# Uniform polychoron

(Redirected from Rectified demitesseract)
Schlegel diagram for the truncated 120-cell with tetrahedral cells visible
orthographic projection of the truncated 120-cell, in the H3 Coxeter plane (D10 symmetry). Only vertices and edges are drawn.

In geometry, a uniform polychoron (plural: uniform polychora) is a polychoron (4-polytope) which is vertex-transitive and whose cells are uniform polyhedra.

This article contains the complete list of 47 non-prismatic convex uniform polychora, and describes three sets of convex prismatic forms, two being infinite.

## History of discovery

• Regular polytopes: (convex faces)
• 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-polychora (star polyhedron cells and/or vertex figures)
• 1852: Ludwig Schläfli also found 4 of the 10 regular star polychora, 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 polychora, 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 [3].
• Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
• 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.[1]
• 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 polychora.[2]
• 1911: Pieter Hendrik Schoute published Analytic treatment of the polytopes regularly derived from the regular polytopes, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on 5-cell, 8-cell/16-cell, and 24-cell.
• 1912: E. L. Elte independently expanded on Gosset's list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets.[3]
• Convex uniform polytopes:
• 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
• Convex uniform polychora:
• 1965: The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex polychoron, the grand antiprism.
• 1966 N.W. 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
• 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope.[4]
• Nonregular uniform star polychora: (similar to the nonconvex uniform polyhedra)
• 2000-2005: In a collaborative search, a total of 1845 uniform polychora (convex and nonconvex) had been identified by Jonathan Bowers and George Olshevsky.[5]

## Regular polychora

The uniform polychora include two special subsets, which satisfy additional requirements:

## Convex uniform polychora

### Enumeration

There are 64 convex uniform polychora, including the 6 regular convex polychora, 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 self-dual regular A4 [3,3,3] group (5-cell) family.
• 9 are in the self-dual regular F4 [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
• 15 are in the regular BC4 [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family)
• 15 are in the regular H4 [3,3,5] group (120-cell/600-cell) family.
• 1 special snub form in the [3,4,3] group (24-cell) family.
• 1 special non-Wythoffian polychoron, the grand antiprism.
• TOTAL: 68 − 4 = 64

These 64 uniform polychora 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 A4 family

The 5-cell has diploid pentachoric [3,3,3] symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.

Facets (cells) are given, grouped in their Coxeter diagram locations by removing specified nodes.

[3,3,3] uniform polytopes
# Johnson 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 (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
Prismatotrhombated 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
[[3,3,3]] uniform polytopes
# Johnson 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 (snip)[6]
ht0,1,2,3{4,3,3}
(2)
(3.3.3.3.3)
(2)
(3.3.3.3)
(4)
(3.3.3)
90 300 270 60

The three polychora 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 a omnisnub 5-cell which is listed for completeness, but is not uniform.

#### Graphs

Three Coxeter plane 2D projections are given, for the A4, A3, A2 Coxeter groups, showing symmetry order 5,4,3, and doubled on even Ak orders to 10,4,6 for symmetric Coxeter diagrams.

The 3D picture are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.

# Johnson Name
Bowers name (and acronym)
Coxeter diagram
and Schläfli
symbols
Coxeter plane graphs Schlegel
diagram
A4
[5]
A3
[4]
A2
[3]
Tetrahedron
centered
Dual tetrahedron
centered
1 5-cell
Pentachoron (pen)

{3,3,3}
2 rectified 5-cell
Rectified pentachoron (rap)

r{3,3,3}
3 truncated 5-cell
Truncated pentachoron (tip)

t{3,3,3}
4 cantellated 5-cell
Small rhombated pentachoron (srip)

rr{3,3,3}
5 *runcinated 5-cell
Small prismatodecachoron (spid)

t0,3{3,3,3}
6 *bitruncated 5-cell
Decachoron (deca)

2t{3,3,3}
7 cantitruncated 5-cell
Great rhombated pentachoron (grip)

tr{3,3,3}
8 runcitruncated 5-cell
Prismatotrhombated pentachoron (prip)

t0,1,3{3,3,3}
9 *omnitruncated 5-cell
Great prismatodecachoron (gippid)

t0,1,2,3{3,3,3}

#### Coordinates

The coordinates of uniform 4-polytopes with pentachoric symmetry can be generated as permutations of simple integers in 5-space, all in hyperplanes with normal vector (1,1,1,1,1). The A4 Coxeter group is palindromic, so repeated polytopes exist in pairs of dual configurations. There are 3 symmetric positions, and 6 pairs making the total 15 permutations of one or more rings. All 15 are listed here in order of binary arithmetic for clarity of the coordinate generation from the rings in each corresponding Coxeter diagram diagram.

