Uniform 4polytope
In geometry, a uniform 4polytope (or uniform polychoron[1]) is a 4polytope which is vertextransitive and whose cells are uniform polyhedra, and faces are regular polygons.
47 nonprismatic convex uniform 4polytopes, one finite set of convex prismatic forms, and two infinite sets of convex prismatic forms have been described. There are also an unknown number of nonconvex 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 4polytopes (star polyhedron cells and/or vertex figures)
 1852: Ludwig Schläfli also found 4 of the 10 regular star 4polytopes, 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 4polytopes, 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 .
 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 SemiRegular Figures in Space of n Dimensions.[2]
 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 4polytopes.[3]
 1911: Pieter Hendrik Schoute published Analytic treatment of the polytopes regularly derived from the regular polytopes, followed BooleStott's notations, enumerating the convex uniform polytopes by symmetry based on 5cell, 8cell/16cell, and 24cell.
 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.[4]
 Convex uniform polytopes:
 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and SemiRegular Polytopes.
 Convex uniform 4polytopes:
 1965: The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy, in their publication FourDimensional Archimedean Polytopes, established by computer analysis, adding only one nonWythoffian convex 4polytope, 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 SemiRegular Polytopes II which included analysis of the unique snub 24cell structure, and the symmetry of the anomalous grand antiprism.
 1998[5]2000: The 4polytopes 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 4polytopes as polychora, like polyhedra for 3polytopes, from the Greek roots poly ("many") and choros ("room" or "space").[6] 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,triprefixes added when the first ring was on the second or third nodes.[7][8]
 2004: A proof that the ConwayGuy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnson's naming system in his listing.[9]
 2008: The Symmetries of Things[10] was published by John H. Conway contains the first printpublished listing of the convex uniform 4polytopes and higher dimensions 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 ijkambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, with all of Johnson's names were included in the book index.
 Nonregular uniform star 4polytopes: (similar to the nonconvex uniform polyhedra)
Regular 4polytopes
Regular 4polytopes are a subset of the uniform 4polytopes, which satisfy additional requirements. Regular 4polytopes can be expressed with Schläfli symbol {p,q,r} have cells of type , faces of type {p}, edge figures {r}, and vertex figures {q,r}.
The existence of a regular 4polytope {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 4polytope is dependent upon an inequality:[13]
The 16 regular 4polytopes, with the property that all cells, faces, edges, and vertices are congruent:
 6 regular convex 4polytopes: 5cell {3,3,3}, 8cell {4,3,3}, 16cell {3,3,4}, 24cell {3,4,3}, 120cell {5,3,3}, and 600cell {3,3,5}.
 10 regular star 4polytopes: {3,5,5/2}, {5/2,5,3}, {5,5/2,5}, {5,3,5/2}, {5/2,3,5}, {5/2,5,5/2}, {5,5/2,3}, {3,5/2,5}, {3,3,5/2}, and {5/2,3,3}.
Convex uniform 4polytopes
Symmetry of uniform 4polytopes in four dimensions
The 16 mirrors of B_{4} can be decomposed into 2 orthogonal groups, 4A_{1} and D_{4}:

The 24 mirrors of F_{4} can be decomposed into 2 orthogonal D_{4} groups:

The 10 mirrors of B_{3}×A_{1} can be decomposed into orthogonal groups, 4A_{1} and D_{3}:

There are 5 fundamental mirror symmetry point group families in 4dimensions: A_{4} =
Each reflective uniform 4polytope 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 evenbranches. 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 4polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.
Weyl group 
Conway Quaternion 
Abstract structure 
Order  Coxeter diagram 
Coxeter notation 
Commutator subgroup 
Coxeter number (h) 
Mirrors m=2h  

