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] |
Properties | convex if both bases are convex |
A duoprism is a 4-dimensional figure, or polychoron, resulting from 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.
The duoprism is a 4-dimensional polytope, convex if both bases are convex. It is bounded by prismic cells.
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[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] Geometry
A 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] Duoantiprisms
Like the antiprisms as alternated prisms, there is a set of 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
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
- 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.