In geometry, a uniform 4polytope (or uniform polychoron)^{[1]} is a 4dimensional polytope which is vertextransitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 nonprismatic convex uniform 4polytopes. 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 nonconvex star forms.
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 {p,q}, 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:^{[15]}
The 16 regular 4polytopes, with the property that all cells, faces, edges, and vertices are congruent:
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} = , B_{4} = , D_{4} = , F_{4} = , H_{4} = .^{[7]} There are also 3 prismatic groups A_{3}A_{1} = , B_{3}A_{1} = , H_{3}A_{1} = , and duoprismatic groups: I_{2}(p)×I_{2}(q) = . Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes.
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 
There are 64 convex uniform 4polytopes, including the 6 regular convex 4polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic prisms.
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:
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 Bowers name (and acronym) 
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]} (pen) 
{3,3,3} 
(4) (3.3.3) 
5  10  10  5  
2  rectified 5cell Rectified pentachoron (rap) 
r{3,3,3} 
(3) (3.3.3.3) 
(2) (3.3.3) 
10  30  30  10  
3  truncated 5cell Truncated pentachoron (tip) 
t{3,3,3} 
(3) (3.6.6) 
(1) (3.3.3) 
10  30  40  20  
4  cantellated 5cell Small rhombated pentachoron (srip) 
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 Great rhombated pentachoron (grip) 
tr{3,3,3} 
(2) (4.6.6) 
(1) (3.4.4) 
(1) (3.6.6) 
20  80  120  60  
8  runcitruncated 5cell Prismatorhombated pentachoron (prip) 
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 Bowers name (and acronym) 
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 Small prismatodecachoron (spid) 
t_{0,3}{3,3,3} 
(2) (3.3.3) 
(6) (3.4.4) 
30  70  60  20  
6  *bitruncated 5cell Decachoron (deca) 
2t{3,3,3} 
(4) (3.6.6) 
10  40  60  30  
9  *omnitruncated 5cell Great prismatodecachoron (gippid) 
t_{0,1,2,3}{3,3,3} 
(2) (4.6.6) 
(2) (4.4.6) 
30  150  240  120  
Nonuniform  omnisnub 5cell Snub decachoron (snad) Snub pentachoron (snip)^{[16]} 
ht_{0,1,2,3}{3,3,3} 
(2) (3.3.3.3.3) 
(2) (3.3.3.3) 
(4) (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.
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.
#  Name (Bowers name and acronym) 
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 Tesseract (tes) 
{4,3,3} 
(4) (4.4.4) 
8  24  32  16  
11  Rectified tesseract (rit)  r{4,3,3} 
(3) (3.4.3.4) 
(2) (3.3.3) 
24  88  96  32  
13  Truncated tesseract (tat)  t{4,3,3} 
(3) (3.8.8) 
(1) (3.3.3) 
24  88  128  64  
14  Cantellated tesseract Small rhombated tesseract (srit) 
rr{4,3,3} 
(2) (3.4.4.4) 
(2) (3.4.4) 
(1) (3.3.3.3) 
56  248  288  96  
15  Runcinated tesseract (also runcinated 16cell) Small disprismatotesseractihexadecachoron (sidpith) 
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) Tesseractihexadecachoron (tah) 
2t{4,3,3} 
(2) (4.6.6) 
(2) (3.6.6) 
24  120  192  96  
18  Cantitruncated tesseract Great rhombated tesseract (grit) 
tr{4,3,3} 
(2) (4.6.8) 
(1) (3.4.4) 
(1) (3.6.6) 
56  248  384  192  
19  Runcitruncated tesseract Prismatorhombated hexadecachoron (proh) 
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) Great disprismatotesseractihexadecachoron (gidpith) 
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 (Bowers style acronym) 
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 (hex) 
= h{4,3,3}={3,3,4} 
(4) (3.3.3) 
(4) (3.3.3) 
16  32  24  8  
[17]  Cantic tesseract = Truncated 16cell (thex) 
= 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 = Rectified tesseract (rit) 
= 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 = Bitruncated tesseract (tah) 
= 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 (rat)  = h_{1}{4,3,3}=r{4,3,3} 
24  88  96  32  
[16]  = Bitruncated tesseract (tah)  = h_{1,2}{4,3,3}=2t{4,3,3} 
24  120  192  96  
[23]  = Rectified 24cell (rico)  = h_{1,3}{4,3,3}=rr{3,3,4} 
48  240  288  96  
[24]  = Truncated 24cell (tico)  = h_{1,2,3}{4,3,3}=tr{3,3,4} 
48  240  384  192 
#  Name (Bowers style acronym) 
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 Snub tesseract (snet)^{[17]} (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 
#  Name (Bowers name and acronym)  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]} (hex) 
{3,3,4} 
(8) (3.3.3) 
16  32  24  8  
[22]  *Rectified 16cell (Same as 24cell) (ico) 
= r{3,3,4} 
(2) (3.3.3.3) 
(4) (3.3.3.3) 
24  96  96  24  
17  Truncated 16cell Truncated hexadecachoron (thex) 
t{3,3,4} 
(1) (3.3.3.3) 
(4) (3.6.6) 
24  96  120  48  
[23]  *Cantellated 16cell (Same as rectified 24cell) (rico) 
= 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 tesseract) (sidpith) 
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 tesseract) (tah) 
2t{3,3,4} 
(2) (4.6.6) 
(2) (3.6.6) 
24  120  192  96  
[24]  *Cantitruncated 16cell (Same as truncated 24cell) (tico) 
= tr{3,3,4} 
(1) (4.6.6) 
(1) (4.4.4) 
(2) (4.6.6) 
48  240  384  192  
20  Runcitruncated 16cell Prismatorhombated tesseract (prit) 
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 tesseract) (gidpith) 
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) (sadi) 
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 Pyritosnub tesseract (pysnet) 
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 
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.
