Duoprism

Set of uniform p-q duoprisms
Type	Prismatic uniform 4-polytopes
Schläfli symbol	{p}×{q}
Coxeter-Dynkin diagram
Cells	p q-gonal prisms, q p-gonal prisms
Faces	pq squares, p q-gons, q p-gons
Edges	2pq
Vertices	pq
Vertex figure	100px disphenoid
Symmetry	[p,2,q], order 4pq
Dual	p-q duopyramid
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-gonal prisms
Faces	p2 squares, 2p p-gons
Edges	2p2
Vertices	p2
Symmetry	[[p,2,p]] = [2p,2+,2p], order 8p2
Dual	p-p duopyramid
Properties	convex, vertex-uniform, Facet-transitive

File:23,29-duoprism stereographic closeup.jpg

A close up inside the 23-29 duoprism projected onto a 3-sphere, and perspective projected to 3-space. As m and n become large, a duoprism approaches the geometry of duocylinder just like a p-gonal prism approaches a cylinder.

In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (n+m)-polytope, where n and m are 2 (polygon) or higher.

The lowest-dimensional duoprisms exist in 4-dimensional space as 4-polytopes being the Cartesian product of two polygons in 2-dimensional Euclidean space. More precisely, it is the set of points:

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

where P₁ and P₂ are the sets of the points contained in the respective polygons. Such a duoprism is convex if both bases are convex, and is bounded by prismatic cells.

Nomenclature

Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from two regular polygons of the same edge length is a uniform duoprism.

A duoprism made of n-polygons and m-polygons is named by prefixing 'duoprism' with the names of the base polygons, for example: a triangular-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 term duoprism is coined by George Olshevsky, shortened from double prism. John Horton Conway proposed a similar name proprism for product prism, a Cartesian product of two or more polytopes of dimension at least two. The duoprisms are proprisms formed from exactly two polytopes.

Example 16-16 duoprism

Schlegel diagram 250px Projection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown.

net 250px The two sets of 16-gonal prisms are shown. The top and bottom faces of the vertical cylinder are connected when folded together in 4D.

Geometry of 4-dimensional duoprisms

A 4-dimensional uniform duoprism is created by the product of a regular n-sided polygon and a regular m-sided polygon with the same edge length. It is bounded by n m-gonal prisms and m n-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.

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

The m-gonal prisms are attached to each other via their m-gonal faces, and form a closed loop. Similarly, the n-gonal prisms are attached to each other via their n-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.

As m and n approach infinity, the corresponding duoprisms approach the duocylinder. As such, duoprisms are useful as non-quadric approximations of the duocylinder.

Nets

100px 3-3	4-4	100px 5-5	100px 6-6	100px 8-8	10-10
100px 3-4	100px 3-5	100px 3-6	100px 4-5	100px 4-6	100px 3-8

Perspective projections

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

Schlegel diagrams

160px	160px
6-prism	6-6 duoprism
A hexagonal 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 a torus, 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

75px 3-3	3-4	75px 3-5	75px 3-6	75px 3-7	75px 3-8
75px 4-3	75px 4-4	75px 4-5	75px 4-6	75px 4-7	75px 4-8
75px 5-3	75px 5-4	75px 5-5	75px 5-6	75px 5-7	75px 5-8
75px 6-3	6-4	75px 6-5	75px 6-6	75px 6-7	6-8
75px 7-3	75px 7-4	75px 7-5	75px 7-6	75px 7-7	75px 7-8
8-3	75px 8-4	75px 8-5	8-6	75px 8-7	75px 8-8

Orthogonal projections

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 the tesseract. The 5,5 projection is identical to the 3D rhombic triacontahedron.

Orthogonal projection wireframes of p-p duoprisms

Odd
3-3		5-5		7-7		9-9
100px		100px		100px		100px	100px
[3]	[6]	[5]	[10]	[7]	[14]	[9]	[18]
Even
4-4 (tesseract)		6-6		8-8		10-10
		100px		100px
[4]	[8]	[6]	[12]	[8]	[16]	[10]	[20]

Related polytopes

A stereographic projection of a rotating duocylinder, divided into a checkerboard surface of squares from the {4,4|n} skew polyhedron

The regular skew polyhedron, {4,4|n}, exists in 4-space as the n² square faces of a n-n duoprism, using all 2n² edges and n² vertices. The 2n n-gonal faces can be seen as removed. (skew polyhedra can be seen in the same way by a n-m duoprism, but these are not regular.)

Duoantiprism

File:Snub p2q verf.png

p-q duoantiprism vertex figure, a gyrobifastigium

Great duoantiprism, stereographic projection, centred on one pentagrammic crossed-antiprism

Like the antiprisms as alternated prisms, there is a set of 4-dimensional duoantiprisms: 4-polytopes that can be created by an alternation operation applied to a duoprism. The alternated vertices create nonregular tetrahedral cells, except for the special case, the 4-4 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 the tesseract , t_0,1,2,3{2,2,2}, with its alternation as the 16-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 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra, known as the great duoantiprism (gudap).

k_22 polytopes

The 3-3 duoprism, -1₂₂, is first in a dimensional series of uniform polytopes, expressed by Coxeter as k₂₂ series. The 3-3 duoprism is the vertex figure for the second, the birectified 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], ${\bar{T}}_7$. Each progressive uniform polytope is constructed from the previous as its vertex figure.

k₂₂ figures in n dimensions

Space	Finite			Euclidean	Hyperbolic
n	4	5	6	7	8
Coxeter group	2A2	A5	E6	${\tilde{E}}_{6}$=E6+	${\bar{T}}_7$=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

See also

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

References

• 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.
• The Fourth Dimension Simply Explained, Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online: The Fourth Dimension Simply Explained—contains a description of duoprisms (double prisms) and duocylinders (double cylinders). Googlebook
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Olshevsky, George, Duoprism at Glossary for Hyperspace.
  • Olshevsky, George, Cartesian product at Glossary for Hyperspace.
  • Catalogue of Convex Polychora, section 6, George Olshevsky.

External links

• The Fourth Dimension Simply Explained—describes duoprisms as "double prisms" and duocylinders as "double cylinders"
• Polygloss - glossary of higher-dimensional terms
• Exploring Hyperspace with the Geometric Product