The number of vertices can be deduced here from the permutations of the number of coordinates, peaking at 5 factorial for the omnitruncated form with 5 unique coordinate values.

Pentachora truncations in 5-space:
# Base point Name
(symmetric name)
Coxeter diagram Vertices
1 (0, 0, 0, 0, 1) 5-cell 5
2 (0, 0, 0, 1, 1) Rectified 5-cell 10
3 (0, 0, 0, 1, 2) Truncated 5-cell 20
4 (0, 0, 1, 1, 1) Birectified 5-cell
(rectified 5-cell)
10
5 (0, 0, 1, 1, 2) Cantellated 5-cell 30
6 (0, 0, 1, 2, 2) Bitruncated 5-cell 30
7 (0, 0, 1, 2, 3) Cantitruncated 5-cell 60
8 (0, 1, 1, 1, 1) Trirectified 5-cell
(5-cell)
5
9 (0, 1, 1, 1, 2) Runcinated 5-cell 20
10 (0, 1, 1, 2, 2) Bicantellated 5-cell
(cantellated 5-cell)
30
11 (0, 1, 1, 2, 3) Runcitruncated 5-cell 60
12 (0, 1, 2, 2, 2) Tritruncated 5-cell
(truncated 5-cell)
20
13 (0, 1, 2, 2, 3) Runcicantellated 5-cell
(runcitruncated 5-cell)
60
14 (0, 1, 2, 3, 3) Bicantitruncated 5-cell
(cantitruncated 5-cell)
60
15 (0, 1, 2, 3, 4) Omnitruncated 5-cell 120

### The BC4 family

This family has diploid hexadecachoric symmetry, [4,3,3], of order 24*16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform polychora which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192.

#### Tesseract truncations

# Johnson 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
10 tesseract or (tes)
8-cell

{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 (srit)
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 16-cell) (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) (tah)

2t{4,3,3}
(2)

(4.6.6)
(2)

(3.6.6)
24 120 192 96
18 Cantitruncated 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 (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) (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
12 Demitesseract
16-cell (hex)
=
h{4,3,3}
(4)

(3.3.3)
(4)

(3.3.3)
16 32 24 8
[17] Cantic tesseract =
h2{4,3,3}
(4)

(6.6.3)
(1)

(3.3.3.3)
24 96 120 48
[11] Runcic tesseract =
h3{4,3,3}
(3)

(3.4.3.4)
(2)

(3.3.3)
24 88 96 32
[16] Runcicantic tesseract =
h2,3{4,3,3}
(2)

(3.4.3.4)
(2)

(3.6.6)
24 96 96 24
Nonuniform omnisnub tesseract (snet)[7]
(Same as the 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

#### 16-cell truncations

# Johnson 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] 16-cell (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 (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 8-cell) (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 8-cell) (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 (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 8-cell) (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
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
(*) Just as rectifying the tetrahedron produces the octahedron, rectifying the 16-cell produces the 24-cell, the regular member of the following family.

The snub 24-cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16-cell or truncated 24-cell, with the half symmetry group [(3,3)+,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices.

#### Graphs

The pictures are drawn as Schlegel diagram perspective projections, centered on the cell at pos. 3, with a consistent orientation, and the 16 cells at position 0 are shown solid, alternately colored.

# Johnson Name
(Bowers style acronym)
Coxeter plane projections Schlegel
diagrams
F4
[12/3]
B4
[8]
B3
[6]
B2
[4]
A3
[4]
Cube
centered
Tetrahedron
centered
10 8-cell
or tesseract (tes)
11 rectified 8-cell (rit)
12 16-cell (hex)
13 truncated 8-cell (tat)
14 cantellated 8-cell (srit)
15 runcinated 8-cell
(also runcinated 16-cell) (sidpith)
16 bitruncated 8-cell
(also bitruncated 16-cell) (tah)
17 truncated 16-cell (thex)
18 cantitruncated 8-cell (grit)
19 runcitruncated 8-cell (proh)
20 runcitruncated 16-cell (prit)
21 omnitruncated 8-cell
(also omnitruncated 16-cell) (gidpith)
[22] *rectified 16-cell
(Same as 24-cell) (ico)
[23] *cantellated 16-cell
(Same as rectified 24-cell) (rico)
[24] *cantitruncated 16-cell
(Same as truncated 24-cell) (tico)
[31] alternated cantitruncated 16-cell
(Same as the snub 24-cell) (sadi)

#### Coordinates

The tesseractic family of polychora are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform polychoron. All coordinates correspond with uniform polychora of edge length 2.

