geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

of 4 dimensions or higher, a proprism is a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

resulting from the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...

of two or more polytopes, each of two dimensions or higher. The term was coined by

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branc ...

for ''product prism''. The dimension of the space of a proprism equals the sum of the dimensions of all its product elements. Proprisms are often seen as ''k''-face elements of

uniform polytope In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude ver ...

Properties

The number of vertices in a proprism is equal to the product of the number of vertices in all the polytopes in the product. The minimum symmetry order of a proprism is the product of the symmetry orders of all the polytopes. A higher symmetry order is possible if polytopes in the product are identical. A proprism is

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

if all its product polytopes are convex.

Double products or duoprisms

In geometry of 4 dimensions or higher, duoprism is a

resulting from the

of two polytopes, each of two dimensions or higher. The Cartesian product of an ''a''-polytope, a ''b''-polytope is an ''(a+b)''-polytope, where ''a'' and ''b'' are 2-polytopes (

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...

) or higher. Most commonly this refers to the product of two polygons in 4-dimensions. In the context of a product of polygons, Henry P. Manning's 1910 work explaining the

fourth dimension Fourth dimension may refer to: Science * Time in physics, the continued progress of existence and events * Four-dimensional space, the concept of a fourth spatial dimension * Spacetime, the unification of time and space as a four-dimensional con ...

called these double prisms.''The Fourth Dimension Simply Explained'', Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online
The Fourth Dimension Simply Explained
mdash;contains a description of duoprisms (double prisms) and duocylinders (double cylinders)
Googlebook
/ref> The

of two

s is the set of points: :

P_1 \times P_2 = \

where ''P₁'' and ''P₂'' are the sets of the points contained in the respective polygons. The smallest is a 3-3 duoprism, made as the product of 2 triangles. If the triangles are regular it can be written as a product of

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...

s, × , and is composed of 9 vertices. The

tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of ei ...

, can be constructed as the duoprism × , the product of two equal-size orthogonal

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

s, composed of 16 vertices. The 5-cube can be constructed as a duoprism × , the product of a square and cube, while the 6-cube can be constructed as the product of two cubes, × .

Triple products

In geometry of 6 dimensions or higher, a triple product is a

resulting from the

of three polytopes, each of two dimensions or higher. The Cartesian product of an ''a''-polytope, a ''b''-polytope, and a ''c''-polytope is an (''a'' + ''b'' + ''c'')-polytope, where ''a'', ''b'' and ''c'' are 2-polytopes (

) or higher. The lowest-dimensional forms are 6-polytopes being the

of three

s. The smallest can be written as × × in

s if they are regular, and contains 27 vertices. This is the product of three

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

s and is a

. The 6-cube, can be constructed as a triple product × × .

References

{{reflist Multi-dimensional geometry Polytopes