Prismatic uniform 4-polytope

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.

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 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 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
• ...

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)
• ...

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.

• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
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