Coordinates for uniform polychora in Tesseract/16-cell family
# Base point Johnson Name
Bowers Name (Bowers style acronym)
Coxeter diagram
1 (0,0,0,1)√2 16-cell
2 (0,0,1,1)√2 Rectified 16-cell
Icositetrachoron (ico)
3 (0,0,1,2)√2 Truncated 16-cell
4 (0,1,1,1)√2 Rectified tesseract (birectified 16-cell)
Rectified tesseract (rit)
5 (0,1,1,2)√2 Cantellated 16-cell
Rectified icositetrachoron (rico)
6 (0,1,2,2)√2 Bitruncated 16-cell
7 (0,1,2,3)√2 cantitruncated 16-cell
Truncated icositetrachoron (tico)
8 (1,1,1,1) Tesseract
Tesseract (tes)
9 (1,1,1,1) + (0,0,0,1)√2 Runcinated tesseract (runcinated 16-cell)
10 (1,1,1,1) + (0,0,1,1)√2 Cantellated tesseract
Small rhombated tesseract (srit)
11 (1,1,1,1) + (0,0,1,2)√2 Runcitruncated 16-cell
Prismatorhombated tesseract (prit)
12 (1,1,1,1) + (0,1,1,1)√2 Truncated tesseract
Truncated tesseract (tat)
13 (1,1,1,1) + (0,1,1,2)√2 Runcitruncated tesseract (runcicantellated 16-cell)
14 (1,1,1,1) + (0,1,2,2)√2 Cantitruncated tesseract
Great rhombated tesseract (grit)
15 (1,1,1,1) + (0,1,2,3)√2 Omnitruncated 16-cell (omnitruncated tesseract)

### The F4 family

This family has diploid icositetrachoric symmetry, [3,4,3], of order 24*48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There are 3 small index subgroups, with the first two isomorphic pairs generating uniform polychora which are also repeated in other families, [3+,4,3], [3,4,3+], and [3,4,3]+, all order 576.

[3,4,3] uniform polychora
# 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
22 24-cell
(Same as rectified 16-cell) (ico)

{3,4,3}
(6)

(3.3.3.3)
24 96 96 24
23 rectified 24-cell
(Same as cantellated 16-cell) (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) (tico)

t{3,4,3}
(3)

(4.6.6)
(1)

(4.4.4)
48 240 384 192
25 cantellated 24-cell (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 (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 (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
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 (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 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 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
(†) The snub 24-cell here, despite its common name, is not analogous to the snub cube; rather, is derived by an alternation of the truncated 24-cell. Its symmetry number is only 576, (the ionic diminished icositetrachoric group, [3+,4,3]).
[[3,4,3]] uniform polychora
# 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
26 *runcinated 24-cell (spic)
t0,3{3,4,3}
(2)

(3.3.3.3)
(6)

(3.4.4)
240 672 576 144
27 *bitruncated 24-cell (cont)
2t{3,4,3}
(4)

(3.8.8)
48 336 576 288
30 *omnitruncated 24-cell (gippic)
t0,1,2,3{3,4,3}
(2)

(4.6.8)
(2)

(4.4.6)
240 1392 2304 1152
Nonuniform omnisnub 24-cell (snico)[8]
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
(*) Like the 5-cell, the 24-cell is self-dual, and so the three asterisked forms have twice as many symmetries, bringing their total to 2304 (extended icositetrachoric symmetry [[3,4,3]]).

