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. 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

Octahedral prisms: BC₃ × A₁

Icosahedral prisms: H₃ × A₁

Duoprisms: ×

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 4''pq'' if ''p''≠''q''; if the factors are both ''p''-gons, the symmetry number is 8''p''². 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'', ath. Zeit. 46 (1940) 380-407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. 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
*

{{DEFAULTSORT:Prismatic Uniform Polychoron
4-polytopes