Duoprism
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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. |
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| 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:
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.
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[edit] 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.
[edit] 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.
[edit] 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.
[edit] 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 |
[edit] See also
[edit] 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).
[edit] External links
- Olshevsky, George, Duoprism at Glossary for Hyperspace.
- Olshevsky, George, Cartesian product at Glossary for Hyperspace.
- Catalogue of Convex Polychora, section 6, George Olshevsky.
- 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
- The word Duoprism is also the name of an LCD monitor. It has no relation to the mathematical use of the term as described here.
