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.[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
[edit]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
[edit]# | 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
[edit]# | 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
[edit]# | 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]
[edit]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
[edit]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
[edit]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.
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
[edit]- 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)".