Prismatic uniform 4-polytope
# Prismatic uniform 4-polytope

In four-dimensional geometry, a prismatic uniform 4-polytope is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group. 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 4-polytopes 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 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytope 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).

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. The symmetry number of a polyhedral prism is twice that of the base polyhedron.

## Tetrahedral prisms: $A_3$ × $A_1$

| # | Johnson Name (Bowers style acronym) | Picture | Coxeter diagram and Schläfli symbols | Cells by type | Cells by type | Cells by type | Element counts | Element counts | Element counts | Element counts |
| - | - | - | - | - | - | - | - | - | - | - |
| # | Johnson Name (Bowers style acronym) | Picture | Coxeter diagram and Schläfli symbols | Cells by type | Cells by type | Cells by type | Cells | Faces | Edges | Vertices |
| 48 | Tetrahedral prism (tepe) | | {3,3}× | 2 3.3.3 | 4 3.4.4 | | 6 | 8 {3} 6 {4} | 16 | 8 |
| 49 | Truncated tetrahedral prism (tuttip) | | t{3,3}× | 2 3.6.6 | 4 3.4.4 | 4 4.4.6 | 10 | 8 {3} 18 {4} 8 {6} | 48 | 24 |
| [51] | Rectified tetrahedral prism (Same as octahedral prism) (ope) | | r{3,3}× | 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}× | 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}× | 2 4.6.6 | 8 3.4.4 | 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 |

## Octahedral prisms: $BC_3$ × $A_1$

| # | Johnson Name (Bowers style acronym) | Picture | Coxeter diagram and Schläfli symbols | Cells by type | Cells by type | Cells by type | Cells by type | Element counts | Element counts | Element counts | Element counts |
| - | - | - | - | - | - | - | - | - | - | - | - |
| # | Johnson Name (Bowers style acronym) | Picture | Coxeter diagram and Schläfli symbols | Cells by type | Cells by type | Cells by type | Cells by type | Cells | Faces | Edges | Vertices |
| [10] | Cubic prism (Same as tesseract) (Same as 4-4 duoprism) (tes) | | {4,3}× | 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}× | 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}× | 2 3.3.3.3 | 8 3.4.4 | | | 10 | 16 {3} 12 {4} | 30 | 12 |
| 52 | Rhombicuboctahedral prism (sircope) | | rr{4,3}× | 2 3.4.4.4 | 8 3.4.4 | 18 4.4.4 | | 28 | 16 {3} 84 {4} | 120 | 96 |
| 53 | Truncated cubic prism (ticcup) | | t{4,3}× | 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}× | 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}× | 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 |

## Icosahedral prisms: $H_3$ × $A_1$

| # | Johnson Name (Bowers style acronym) | Picture | Coxeter diagram and Schläfli symbols | Cells by type | Cells by type | Cells by type | Cells by type | Element counts | Element counts | Element counts | Element counts |
| - | - | - | - | - | - | - | - | - | - | - | - |
| # | Johnson Name (Bowers style acronym) | Picture | Coxeter diagram and Schläfli symbols | Cells by type | Cells by type | Cells by type | Cells by type | Cells | Faces | Edges | Vertices |
| 57 | Dodecahedral prism (dope) | | {5,3}× | 2 5.5.5 | 12 4.4.5 | | | 14 | 30 {4} 24 {5} | 80 | 40 |
| 58 | Icosidodecahedral prism (iddip) | | r{5,3}× | 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}× | 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}× | 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}× | 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}× | 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}× | 2 4.6.4.10 | 30 4.4.4 | 20 4.4.6 | 12 4.4.10 | 64 | 240 {4} 40 {6} 24 {5} | 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 {10} | 360 | 120 |

## Duoprisms: [p] × [q]

Set of uniform p,q duoprisms :

| 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 |

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

Their Coxeter diagram is of the form

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 $8p^2.$ 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 with the exception of the great duoantiprism.

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

* 3-3 duoprism - - 6 triangular prisms
* 3-4 duoprism - - 3 cubes, 4 triangular prisms
* 4-4 duoprism - - 8 cubes (same as tesseract)
* 3-5 duoprism - - 3 pentagonal prisms, 5 triangular prisms
* 4-5 duoprism - - 4 pentagonal prisms, 5 cubes
* 5-5 duoprism - - 10 pentagonal prisms
* 3-6 duoprism - - 3 hexagonal prisms, 6 triangular prisms
* 4-6 duoprism - - 4 hexagonal prisms, 6 cubes
* 5-6 duoprism - - 5 hexagonal prisms, 6 pentagonal prisms
* 6-6 duoprism - - 12 hexagonal prisms

## Polygonal prismatic prisms

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)

* Triangular prismatic prism - - 3 cubes and 4 triangular prisms - (same as 3-4 duoprism)
* Square prismatic prism - - 4 cubes and 4 cubes - (same as 4-4 duoprism and same as tesseract)
* Pentagonal prismatic prism - - 5 cubes and 4 pentagonal prisms - (same as 4-5 duoprism)
* Hexagonal prismatic prism - - 6 cubes and 4 hexagonal prisms - (same as 4-6 duoprism)
* Heptagonal prismatic prism - - 7 cubes and 4 heptagonal prisms - (same as 4-7 duoprism)
* Octagonal prismatic prism - - 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) - - 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}× |
| Net | | | | | | | | |

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

