Duoprism

A duoprism is a 4-Dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. More precisely, it is the set of points:

:P_1 imes 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.

The duoprism is a convex 4-dimensional polytope bounded by prism_(geometry) cell_(mathematics).

=Geometry=

A duoprism created by the product of a regular n -sided polygon and a regular m -sided polygon is bounded by n m -gonal Prism_(geometry) 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 tetragonal prisms (Cube_(geometry)), and is identical to the hypercube.

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.

=Nomenclature=

The term duoprism is coined by George Olshevsky. It is a subset of the prismatic polychora. In Olshevsky s usage, a duoprism made of n -polygons and m -polygons is named by prefixing duoprism with the names of the base polygons, for example: the 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.

Another name for the duoprism is the double prism. However, this may be confused with other, unrelated uses of term, such as [http://scienceworld.wolfram.com/physics/FresnelsDoublePrism.html Fresnel s double prism].

=See also=

*Polytope and Polychoron *Convex regular polychoron *Duocylinder *Hypercube

=External links=

*The word Duoprism is also the name of an [http://www.theapplecollection.com/design/pcreleased/Duoprism.html LCD monitor]. It has no relation to the mathematical use of the term as described here.

=References=

*[http://members.aol.com/Polycell/names.html Nomenclature of Polychora] (the description of duoprisms is near the middle of the page) *[http://etext.lib.virginia.edu/etcbin/toccer-new2id=ManFour.sgm&images=images/modeng&data=/texts/english/modeng/parsed&tag=public&part=all The Fourth Dimension Simply Explained]—describes duoprisms as double prisms and duocylinders as double cylinders *[http://members.aol.com/Polycell/section6.html Catalogue of Convex Polychora, section 6] *[http://www.geocities.com/os2fan2/gloss.htm Polygloss] - glossary of higher-dimensional terms