In geometry, a uniform 4-polytope (or uniform polychoron)[1] 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.
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:[15]
The 16 regular 4-polytopes, with the property that all cells, faces, edges, and vertices are congruent:
The 24 mirrors of F4 can be decomposed into 2 orthogonal D4 groups:
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The 10 mirrors of B3×A1 can be decomposed into orthogonal groups, 4A1 and D3:
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There are 5 fundamental mirror symmetry point group families in 4-dimensions: A4 = , B4 = , D4 = , F4 = , H4 = .[7] There are also 3 prismatic groups A3A1 = , B3A1 = , H3A1 = , and duoprismatic groups: I2(p)×I2(q) = . Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes.
Each reflective uniform 4-polytope can be constructed in one or more reflective point group in 4 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,a], have an extended symmetry, [[a,b,a]], doubling the symmetry order. This includes [3,3,3], [3,4,3], and [p,2,p]. Uniform polytopes in these group with symmetric rings contain this extended symmetry.
If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.
Weyl group |
Conway Quaternion |
Abstract structure |
Order | Coxeter diagram |
Coxeter notation |
Commutator subgroup |
Coxeter number (h) |
Mirrors m=2h | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Irreducible | ||||||||||||
A4 | +1/60[I×I].21 | S5 | 120 | [3,3,3] | [3,3,3]+ | 5 | 10 | |||||
D4 | ±1/3[T×T].2 | 1/2.2S4 | 192 | [31,1,1] | [31,1,1]+ | 6 | 12 | |||||
B4 | ±1/6[O×O].2 | 2S4 = S2≀S4 | 384 | [4,3,3] | 8 | 4 | 12 | |||||
F4 | ±1/2[O×O].23 | 3.2S4 | 1152 | [3,4,3] | [3+,4,3+] | 12 | 12 | 12 | ||||
H4 | ±[I×I].2 | 2.(A5×A5).2 | 14400 | [5,3,3] | [5,3,3]+ | 30 | 60 | |||||
Prismatic groups | ||||||||||||
A3A1 | +1/24[O×O].23 | S4×D1 | 48 | [3,3,2] = [3,3]×[ ] | [3,3]+ | - | 6 | 1 | ||||
B3A1 | ±1/24[O×O].2 | S4×D1 | 96 | [4,3,2] = [4,3]×[ ] | - | 3 | 6 | 1 | ||||
H3A1 | ±1/60[I×I].2 | A5×D1 | 240 | [5,3,2] = [5,3]×[ ] | [5,3]+ | - | 15 | 1 | ||||
Duoprismatic groups (Use 2p,2q for even integers) | ||||||||||||
I2(p)I2(q) | ±1/2[D2p×D2q] | Dp×Dq | 4pq | [p,2,q] = [p]×[q] | [p+,2,q+] | - | p | q | ||||
I2(2p)I2(q) | ±1/2[D4p×D2q] | D2p×Dq | 8pq | [2p,2,q] = [2p]×[q] | - | p | p | q | ||||
I2(2p)I2(2q) | ±1/2[D4p×D4q] | D2p×D2q | 16pq | [2p,2,2q] = [2p]×[2q] | - | p | p | q | q |
There are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic prisms.
These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.
In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms:
The 5-cell 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 | 5-cell Pentachoron[7] (pen) |
{3,3,3} |
(4) (3.3.3) |
5 | 10 | 10 | 5 | ||||
2 | rectified 5-cell Rectified pentachoron (rap) |
r{3,3,3} |
(3) (3.3.3.3) |
(2) (3.3.3) |
10 | 30 | 30 | 10 | |||
3 | truncated 5-cell Truncated pentachoron (tip) |
t{3,3,3} |
(3) (3.6.6) |
(1) (3.3.3) |
10 | 30 | 40 | 20 | |||
4 | cantellated 5-cell 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 5-cell 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 5-cell Prismatorhombated pentachoron (prip) |
t0,1,3{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. 3-0 (10) |
Pos. 1-2 (20) |
Alt | Cells | Faces | Edges | Vertices | ||||
5 | *runcinated 5-cell Small prismatodecachoron (spid) |
t0,3{3,3,3} |
(2) (3.3.3) |
(6) (3.4.4) |
30 | 70 | 60 | 20 | ||
6 | *bitruncated 5-cell Decachoron (deca) |
2t{3,3,3} |
(4) (3.6.6) |
10 | 40 | 60 | 30 | |||
9 | *omnitruncated 5-cell Great prismatodecachoron (gippid) |
t0,1,2,3{3,3,3} |
(2) (4.6.6) |
(2) (4.4.6) |
30 | 150 | 240 | 120 | ||
Nonuniform | omnisnub 5-cell Snub decachoron (snad) Snub pentachoron (snip)[16] |
ht0,1,2,3{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 4-polytopes 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 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 [[3,3,3]]+, order 120, defining an omnisnub 5-cell 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, 24=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192.