Irreducible  
A_{4}  +1/60[I×I].21  S_{5}  120  [3,3,3]  [3,3,3]^{+}  5  10  
D_{4}  ±1/3[T×T].2  1/2.^{2}S_{4}  192  [3^{1,1,1}]  [3^{1,1,1}]^{+}  6  12  
B_{4}  ±1/6[O×O].2  ^{2}S_{4} = S_{2}≀S_{4}  384  [4,3,3]  8  4  12  
F_{4}  ±1/2[O×O].2_{3}  3.^{2}S_{4}  1152  [3,4,3]  [3^{+},4,3^{+}]  12  12  12  
H_{4}  ±[I×I].2  2.(A_{5}×A_{5}).2  14400  [5,3,3]  [5,3,3]^{+}  30  60  
Prismatic groups  
A_{3}A_{1}  +1/24[O×O].2_{3}  S_{4}×D_{1}  48  [3,3,2] = [3,3]×[ ]  [3,3]^{+}    6  1  
B_{3}A_{1}  ±1/24[O×O].2  S_{4}×D_{1}  96  [4,3,2] = [4,3]×[ ]    3  6  1  
H_{3}A_{1}  ±1/60[I×I].2  A_{5}×D_{1}  240  [5,3,2] = [5,3]×[ ]  [5,3]^{+}    15  1  
Duoprismatic groups (Use 2p,2q for even integers)  
I_{2}(p)I_{2}(q)  ±1/2[D_{2p}×D_{2q}]  D_{p}×D_{q}  4pq  [p,2,q] = [p]×[q]  [p^{+},2,q^{+}]    p  q  
I_{2}(2p)I_{2}(q)  ±1/2[D_{4p}×D_{2q}]  D_{2p}×D_{q}  8pq  [2p,2,q] = [2p]×[q]    p  p  q  
I_{2}(2p)I_{2}(2q)  ±1/2[D_{4p}×D_{4q}]  D_{2p}×D_{2q}  16pq  [2p,2,2q] = [2p]×[2q]    p  p  q  q 
Enumeration
There are 64 convex uniform 4polytopes, including the 6 regular convex 4polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms.
 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 selfdual regular A_{4} [3,3,3] group (5cell) family.
 9 are in the selfdual regular F_{4} [3,4,3] group (24cell) family. (Excluding snub 24cell)
 15 are in the regular B_{4} [3,3,4] group (tesseract/16cell) family (3 overlap with 24cell family)
 15 are in the regular H_{4} [3,3,5] group (120cell/600cell) family.
 1 special snub form in the [3,4,3] group (24cell) family.
 1 special nonWythoffian 4polytopes, the grand antiprism.
 TOTAL: 68 − 4 = 64
These 64 uniform 4polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.
In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms:
 Set of uniform antiprismatic prisms  sr{p,2}×{ }  Polyhedral prisms of two antiprisms.
 Set of uniform duoprisms  {p}×{q}  A product of two polygons.
The A_{4} family
The 5cell has diploid pentachoric [3,3,3] symmetry,[7] 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.
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 3 (5) 
Pos. 2 (10) 
Pos. 1 (10) 
Pos. 0 (5) 
Cells  Faces  Edges  Vertices  
1  5cell pentachoron[7] 
{3,3,3} 
(4) (3.3.3) 
5  10  10  5  
2  rectified 5cell  r{3,3,3} 
(3) (3.3.3.3) 
(2) (3.3.3) 
10  30  30  10  
3  truncated 5cell  t{3,3,3} 
(3) (3.6.6) 
(1) (3.3.3) 
10  30  40  20  
4  cantellated 5cell  rr{3,3,3} 
(2) (3.4.3.4) 
(2) (3.4.4) 
(1) (3.3.3.3) 
20  80  90  30  
7  cantitruncated 5cell  tr{3,3,3} 
(2) (4.6.6) 
(1) (3.4.4) 
(1) (3.6.6) 
20  80  120  60  
8  runcitruncated 5cell  t_{0,1,3}{3,3,3} 
(1) (3.6.6) 
(2) (4.4.6) 
(1) (3.4.4) 
(1) (3.4.3.4) 
30  120  150  60 
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 30 (10) 
Pos. 12 (20) 
Alt  Cells  Faces  Edges  Vertices  
5  *runcinated 5cell  t_{0,3}{3,3,3} 
(2) (3.3.3) 
(6) (3.4.4) 
30  70  60  20  
6  *bitruncated 5cell decachoron 
2t{3,3,3} 
(4) (3.6.6) 
10  40  60  30  
9  *omnitruncated 5cell  t_{0,1,2,3}{3,3,3} 
(2) (4.6.6) 
(2) (4.4.6) 
30  150  240  120  
Nonuniform  omnisnub 5cell[14]  ht_{0,1,2,3}{3,3,3} 
(3.3.3.3.3) 
(3.3.3.3) 
(3.3.3) 
90  300  270  60 
The three uniform 4polytopes forms marked with an asterisk, *, have the higher extended pentachoric symmetry, of order 240, [[3,3,3]] because the element corresponding to any element of the underlying 5cell 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 [[3,3,3]]^{+}, order 120, defining an omnisnub 5cell which is listed for completeness, but is not uniform.
The B_{4} family
This family has diploid hexadecachoric symmetry,[7] [4,3,3], of order 24×16=384: 4!=24 permutations of the four axes, 2^{4}=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4polytopes which are also repeated in other families, [1^{+},4,3,3], [4,(3,3)^{+}], and [4,3,3]^{+}, all order 192.
Tesseract truncations
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 3 (8) 
Pos. 2 (24) 
Pos. 1 (32) 
Pos. 0 (16) 
Cells  Faces  Edges  Vertices  
10  tesseract or 8cell 
{4,3,3} 
(4) (4.4.4) 
8  24  32  16  
11  Rectified tesseract  r{4,3,3} 
(3) (3.4.3.4) 
(2) (3.3.3) 
24  88  96  32  
13  Truncated tesseract  t{4,3,3} 
(3) (3.8.8) 
(1) (3.3.3) 
24  88  128  64  
14  Cantellated tesseract  rr{4,3,3} 
(1) (3.4.4.4) 
(2) (3.4.4) 
(1) (3.3.3.3) 
56  248  288  96  
15  Runcinated tesseract (also runcinated 16cell) 
t_{0,3}{4,3,3} 
(1) (4.4.4) 
(3) (4.4.4) 
(3) (3.4.4) 
(1) (3.3.3) 
80  208  192  64  
16  Bitruncated tesseract (also bitruncated 16cell) 
2t{4,3,3} 
(2) (4.6.6) 
(2) (3.6.6) 
24  120  192  96  
18  Cantitruncated tesseract  tr{4,3,3} 
(2) (4.6.8) 
(1) (3.4.4) 
(1) (3.6.6) 
56  248  384  192  
19  Runcitruncated tesseract  t_{0,1,3}{4,3,3} 
(1) (3.8.8) 
(2) (4.4.8) 
(1) (3.4.4) 
(1) (3.4.3.4) 
80  368  480  192  
21  Omnitruncated tesseract (also omnitruncated 16cell) 
t_{0,1,2,3}{3,3,4} 
(1) (4.6.8) 
(1) (4.4.8) 
(1) (4.4.6) 
(1) (4.6.6) 
80  464  768  384 
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 3 (8) 
Pos. 2 (24) 
Pos. 1 (32) 
Pos. 0 (16) 
Alt  Cells  Faces  Edges  Vertices  
12  Half tesseract Demitesseract 16cell 
h{4,3,3}={3,3,4} 
(4) (3.3.3) 
(4) (3.3.3) 
16  32  24  8  
[17]  Cantic tesseract (Or truncated 16cell) 
h_{2}{4,3,3}=t{4,3,3} 
(4) (6.6.3) 
(1) (3.3.3.3) 
24  96  120  48  
[11]  Runcic tesseract (Or rectified tesseract) 
h_{3}{4,3,3}=r{4,3,3} 
(3) (3.4.3.4) 
(2) (3.3.3) 
24  88  96  32  
[16]  Runcicantic tesseract (Or bitruncated tesseract) 
h_{2,3}{4,3,3}=2t{4,3,3} 
(2) (3.4.3.4) 
(2) (3.6.6) 
24  120  192  96  
[11]  (rectified tesseract)  h_{1}{4,3,3}=r{4,3,3} 
24  88  96  32  
[16]  (bitruncated tesseract)  h_{1,2}{4,3,3}=2t{4,3,3} 
24  120  192  96  
[23]  (rectified 24cell)  h_{1,3}{4,3,3}=rr{3,3,4} 
48  240  288  96  
[24]  (truncated 24cell)  h_{1,2,3}{4,3,3}=tr{3,3,4} 
48  240  384  192 
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 3 (8) 
Pos. 2 (24) 
Pos. 1 (32) 
Pos. 0 (16) 
Alt  Cells  Faces  Edges  Vertices  
Nonuniform  omnisnub tesseract[15] (Or omnisnub 16cell) 
ht_{0,1,2,3}{4,3,3} 
(1) (3.3.3.3.4) 
(1) (3.3.3.4) 
(1) (3.3.3.3) 
(1) (3.3.3.3.3) 
(4) (3.3.3) 
272  944  864  192 
16cell truncations
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 3 (8) 
Pos. 2 (24) 
Pos. 1 (32) 
Pos. 0 (16) 
Alt  Cells  Faces  Edges  Vertices  
[12]  16cell, hexadecachoron[7]  {3,3,4} 
(8) (3.3.3) 
16  32  24  8  
[22]  *rectified 16cell (Same as 24cell) 
r{3,3,4} 
(2) (3.3.3.3) 
(4) (3.3.3.3) 
24  96  96  24  
17  truncated 16cell  t{3,3,4} 
(1) (3.3.3.3) 
(4) (3.6.6) 
24  96  120  48  
[23]  *cantellated 16cell (Same as rectified 24cell) 
rr{3,3,4} 
(1) (3.4.3.4) 
(2) (4.4.4) 
(2) (3.4.3.4) 
48  240  288  96  
[15]  runcinated 16cell (also runcinated 8cell) 
t_{0,3}{3,3,4} 
(1) (4.4.4) 
(3) (4.4.4) 
(3) (3.4.4) 
(1) (3.3.3) 
80  208  192  64  
[16]  bitruncated 16cell (also bitruncated 8cell) 
2t{3,3,4} 
(2) (4.6.6) 
(2) (3.6.6) 
24  120  192  96  
[24]  *cantitruncated 16cell (Same as truncated 24cell) 
tr{3,3,4} 
(1) (4.6.6) 
(1) (4.4.4) 
(2) (4.6.6) 
48  240  384  192  
20  runcitruncated 16cell  t_{0,1,3}{3,3,4} 
(1) (3.4.4.4) 
(1) (4.4.4) 
(2) (4.4.6) 
(1) (3.6.6) 
80  368  480  192  
[21]  omnitruncated 16cell (also omnitruncated 8cell) 
t_{0,1,2,3}{3,3,4} 
(1) (4.6.8) 
(1) (4.4.8) 
(1) (4.4.6) 
(1) (4.6.6) 
80  464  768  384  
[31]  alternated cantitruncated 16cell (Same as the snub 24cell) 
sr{3,3,4} 
(1) (3.3.3.3.3) 
(1) (3.3.3) 
(2) (3.3.3.3.3) 
(4) (3.3.3) 
144  480  432  96  
Nonuniform  Runcic snub rectified 16cell  sr_{3}{3,3,4} 
(1) (3.4.4.4) 
(2) (3.4.4) 
(1) (4.4.4) 
(1) (3.3.3.3.3) 
(2) (3.4.4) 
176  656  672  192 
 (*) Just as rectifying the tetrahedron produces the octahedron, rectifying the 16cell produces the 24cell, the regular member of the following family.
The snub 24cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16cell or truncated 24cell, 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_{4} family
This family has diploid icositetrachoric symmetry,[7] [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 4polytopes which are also repeated in other families, [3^{+},4,3], [3,4,3^{+}], and [3,4,3]^{+}, all order 576.
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 3 (24) 
Pos. 2 (96) 
Pos. 1 (96) 
Pos. 0 (24) 
Cells  Faces  Edges  Vertices  
22  24cell, icositetrachoron[7] (Same as rectified 16cell) 
{3,4,3} 
(6) (3.3.3.3) 
24  96  96  24  
23  rectified 24cell (Same as cantellated 16cell) 
r{3,4,3} 
(3) (3.4.3.4) 
(2) (4.4.4) 
48  240  288  96  
24  truncated 24cell (Same as cantitruncated 16cell) 
t{3,4,3} 
(3) (4.6.6) 
(1) (4.4.4) 
48  240  384  192  
25  cantellated 24cell  rr{3,4,3} 
(2) (3.4.4.4) 
(2) (3.4.4) 
(1) (3.4.3.4) 
144  720  864  288  
28  cantitruncated 24cell  tr{3,4,3} 
(2) (4.6.8) 
(1) (3.4.4) 
(1) (3.8.8) 
144  720  1152  576  
29  runcitruncated 24cell  t_{0,1,3}{3,4,3} 
(1) (4.6.6) 
(2) (4.4.6) 
(1) (3.4.4) 
(1) (3.4.4.4) 
240  1104  1440  576 
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 3 (24) 
Pos. 2 (96) 
Pos. 1 (96) 
Pos. 0 (24) 
Alt  Cells  Faces  Edges  Vertices  
31  †snub 24cell  s{3,4,3} 
(3) (3.3.3.3.3) 
(1) (3.3.3) 
(4) (3.3.3) 
144  480  432  96  
Nonuniform  runcic snub 24cell  s_{3}{3,4,3} 
(1) (3.3.3.3.3) 
(2) (3.4.4) 
(1) (3.6.6) 
(3) Tricup 
240  960  1008  288  
[25]  cantic snub 24cell (Same as cantellated 24cell) 
s_{2}{3,4,3} 
(2) (3.4.4.4) 
(1) (3.4.3.4) 
(2) (3.4.4) 
144  720  864  288  
[29]  runcicantic snub 24cell (Same as runcitruncated 24cell) 
s_{2,3}{3,4,3} 
(1) (4.6.6) 
(1) (3.4.4) 
(1) (3.4.4.4) 
(2) (4.4.6) 
240  1104  1440  576 
 (†) The snub 24cell here, despite its common name, is not analogous to the snub cube; rather, is derived by an alternation of the truncated 24cell. Its symmetry number is only 576, (the ionic diminished icositetrachoric group, [3^{+},4,3]).
Like the 5cell, the 24cell is selfdual, and so the following three forms have twice as many symmetries, bringing their total to 2304 (extended icositetrachoric symmetry [[3,4,3]]).
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 30 (48) 
Pos. 21 (192) 
Cells  Faces  Edges  Vertices  
26  runcinated 24cell  t_{0,3}{3,4,3} 
(2) (3.3.3.3) 
(6) (3.4.4) 
240  672  576  144  
27  bitruncated 24cell tetracontoctachoron 
2t{3,4,3} 
(4) (3.8.8) 
48  336  576  288  
30  omnitruncated 24cell  t_{0,1,2,3}{3,4,3} 
(2) (4.6.8) 
(2) (4.4.6) 
240  1392  2304  1152 
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 30 (48) 
Pos. 21 (192) 
Alt  Cells  Faces  Edges  Vertices  
Nonuniform  omnisnub 24cell[16]  ht_{0,1,2,3}{3,4,3} 
(2) (3.3.3.3.4) 
(2) (3.3.3.3) 
(4) (3.3.3) 
816  2832  2592  576 
The H_{4} family
This family has diploid hexacosichoric symmetry,[7] [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.
120cell truncations
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cell counts by location  Element counts  