#### Graphs

# Name
Coxeter diagram
Schläfli symbol
Graph
Schlegel
diagram
Orthogonal
Projection
F4
[12]
B4
[8]
B3
[6]
B2
[4]
Octahedron
centered
Dual octahedron
centered
Octahedron
centered
22 24-cell (ico)
(rectified 16-cell)

{3,4,3}
23 rectified 24-cell (rico)
(cantellated 16-cell)

t1{3,4,3}
24 truncated 24-cell (tico)
(cantitruncated 16-cell)

t0,1{3,4,3}
25 cantellated 24-cell (srico)

t0,2{3,4,3}
26 *runcinated 24-cell (spic)

t0,3{3,4,3}
27 *bitruncated 24-cell (cont)

t1,2{3,4,3}
28 cantitruncated 24-cell (grico)

t0,1,2{3,4,3}
29 runcitruncated 24-cell (prico)

t0,1,3{3,4,3}
30 *omnitruncated 24-cell (gippic)

t0,1,2,3{3,4,3}

s{3,4,3}
- Runcic snub 24-cell (prissi)

s3{3,4,3}

#### Coordinates

Vertex coordinates for all 15 forms are given below, including dual configurations from the two regular 24-cells. (The dual configurations are named in bold.) Active rings in the first and second nodes generate points in the first column. Active rings in the third and fourth nodes generate the points in the second column. The sum of each of these points are then permutated by coordinate positions, and sign combinations. This generates all vertex coordinates. Edge lengths are 2.

The only exception is the snub 24-cell, which is generated by half of the coordinate permutations, only an even number of coordinate swaps. φ=(√5+1)/2.

24-cell family coordinates
Base point(s)
t(0,1)
Base point(s)
t(2,3)
Schläfli symbol Name
Coxeter diagram

(0,0,1,1)√2 t0{3,4,3} 24-cell
(0,1,1,2)√2 t1{3,4,3} Rectified 24-cell
(0,1,2,3)√2 t0,1{3,4,3} Truncated 24-cell
(0,1,φ,φ+1)√2 s{3,4,3} Snub 24-cell

(0,2,2,2)
(1,1,1,3)
t2{3,4,3} Birectified 24-cell
(Rectified 24-cell)
(0,2,2,2) +
(1,1,1,3) +
(0,0,1,1)√2
"
t0,2{3,4,3} Cantellated 24-cell
(0,2,2,2) +
(1,1,1,3) +
(0,1,1,2)√2
"
t1,2{3,4,3} Bitruncated 24-cell
(0,2,2,2) +
(1,1,1,3) +
(0,1,2,3)√2
"
t0,1,2{3,4,3} Cantitruncated 24-cell

(0,0,0,2)
(1,1,1,1)
t3{3,4,3} Trirectified 24-cell
(24-cell)
(0,0,0,2) +
(1,1,1,1) +
(0,0,1,1)√2
"
t0,3{3,4,3} Runcinated 24-cell
(0,0,0,2) +
(1,1,1,1) +
(0,1,1,2)√2
"
t1,3{3,4,3} bicantellated 24-cell
(Cantellated 24-cell)
(0,0,0,2) +
(1,1,1,1) +
(0,1,2,3)√2
"
t0,1,3{3,4,3} Runcitruncated 24-cell

(1,1,1,5)
(1,3,3,3)
(2,2,2,4)
t2,3{3,4,3} Tritruncated 24-cell
(Truncated 24-cell)
(1,1,1,5) +
(1,3,3,3) +
(2,2,2,4) +
(0,0,1,1)√2
"
"
t0,2,3{3,4,3} Runcicantellated 24-cell
(Runcitruncated 24-cell)
(1,1,1,5) +
(1,3,3,3) +
(2,2,2,4) +
(0,1,1,2)√2
"
"
t1,2,3{3,4,3} Bicantitruncated 24-cell
(Cantitruncated 24-cell)
(1,1,1,5) +
(1,3,3,3) +
(2,2,2,4) +
(0,1,2,3)√2
"
"
t0,1,2,3{3,4,3} Omnitruncated 24-cell

### The H4 family

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

#### 120-cell truncations

# Johnson Name
(Bowers style 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 (hi)
{5,3,3}
(4)

(5.5.5)
120 720 1200 600
33 rectified 120-cell (rahi)
r{5,3,3}
(3)

(3.5.3.5)
(2)

(3.3.3)
720 3120 3600 1200
36 truncated 120-cell (thi)
t{5,3,3}
(3)

(3.10.10)
(1)

(3.3.3)
720 3120 4800 2400
37 cantellated 120-cell (srahi)
rr{5,3,3}
(1)

(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) (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) (xhi)

2t{5,3,3}
(2)

(5.6.6)
(2)

(3.6.6)
720 4320 7200 3600
42 cantitruncated 120-cell (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 (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) (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 (snahi)[9]
(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

