# 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 | |

Type | Prismatic uniform 4-polytopes |

Schläfli symbol | {p}×{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 |

Symmetry | [p,2,q], order 4pq |

Dual | p-q duopyramid |

Properties | convex, vertex-uniform |

Set of uniform p-p duoprisms | |

Type | Prismatic uniform 4-polytope |

Schläfli symbol | {p}×{p} |

Coxeter-Dynkin diagram | |

Cells | 2p p-gonal prisms |

Faces | p^{2} squares,2p p-gons |

Edges | 2p^{2} |

Vertices | p^{2} |

Symmetry | [[p,2,p]] = [2p,2^{+},2p], order 8p^{2} |

Dual | p-p duopyramid |

Properties | convex, 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 *P _{1}* and

*P*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.

_{2}## 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 2*n*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.

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.

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 A^{3} Coxeter plane of the tesseract. The 5,5 projection is identical to the 3D rhombic triacontahedron.

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] |

## Related polytopes

The regular skew polyhedron, {4,4|n}, exists in 4-space as the n^{2} square faces of a *n-n duoprism*, using all 2n^{2} edges and n^{2} vertices. The 2*n* *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 _{0,1,2,3}{p,2,q}, can be alternated into _{0,1,2,3}{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 _{0,1,2,3}{2,2,2}, with its alternation as the 16-cell,

The only nonconvex uniform solution is p=5, q=5/3, ht_{0,1,2,3}{5,2,5/3},

### 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 p^{2} rectangular trapezoprisms (a cube with *D _{2d}* 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, -1_{22}, is first in a dimensional series of uniform polytopes, expressed by Coxeter as k_{22} series. The 3-3 duoprism is the vertex figure for the second, the birectified 5-simplex. The fourth figure is a Euclidean honeycomb, 2_{22}, and the final is a paracompact hyperbolic honeycomb, 3_{22}, with Coxeter group [3^{2,2,3}],
. Each progressive uniform polytope is constructed from the previous as its vertex figure.

Space | Finite | Euclidean | Hyperbolic | ||
---|---|---|---|---|---|

n | 4 | 5 | 6 | 7 | 8 |

Coxeter group |
A_{2}A_{2} |
E_{6} |
=E_{6}^{+} |
=E_{6}^{++} | |

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 | −1_{22} |
0_{22} |
1_{22} |
2_{22} |
3_{22} |

## See also

## Notes

- Jonathan Bowers - Miscellaneous Uniform Polychora 965. Gudap
- http://www.polychora.com/12GudapsMovie.gif Animation of cross sections

## 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.

- Coxeter, H. S. M.
*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). 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