Pos. 3 (120) 
Pos. 2 (720) 
Pos. 1 (1200) 
Pos. 0 (600) 
Alt  Cells  Faces  Edges  Vertices  
32  120cell (hecatonicosachoron or dodecacontachoron)[7] 
{5,3,3} 
(4) (5.5.5) 
120  720  1200  600  
33  rectified 120cell  r{5,3,3} 
(3) (3.5.3.5) 
(2) (3.3.3) 
720  3120  3600  1200  
36  truncated 120cell  t{5,3,3} 
(3) (3.10.10) 
(1) (3.3.3) 
720  3120  4800  2400  
37  cantellated 120cell  rr{5,3,3} 
(1) (3.4.5.4) 
(2) (3.4.4) 
(1) (3.3.3.3) 
1920  9120  10800  3600  
38  runcinated 120cell (also runcinated 600cell) 
t_{0,3}{5,3,3} 
(1) (5.5.5) 
(3) (4.4.5) 
(3) (3.4.4) 
(1) (3.3.3) 
2640  7440  7200  2400  
39  bitruncated 120cell (also bitruncated 600cell) 
2t{5,3,3} 
(2) (5.6.6) 
(2) (3.6.6) 
720  4320  7200  3600  
42  cantitruncated 120cell  tr{5,3,3} 
(2) (4.6.10) 
(1) (3.4.4) 
(1) (3.6.6) 
1920  9120  14400  7200  
43  runcitruncated 120cell  t_{0,1,3}{5,3,3} 
(1) (3.10.10) 
(2) (4.4.10) 
(1) (3.4.4) 
(1) (3.4.3.4) 
2640  13440  18000  7200  
46  omnitruncated 120cell (also omnitruncated 600cell) 
t_{0,1,2,3}{5,3,3} 
(1) (4.6.10) 
(1) (4.4.10) 
(1) (4.4.6) 
(1) (4.6.6) 
2640  17040  28800  14400  
Nonuniform  omnisnub 120cell[17] (Same as the omnisnub 600cell) 
ht_{0,1,2,3}{5,3,3} 
(3.3.3.3.5) 
(3.3.3.5) 
(3.3.3.3) 
(3.3.3.3.3) 
(3.3.3) 
9840  35040  32400  7200 
600cell truncations
#  Name  Vertex figure 
Coxeter diagram and Schläfli symbols 
Symmetry  Cell counts by location  Element counts  