#### 600-cell truncations

# Johnson Name
(Bowers style 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)
Cells Faces Edges Vertices
35 600-cell (ex)
{3,3,5}
(20)

(3.3.3)
600 1200 720 120
34 rectified 600-cell (rox)
r{3,3,5}
(2)

(3.3.3.3.3)
(5)

(3.3.3.3)
720 3600 3600 720
41 truncated 600-cell (tex)
t{3,3,5}
(1)

(3.3.3.3.3)
(5)

(3.6.6)
720 3600 4320 1440
40 cantellated 600-cell (srix)
rr{3,3,5}
(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}
(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}
(2)

(5.6.6)
(2)

(3.6.6)
720 4320 7200 3600
45 cantitruncated 600-cell (grix)
tr{3,3,5}
(1)

(5.6.6)
(1)

(4.4.5)
(2)

(4.6.6)
1440 8640 14400 7200
44 runcitruncated 600-cell (prahi)
t0,1,3{3,3,5}
(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}
(1)

(4.6.10)
(1)

(4.4.10)
(1)

(4.4.6)
(1)

(4.6.6)
2640 17040 28800 14400

#### Graphs

# Johnson Name
(Bowers style Acronym)
Coxeter plane projections Schlegel diagrams
F4
[12]
[20] H4
[30]
H3
[10]
A3
[4]
A2
[3]
Dodecahedron
centered
Tetrahedron
centered
32 120-cell (hi)
33 rectified 120-cell (rahi)
34 rectified 600-cell (rox)
35 600-cell (ex)
36 truncated 120-cell (thi)
37 cantellated 120-cell (srahi)
38 runcinated 120-cell
(also runcinated 600-cell) (sidpixhi)
39 bitruncated 120-cell
(also bitruncated 600-cell) (xhi)
40 cantellated 600-cell (srix)
41 truncated 600-cell (tex)
42 cantitruncated 120-cell (grahi)
43 runcitruncated 120-cell (prix)
44 runcitruncated 600-cell (prahi)
45 cantitruncated 600-cell (grix)
46 omnitruncated 120-cell
(also omnitruncated 600-cell) (gidpixhi)

### The D4 family

This demitesseract family, [31,1,1], introduces no new uniform polychora, 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 polychora, [31,1,1]+, order 96.

[31,1,1] uniform polychora
# Johnson 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{4,3,4}
(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.

[3[31,1,1]] uniform polychora
# Johnson 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 polychoron, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the snub cube and the snub dodecahedron.

#### Graphs

# Johnson Name (Bowers style acronym)
Coxeter diagram
Coxeter plane projections Schlegel diagrams Parallel
3D
B4
[8/2]
D4
[6]
D3
[2]
Cube
centered
Tetrahedron
centered
D4
[6]
[12] demitesseract
(Same as 16-cell) (hex)
or
h{4,3,3} = {3,31,1}
[17] truncated demitesseract (thex)
or
h2{4,3,3} = t{3,31,1}
[11] birectified demitesseract
(Same as rectified tesseract) (rit)
or
h3{4,3,3} = 2r{3,31,1}
[16] bitruncated demitesseract
(Same as bitruncated tesseract) (tah)
or
h2,3{4,3,3} = 2t{3,31,1}
[22] rectified demitesseract
(Same as 24-cell) (ico)

{31,1,1}
[23] Cantellated demitesseract
(Same as rectified 24-cell) (rico)

r{31,1,1}
[24] cantitruncated demitesseract
(Same as truncated 24-cell) (tico)

t{31,1,1} = tr{3,31,1}
[31] Snub demitesseract

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

#### Coordinates

The base point can generate the coordinates of the polytope by taking all coordinate permutations and sign combinations. The edges' length will be √2. Some polytopes have two possible generator points. Points are prefixed by Even to imply only an even count of sign permutations should be included.

