Set of uniformp-q duoprisms
Type	Prismatic uniform 4-polytopes
Schläfli symbol	{p}×{q}
Coxeter-Dynkin diagram
Cells	p q-gonalprisms, q p-gonal prisms
Faces	pqsquares, p q-gons, q p-gons
Edges	2pq
Vertices	pq
Vertex figure	disphenoid
Symmetry	[p,2,q], order4pq
Dual	p-qduopyramid
Properties	convex,vertex-uniform
Set of uniform p-p duoprisms
Type	Prismatic uniform 4-polytope
Schläfli symbol	{p}×{p}
Coxeter-Dynkin diagram
Cells	2p p-gonalprisms
Faces	p2squares, 2p p-gons
Edges	2p2
Vertices	p2
Symmetry	[p,2,p] = [2p,2+,2p], order8p2
Dual	p-pduopyramid
Properties	convex,vertex-uniform,Facet-transitive

Ingeometry of 4 dimensions or higher, adouble prism^[1] orduoprism is apolytope resulting from theCartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of ann-polytope and anm-polytope is an(n+m)-polytope, wheren andm are dimensions of 2 (polygon) or higher.

The lowest-dimensionalduoprisms exist in4-dimensional space as4-polytopes being the Cartesian product of two polygons in 2-dimensionalEuclidean space. More precisely, it is theset of points:

${\displaystyle P_1\times P_2=\{(x,y,z,w)|(x,y)\in P_1,(z,w)\in P_2\}}$

whereP₁ andP₂ are the sets of the points contained in the respective polygons. Such a duoprism isconvex if both bases are convex, and is bounded byprismatic cells.

Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from tworegular polygons of the same edge length is auniform duoprism.

A duoprism made ofn-polygons andm-polygons is named by prefixing 'duoprism' with the names of the base polygons, for example: atriangular-pentagonal duoprism is the Cartesian product of a triangle and a pentagon.

An alternative, more concise way of specifying a particular duoprism is by prefixing with numbers denoting the base polygons, for example: 3,5-duoprism for the triangular-pentagonal duoprism.

Other alternative names:

• q-gonal-p-gonal prism
• q-gonal-p-gonal double prism
• q-gonal-p-gonal hyperprism

The termduoprism is coined by George Olshevsky, shortened fromdouble prism.John Horton Conway proposed a similar nameproprism forproduct prism, a Cartesian product of two or more polytopes of dimension at least two. The duoprisms are proprisms formed from exactly two polytopes.

Schlegel diagram Projection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown.

net The two sets of 16-gonal prisms are shown. The top and bottom faces of the verticalcylinder are connected when folded together in 4D.

A 4-dimensionaluniform duoprism is created by the product of a regularn-sidedpolygon and a regularm-sided polygon with the same edge length. It is bounded bynm-gonalprisms andmn-gonal prisms. For example, the Cartesian product of a triangle and a hexagon is a duoprism bounded by 6 triangular prisms and 3 hexagonal prisms.

• Whenm andn are identical, the resulting duoprism is bounded by 2n identicaln-gonal prisms. For example, the Cartesian product of two triangles is a duoprism bounded by 6 triangular prisms.
• Whenm andn are identically 4, the resulting duoprism is bounded by 8 square prisms (cubes), and is identical to thetesseract.

Them-gonal prisms are attached to each other via theirm-gonal faces, and form a closed loop. Similarly, then-gonal prisms are attached to each other via theirn-gonal faces, and form a second loop perpendicular to the first. These two loops are attached to each other via their square faces, and are mutually perpendicular.

Asm andn approach infinity, the corresponding duoprisms approach theduocylinder. As such, duoprisms are useful as non-quadric approximations of the duocylinder.

3-3
3-4	4-4
3-5	4-5	5-5
3-6	4-6	5-6	6-6
3-7	4-7	5-7	6-7	7-7
3-8	4-8	5-8	6-8	7-8	8-8
3-9	4-9	5-9	6-9	7-9	8-9	9-9
3-10	4-10	5-10	6-10	7-10	8-10	9-10	10-10

A cell-centered perspective projection makes a duoprism look like atorus, with two sets of orthogonal cells, p-gonal and q-gonal prisms.

Schlegel diagrams

6-prism	6-6 duoprism
Ahexagonal prism, projected into the plane by perspective, centered on a hexagonal face, looks like a double hexagon connected by (distorted)squares. Similarly a 6-6 duoprism projected into 3D approximates atorus, hexagonal both in plan and in section.

The p-q duoprisms are identical to the q-p duoprisms, but look different in these projections because they are projected in the center of different cells.