Pos. 3 (120) 
Pos. 2 (720) 
Pos. 1 (1200) 
Pos. 0 (600) 
Cells  Faces  Edges  Vertices  
35  600cell, hexacosichoron[7]  {3,3,5} 
[5,3,3] order 14400 
(20) (3.3.3) 
600  1200  720  120  
[47]  20diminished 600cell (grand antiprism) 
Nonwythoffian construction 
[[10,2<sup>+</sup>,10]] order 400 Index 36 
(2) (3.3.3.5) 
(12) (3.3.3) 
320  720  500  100  
[31]  24diminished 600cell (snub 24cell) 
Nonwythoffian construction 
[3^{+},4,3] order 576 index 25 
(3) (3.3.3.3.3) 
(5) (3.3.3) 
144  480  432  96  
Nonuniform  bi24diminished 600cell  Nonwythoffian construction 
order 144 index 100 
(6) tdi 
48  192  216  72  
34  rectified 600cell  r{3,3,5} 
[5,3,3]  (2) (3.3.3.3.3) 
(5) (3.3.3.3) 
720  3600  3600  720  
Nonuniform  120diminished rectified 600cell  Nonwythoffian construction 
order 1200 index 12 
(2) 3.3.3.5 
(2) 4.4.5 
(5) P4 
840  2640  2400  600  
41  truncated 600cell  t{3,3,5} 
[5,3,3]  (1) (3.3.3.3.3) 
(5) (3.6.6) 
720  3600  4320  1440  
40  cantellated 600cell  rr{3,3,5} 
[5,3,3]  (1) (3.5.3.5) 
(2) (4.4.5) 
(1) (3.4.3.4) 
1440  8640  10800  3600  
[38]  runcinated 600cell (also runcinated 120cell) 
t_{0,3}{3,3,5} 
[5,3,3]  (1) (5.5.5) 
(3) (4.4.5) 
(3) (3.4.4) 
(1) (3.3.3) 
2640  7440  7200  2400  
[39]  bitruncated 600cell (also bitruncated 120cell) 
2t{3,3,5} 
[5,3,3]  (2) (5.6.6) 
(2) (3.6.6) 
720  4320  7200  3600  
45  cantitruncated 600cell  tr{3,3,5} 
[5,3,3]  (1) (5.6.6) 
(1) (4.4.5) 
(2) (4.6.6) 
1440  8640  14400  7200  
44  runcitruncated 600cell  t_{0,1,3}{3,3,5} 
[5,3,3]  (1) (3.4.5.4) 
(1) (4.4.5) 
(2) (4.4.6) 
(1) (3.6.6) 
2640  13440  18000  7200  
[46]  omnitruncated 600cell (also omnitruncated 120cell) 
t_{0,1,2,3}{3,3,5} 
[5,3,3]  (1) (4.6.10) 
(1) (4.4.10) 
(1) (4.4.6) 
(1) (4.6.6) 
2640  17040  28800  14400 
The D_{4} family
This demitesseract family, [3^{1,1,1}], introduces no new uniform 4polytopes, 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, 2^{4}=16 for reflection in each axis. There is one small index subgroups that generating uniform 4polytopes, [3^{1,1,1}]^{+}, order 96.
#  Name  Vertex figure 
Coxeter diagram 
Cell counts by location  Element counts  

Pos. 0 (8) 
Pos. 2 (24) 
Pos. 1 (8) 
Pos. 3 (8) 
Pos. Alt (96) 
3  2  1  0  
[12]  demitesseract half tesseract (Same as 16cell) 
h{4,3,3} 
(4) (3.3.3) 
(4) (3.3.3) 
16  32  24  8  
[17]  cantic tesseract (Same as truncated 16cell) 
h_{2}{4,3,3} 
(1) (3.3.3.3) 
(2) (3.6.6) 
(2) (3.6.6) 
24  96  120  48  
[11]  runcic tesseract (Same as rectified tesseract) 
h_{3}{4,3,3} 
(1) (3.3.3) 
(1) (3.3.3) 
(3) (3.4.3.4) 
24  88  96  32  
[16]  runcicantic tesseract (Same as bitruncated tesseract) 
h_{2,3}{4,3,3} 
(1) (3.6.6) 
(1) (3.6.6) 
(2) (4.6.6) 
24  96  96  24 
When the 3 bifurcated branch nodes are identically ringed, the symmetry can be increased by 6, as [3[3^{1,1,1}]] = [3,4,3], and thus these polytopes are repeated from the 24cell family.
#  Name  Vertex figure 
Coxeter diagram 
Cell counts by location  Element counts  

Pos. 0,1,3 (24) 
Pos. 2 (24) 
Pos. Alt (96) 
3  2  1  0  
[22]  rectified 16cell) (Same as 24cell) 
{3^{1,1,1}} = r{3,3,4} = {3,4,3} 
(6) (3.3.3.3) 
48  240  288  96  
[23]  cantellated 16cell (Same as rectified 24cell) 
r{3^{1,1,1}} = rr{3,3,4} = r{3,4,3} 
(3) (3.4.3.4) 
(2) (4.4.4) 
24  120  192  96  
[24]  cantitruncated 16cell (Same as truncated 24cell) 
t{3^{1,1,1}} = tr{3,3,4} = t{3,4,3} 
(3) (4.6.6) 
(1) (4.4.4) 
48  240  384  192  
[31]  snub 24cell  s{3^{1,1,1}} = sr{3,3,4} = s{3,4,3} 
(3) (3.3.3.3.3) 
(1) (3.3.3) 
(4) (3.3.3) 
144  480  432  96 
Here again the snub 24cell, with the symmetry group [3^{1,1,1}]^{+} this time, represents an alternated truncation of the truncated 24cell creating 96 new tetrahedra at the position of the deleted vertices. In contrast to its appearance within former groups as partly snubbed 4polytope, 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 nonWythoffian uniform convex 4polytope, known as the grand antiprism, consisting of 20 pentagonal antiprisms forming two perpendicular rings joined by 300 tetrahedra. It is loosely analogous to the threedimensional 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<sup>+</sup>,10]], order 400.
#  Name  Picture  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cells by type  Element counts  Net  

Cells  Faces  Edges  Vertices  
47  grand antiprism  No symbol  300 (3.3.3) 
20 (3.3.3.5) 
320  20 {5} 700 {3} 
500  100 
Prismatic uniform 4polytopes
A prismatic polytope is a Cartesian product of two polytopes of lower dimension; familiar examples are the 3dimensional prisms, which are products of a polygon and a line segment. The prismatic uniform 4polytopes 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 3dimensional prisms and antiprisms.
 Duoprisms: products of two polygons.
Convex polyhedral prisms
The most obvious family of prismatic 4polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4polytopes 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 cubeprism, is listed above as the tesseract).
There are 18 convex polyhedral prisms created from 5 Platonic solids and 13 Archimedean solids as well as for the infinite families of threedimensional prisms and antiprisms. The symmetry number of a polyhedral prism is twice that of the base polyhedron.
Tetrahedral prisms: A_{3} × A_{1}
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 4polytope.
#  Name  Picture  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cells by type  Element counts  Net  

Cells  Faces  Edges  Vertices  
48  Tetrahedral prism  {3,3}×{ } t_{0,3}{3,3,2} 
2 3.3.3 
4 3.4.4 
6  8 {3} 6 {4} 
16  8  
49  Truncated tetrahedral prism  t{3,3}×{ } t_{0,1,3}{3,3,2} 
2 3.6.6 
4 3.4.4 
4 4.4.6 
10  8 {3} 18 {4} 8 {6} 
48  24 
#  Name  Picture  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cells by type  Element counts  Net  

Cells  Faces  Edges  Vertices  
[51]  Rectified tetrahedral prism (Same as octahedral prism) 
r{3,3}×{ } t_{1,3}{3,3,2} 
2 3.3.3.3 
4 3.4.4 
6  16 {3} 12 {4} 
30  12  
[50]  Cantellated tetrahedral prism (Same as cuboctahedral prism) 
rr{3,3}×{ } t_{0,2,3}{3,3,2} 
2 3.4.3.4 
8 3.4.4 
6 4.4.4 
16  16 {3} 36 {4} 
60  24  
[54]  Cantitruncated tetrahedral prism (Same as truncated octahedral prism) 
tr{3,3}×{ } t_{0,1,2,3}{3,3,2} 
2 4.6.6 
8 6.4.4 
6 4.4.4 
16  48 {4} 16 {6} 
96  48  
[59]  Snub tetrahedral prism (Same as icosahedral prism) 
sr{3,3}×{ } 
2 3.3.3.3.3 
20 3.4.4 
22  40 {3} 30 {4} 
72  24  
Nonuniform  omnisnub tetrahedral antiprism  2 3.3.3.3.3 
8 3.3.3.3 
6+24 3.3.3 
40  16+96 {3}  96  24 
Octahedral prisms: B_{3} × A_{1}
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 4polytopes below. Symmetries are [(4,3)^{+},2], [1^{+},4,3,2], [4,3,2^{+}], [4,3^{+},2], [4,(3,2)^{+}], and [4,3,2]^{+}.
#  Name  Picture  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cells by type  Element counts  Net  

