# 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 p2 squares,2p p-gons Edges 2p2 Vertices p2 Symmetry [[p,2,p]] = [2p,2+,2p], order 8p2 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:

$P_{1}\times P_{2}=\{(x,y,z,w)|(x,y)\in P_{1},(z,w)\in P_{2}\}$ where P1 and P2 are the sets of the points contained in the respective polygons. 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 diagramProjection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown. netThe 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.

6-prism 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 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
       
Even
4-4 (tesseract) 6-6 8-8 10-10
       

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).

### 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], ${\bar {T}}_{7}$ . 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 ${\tilde {E}}_{6}$ =E6+ ${\bar {T}}_{7}$ =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

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