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.

Set of uniform $p-q$ duoprisms
Type	Prismatic uniform 4-polytopes
Schläfli symbol	${p}\times{q}$
Coxeter-Dynkin diagram
Cells	$p q$ -gonal prisms, $q p$ -gonal prisms
Faces	$pq$ squares, $p q$ -gons, $q p$ -gons
Edges	$2 pq$
Vertices	$pq$
Vertex figure	disphenoid
Symmetry	$[p,2, q]$ , order $4 pq$
Dual	$p-q$ duopyramid
Properties	convex, vertex-uniform

Set of uniform p-p duoprisms
Type	Prismatic uniform 4-polytope
Schläfli symbol	${p}\times{p}$
Coxeter-Dynkin diagram
Cells	$2 p p$ -gonal prisms
Faces	$p 2$ squares, $2 p p$ -gons
Edges	$2 p 2$
Vertices	$p 2$
Symmetry	$[p,2, p] = [2 p,2 +,2 p],$ order $8 p 2$
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:

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.

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