Cells  Faces  Edges  Vertices  
[10]  Cubic prism (Same as tesseract) (Same as 44 duoprism) 
{4,3}×{ } t_{0,3}{4,3,2} 
2 4.4.4 
6 4.4.4 
8  24 {4}  32  16  
50  Cuboctahedral prism (Same as cantellated tetrahedral prism) 
r{4,3}×{ } t_{1,3}{4,3,2} 
2 3.4.3.4 
8 3.4.4 
6 4.4.4 
16  16 {3} 36 {4}  60  24  
51  Octahedral prism (Same as rectified tetrahedral prism) (Same as triangular antiprismatic prism) 
{3,4}×{ } t_{2,3}{4,3,2} 
2 3.3.3.3 
8 3.4.4 
10  16 {3} 12 {4}  30  12  
52  Rhombicuboctahedral prism  rr{4,3}×{ } t_{0,2,3}{4,3,2} 
2 3.4.4.4 
8 3.4.4 
18 4.4.4 
28  16 {3} 84 {4}  120  48  
53  Truncated cubic prism  t{4,3}×{ } t_{0,1,3}{4,3,2} 
2 3.8.8 
8 3.4.4 
6 4.4.8 
16  16 {3} 36 {4} 12 {8}  96  48  
54  Truncated octahedral prism (Same as cantitruncated tetrahedral prism) 
t{3,4}×{ } t_{1,2,3}{4,3,2} 
2 4.6.6 
6 4.4.4 
8 4.4.6 
16  48 {4} 16 {6}  96  48  
55  Truncated cuboctahedral prism  tr{4,3}×{ } t_{0,1,2,3}{4,3,2} 
2 4.6.8 
12 4.4.4 
8 4.4.6 
6 4.4.8 
28  96 {4} 16 {6} 12 {8}  192  96  
56  Snub cubic prism  sr{4,3}×{ } 
2 3.3.3.3.4 
32 3.4.4 
6 4.4.4 
40  64 {3} 72 {4}  144  48  
[48]  Tetrahedral prism  h{4,3}×{ } 
2 3.3.3 
4 3.4.4 
6  8 {3} 6 {4}  16  8  
[49]  Truncated tetrahedral prism  h_{2}{4,3}×{ } 
2 3.3.6 
4 3.4.4 
4 4.4.6 
6  8 {3} 6 {4}  16  8  
[50]  Cuboctahedral prism  rr{3,3}×{ } 
2 3.4.3.4 
8 3.4.4 
6 4.4.4 
16  16 {3} 36 {4}  60  24  
[52]  Rhombicuboctahedral prism  s_{2}{3,4}×{ } 
2 3.4.4.4 
8 3.4.4 
18 4.4.4 
28  16 {3} 84 {4}  120  48  
[54]  Truncated octahedral prism  tr{3,3}×{ } 
2 4.6.6 
6 4.4.4 
8 4.4.6 
16  48 {4} 16 {6}  96  48  
[59]  Icosahedral prism  s{3,4}×{ } 
2 3.3.3.3.3 
20 3.4.4 
22  40 {3} 30 {4}  72  24  
[12]  16cell  s{2,4,3} 
2+6+8 3.3.3.3 
16  32 {3}  24  8  
Nonuniform  Omnisnub tetrahedral antiprism  sr{2,3,4} 
2 3.3.3.3.3 
8 3.3.3.3 
6+24 3.3.3 
40  16+96 {3}  96  24  
Nonuniform  Omnisnub cubic antiprism  2 3.3.3.3.4 
12+48 3.3.3 
8 3.3.3.3 
6 3.3.3.4 
76  16+192 {3} 12 {4}  192  48  
Nonuniform  Runcic snub cubic hosochoron  s_{3}{2,4,3} 
2 3.6.6 
6 3.3.3 
8 triangular cupola 
16  52  60  24 
Icosahedral prisms: H_{3} × A_{1}
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.
#  Name  Picture  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cells by type  Element counts  Net  

Cells  Faces  Edges  Vertices  
57  Dodecahedral prism  {5,3}×{ } t_{0,3}{5,3,2} 
2 5.5.5 
12 4.4.5 
14  30 {4} 24 {5} 
80  40  
58  Icosidodecahedral prism  r{5,3}×{ } t_{1,3}{5,3,2} 
2 3.5.3.5 
20 3.4.4 
12 4.4.5 
34  40 {3} 60 {4} 24 {5} 
150  60  
59  Icosahedral prism (same as snub tetrahedral prism) 
{3,5}×{ } t_{2,3}{5,3,2} 
2 3.3.3.3.3 
20 3.4.4 
22  40 {3} 30 {4} 
72  24  
60  Truncated dodecahedral prism  t{5,3}×{ } t_{0,1,3}{5,3,2} 
2 3.10.10 
20 3.4.4 
12 4.4.10 
34  40 {3} 90 {4} 24 {10} 
240  120  
61  Rhombicosidodecahedral prism  rr{5,3}×{ } t_{0,2,3}{5,3,2} 
2 3.4.5.4 
20 3.4.4 
30 4.4.4 
12 4.4.5 
64  40 {3} 180 {4} 24 {5} 
300  120  
62  Truncated icosahedral prism  t{3,5}×{ } t_{1,2,3}{5,3,2} 
2 5.6.6 
12 4.4.5 
20 4.4.6 
34  90 {4} 24 {5} 40 {6} 
240  120  
63  Truncated icosidodecahedral prism  tr{5,3}×{ } t_{0,1,2,3}{5,3,2} 
2 4.6.10 
30 4.4.4 
20 4.4.6 
12 4.4.10 
64  240 {4} 40 {6} 24 {10} 
480  240  
64  Snub dodecahedral prism  sr{5,3}×{ } 
2 3.3.3.3.5 
80 3.4.4 
12 4.4.5 
94  160 {3} 150 {4} 24 {5} 
360  120  
Nonuniform  Omnisnub dodecahedral antiprism  2 3.3.3.3.5 
30+120 3.3.3 
20 3.3.3.3 
12 3.3.3.5 
184  20+240 {3} 24 {5}  220  120 
Duoprisms: [p] × [q]
The second is the infinite family of uniform duoprisms, products of two regular polygons. A duoprism's CoxeterDynkin diagram is
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 threedimensional prism. The symmetry number of a duoprism whose factors are a pgon and a qgon (a "p,qduoprism") is 4pq if p≠q; if the factors are both pgons, the symmetry number is 8p^{2}. The tesseract can also be considered a 4,4duoprism.
The elements of a p,qduoprism (p ≥ 3, q ≥ 3) are:
 Cells: p qgonal prisms, q pgonal prisms
 Faces: pq squares, p qgons, q pgons
 Edges: 2pq
 Vertices: pq
There is no uniform analogue in four dimensions to the infinite family of threedimensional antiprisms.
Infinite set of pq duoprism 
Name  Coxeter graph  Cells  Images  Net 

