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.
Type Prismatic uniform polychoron
Schläfli symbol {p}x{q}
Coxeter-Dynkin diagram Image:CDW ring.pngImage:CDW p.pngImage:CDW dot.pngImage:CDW 2.pngImage:CDW ring.pngImage:CDW q.pngImage:CDW dot.png
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 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.
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.

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:

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.

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 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 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 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

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