki><br /><br />(order 2304)||3

|₍₂₆₎ | ₍₂₇₎ | ₍₃₀₎

|BGCOLOR="#e0f0e0"|[2⁺[3,4,3<nowiki>]]</nowiki>⁺<br />(order 1152)||BGCOLOR="#e0f0e0"|(1)

|BGCOLOR="#e0f0e0"|₍₋₎

|- align=center

|[5,3,3]<br />||[5,3,3]<br /><br />(order 14400)||15

| ₍₃₂₎ | ₍₃₃₎ | ₍₃₄₎ | ₍₃₅₎ | ₍₃₆₎<br />₍₃₇₎ | ₍₃₈₎ | ₍₃₉₎ | ₍₄₀₎ | ₍₄₁₎<br />₍₄₂₎ | ₍₄₃₎ | ₍₄₄₎ | ₍₄₅₎ | ₍₄₆₎

|BGCOLOR="#e0f0e0"|[5,3,3]⁺<br />(order 7200)||BGCOLOR="#e0f0e0"|(1)

|BGCOLOR="#e0f0e0"|₍₋₎

|- align=center

|rowspan=4|[3,2,3]<br />||[3,2,3]<br /><br />(order 36)||0

| (none)

|[3,2,3]⁺<br />(order 18)||0

| (none)

|- align=center

|[2⁺[3,2,3<nowiki>]]</nowiki><br /><br />(order 72)||0

|

| [2⁺[3,2,3<nowiki>]]</nowiki>⁺<br />(order 36)|| 0

| (none)

|- align=center

||<nowiki>[[</nowiki>3],2,3]=[6,2,3]<br /> = <br />(order 72)||1

|

|BGCOLOR="#e0f0e0"|[1[3,2,3<nowiki>]]</nowiki>=<nowiki>[[</nowiki>3],2,3]⁺=[6,2,3]⁺<br />(order 36)||BGCOLOR="#e0f0e0"|(1)

|BGCOLOR="#e0f0e0"|

|- align=center

||[(2⁺,4)[3,2,3<nowiki>]]</nowiki>=[2⁺[6,2,6<nowiki>]]</nowiki><br /> = <br />(order 288)||1

|

|BGCOLOR="#e0f0e0"|[(2⁺,4)[3,2,3<nowiki>]]</nowiki>⁺=[2⁺[6,2,6<nowiki>]]</nowiki>⁺<br />(order 144)||BGCOLOR="#e0f0e0"|(1)

|BGCOLOR="#e0f0e0"|

|- align=center BGCOLOR="#e0f0e0"

|rowspan=5|[4,2,4]<br />||[4,2,4]<br /><br />(order 64)||0

| (none)

|[4,2,4]⁺<br />(order 32)||0

| (none)

|- align=center BGCOLOR="#e0f0e0"

|[2⁺[4,2,4<nowiki>]]</nowiki><br /><br />(order 128)||0

| (none)

|[2⁺[(4,2⁺,4,2⁺)<nowiki>]]</nowiki><br />(order 64)||0

| (none)

|- align=center BGCOLOR="#e0f0e0"

|[(3,3)[4,2*,4<nowiki>]]</nowiki>=[4,3,3]<br /> = <br />(order 384)||(1)

| ₍₁₀₎

|BGCOLOR="#e0f0e0"|[(3,3)[4,2*,4<nowiki>]]</nowiki>⁺=[4,3,3]⁺<br />(order 192)||BGCOLOR="#e0f0e0"|(1)

|BGCOLOR="#e0f0e0"| ₍₁₂₎

|- align=center BGCOLOR="#e0f0e0"

||<nowiki>[[</nowiki>4],2,4]=[8,2,4]<br /> = <br />(order 128)||(1)

|

|BGCOLOR="#e0f0e0"|[1[4,2,4<nowiki>]]</nowiki>=<nowiki>[[</nowiki>4],2,4]⁺=[8,2,4]⁺<br />(order 64)||BGCOLOR="#e0f0e0"|(1)

|

|- align=center BGCOLOR="#e0f0e0"

||[(2⁺,4)[4,2,4<nowiki>]]</nowiki>=[2⁺[8,2,8<nowiki>]]</nowiki><br /> = <br />(order 512)||(1)

|

|BGCOLOR="#e0f0e0"|[(2⁺,4)[4,2,4<nowiki>]]</nowiki>⁺=[2⁺[8,2,8<nowiki>]]</nowiki>⁺<br />(order 256)||BGCOLOR="#e0f0e0"|(1)

|BGCOLOR="#e0f0e0"|

|}

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<!--2191 – 64 = 2127--> known uniform star polychora at present (not counting the infinite set of duoprisms based on star polygons). There is currently no proof of the set's completeness.

See also

• Convex uniform honeycomb - related infinite 4-polytopes in Euclidean 3-space.
• Convex uniform honeycombs in hyperbolic space - related infinite 4-polytopes in Hyperbolic 3-space.
• Finite regular skew polyhedra of 4-space
• Paracompact uniform honeycombs

References

• B. Grünbaum Convex Polytopes, New York, London: Springer, 2003. . <br />Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.
• H.S.M. Coxeter:
• H.S.M. Coxeter, M.S. Longuet-Higgins and J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, 1954
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover, New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
• (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. pp. 92, 122.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 26)
• John H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, pp. 38–39, 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, pp. 329–344, (2010)

External links

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• Convex uniform 4-polytopes
• Uniform, convex polytopes in four dimensions, Marco Möller . Includes alternative names for these figures, including those from Jonathan Bowers, George Olshevsky, and Norman Johnson.
• Regular and semi-regular convex polytopes a short historical overview
• Java3D Applets with sources
• Nonconvex uniform 4-polytopes
• Uniform polychora by Jonathan Bowers
• Stella4D Stella (software) produces interactive views of known uniform polychora including the 64 convex forms and the infinite prismatic families.
• 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]