 Duoprism

Set of uniform p,qduoprisms
Example 16,16duoprism
Schlegel diagram
Projection from the center of one 16gonal prism, and all but one of the opposite 16gonal prisms are shown.Type Prismatic uniform polychoron Schläfli symbol {p}x{q} CoxeterDynkin diagram Cells p qgonal prisms,
q pgonal prismsFaces pq squares,
p qgons,
q pgonsEdges 2pq Vertices pq Vertex figure
disphenoid tetrahedronSymmetry [p,2,q], order 4pq
[[p,2,p]], order 8p^{2}, p=qDual Duopyramid Properties convex if both bases are convex 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 npolytope and an mpolytope is an (n+m)polytope, where n and m are 2 (polygon) or higher.
The lowest dimensional duoprisms exist in 4dimensional space as polychora (4polytopes) being the Cartesian product of two polygons in 2dimensional Euclidean space. More precisely, it is the set of points:
where P_{1} and P_{2} 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.
Contents
Nomenclature
Fourdimensional 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 npolygons and mpolygons is named by prefixing 'duoprism' with the names of the base polygons, for example: a triangularpentagonal 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: 3,5duoprism for the triangularpentagonal duoprism.
Other alternative names:
 qgonalpgonal prism
 qgonalpgonal double prism
 qgonalpgonal hyperprism
The term duoprism is coined by George Olshevsky, shortened from double prism. Conway proposed a similar name proprism for product prism.
Geometry of 4dimensional duoprisms
A 4dimensional uniform duoprism is created by the product of a regular nsided polygon and a regular msided polygon with the same edge length. It is bounded by n mgonal prisms and m ngonal 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 ngonal 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 mgonal prisms are attached to each other via their mgonal faces, and form a closed loop. Similarly, the ngonal prisms are attached to each other via their ngonal 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 nonquadric approximations of the duocylinder.
Polychoral duoantiprisms
Like the antiprisms as alternated prisms, there is a set of 4dimensional 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 44 duoprism (tesseract) which creates the uniform (and regular) 16cell. The 16cell 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 pq duoprisms are identical to the qp duoprisms, but look different because they are projected in the center of different cells.
6prism 66duoprism 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 66 duoprism projected into 3D approximates a torus, hexagonal both in plan and in section.
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88Related polytopes
The regular skew polyhedron, {4,4n}, exists in 4space as the n^{2} square faces of a nn duoprism, using all 2n^{2} edges and n^{2} vertices. The 2n ngonal faces can be seen as removed. (skew polyhedra can be seen in the same way by a nm 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, 3362, 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 GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (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 higherdimensional terms
 Exploring Hyperspace with the Geometric Product
Categories: Fourdimensional geometry
 Algebraic topology
 Polychora
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