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 (Same as rectified 16cell) Icositetrachoron^{[7]} (ico) 
{3,4,3} 
(6) (3.3.3.3) 
24  96  96  24  
23  rectified 24cell (Same as cantellated 16cell) Rectified icositetrachoron (rico) 
r{3,4,3} 
(3) (3.4.3.4) 
(2) (4.4.4) 
48  240  288  96  
24  truncated 24cell (Same as cantitruncated 16cell) Truncated icositetrachoron (tico) 
t{3,4,3} 
(3) (4.6.6) 
(1) (4.4.4) 
48  240  384  192  
25  cantellated 24cell Small rhombated icositetrachoron (srico) 
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 Great rhombated icositetrachoron (grico) 
tr{3,4,3} 
(2) (4.6.8) 
(1) (3.4.4) 
(1) (3.8.8) 
144  720  1152  576  
29  runcitruncated 24cell Prismatorhombated icositetrachoron (prico) 
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 Snub disicositetrachoron (sadi) 
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 Prismatorhombisnub icositetrachoron (prissi) 
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) (srico) 
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) (prico) 
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 
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 Small prismatotetracontoctachoron (spic) 
t_{0,3}{3,4,3} 
(2) (3.3.3.3) 
(6) (3.4.4) 
240  672  576  144  
27  bitruncated 24cell Tetracontoctachoron (cont) 
2t{3,4,3} 
(4) (3.8.8) 
48  336  576  288  
30  omnitruncated 24cell Great prismatotetracontoctachoron (gippic) 
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 Snub tetracontoctachoron (snoc) Snub icositetrachoron (sni)^{[18]} 
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 
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.
#  Name (Bowers name and acronym) 
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]} Hecatonicosachoron (hi) 
{5,3,3} 
(4) (5.5.5) 
120  720  1200  600  
33  rectified 120cell Rectified hecatonicosachoron (rahi) 
r{5,3,3} 
(3) (3.5.3.5) 
(2) (3.3.3) 
720  3120  3600  1200  
36  truncated 120cell Truncated hecatonicosachoron (thi) 
t{5,3,3} 
(3) (3.10.10) 
(1) (3.3.3) 
720  3120  4800  2400  
37  cantellated 120cell Small rhombated hecatonicosachoron (srahi) 
rr{5,3,3} 
(2) (3.4.5.4) 
(2) (3.4.4) 
(1) (3.3.3.3) 
1920  9120  10800  3600  
38  runcinated 120cell (also runcinated 600cell) Small disprismatohexacosihecatonicosachoron (sidpixhi) 
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) Hexacosihecatonicosachoron (xhi) 
2t{5,3,3} 
(2) (5.6.6) 
(2) (3.6.6) 
720  4320  7200  3600  
42  cantitruncated 120cell Great rhombated hecatonicosachoron (grahi) 
tr{5,3,3} 
(2) (4.6.10) 
(1) (3.4.4) 
(1) (3.6.6) 
1920  9120  14400  7200  
43  runcitruncated 120cell Prismatorhombated hexacosichoron (prix) 
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) Great disprismatohexacosihecatonicosachoron (gidpixhi) 
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 Snub hecatonicosachoron (snixhi)^{[19]} (Same as the omnisnub 600cell) 
ht_{0,1,2,3}{5,3,3} 
(1) (3.3.3.3.5) 
(1) (3.3.3.5) 
(1) (3.3.3.3) 
(1) (3.3.3.3.3) 
(4) (3.3.3) 
9840  35040  32400  7200 
#  Name (Bowers style acronym) 
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]} (ex) 
{3,3,5} 
[5,3,3] order 14400 
(20) (3.3.3) 
600  1200  720  120  
[47]  20diminished 600cell = Grand antiprism (gap) 
Nonwythoffian construction 
[[10,2^{+},10]] order 400 Index 36 
(2) (3.3.3.5) 
(12) (3.3.3) 
320  720  500  100  
[31]  24diminished 600cell = Snub 24cell (sadi) 
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 Biicositetradiminished hexacosichoron (bidex) 
Nonwythoffian construction 
order 144 index 100 
(6) tdi 
48  192  216  72  
34  rectified 600cell Rectified hexacosichoron (rox) 
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 Swirlprismatodiminished rectified hexacosichoron (spidrox) 
Nonwythoffian construction 
order 1200 index 12 
(2) 3.3.3.5 
(2) 4.4.5 
(5) P4 
840  2640  2400  600  
41  truncated 600cell Truncated hexacosichoron (tex) 
t{3,3,5} 
[5,3,3]  (1) (3.3.3.3.3) 
(5) (3.6.6) 
720  3600  4320  1440  
40  cantellated 600cell Small rhombated hexacosichoron (srix) 
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) (sidpixhi) 
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) (xhi) 
2t{3,3,5} 
[5,3,3]  (2) (5.6.6) 
(2) (3.6.6) 
720  4320  7200  3600  
45  cantitruncated 600cell Great rhombated hexacosichoron (grix) 
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 Prismatorhombated hecatonicosachoron (prahi) 
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) (gidpixhi) 
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 
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 (Bowers style acronym)  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) (hex) 
= h{4,3,3} 
(4) (3.3.3) 
(4) (3.3.3) 
16  32  24  8  
[17]  cantic tesseract (Same as truncated 16cell) (thex) 
= 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) (rit) 
= 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) (tah) 
= 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 (Bowers style acronym)  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) (ico) 
= = = {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) (rico) 
= = = 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) (tico) 
= = = 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 (sadi)  = = = 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.
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^{+},10]], order 400.
#  Name (Bowers style acronym)  Picture  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cells by type  Element counts  Net  

