Prismatic uniform polychoron

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A cubic prism, {4,3}×{}, is a lower symmetry construction of the regular tesseract, {4,3,3}, as a prism of two parallel cubes, as seen in this Schlegel diagram

In four-dimensional geometry, a prismatic uniform polytope is a uniform polychoron with a nonconnected Coxeter diagram symmetry group.[citation needed] These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category called duoprisms, constructed as a product of two regular polygons.

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: product of two regular 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

# Johnson Name (Bowers style acronym) Picture Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
Cells Faces Edges Vertices
48 Tetrahedral prism (tepe) Tetrahedral prism.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{3,3}×{}
2 Tetrahedron.png
3.3.3
4 Triangular prism.png
3.4.4
6 8 {3}
6 {4}
16 8
49 Truncated tetrahedral prism (tuttip) Truncated tetrahedral prism.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{3,3}×{}
2 Truncated tetrahedron.png
3.6.6
4 Triangular prism.png
3.4.4
4 Hexagonal prism.png
4.4.6
10 8 {3}
18 {4}
8 {6}
48 24
[51] Rectified tetrahedral prism
(Same as octahedral prism) (ope)
Octahedral prism.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{3,3}×{}
2 Octahedron.png
3.3.3.3
4 Triangular prism.png
3.4.4
6 16 {3}
12 {4}
30 12
[50] Cantellated tetrahedral prism
(Same as cuboctahedral prism) (cope)
Cuboctahedral prism.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
rr{3,3}×{}
2 Cuboctahedron.png
3.4.3.4
8 Triangular prism.png
3.4.4
6 Hexahedron.png
4.4.4
16 16 {3}
36 {4}
60 24
[54] Cantitruncated tetrahedral prism
(Same as truncated octahedral prism) (tope)
Truncated octahedral prism.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{3,3}×{}
2 Truncated octahedron.png
4.6.6
8 Hexagonal prism.png
3.4.4
6 Hexahedron.png
4.4.4
16 48 {4}
16 {6}
96 48
[59] Snub tetrahedral prism
(Same as icosahedral prism) (ipe)
Icosahedral prism.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
sr{3,3}×{}
2 Icosahedron.png
3.3.3.3.3
20 Triangular prism.png
3.4.4
22 40 {3}
30 {4}
72 24

Octahedral prisms: BC3 × A1

# Johnson Name (Bowers style acronym) Picture 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)
Schlegel wireframe 8-cell.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{4,3}×{}
2 Hexahedron.png
4.4.4
6 Hexahedron.png
4.4.4
8 24 {4} 32 16
50 Cuboctahedral prism
(Same as cantellated tetrahedral prism) (cope)
Cuboctahedral prism.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{4,3}×{}
2 Cuboctahedron.png
3.4.3.4
8 Triangular prism.png
3.4.4
6 Hexahedron.png
4.4.4
16 16 {3}
36 {4}
60 24
51 Octahedral prism
(Same as rectified tetrahedral prism)
(Same as triangular antiprismatic prism) (ope)
Octahedral prism.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{3,4}×{}
2 Octahedron.png
3.3.3.3
8 Triangular prism.png
3.4.4
10 16 {3}
12 {4}
30 12
52 Rhombicuboctahedral prism (sircope) Rhombicuboctahedral prism.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
rr{4,3}×{}
2 Small rhombicuboctahedron.png
3.4.4.4
8 Triangular prism.png
3.4.4
18 Hexahedron.png
4.4.4
28 16 {3}
84 {4}
120 96
53 Truncated cubic prism (ticcup) Truncated cubic prism.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{4,3}×{}
2 Truncated hexahedron.png
3.8.8
8 Triangular prism.png
3.4.4
6 Octagonal prism.png
4.4.8
16 16 {3}
36 {4}
12 {8}
96 48
54 Truncated octahedral prism
(Same as cantitruncated tetrahedral prism) (tope)
Truncated octahedral prism.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{3,4}×{}
2 Truncated octahedron.png
4.6.6
6 Hexahedron.png
4.4.4
8 Hexagonal prism.png
4.4.6
16 48 {4}
16 {6}
96 48
55 Truncated cuboctahedral prism (gircope) Truncated cuboctahedral prism.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{4,3}×{}
2 Great rhombicuboctahedron.png
4.6.8
12 Hexahedron.png
4.4.4
8 Hexagonal prism.png
4.4.6
6 Octagonal prism.png
4.4.8
28 96 {4}
16 {6}
12 {8}
192 96
56 Snub cubic prism (sniccup) Snub cubic prism.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
sr{4,3}×{}
2 Snub hexahedron.png
3.3.3.3.4
32 Triangular prism.png
3.4.4
6 Hexahedron.png
4.4.4
40 64 {3}
72 {4}
144 48

