# Duoprism

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 |
In geometry of 4 dimensions or higher, a double prism or 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 dimensions of 2 (polygon) or higher.
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:
    $P_{1}\times P_{2}=\{(x,y,z,w)|(x,y)\in P_{1},(z,w)\in P_{2}\}$
where P$_1$ and P$_2$ 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 |
| 3-4 | 4-4 |
| 3-5 | 4-5 | 5-5 |
| 3-6 | 4-6 | 5-6 | 6-6 |
| 3-7 | 4-7 | 5-7 | 6-7 | 7-7 |
| 3-8 | 4-8 | 5-8 | 6-8 | 7-8 | 8-8 |
| 3-9 | 4-9 | 5-9 | 6-9 | 7-9 | 8-9 | 9-9 |
| 3-10 | 4-10 | 5-10 | 6-10 | 7-10 | 8-10 | 9-10 | 10-10 |

### 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 A$_3$ 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 | 3-3 | 3-3 | 5-5 | 5-5 | 5-5 | 7-7 | 7-7 | 7-7 | 9-9 | 9-9 | 9-9 |
| [3] | [3] | [6] | [5] | [5] | [10] | [7] | [7] | [14] | [9] | [9] | [18] |
| Even ||||||||||||
| 4-4 (tesseract) | 4-4 (tesseract) | 4-4 (tesseract) | 6-6 | 6-6 | 6-6 | 8-8 | 8-8 | 8-8 | 10-10 | 10-10 | 10-10 |
| [4] | [4] | [8] | [6] | [6] | [12] | [8] | [8] | [16] | [10] | [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 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 , t$_{0,1,2,3}${p,2,q}, can be alternated into , ht$_{0,1,2,3}${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 , t$_{0,1,2,3}${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, ht$_{0,1,2,3}${5,2,5/3}, , constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra, known as the great duoantiprism (gudap).

### 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 sphenocorona.

### 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$^{2,2,3}$], ${\bar {T}}_{7}$. Each progressive uniform polytope is constructed from the previous as its vertex figure.
k$_{22}$ figures in n dimensions :
| Space | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 |
| Coxeter group | A$_2$A$_2$ | E$_6$ | ${\tilde {E}}_{6}$=E$_6$$^+$ | ${\bar {T}}_{7}$=E$_6$$^{++}$ |
| Coxeter diagram | | | | | |
| Symmetry | [[3$^{2,2,-1}$]] | [[3$^{2,2,0}$]] | [[3$^{2,2,1}$]] | [[3$^{2,2,2}$]] | [[3$^{2,2,3}$]] |
| Order | 72 | 1440 | 103,680 | ∞ | ∞ |
| Graph | | | | ∞ | ∞ |
| Name | −1$_{22}$ | 0$_{22}$ | 1$_{22}$ | 2$_{22}$ | 3$_{22}$ |