Cells  Faces  Edges  Vertices  
47  grand antiprism (gap)  No symbol  300 (3.3.3) 
20 (3.3.3.5) 
320  20 {5} 700 {3} 
500  100 
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:
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).^{[citation needed]}
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.^{[citation needed]} The symmetry number of a polyhedral prism is twice that of the base polyhedron.
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 (Bowers style acronym)  Picture  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cells by type  Element counts  Net  

Cells  Faces  Edges  Vertices  
48  Tetrahedral prism (tepe)  {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 (tuttip)  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 (Bowers style acronym)  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) (ope) 
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) (cope) 
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) (tope) 
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) (ipe) 
sr{3,3}×{ } 
2 3.3.3.3.3 
20 3.4.4 
22  40 {3} 30 {4} 
72  24  
Nonuniform  omnisnub tetrahedral antiprism Pyritohedral icosahedral antiprism (pikap) 
2 3.3.3.3.3 
8 3.3.3.3 
6+24 3.3.3 
40  16+96 {3}  96  24 
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 (Bowers style acronym)  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) (tes) 
{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) (cope) 
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) (ope) 
{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 (sircope)  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 (ticcup)  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) (tope) 
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 (gircope)  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 (sniccup)  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 (tepe)  h{4,3}×{ } 
2 3.3.3 
4 3.4.4 
6  8 {3} 6 {4} 
16  8  
[49]  Truncated tetrahedral prism (tuttip)  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 (cope)  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 (sircope)  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 (tope)  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 (ipe)  s{3,4}×{ } 
2 3.3.3.3.3 
20 3.4.4 
22  40 {3} 30 {4} 
72  24  
[12]  16cell (hex)  s{2,4,3} 
2+6+8 3.3.3.3 
16  32 {3}  24  8  
Nonuniform  Omnisnub tetrahedral antiprism = Pyritohedral icosahedral antiprism (pikap) 
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  Edgesnub octahedral hosochoron Pyritosnub alterprism (pysna) 
sr_{3}{2,3,4} 
2 3.4.4.4 
6 4.4.4 
8 3.3.3.3 
24 3.4.4 
40  16+48 {3} 12+12+24+24 {4} 
144  48  
Nonuniform  Omnisnub cubic antiprism Snub cubic antiprism (sniccap) 
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 Truncated tetrahedral alterprism (tuta) 
s_{3}{2,4,3} 
2 3.6.6 
6 3.3.3 
8 triangular cupola 
16  52  60  24 
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 (Bowers name and acronym)  Picture  Vertex figure 
Coxeter diagram and Schläfli symbols 
Cells by type  Element counts  Net  