Icosahedral prisms: H3 × A1

# Johnson Name (Bowers style acronym) Picture Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
Cells Faces Edges Vertices
57 Dodecahedral prism (dope) Dodecahedral prism.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{5,3}×{}
2 Dodecahedron.png
5.5.5
12 Pentagonal prism.png
4.4.5
14 30 {4}
24 {5}
80 40
58 Icosidodecahedral prism (iddip) Icosidodecahedral prism.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{5,3}×{}
2 Icosidodecahedron.png
3.5.3.5
20 Triangular prism.png
3.4.4
12 Pentagonal prism.png
4.4.5
34 40 {3}
60 {4}
24 {5}
150 60
59 Icosahedral prism
(same as snub tetrahedral prism) (ipe)
Icosahedral prism.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{3,5}×{}
2 Icosahedron.png
3.3.3.3.3
20 Triangular prism.png
3.4.4
22 40 {3}
30 {4}
72 24
60 Truncated dodecahedral prism (tiddip) Truncated dodecahedral prism.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{5,3}×{}
2 Truncated dodecahedron.png
3.10.10
20 Triangular prism.png
3.4.4
12 Decagonal prism.png
4.4.5
34 40 {3}
90 {4}
24 {10}
240 120
61 Rhombicosidodecahedral prism (sriddip) Rhombicosidodecahedral prism.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
rr{5,3}×{}
2 Small rhombicosidodecahedron.png
3.4.5.4
20 Triangular prism.png
3.4.4
30 Hexahedron.png
4.4.4
12 Pentagonal prism.png
4.4.5
64 40 {3}
180 {4}
24 {5}
300 120
62 Truncated icosahedral prism (tipe) Truncated icosahedral prism.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{3,5}×{}
2 Truncated icosahedron.png
5.6.6
12 Pentagonal prism.png
4.4.5
20 Hexagonal prism.png
4.4.6
34 90 {4}
24 {5}
40 {6}
240 120
63 Truncated icosidodecahedral prism (griddip) Truncated icosidodecahedral prism.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{5,3}×{}
2 Great rhombicosidodecahedron.png
4.6.4.10
30 Hexahedron.png
4.4.4
20 Hexagonal prism.png
4.4.6
12 Decagonal prism.png
4.4.10
64 240 {4}
40 {6}
24 {5}
480 240
64 Snub dodecahedral prism (sniddip) Snub dodecahedral prism.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
sr{5,3}×{}
2 Snub dodecahedron ccw.png
3.3.3.3.5
80 Triangular prism.png
3.4.4
12 Pentagonal prism.png
4.4.5
94 240 {4}
40 {6}
24 {10}
360 120

Duoprisms: [p] × [q]

Set of uniform p,q duoprisms
3-3 duoprism.png
3-3
3-4 duoprism.png
3-4
3-5 duoprism.png
3-5
3-6 duoprism.png
3-6
3-7 duoprism.png
3-7
3-8 duoprism.png
3-8
4-3 duoprism.png
4-3
4-4 duoprism.png
4-4
4-5 duoprism.png
4-5
4-6 duoprism.png
4-6
4-7 duoprism.png
4-7
4-8 duoprism.png
4-8
5-3 duoprism.png
5-3
5-4 duoprism.png
5-4
5-5 duoprism.png
5-5
5-6 duoprism.png
5-6
5-7 duoprism.png
5-7
5-8 duoprism.png
5-8
6-3 duoprism.png
6-3
6-4 duoprism.png
6-4
6-5 duoprism.png
6-5
6-6 duoprism.png
6-6
6-7 duoprism.png
6-7
6-8 duoprism.png
6-8
7-3 duoprism.png
7-3
7-4 duoprism.png
7-4
7-5 duoprism.png
7-5
7-6 duoprism.png
7-6
7-7 duoprism.png
7-7
7-8 duoprism.png
7-8
8-3 duoprism.png
8-3
8-4 duoprism.png
8-4
8-5 duoprism.png
8-5
8-6 duoprism.png
8-6
8-7 duoprism.png
8-7
8-8 duoprism.png
8-8

The second is the infinite family of uniform duoprisms, products of two regular polygons.

Their Coxeter diagram is of the form CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node.png

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 - CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node.png - p q-gonal prisms, q p-gonal prisms:

  • 3-3 duoprism - CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png - 6 triangular prisms
  • 3-4 duoprism - CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png - 3 cubes, 4 triangular prisms
  • 4-4 duoprism - CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png - 8 cubes (same as tesseract)
  • 3-5 duoprism - CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 5.pngCDel node.png - 3 pentagonal prisms, 5 triangular prisms
  • 4-5 duoprism - CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 5.pngCDel node.png - 4 pentagonal prisms, 5 cubes
  • 5-5 duoprism - CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 5.pngCDel node.png - 10 pentagonal prisms
  • 3-6 duoprism - CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png - 3 hexagonal prisms, 6 triangular prisms
  • 4-6 duoprism - CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png - 4 hexagonal prisms, 6 cubes
  • 5-6 duoprism - CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png - 5 hexagonal prisms, 6 pentagonal prisms
  • 6-6 duoprism - CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png - 12 hexagonal prisms
  • ...

Polygonal prismatic prisms

The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png - p cubes and 4 p-gonal prisms - (All are the same as 4-p duoprism)

  • Triangular prismatic prism - CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png - 3 cubes and 4 triangular prisms - (same as 3-4 duoprism)
  • Square prismatic prism - CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png - 4 cubes and 4 cubes - (same as 4-4 duoprism and same as tesseract)
  • Pentagonal prismatic prism - CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png - 5 cubes and 4 pentagonal prisms - (same as 4-5 duoprism)
  • Hexagonal prismatic prism - CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png - 6 cubes and 4 hexagonal prisms - (same as 4-6 duoprism)
  • Heptagonal prismatic prism - CDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png - 7 cubes and 4 heptagonal prisms - (same as 4-7 duoprism)
  • Octagonal prismatic prism - CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png - 8 cubes and 4 octagonal prisms - (same as 4-8 duoprism)
  • ...

Uniform antiprismatic prism

The infinite sets of uniform antiprismatic prisms or antiduoprisms are constructed from two parallel uniform antiprisms: (p≥3) - CDel node h.pngCDel p.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node 1.png - 2 p-gonal antiprisms, connected by 2 p-gonal prisms and 2p triangular prisms.

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

References

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds