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:
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.
| Johnson Name (Bowers style acronym) | Picture | Coxeter diagram and Schläfli symbols | Cells by type | Element counts | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Cells | Faces | Edges | Vertices | ||||||||
| 48 | Tetrahedral prism (tepe) | × | 2 3.3.3 | 4 3.4.4 | 6 | 8 6 | 16 | 8 | |||
| 49 | Truncated tetrahedral prism (tuttip) | t× | 2 3.6.6 | 4 3.4.4 | 4 4.4.6 | 10 | 8 18 8 | 48 | 24 | ||
| [51] | Rectified tetrahedral prism (Same as octahedral prism) (ope) | r× | 2 3.3.3.3 | 4 3.4.4 | 6 | 16 12 | 30 | 12 | |||
| [50] | Cantellated tetrahedral prism (Same as cuboctahedral prism) (cope) | rr× | 2 3.4.3.4 | 8 3.4.4 | 6 4.4.4 | 16 | 16 36 | 60 | 24 | ||
| [54] | Cantitruncated tetrahedral prism (Same as truncated octahedral prism) (tope) | tr× | 8 3.4.4 | 6 4.4.4 | 16 | 48 16 | 96 | 48 | |||
| [59] | Snub tetrahedral prism (Same as icosahedral prism) (ipe) | sr× | 2 3.3.3.3.3 | 20 3.4.4 | 22 | 40 30 | 72 | 24 | |||
| 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) | × | 2 4.4.4 | 6 4.4.4 | 8 | 24 | 32 | 16 | ||||
| 50 | Cuboctahedral prism (Same as cantellated tetrahedral prism) (cope) | r× | 8 3.4.4 | 6 4.4.4 | 16 | 16 36 | 60 | 24 | ||||
| 51 | Octahedral prism (Same as rectified tetrahedral prism) (Same as triangular antiprismatic prism) (ope) | × | 2 3.3.3.3 | 8 3.4.4 | 10 | 16 12 | 30 | 12 | ||||
| 52 | Rhombicuboctahedral prism (sircope) | rr× | 8 3.4.4 | 18 4.4.4 | 28 | 16 84 | 120 | 96 | ||||
| 53 | Truncated cubic prism (ticcup) | t× | 2 3.8.8 | 8 3.4.4 | 6 4.4.8 | 16 | 16 36 12 | 96 | 48 | |||
| 54 | Truncated octahedral prism (Same as cantitruncated tetrahedral prism) (tope) | t× | 2 4.6.6 | 6 4.4.4 | 8 4.4.6 | 16 | 48 16 | 96 | 48 | |||
| 55 | Truncated cuboctahedral prism (gircope) | tr× | 2 4.6.8 | 12 4.4.4 | 8 4.4.6 | 6 4.4.8 | 28 | 96 16 12 | 192 | 96 | ||
| 56 | Snub cubic prism (sniccup) | sr× | 2 3.3.3.3.4 | 32 3.4.4 | 6 4.4.4 | 40 | 64 72 | 144 | 48 | |||
| Johnson Name (Bowers style acronym) | Picture | Coxeter diagram and Schläfli symbols | Cells by type | Element counts | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cells | Faces | Edges | Vertices | |||||||||
| 57 | Dodecahedral prism (dope) | × | 2 5.5.5 | 12 4.4.5 | 14 | 30 24 | 80 | 40 | ||||
| 58 | Icosidodecahedral prism (iddip) | r× | 2 3.5.3.5 | 20 3.4.4 | 12 4.4.5 | 34 | 40 60 24 | 150 | 60 | |||
| 59 | Icosahedral prism (same as snub tetrahedral prism) (ipe) | × | 20 3.4.4 | 22 | 40 30 | 72 | 24 | |||||
| 60 | Truncated dodecahedral prism (tiddip) | t× | 2 3.10.10 | 20 3.4.4 | 12 4.4.5 | 34 | 40 90 24 | 240 | 120 | |||
| 61 | Rhombicosidodecahedral prism (sriddip) | rr× | 2 3.4.5.4 | 20 3.4.4 | 30 4.4.4 | 12 4.4.5 | 64 | 40 180 24 | 300 | 120 | ||
| 62 | Truncated icosahedral prism (tipe) | t× | 12 4.4.5 | 20 4.4.6 | 34 | 90 24 40 | 240 | 120 | ||||
| 63 | Truncated icosidodecahedral prism (griddip) | tr× | 30 4.4.4 | 20 4.4.6 | 12 4.4.10 | 64 | 240 40 24 | 480 | 240 | |||
| 64 | Snub dodecahedral prism (sniddip) | sr× | 2 3.3.3.3.5 | 80 3.4.4 | 12 4.4.5 | 94 | 240 40 24 | 360 | 120 | |||
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:
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:
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)
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.