Prismatic uniform 4-polytope

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 4-polytope is a uniform 4-polytope 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 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).^{[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)		{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: 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)		{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: 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)		{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 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
• ...

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.

References[]

• 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
  • (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation
• Klitzing, Richard. "4D uniform polytopes (polychora)".