Cells  Faces  Edges  Vertices  
57  Dodecahedral prism (dope)  {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 (iddip)  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) (ipe) 
{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 (tiddip)  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 (sriddip)  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 (tipe)  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 (griddip)  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 (sniddip)  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 Snub dodecahedral antiprism (sniddap) 
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 
The second is the infinite family of uniform duoprisms, products of two regular polygons. A duoprism's CoxeterDynkin diagram is . Its vertex figure is a disphenoid tetrahedron, .
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 extended fvector of {p}×{q} is (p,p,1)*(q,q,1) = (pq,2pq,pq+p+q,p+q).
There is no uniform analogue in four dimensions to the infinite family of threedimensional antiprisms.
Infinite set of pq duoprism   p qgonal prisms, q pgonal prisms:
Name  Coxeter graph  Cells  Images  Net 

33 duoprism (triddip)  3+3 triangular prisms  
34 duoprism (tisdip)  3 cubes 4 triangular prisms 

44 duoprism (tes) (same as tesseract) 
4+4 cubes  
35 duoprism (trapedip)  3 pentagonal prisms 5 triangular prisms 

45 duoprism (squipdip)  4 pentagonal prisms 5 cubes 

55 duoprism (pedip)  5+5 pentagonal prisms  
36 duoprism (thiddip)  3 hexagonal prisms 6 triangular prisms 

46 duoprism (shiddip)  4 hexagonal prisms 6 cubes 

56 duoprism (phiddip)  5 hexagonal prisms 6 pentagonal prisms 

66 duoprism (hiddip)  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 
Alternations are possible. = 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 16cell. p=5, q=5/3 is the only nonconvex case that can be made uniform, giving the socalled great duoantiprism. gives the p2qgonal prismantiprismoid (an edgealternation of the 2p4q duoprism), but this cannot be made uniform in any cases.^{[20]}
The infinite set of uniform prismatic prisms overlaps with the 4p duoprisms: (p≥3)   p cubes and 4 pgonal prisms  (All are the same as 4p duoprism) The second polytope in the series is a lower symmetry of the regular tesseract, {4}×{4}.
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 
The infinite sets of uniform antiprismatic prisms are constructed from two parallel uniform antiprisms): (p≥2)   2 pgonal antiprisms, connected by 2 pgonal prisms and 2p triangular prisms.
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.
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 24cell, , 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 vertextransitive 4polytopes 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.^{[21]}
Wythoff constructions with alternations produce vertextransitive 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.^{[22]} 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.
Coxeter diagram 
s_{3}{2,4,3},  s_{3}{3,4,3},  Others  

Relation  24 of 48 vertices of rhombicuboctahedral prism 
288 of 576 vertices of runcitruncated 24cell 
72 of 120 vertices of 600cell 
600 of 720 vertices of rectified 600cell 
Projection  Two rings of pyramids  
Net  runcic snub cubic hosochoron^{[23]}^{[24]} 
runcic snub 24cell^{[25]}^{[26]} 
^{[27]}^{[28]}^{[29]}  ^{[30]}^{[31]} 
Cells  
Vertex figure 
(1) 3.4.3.4: triangular cupola (2) 3.4.6: triangular cupola (1) 3.3.3: tetrahedron (1) 3.6.6: truncated tetrahedron 
(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 
(2) 3.3.3.5: tridiminished icosahedron (4) 3.5.5: tridiminished icosahedron 
(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 
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} 