Duoprism

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

×

P

2

=

{

(

x

,

y

,

z

,

w

)

|

(

x

,

y

)

∈

P

1

,

(

z

,

w

)

∈

P

2

}

{\displaystyle 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.