# 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.

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}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.

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.

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.

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.

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

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.

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.

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.

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.

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.