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
Symmetry	[p,2,q], order 4pq [[p,2,p]], order 8p2, p=q
Dual	Duopyramid
Properties	convex if both bases are convex

A net for 16-16 duoprism. 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.

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 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, shortened from double prism. Conway proposed a similar name proprism for product prism.

Geometry of 4-dimensional duoprisms

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.

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.

See also grand antiprism.

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.

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.

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

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.)

See also

• Polytope and polychoron
• Convex regular polychoron
• 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 0486409198 (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