Duoprism

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 n-polytope and an m-polytope is an (n+m)-polytope, where n and m are 2 (polygon) or higher.

Set of uniform p-q duoprisms
TypePrismatic uniform 4-polytopes
Schläfli symbol{p}×{q}
Coxeter-Dynkin diagram
Cellsp q-gonal prisms,
q p-gonal prisms
Facespq squares,
p q-gons,
q p-gons
Edges2pq
Verticespq
Vertex figure
disphenoid
Symmetry[p,2,q], order 4pq
Dualp-q duopyramid
Propertiesconvex, vertex-uniform
 
Set of uniform p-p duoprisms
TypePrismatic uniform 4-polytope
Schläfli symbol{p}×{p}
Coxeter-Dynkin diagram
Cells2p p-gonal prisms
Facesp2 squares,
2p p-gons
Edges2p2
Verticesp2
Symmetry[[p,2,p]] = [2p,2+,2p], order 8p2
Dualp-p duopyramid
Propertiesconvex, vertex-uniform, Facet-transitive

The lowest-dimensional duoprisms exist in 4-dimensional space as 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.

Nomenclature

Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from two regular polygons of the same edge length 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, shortened from double prism. John Horton Conway proposed a similar name proprism for product prism, a Cartesian product of two or more polytopes of dimension at least two. The duoprisms are proprisms formed from exactly two polytopes.

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.
net

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.

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.

Nets


3-3

4-4

5-5

6-6

8-8

10-10

3-4

3-5

3-6

4-5

4-6

3-8

Perspective projections

A cell-centered perspective projection makes a duoprism look like a torus, with two sets of orthogonal cells, p-gonal and q-gonal prisms.

Schlegel diagrams
6-prism 6-6 duoprism
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 6-6 duoprism projected into 3D approximates a torus, hexagonal both in plan and in section.

The p-q duoprisms are identical to the q-p duoprisms, but look different in these projections because they are projected in the center of different cells.

Schlegel diagrams

3-3

3-4

3-5

3-6

3-7

3-8

4-3

4-4

4-5

4-6

4-7

4-8

5-3

5-4

5-5

5-6

5-7

5-8

6-3

6-4

6-5

6-6

6-7

6-8

7-3

7-4

7-5

7-6

7-7

7-8

8-3

8-4

8-5

8-6

8-7

8-8

Orthogonal projections

Vertex-centered orthogonal projections of p-p duoprisms project into [2n] symmetry for odd degrees, and [n] for even degrees. There are n vertices projected into the center. For 4,4, it represents the A3 Coxeter plane of the tesseract. The 5,5 projection is identical to the 3D rhombic triacontahedron.

Orthogonal projection wireframes of p-p duoprisms
Odd
3-3 5-5 7-7 9-9
[3] [6] [5] [10] [7] [14] [9] [18]
Even
4-4 (tesseract) 6-6 8-8 10-10
[4] [8] [6] [12] [8] [16] [10] [20]

The regular skew polyhedron, {4,4|n}, exists in 4-space as the n2 square faces of a n-n duoprism, using all 2n2 edges and n2 vertices. The 2n n-gonal faces can be seen as removed. (skew polyhedra can be seen in the same way by a n-m duoprism, but these are not regular.)

Duoantiprism

Like the antiprisms as alternated prisms, there is a set of 4-dimensional duoantiprisms: 4-polytopes that can be created by an alternation operation applied to a duoprism. 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.

The duoprisms , t0,1,2,3{p,2,q}, can be alternated into , ht0,1,2,3{p,2,q}, the "duoantiprisms", which cannot be made uniform in general. The only convex uniform solution is the trivial case of p=q=2, which is a lower symmetry construction of the tesseract , t0,1,2,3{2,2,2}, with its alternation as the 16-cell, , s{2}s{2}.

The only nonconvex uniform solution is p=5, q=5/3, ht0,1,2,3{5,2,5/3}, , constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra, known as the great duoantiprism (gudap).[1][2]

Ditetragoltriates

Also related are the ditetragoltriates or octagoltriates, formed by taking the octagon (considered to be a ditetragon or a truncated square) to a p-gon. The octagon of a p-gon can be clearly defined if one assumes that the octagon is the convex hull of two perpendicular rectangles; then the p-gonal ditetragoltriate is the convex hull of two p-p duoprisms (where the p-gons are similar but not congruent, having different sizes) in perpendicular orientations. The resulting polychoron is isogonal and has 2p p-gonal prisms and p2 rectangular trapezoprisms (a cube with D2d symmetry) but cannot be made uniform. The vertex figure is a triangular bipyramid.

Double antiprismoids

Like the duoantiprisms as alternated duoprisms, there is a set of p-gonal double antiprismoids created by alternating the 2p-gonal ditetragoltriates, creating p-gonal antiprisms and tetrahedra while reinterpreting the non-corealmic triangular bipyramidal spaces as two tetrahedra. The resulting figure is generally not uniform except for two cases: the grand antiprism and its conjugate, the pentagrammic double antiprismoid (with p = 5 and 5/3 respectively), represented as the alternation of a decagonal or decagrammic ditetragoltriate. The vertex figure is a variant of the dissected regular icosahedron.

k_22 polytopes

The 3-3 duoprism, -122, is first in a dimensional series of uniform polytopes, expressed by Coxeter as k22 series. The 3-3 duoprism is the vertex figure for the second, the birectified 5-simplex. The fourth figure is a Euclidean honeycomb, 222, and the final is a paracompact hyperbolic honeycomb, 322, with Coxeter group [32,2,3], . Each progressive uniform polytope is constructed from the previous as its vertex figure.

k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 E6 =E6+ =E6++
Coxeter
diagram
Symmetry [[3<sup>2,2,-1</sup>]] [[3<sup>2,2,0</sup>]] [[3<sup>2,2,1</sup>]] [[3<sup>2,2,2</sup>]] [[3<sup>2,2,3</sup>]]
Order 72 1440 103,680
Graph
Name 122 022 122 222 322

See also

Notes

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 0-486-40919-8 (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, 33-62, 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 Explainedcontains a description of duoprisms (double prisms) and duocylinders (double cylinders). Googlebook
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
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