33 duoprism  3+3 triangular prisms  
34 duoprism  3 cubes 4 triangular prisms  
44 duoprism (same as tesseract) 
4+4 cubes  
35 duoprism  3 pentagonal prisms 5 triangular prisms  
45 duoprism  4 pentagonal prisms 5 cubes  
55 duoprism  5+5 pentagonal prisms  
36 duoprism  3 hexagonal prisms 6 triangular prisms  
46 duoprism  4 hexagonal prisms 6 cubes  
56 duoprism  5 hexagonal prisms 6 pentagonal prisms  
66 duoprism  6+6 hexagonal prisms 
33 
34 
35 
36 
37 
38 
43 
44 
45 
46 
47 
48 
53 
54 
55 
56 
57 
58 
63 
64 
65 
66 
67 
68 
73 
74 
75 
76 
77 
78 
83 
84 
85 
86 
87 
88 
Polygonal prismatic prisms: [p] × [ ] × [ ]
The infinite set of uniform prismatic prisms overlaps with the 4p duoprisms: (p≥3) 
Name  {3}×{4}  {4}×{4}  {5}×{4}  {6}×{4}  {7}×{4}  {8}×{4}  {p}×{4} 

Coxeter diagrams 

Image  
Cells  3 {4}×{} 4 {3}×{} 
4 {4}×{} 4 {4}×{} 
5 {4}×{} 4 {5}×{} 
6 {4}×{} 4 {6}×{} 
7 {4}×{} 4 {7}×{} 
8 {4}×{} 4 {8}×{} 
p {4}×{} 4 {p}×{} 
Net 
Polygonal antiprismatic prisms: [p] × [ ] × [ ]
The infinite sets of uniform antiprismatic prisms are constructed from two parallel uniform antiprisms): (p≥2) 
Name  s{2,2}×{}  s{2,3}×{}  s{2,4}×{}  s{2,5}×{}  s{2,6}×{}  s{2,7}×{}  s{2,8}×{}  s{2,p}×{} 

Coxeter diagram 

Image  
Vertex figure 

Cells  2 s{2,2} (2) {2}×{}={4} 4 {3}×{} 
2 s{2,3} 2 {3}×{} 6 {3}×{} 
2 s{2,4} 2 {4}×{} 8 {3}×{} 
2 s{2,5} 2 {5}×{} 10 {3}×{} 
2 s{2,6} 2 {6}×{} 12 {3}×{} 
2 s{2,7} 2 {7}×{} 14 {3}×{} 
2 s{2,8} 2 {8}×{} 16 {3}×{} 
2 s{2,p} 2 {p}×{} 2p {3}×{} 
Net 
A pgonal antiprismatic prism has 4p triangle, 4p square and 4 pgon faces. It has 10p edges, and 4p vertices.
Nonuniform alternations
Coxeter showed only two uniform solutions for rank 4 Coxeter groups with all rings alternated (shown with empty circle nodes). The first is
Other alternations, such as
Geometric derivations for 46 nonprismatic Wythoffian uniform polychora
The 46 Wythoffian 4polytopes include the six convex regular 4polytopes. 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.
Summary chart of truncation operations 
Example locations of kaleidoscopic generator point on fundamental domain. 
The geometric operations that derive the 40 uniform 4polytopes from the regular 4polytopes are truncating operations. A 4polytope 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 CoxeterDynkin 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.
Operation  Schläfli symbol  Symmetry  Coxeter diagram  Description 

Parent  t_{0}{p,q,r}  [p,q,r]  Original regular form {p,q,r}  
Rectification  t_{1}{p,q,r}  Truncation operation applied until the original edges are degenerated into points.  
Birectification (Rectified dual) 
t_{2}{p,q,r}  Face are fully truncated to points. Same as rectified dual.  
Trirectification (dual) 
t_{3}{p,q,r}  Cells are truncated to points. Regular dual {r,q,p}  
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.  
Bitruncation  t_{1,2}{p,q,r}  A truncation between a rectified form and the dual rectified form.  
Tritruncation  t_{2,3}{p,q,r}  Truncated dual {r,q,p}.  
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.  
Bicantellation  t_{1,3}{p,q,r}  Cantellated dual {r,q,p}.  
Runcination (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.  
Cantitruncation  t_{0,1,2}{p,q,r}  Both the cantellation and truncation operations applied together.  
Bicantitruncation  t_{1,2,3}{p,q,r}  Cantitruncated dual {r,q,p}.  
Runcitruncation  t_{0,1,3}{p,q,r}  Both the runcination and truncation operations applied together.  
Runcicantellation  t_{0,1,3}{p,q,r}  Runcitruncated dual {r,q,p}.  
Omnitruncation (runcicantitruncation) 
t_{0,1,2,3}{p,q,r}  Application of all three operators.  
Half  h{2p,3,q}  [1^{+},2p,3,q] =[(3,p,3),q] 
Alternation of  
Cantic  h_{2}{2p,3,q}  Same as  
Runcic  h_{3}{2p,3,q}  Same as  
Runcicantic  h_{2,3}{2p,3,q}  Same as  
Quarter  q{2p,3,2q}  [1^{+},2p,3,2q,1^{+}]  Same as  
Snub  s{p,2q,r}  [p^{+},2q,r]  Alternated truncation  
Cantic snub  s_{2}{p,2q,r}  Cantellated alternated truncation  
Runcic snub  s_{3}{p,2q,r}  Runcinated alternated truncation  
Runcicantic snub  s_{2,3}{p,2q,r}  Runcicantellated alternated truncation  
Snub rectified  sr{p,q,2r}  [(p,q)^{+},2r]  Alternated truncated rectification  
ht_{0,3}{2p,q,2r}  [(2p,q,2r,2^{+})]  Alternated runcination  
Bisnub  2s{2p,q,2r}  [2p,q^{+},2r]  Alternated bitruncation  
Omnisnub  ht_{0,1,2,3}{p,q,r}  [p,q,r]^{+}  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 16cell, or the 120cell and 600cell), then bitruncating, runcinating or omnitruncating either produces the same figure as the same operation to the other. Thus where only the participle appears in the table it should be understood to apply to either parent.
Summary of constructions by extended symmetry
The 46 uniform polychora constructed from the A_{4}, B_{4}, F_{4}, H_{4} symmetry are given in this table by their full extended symmetry and Coxeter diagrams. Alternations are grouped by their chiral symmetry. All alternations are given, although the snub 24cell, with its 3 family of constructions is the only one that is uniform. Counts in parenthesis are either repeats or nonuniform. The Coxeter diagrams are given with subscript indices 1 through 46. The 33 and 44 duoprismatic family is included, the second for its relation to the B_{4} family.
Coxeter group  Extended symmetry 
Polychora  Chiral extended symmetry 
Alternation honeycombs  

