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: A3 × A1 # Johnson Name (Bowers style acronym) PictureCoxeter diagramand SchläflisymbolsCells by typeElement counts Cells Faces Edges Vertices48Tetrahedral prism (tepe){3,3}×{}2 3.3.34 3.4.468 {3}6 {4}16849Truncated tetrahedral prism (tuttip)t{3,3}×{}2 3.6.64 3.4.44 4.4.6108 {3}18 {4}8 {6}4824Rectified tetrahedral prism(Same as octahedral prism) (ope)r{3,3}×{}2 3.3.3.34 3.4.4616 {3}12 {4}3012Cantellated tetrahedral prism(Same as cuboctahedral prism) (cope) rr{3,3}×{}2 3.4.3.48 3.4.46 4.4.41616 {3}36 {4}6024Cantitruncated tetrahedral prism(Same as truncated octahedral prism) (tope)tr{3,3}×{}2 4.6.6 8 3.4.46 4.4.41648 {4}16 {6}9648Snub tetrahedral prism(Same as icosahedral prism) (ipe)sr{3,3}×{}2 3.3.3.3.320 3.4.42240 {3}30 {4}7224
Octahedral prisms: BC3 × A1 # Johnson Name (Bowers style acronym) PictureCoxeter diagramand SchläflisymbolsCells by typeElement counts Cells Faces Edges VerticesCubic prism(Same as tesseract)(Same as 4-4 duoprism) (tes){4,3}×{}2 4.4.46 4.4.4824 {4}321650Cuboctahedral 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 2451Octahedral prism(Same as rectified tetrahedral prism)(Same as triangular antiprismatic prism) (ope){3,4}×{}2 3.3.3.38 3.4.41016 {3}12 {4}301252Rhombicuboctahedral prism (sircope)rr{4,3}×{} 2 3.4.4.4 8 3.4.4 18 4.4.4 28 16 {3}84 {4} 120 9653Truncated 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 4854Truncated 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 4855Truncated 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 9656Snub 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) PictureCoxeter diagramand SchläflisymbolsCells by typeElement counts Cells Faces Edges Vertices57Dodecahedral prism (dope){5,3}×{} 2 5.5.512 4.4.5 14 30 {4}24 {5} 80 4058Icosidodecahedral prism (iddip)r{5,3}×{}2 3.5.3.520 3.4.412 4.4.5 34 40 {3}60 {4}24 {5} 150 6059Icosahedral 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 2460Truncated dodecahedral prism (tiddip)t{5,3}×{}2 3.10.1020 3.4.412 4.4.5 34 40 {3}90 {4}24 {10} 240 12061Rhombicosidodecahedral prism (sriddip)rr{5,3}×{}2 3.4.5.420 3.4.430 4.4.412 4.4.5 64 40 {3}180 {4}24 {5} 300 12062Truncated icosahedral prism (tipe)t{3,5}×{}2 5.6.6 12 4.4.520 4.4.6 34 90 {4}24 {5}40 {6} 240 12063Truncated icosidodecahedral prism (griddip)tr{5,3}×{}2 4.6.4.10 30 4.4.420 4.4.612 4.4.10 64 240 {4}40 {6}24 {5} 480 24064Snub dodecahedral prism (sniddip)sr{5,3}×{}2 3.3.3.3.580 3.4.412 4.4.5 94 240 {4}40 {6}24 {10} 360 120
Duoprisms: [p] × [q] + Set of uniform p,q duoprisms3-33-43-53-63-73-84-34-44-54-64-74-85-35-45-55-65-75-86-36-46-56-66-76-87-37-47-57-67-77-88-38-48-58-68-78-8
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.
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, (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