# | 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 8-cell 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 16-cell) Small disprismatotesseractihexadecachoron (sidpith) |
t0,3{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 16-cell) 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) |
t0,1,3{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 16-cell) Great disprismatotesseractihexadecachoron (gidpith) |
t0,1,2,3{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 = 16-cell (hex) |
= h{4,3,3}={3,3,4} |
(4) (3.3.3) |
(4) (3.3.3) |
16 | 32 | 24 | 8 | ||||
[17] | Cantic tesseract = Truncated 16-cell (thex) |
= h2{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) |
= h3{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) |
= h2,3{4,3,3}=2t{4,3,3} |
(2) (3.4.3.4) |
(2) (3.6.6) |
24 | 120 | 192 | 96 | ||||
[11] | = Rectified tesseract (rat) | = h1{4,3,3}=r{4,3,3} |
24 | 88 | 96 | 32 | ||||||
[16] | = Bitruncated tesseract (tah) | = h1,2{4,3,3}=2t{4,3,3} |
24 | 120 | 192 | 96 | ||||||
[23] | = Rectified 24-cell (rico) | = h1,3{4,3,3}=rr{3,3,4} |
48 | 240 | 288 | 96 | ||||||
[24] | = Truncated 24-cell (tico) | = h1,2,3{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 16-cell) |
ht0,1,2,3{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] | 16-cell Hexadecachoron[7] (hex) |
{3,3,4} |
(8) (3.3.3) |
16 | 32 | 24 | 8 | |||||
[22] | *Rectified 16-cell (Same as 24-cell) (ico) |
= r{3,3,4} |
(2) (3.3.3.3) |
(4) (3.3.3.3) |
24 | 96 | 96 | 24 | ||||
17 | Truncated 16-cell Truncated hexadecachoron (thex) |
t{3,3,4} |
(1) (3.3.3.3) |
(4) (3.6.6) |
24 | 96 | 120 | 48 | ||||
[23] | *Cantellated 16-cell (Same as rectified 24-cell) (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 16-cell (also runcinated tesseract) (sidpith) |
t0,3{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 16-cell (also bitruncated tesseract) (tah) |
2t{3,3,4} |
(2) (4.6.6) |
(2) (3.6.6) |
24 | 120 | 192 | 96 | ||||
[24] | *Cantitruncated 16-cell (Same as truncated 24-cell) (tico) |
= tr{3,3,4} |
(1) (4.6.6) |
(1) (4.4.4) |
(2) (4.6.6) |
48 | 240 | 384 | 192 | |||
20 | Runcitruncated 16-cell Prismatorhombated tesseract (prit) |
t0,1,3{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 16-cell (also omnitruncated tesseract) (gidpith) |
t0,1,2,3{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 16-cell (Same as the snub 24-cell) (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 16-cell Pyritosnub tesseract (pysnet) |
sr3{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 24-cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16-cell or truncated 24-cell, with the half symmetry group [(3,3)+,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices.
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 4-polytopes 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 | 24-cell (Same as rectified 16-cell) Icositetrachoron[7] (ico) |
{3,4,3} |
(6) (3.3.3.3) |
24 | 96 | 96 | 24 | ||||
23 | rectified 24-cell (Same as cantellated 16-cell) Rectified icositetrachoron (rico) |
r{3,4,3} |
(3) (3.4.3.4) |
(2) (4.4.4) |
48 | 240 | 288 | 96 | |||
24 | truncated 24-cell (Same as cantitruncated 16-cell) Truncated icositetrachoron (tico) |
t{3,4,3} |
(3) (4.6.6) |
(1) (4.4.4) |
48 | 240 | 384 | 192 | |||
25 | cantellated 24-cell 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 24-cell 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 24-cell Prismatorhombated icositetrachoron (prico) |
t0,1,3{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 24-cell 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 24-cell Prismatorhombisnub icositetrachoron (prissi) |
s3{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 24-cell (Same as cantellated 24-cell) (srico) |
s2{3,4,3} |
(2) (3.4.4.4) |
(1) (3.4.3.4) |
(2) (3.4.4) |
144 | 720 | 864 | 288 | |||
[29] | runcicantic snub 24-cell (Same as runcitruncated 24-cell) (prico) |
s2,3{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 5-cell, the 24-cell is self-dual, and so the following three forms have twice as many symmetries, bringing their total to 2304 (extended icositetrachoric symmetry [[3,4,3]]).
