Duoprism
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Set of uniform p,q-duoprisms Example 23,29-duoprism |
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| Type | Prismatic uniform polychoron |
| Cells | p q-gonal prisms, q p-gonal prisms |
| Faces | pq squares, p q-gons, q p-gons |
| Edges | 2pq |
| Vertices | pq |
| Vertex configuration | tetrahedron |
| Symmetry group | [p]x[q] |
| Schläfli symbol | {p}x{q} |
| Properties | convex |
A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 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.
The duoprism is a convex 4-dimensional polytope bounded by prismic cells.
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[edit] Geometry
A uniform duoprism is created by the product of a regular n-sided polygon and a regular m-sided polygon 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 tetragonal prisms (cubes), and is identical to the hypercube.
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] Nomenclature
The term duoprism is coined by George Olshevsky. It is a subset of the prismatic polychora. In Olshevsky's usage, a duoprism made of n-polygons and m-polygons is named by prefixing 'duoprism' with the names of the base polygons, for example: the 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
[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 and duocylinders (double cylinders)
[edit] External links
- Nomenclature of Polychora (the description of duoprisms is near the middle of the page)
- The Fourth Dimension Simply Explained—describes duoprisms as "double prisms" and duocylinders as "double cylinders"
- Catalogue of Convex Polychora, section 6
- Polygloss - glossary of higher-dimensional terms
- 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.
