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Duoprism

Set of uniform p,q-duoprisms Example 16,16-duoprism Schlegel diagram Projection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown.
Type	Prismatic uniform polychoron
Schläfli symbol	{p}x{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	disphenoid tetrahedron Example for 16-16 duoprism (Each edge is part of a face at the central vertex with a given number of sides)
Symmetry group	[p]x[q] or [p,2,q]
Properties	convex if both bases are convex

In geometry, a duoprism is a polytope resulting from the Cartesian product of two polytopes 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 polychora (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 polychora. A duoprism constructed from two regular polygons of the same size 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. Conway proposed a similar name proprism for product prism.

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.

Polychoral duoantiprisms

Like the antiprisms as alternated prisms, there is a set of 4-dimensional duoantiprisms polychorons that can be created by an alternation operation applied to a duoprism. However most are not uniform. 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.

Images of uniform polychoral duoprisms

All of these images are Schlegel diagrams with one cell shown. The p-q duoprisms are identical to the q-p duoprisms, but look different because they are projected in the center of different cells.

3-3	3-4	3-5	3-6
4-3	4-4	4-5	4-6
5-3	5-4	5-5	5-6
6-3	6-4	6-5	6-6

References

Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
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).