# Name(s) Base point Johnson and Bowers Names Coxeter diagrams
D4 B4 F4
[12] t3γ4 = β4 (0,0,0,2) 16-cell
[22] t2γ4 = t1β4 (0,0,2,2) Rectified 16-cell
[17] t2,3γ4 = t0,1β4 (0,0,2,4) Truncated 16-cell
[11] t1γ4 = t2β4 (0,2,2,2) Cantellated 16-cell
[23] t1,3γ4 = t0,2β4 (0,2,2,4) Cantellated 16-cell
[16] t1,2γ4 = t1,2β4 (0,2,4,4) Bitruncated 16-cell
[24] t1,2,3γ = t0,1,2β4 (0,2,4,6) Cantitruncated 16-cell
[12] 4 Even (1,1,1,1) 16-cell
[17] h2γ4 Even (1,1,3,3) Truncated 16-cell
[11] h3γ4 Even (1,1,1,3) Cantellated 16-cell
[16] h2,3γ4 Even (1,3,3,3) Cantitruncated 16-cell
[31] s{31,1,1} (0,1,φ,φ+1)/√2 Snub 24-cell

### The grand antiprism

There is one non-Wythoffian uniform convex polychoron, 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.

# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
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

### Prismatic uniform polychora

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 polychora consist of two infinite families:

• Polyhedral prisms: products of a line segment and a uniform polyhedron. This family is infinite because it includes prisms built on 3-dimensional prisms and antiprisms.
• Duoprisms: products of two polygons.

#### Convex polyhedral prisms

The most obvious family of prismatic polychora is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a polychoron 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.

#### Tetrahedral prisms: A3 × A1

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 polychoron.

[3,3,2] uniform polychora
# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
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
[[3,3],2] uniform polychora
# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
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
$s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}$
2
3.3.3.3.3
8
3.3.3.3
6+24
3.3.3
40 16+96 {3} 96 24

#### Octahedral prisms: BC3 × A1

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

# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
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
[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
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
$s\left\{\begin{array}{l}4\\3\\2\end{array}\right\}$
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
(Scaliform)
Runcic snub cubic hosochoron
Truncated tetrahedral cupoliprism (tutcup)

s3{2,4,3}
2
3.6.6
6
3.3.3
8
triangular cupola
16 52 60 24

#### Icosahedral prisms: H3 × A1

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.

# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
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
4.4.5
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
94 240 {4}
40 {6}
24 {5}
360 120
Nonuniform Omnisnub dodecahedral antiprism
$s\left\{\begin{array}{l}5\\3\\2\end{array}\right\}$
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 simplest of the duoprisms, the 3,3-duoprism, in Schlegel diagram, one of 6 triangular prism cells shown.

The second is the infinite family of uniform duoprisms, products of two regular polygons. A duoprism's Coxeter-Dynkin diagram is . Its vertex figure is an 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 pq; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism.

The elements of a p,q-duoprism (p ≥ 3, q ≥ 3) are:

• Cells: p q-gonal prisms, q p-gonal prisms
• Faces: pq squares, p q-gons, q p-gons
• Edges: 2pq
• Vertices: pq

There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms.

Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms:

Name Coxeter graph Cells Images
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

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

The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - - p cubes and 4 p-gonal prisms - (All are the same as 4-p duoprism)

Convex p-gonal prismatic prisms
Name {3}×{}×{} {4}×{}×{} {5}×{}×{} {6}×{}×{} {7}×{}×{} {8}×{}×{} {p}×{}×{}
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}×{}

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.

Convex p-gonal antiprismatic 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}×{}

A p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon faces. It has 10p edges, and 4p vertices.

### Nonuniform alternations

 ht0,1,2,3{3,3,3} ht0,1,2,3{4,3,3} ht0,1,2,3{5,3,3} $s\left\{\begin{array}{l}2\\2\\2\end{array}\right\}$ ht0,1,2,3{p,2,q} ht0,1,2,3{3,4,3} $s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}$ $s\left\{\begin{array}{l}4\\3\\2\end{array}\right\}$ $s\left\{\begin{array}{l}5\\3\\2\end{array}\right\}$ s3{2,4,3} s3{3,4,3} sr3{3,3,4}

There are a number of alternations of the uniform polychora that can not be made uniform as they have too many parameters to satisfy.

Four snubs are not uniform unlike their 3-dimensional analogies. Only the snub 24-cell is uniform, although it is more accurately called a semisnub 24-cell or snub demitesseract for being an omnisnub of the bifurcating family D4 with the demitesseract as the alternated tesseract.

The polyhedral prisms , can be alternated into , but do not generate uniform solutions.