[3,3,3]  [3,3,3] (order 120)  6  
[2^{+}[3,3,3]] (order 240)  3  [2^{+}[3,3,3]]^{+} (order 120)  (1)  
[3,3^{1,1}]  [3,3^{1,1}] (order 192)  0  (none)  
[1[3,3^{1,1}]]=[4,3,3] (order 384)  (4)  
[3[3^{1,1,1}]]=[3,4,3] (order 1152)  (3)  [3[3,3^{1,1}]]^{+} =[3,4,3]^{+} (order 576)  (1)  
[4,3,3] 
[3[1^{+},4,3,3]]=[3,4,3] (order 1152)  (3)  
[4,3,3] (order 384)  12  [1^{+},4,3,3]^{+} (order 96)  (2)  
[4,3,3]^{+} (order 192)  (1)  
[3,4,3]  [3,4,3] (order 1152)  6  [2^{+}[3^{+},4,3^{+}]] (order 576)  1  
[2^{+}[3,4,3]] (order 2304)  3  [2^{+}[3,4,3]]^{+} (order 1152)  (1)  
[5,3,3]  [5,3,3] (order 14400)  15  [5,3,3]^{+} (order 7200)  (1)  
[3,2,3]  [3,2,3] (order 36)  0  (none)  [3,2,3]^{+} (order 18)  0  (none) 
[2^{+}[3,2,3]] (order 72)  0  [2^{+}[3,2,3]]^{+} (order 36)  0  (none)  
[[3],2,3]=[6,2,3] (order 72)  1  [1[3,2,3]]=[[3],2,3]^{+}=[6,2,3]^{+} (order 36)  (1)  
[(2^{+},4)[3,2,3]]=[2^{+}[6,2,6]] (order 288)  1  [(2^{+},4)[3,2,3]]^{+}=[2^{+}[6,2,6]]^{+} (order 144)  (1)  
[4,2,4]  [4,2,4] (order 64)  0  (none)  [4,2,4]^{+} (order 32)  0  (none) 
[2^{+}[4,2,4]] (order 128)  0  (none)  [2^{+}[(4,2^{+},4,2^{+})]] (order 64)  0  (none)  
[(3,3)[4,2*,4]]=[4,3,3] (order 384)  (1)  [(3,3)[4,2*,4]]^{+}=[4,3,3]^{+} (order 192)  (1)  
[[4],2,4]=[8,2,4] (order 128)  (1)  [1[4,2,4]]=[[4],2,4]^{+}=[8,2,4]^{+} (order 64)  (1)  
[(2^{+},4)[4,2,4]]=[2^{+}[8,2,8]] (order 512)  (1)  [(2^{+},4)[4,2,4]]^{+}=[2^{+}[8,2,8]]^{+} (order 256)  (1) 
See also
 Regular skew polyhedron#Finite regular skew polyhedra of 4space
 Convex uniform honeycomb  related infinite 4polytopes in Euclidean 3space.
 Convex uniform honeycombs in hyperbolic space  related infinite 4polytopes in Hyperbolic 3space.
 Paracompact uniform honeycombs
Notes
 N.W. Johnson: Geometries and Transformations, (2018) ISBN 9781107103405 Chapter 11: Finite Symmetry Groups, 11.1 Polytopes and Honeycombs, p.224
 T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
 "Archived copy" (PDF). Archived from the original (PDF) on 20091229. Retrieved 20100813.CS1 maint: archived copy as title (link)
 Elte (1912)
 https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html December 6, 1998 oldest archive
 The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes By David Darling, (2004) ASIN: B00SB4TU58
 Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups
 Uniform Polytopes in Four Dimensions, George Olshevsky.
 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope (in German)
 Conway (2008)
 Convex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract
 "Uniform Polychora". www.polytope.net. Retrieved 20191207.
 Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135
 http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm
 http://www.bendwavy.org/klitzing/incmats/s3s3s4s.htm
 http://www.bendwavy.org/klitzing/incmats/s3s4s3s.htm
 http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm
 H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) p. 582588 2.7 The fourdimensional analogues of the snub cube
References
 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
 Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen, ISBN 141817968X
 H.S.M. Coxeter:
 H.S.M. Coxeter, M.S. LonguetHiggins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londen, 1954
 Schoute, Pieter Hendrik (1911), "Analytic treatment of the polytopes regularly derived from the regular polytopes", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, 11 (3): 87 pp. Googlebook, 370381
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, SpringerVerlag. New York. 1980 p92, p122.
 J.H. Conway and M.J.T. Guy: FourDimensional 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
 B. Grünbaum Convex polytopes, New York ; London : Springer, c2003. ISBN 0387004246.
Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.  John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26)
 Richard Klitzing, Snubs, alternated facetings, and StottCoxeterDynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329344, (2010)
External links
 Convex uniform 4polytopes
 Uniform, convex polytopes in four dimensions:, Marco Möller (in German)
 2004 Dissertation Fourdimensional Archimedean polytopes (in German)
 Uniform Polytopes in Four Dimensions, George Olshevsky.
 Convex uniform polychora based on the pentachoron, George Olshevsky.
 Convex uniform polychora based on the tesseract/16cell, George Olshevsky.
 Convex uniform polychora based on the 24cell, George Olshevsky.
 Convex uniform polychora based on the 120cell/600cell, George Olshevsky.
 Anomalous convex uniform polychoron: (grand antiprism), George Olshevsky.
 Convex uniform prismatic polychora, George Olshevsky.
 Uniform polychora derived from glomeric tetrahedron B4, George Olshevsky.
 Regular and semiregular convex polytopes a short historical overview
 Java3D Applets with sources
 Uniform, convex polytopes in four dimensions:, Marco Möller (in German)
 Nonconvex uniform 4polytopes
 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.
 Klitzing, Richard. "4D uniform polytopes".
 4DPolytopes 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 AlAjmi (2012)