# | Name | Vertex figure |
Coxeter diagram and Schläfli symbols |
Cell counts by location | Element counts | ||||
---|---|---|---|---|---|---|---|---|---|
Pos. 3-0 (48) |
Pos. 2-1 (192) |
Cells | Faces | Edges | Vertices | ||||
26 | runcinated 24-cell Small prismatotetracontoctachoron (spic) |
t0,3{3,4,3} |
(2) (3.3.3.3) |
(6) (3.4.4) |
240 | 672 | 576 | 144 | |
27 | bitruncated 24-cell Tetracontoctachoron (cont) |
2t{3,4,3} |
(4) (3.8.8) |
48 | 336 | 576 | 288 | ||
30 | omnitruncated 24-cell Great prismatotetracontoctachoron (gippic) |
t0,1,2,3{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. 3-0 (48) |
Pos. 2-1 (192) |
Alt | Cells | Faces | Edges | Vertices | ||||
Nonuniform | omnisnub 24-cell Snub tetracontoctachoron (snoc) Snub icositetrachoron (sni)[18] |
ht0,1,2,3{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 | 120-cell (hecatonicosachoron or dodecacontachoron)[7] Hecatonicosachoron (hi) |
{5,3,3} |
(4) (5.5.5) |
120 | 720 | 1200 | 600 | |||||
33 | rectified 120-cell Rectified hecatonicosachoron (rahi) |
r{5,3,3} |
(3) (3.5.3.5) |
(2) (3.3.3) |
720 | 3120 | 3600 | 1200 | ||||
36 | truncated 120-cell Truncated hecatonicosachoron (thi) |
t{5,3,3} |
(3) (3.10.10) |
(1) (3.3.3) |
720 | 3120 | 4800 | 2400 | ||||
37 | cantellated 120-cell 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 120-cell (also runcinated 600-cell) Small disprismatohexacosihecatonicosachoron (sidpixhi) |
t0,3{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 120-cell (also bitruncated 600-cell) Hexacosihecatonicosachoron (xhi) |
2t{5,3,3} |
(2) (5.6.6) |
(2) (3.6.6) |
720 | 4320 | 7200 | 3600 | ||||
42 | cantitruncated 120-cell 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 120-cell Prismatorhombated hexacosichoron (prix) |
t0,1,3{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 120-cell (also omnitruncated 600-cell) Great disprismatohexacosihecatonicosachoron (gidpixhi) |
t0,1,2,3{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 120-cell Snub hecatonicosachoron (snixhi)[19] (Same as the omnisnub 600-cell) |
ht0,1,2,3{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 | 600-cell Hexacosichoron[7] (ex) |
{3,3,5} |
[5,3,3] order 14400 |
(20) (3.3.3) |
600 | 1200 | 720 | 120 | ||||
[47] | 20-diminished 600-cell = 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] | 24-diminished 600-cell = Snub 24-cell (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 | bi-24-diminished 600-cell Bi-icositetradiminished hexacosichoron (bidex) |
Nonwythoffian construction |
order 144 index 100 |
(6) tdi |
48 | 192 | 216 | 72 | ||||
34 | rectified 600-cell 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 | 120-diminished rectified 600-cell 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 600-cell 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 600-cell 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 600-cell (also runcinated 120-cell) (sidpixhi) |
t0,3{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 600-cell (also bitruncated 120-cell) (xhi) |
2t{3,3,5} |
[5,3,3] | (2) (5.6.6) |
(2) (3.6.6) |
720 | 4320 | 7200 | 3600 | |||
45 | cantitruncated 600-cell 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 600-cell Prismatorhombated hecatonicosachoron (prahi) |
t0,1,3{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 600-cell (also omnitruncated 120-cell) (gidpixhi) |
t0,1,2,3{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, [31,1,1], introduces no new uniform 4-polytopes, but it is worthy to repeat these alternative constructions. This family has order 12×16=192: 4!/2=12 permutations of the four axes, half as alternated, 24=16 for reflection in each axis. There is one small index subgroups that generating uniform 4-polytopes, [31,1,1]+, order 96.