1. Omnisnub tetrahedral antiprism, $s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}$ , 2 icosahedrons connected by 6 tetrahedrons, and 8 octahedrons, with 24 tetrahedra in the alternated gaps.
2. Omnisnub cubic antiprism, $s\left\{\begin{array}{l}4\\3\\2\end{array}\right\}$ , 2 snub cubes connected by 12 tetrahedrons, 6 square antiprisms, and 8 octahedrons, with 48 tetrahedra in the alternated gaps.
3. Omnisnub dodecahedral antiprism, $s\left\{\begin{array}{l}5\\3\\2\end{array}\right\}$ , 2 snub dodecahedrons connected by 30 tetrahedrons, 12 pentagonal antiprisms, and 20 octahedrons, with 120 tetrahedra in the alternated gaps.
4. Truncated tetrahedral cupoliprism (tutcup), s3{2,4,3}, , from 2 truncated tetrahedra, 6 tetrahedra, and 8 triangular cupolae in the gaps, for a total of 16 cells, 52 faces, 60 edges, and 24 vertices. It is vertex-transitive, and equilateral, but not uniform, due to the cupolae. It has symmetry [2+,4,3], order 48.[11]

The duoprisms , t0,1,2,3{p,2,q}, can be alternated into , ht0,1,2,3{p,2,q}, called duoantiprisms, which cannot be made uniform in general. The only convex uniform solution is the trivial case of p=q=2, which is a lower symmetry construction of the tesseract , t0,1,2,3{2,2,2} = t{21,1,1}, with its alternation as the 16-cell, , s{21,1,1} = $s\left\{\begin{array}{l}2\\2\\2\end{array}\right\}$.

### Geometric derivations for 46 nonprismatic Wythoffian uniform polychora

The 46 Wythoffian polychora include the six convex regular polychora. 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 polychora from the regular polychora are truncating operations. A polychoron 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} Both the cantellation and truncation operations applied together.
Bicantitruncation t1,2,3{p,q,r} Cantitruncated dual {r,q,p}.
Runcitruncation t0,1,3{p,q,r} Both the runcination and truncation operations applied together.
Runcicantellation t0,1,3{p,q,r} Runcitruncated dual {r,q,p}.
Omnitruncation
(runcicantitruncation)
t0,1,2,3{p,q,r} Application of all three operators.
Half h{2p,3,q} [1+,2p,3,q] Alternation of , same as
Cantic h2{2p,3,q} Same as
Runcic h3{2p,3,q} Same as
Runcicantic h2,3{2p,3,q} Same as
Quarter q{2p,3,2q} [1+,2p,3,2r,1+] Same as
Snub s{p,2q,r} [p+,2q,r] Alternated truncation
Cantic snub s2{p,2q,r} Cantellated alternated truncation
Runcic snub s3{p,2q,r} Runcinated alternated truncation
Runcicantic snub s2,3{p,2q,r} Runcicantellated alternated truncation
Snub rectified sr{p,q,2r} [(p,q)+,2r] Alternated truncated rectification
ht0,3{2p,q,2r} [(2p,q,2r,2+)] Alternated runcination
Bisnub ht1,2{2p,q,2r} [2p,q+,2r] Alternated bitruncation
Omnisnub ht0,1,2,3{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 16-cell, or the 120-cell and 600-cell), then bitruncating, runcinating or omnitruncating either produces the same figure as the same operation to the other. Thus where only the participle appears in the table it should be understood to apply to either parent.

#### Summary of constructions by extended symmetry

The 46 uniform polychora constructed from the A4, BC4, F4, H4 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 24-cell, 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 3-3 and 4-4 duoprismatic family is included, the second for its relation to the BC4 family.

Coxeter group Extended
symmetry
Polychora Chiral
extended
symmetry
Alternation honeycombs
[3,3,3]
[3,3,3]

(order 120)
6 1 | 2 | 3
4 | 7 | 8
[2+[3,3,3]]

(order 240)
3 5| 6 | 9 [2+[3,3,3]]+
(order 120)
(1) -
[3,31,1]
[3,31,1]

(order 192)
0 (none)
[1[3,31,1]]=[4,3,3]
=
(order 384)
(4) 12 | 17 | 11 | 16
[3[31,1,1]]=[3,4,3]
=
(order 1152)
(3) 22 | 23 | 24 [3[3,31,1]]+
=[3,4,3]+
(order 576)
(1) 31, -
[4,3,3]
[3[1+,4,3,3]]=[3,4,3]
=
(order 1152)
(3) 22 | 23 | 24
[4,3,3]

(order 384)
12 10 | 11 | 12 | 13 | 14
15 |