Schlegel diagrams

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

Vertex-centered orthogonal projections of p-p duoprisms project into [2n] symmetry for odd degrees, and [n] for even degrees. There are n vertices projected into the center. For 4,4, it represents the A₃ Coxeter plane of thetesseract. The 5,5 projection is identical to the 3Drhombic triacontahedron.

Orthogonal projection wireframes of p-p duoprisms

Odd
3-3			5-5			7-7			9-9
[3]		[6]	[5]		[10]	[7]		[14]	[9]		[18]
Even
4-4 (tesseract)			6-6			8-8			10-10
[4]		[8]	[6]		[12]	[8]		[16]	[10]		[20]

Theregular skew polyhedron, {4,4|n}, exists in 4-space as the n² square faces of an-n duoprism, using all 2n² edges and n² vertices. The 2nn-gonal faces can be seen as removed. (skew polyhedra can be seen in the same way by a n-m duoprism, but these are notregular.)

Like theantiprisms as alternatedprisms, there is a set of 4-dimensional duoantiprisms:4-polytopes that can be created by analternation operation applied to a duoprism. The alternated vertices create nonregular tetrahedral cells, except for the special case, the4-4 duoprism (tesseract) which creates the uniform (and regular)16-cell. The 16-cell is the only convex uniform duoantiprism.

The duoprisms, t_0,1,2,3{p,2,q}, can be alternated into, ht_0,1,2,3{p,2,q}, the "duoantiprisms", which cannot be made uniform in general. The only convex uniform solution is the trivial case of p=q=2, which is a lower symmetry construction of thetesseract, t_0,1,2,3{2,2,2}, with its alternation as the16-cell,, s{2}s{2}.

The only nonconvex uniform solution is p=5, q=5/3, ht_0,1,2,3{5,2,5/3},, constructed from 10pentagonal antiprisms, 10pentagrammic crossed-antiprisms, and 50 tetrahedra, known as thegreat duoantiprism (gudap).^[2][3]

Also related are the ditetragoltriates or octagoltriates, formed by taking theoctagon (considered to be a ditetragon or a truncated square) to a p-gon. Theoctagon of a p-gon can be clearly defined if one assumes that the octagon is the convex hull of two perpendicularrectangles; then the p-gonal ditetragoltriate is the convex hull of two p-p duoprisms (where the p-gons are similar but not congruent, having different sizes) in perpendicular orientations. The resulting polychoron is isogonal and has 2p p-gonal prisms and p² rectangular trapezoprisms (acube withD_2d symmetry) but cannot be made uniform. The vertex figure is atriangular bipyramid.

Like the duoantiprisms as alternated duoprisms, there is a set of p-gonal double antiprismoids created by alternating the 2p-gonal ditetragoltriates, creating p-gonal antiprisms and tetrahedra while reinterpreting the non-corealmic triangular bipyramidal spaces as two tetrahedra. The resulting figure is generally not uniform except for two cases: thegrand antiprism and its conjugate, the pentagrammic double antiprismoid (with p = 5 and 5/3 respectively), represented as the alternation of a decagonal or decagrammic ditetragoltriate. The vertex figure is a variant of thesphenocorona.

The3-3 duoprism, -1₂₂, is first in a dimensional series of uniform polytopes, expressed byCoxeter as k₂₂ series. The 3-3 duoprism is the vertex figure for the second, thebirectified 5-simplex. The fourth figure is a Euclidean honeycomb,2₂₂, and the final is a paracompact hyperbolic honeycomb, 3₂₂, with Coxeter group [3^2,2,3],${\displaystyle {\overline{T}}_7}$. Each progressiveuniform polytope is constructed from the previous as itsvertex figure.

k₂₂ figures in n dimensions

Space	Finite			Euclidean	Hyperbolic
n	4	5	6	7	8
Coxeter group	A2A2	E6	${\displaystyle {\tilde{E}}_6}$=E6+	${\displaystyle {\overline{T}}_7}$=E6++
Coxeter diagram
Symmetry	[[32,2,-1]]	[[32,2,0]]	[[32,2,1]]	[[32,2,2]]	[[32,2,3]]
Order	72	1440	103,680	∞
Graph				∞	∞
Name	−122	022	122	222	322

• Polytope and4-polytope
• Convex regular 4-polytope
• Duocylinder
• Tesseract

• Regular Polytopes,H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
• Coxeter,The Beauty of Geometry: Twelve Essays, Dover Publications, 1999,ISBN 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
  • Coxeter, H. S. M.Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,The Symmetries of Things 2008,ISBN 978-1-56881-220-5 (Chapter 26)
• N.W. Johnson:The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

• Uniform 4-polytopes

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