# | 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 16-cell) (hex) |
= h{4,3,3} |
(4) (3.3.3) |
(4) (3.3.3) |
16 | 32 | 24 | 8 | ||||
[17] | cantic tesseract (Same as truncated 16-cell) (thex) |
= h2{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) |
= h3{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) |
= h2,3{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[31,1,1]] = [3,4,3], and thus these polytopes are repeated from the 24-cell 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 16-cell (Same as 24-cell) (ico) |
= = = {31,1,1} = r{3,3,4} = {3,4,3} |
(6) (3.3.3.3) |
48 | 240 | 288 | 96 | |||
[23] | cantellated 16-cell (Same as rectified 24-cell) (rico) |
= = = r{31,1,1} = rr{3,3,4} = r{3,4,3} |
(3) (3.4.3.4) |
(2) (4.4.4) |
24 | 120 | 192 | 96 | ||
[24] | cantitruncated 16-cell (Same as truncated 24-cell) (tico) |
= = = t{31,1,1} = tr{3,3,4} = t{3,4,3} |
(3) (4.6.6) |
(1) (4.4.4) |
48 | 240 | 384 | 192 | ||
[31] | snub 24-cell (sadi) | = = = s{31,1,1} = 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 24-cell, with the symmetry group [31,1,1]+ this time, represents an alternated truncation of the truncated 24-cell creating 96 new tetrahedra at the position of the deleted vertices. In contrast to its appearance within former groups as partly snubbed 4-polytope, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the snub cube and the snub dodecahedron.
There is one non-Wythoffian uniform convex 4-polytope, known as the grand antiprism, consisting of 20 pentagonal antiprisms forming two perpendicular rings joined by 300 tetrahedra. It is loosely analogous to the three-dimensional antiprisms, which consist of two parallel polygons joined by a band of triangles. Unlike them, however, the grand antiprism is not a member of an infinite family of uniform polytopes.
Its symmetry is the ionic diminished Coxeter group, [[10,2+,10]], order 400.
# | 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 3-dimensional prisms, which are products of a polygon and a line segment. The prismatic uniform 4-polytopes consist of two infinite families:
The most obvious family of prismatic 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytopes are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract).[citation needed]
There are 18 convex polyhedral prisms created from 5 Platonic solids and 13 Archimedean solids as well as for the infinite families of three-dimensional prisms and antiprisms.[citation needed] 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 4-polytope.
# | 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}×{ } t0,3{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}×{ } t0,1,3{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}×{ } t1,3{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}×{ } t0,2,3{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}×{ } t0,1,2,3{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 4-polytopes below. Symmetries are [(4,3)+,2], [1+,4,3,2], [4,3,2+], [4,3+,2], [4,(3,2)+], and [4,3,2]+.
# | 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 4-4 duoprism) (tes) |
{4,3}×{ } t0,3{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}×{ } t1,3{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}×{ } t2,3{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}×{ } t0,2,3{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}×{ } t0,1,3{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}×{ } t1,2,3{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}×{ } t0,1,2,3{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) | h2{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) | s2{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] | 16-cell (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 | Edge-snub octahedral hosochoron Pyritosnub alterprism (pysna) |
sr3{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) |
s3{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}×{ } t0,3{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}×{ } t1,3{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}×{ } t2,3{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}×{ } t0,1,3{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}×{ } t0,2,3{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}×{ } t1,2,3{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}×{ } t0,1,2,3{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 Coxeter-Dynkin 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 three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if p≠q; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism.
The extended f-vector of {p}×{q} is (p,p,1)*(q,q,1) = (pq,2pq,pq+p+q,p+q).
There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms.
Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms:
Name | Coxeter graph | Cells | Images | Net |
---|---|---|---|---|
3-3 duoprism (triddip) | 3+3 triangular prisms | |||
3-4 duoprism (tisdip) | 3 cubes 4 triangular prisms |
|||
4-4 duoprism (tes) (same as tesseract) |
4+4 cubes | |||
3-5 duoprism (trapedip) | 3 pentagonal prisms 5 triangular prisms |
|||
4-5 duoprism (squipdip) | 4 pentagonal prisms 5 cubes |
|||
5-5 duoprism (pedip) | 5+5 pentagonal prisms | |||
3-6 duoprism (thiddip) | 3 hexagonal prisms 6 triangular prisms |
|||
4-6 duoprism (shiddip) | 4 hexagonal prisms 6 cubes |
|||
5-6 duoprism (phiddip) | 5 hexagonal prisms 6 pentagonal prisms |
|||
6-6 duoprism (hiddip) | 6+6 hexagonal prisms |
3-3 |
3-4 |
3-5 |
3-6 |
3-7 |
3-8 |
4-3 |
4-4 |
4-5 |
4-6 |
4-7 |
4-8 |
5-3 |
5-4 |
5-5 |
5-6 |
5-7 |
5-8 |
6-3 |
6-4 |
6-5 |
6-6 |
6-7 |
6-8 |
7-3 |
7-4 |
7-5 |
7-6 |
7-7 |
7-8 |
8-3 |
8-4 |
8-5 |
8-6 |
8-7 |
8-8 |
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 16-cell. p=5, q=5/3 is the only nonconvex case that can be made uniform, giving the so-called great duoantiprism. gives the p-2q-gonal prismantiprismoid (an edge-alternation of the 2p-4q duoprism), but this cannot be made uniform in any cases.[20]
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}.
Name | {3}×{4} | {4}×{4} | {5}×{4} | {6}×{4} | {7}×{4} | {8}×{4} | {p}×{4} |
---|---|---|---|---|---|---|---|
Coxeter diagrams |
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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 p-gonal antiprisms, connected by 2 p-gonal 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 |
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Image | ||||||||
Vertex figure |
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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 p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon 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{21,1,1} 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] = [31,1,1], order 192). The second is , s{31,1,1}, which is an index 6 subgroup (symmetry [31,1,1]+, 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.[21]
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.[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 |
s3{2,4,3}, | s3{3,4,3}, | Others | |
---|---|---|---|---|
Relation | 24 of 48 vertices of rhombicuboctahedral prism |
288 of 576 vertices of runcitruncated 24-cell |
72 of 120 vertices of 600-cell |
600 of 720 vertices of rectified 600-cell |
Projection | Two rings of pyramids | |||
Net | runcic snub cubic hosochoron[23][24] |
runcic snub 24-cell[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 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.
Summary chart of truncation operations |
Example locations of kaleidoscopic generator point on fundamental domain. |
The geometric operations that derive the 40 uniform 4-polytopes from the regular 4-polytopes are truncating operations. A 4-polytope may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below.
The Coxeter-Dynkin diagram shows the four mirrors of the Wythoffian kaleidoscope as nodes, and the edges between the nodes are labeled by an integer showing the angle between the mirrors (π/n radians or 180/n degrees). Circled nodes show which mirrors are active for each form; a mirror is active with respect to a vertex that does not lie on it.
Operation | Schläfli symbol | Symmetry | Coxeter diagram | Description |
---|---|---|---|---|
Parent | t0{p,q,r} | [p,q,r] | Original regular form {p,q,r} | |
Rectification | t1{p,q,r} | Truncation operation applied until the original edges are degenerated into points. | ||
Birectification (Rectified dual) |
t2{p,q,r} | Face are fully truncated to points. Same as rectified dual. | ||
Trirectification (dual) |
t3{p,q,r} | Cells are truncated to points. Regular dual {r,q,p} | ||
Truncation | t0,1{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 | t1,2{p,q,r} | A truncation between a rectified form and the dual rectified form. | ||
Tritruncation | t2,3{p,q,r} | Truncated dual {r,q,p}. | ||
Cantellation | t0,2{p,q,r} | A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form. | ||
Bicantellation | t1,3{p,q,r} | Cantellated dual {r,q,p}. | ||
Runcination (or expansion) |
t0,3{p,q,r} | A truncation applied to the cells, faces and edges; defines a progression between a regular form and the dual. | ||
Cantitruncation | t0,1,2{p,q,r} |