geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

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 and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

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. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...

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 Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...

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 The symmetry number or symmetry order of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object, that is, it is the order of its symmetry group. The object can be a molecule, crystal lattice ...

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.

f-vectors

An f-vector is a number of ''k''-face elements in a polytope from ''k''=0 (points) to ''k''=''n''−1 (facets). An extended f-vector can also include ''k''=−1 (nullitope), or ''k''=''n'' (body). Prism products include the body element. (The dual to prism products includes the nullitope, while pyramid products include both.) The f-vector of prism product, A×B, can be computed as (f_A,1)*(f_B,1), like polynomial multiplication polynomial coefficients. For example for product of a triangle, f=(3,3), and dion, f=(2) makes a

triangular prism In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...

with 6 vertices, 9 edges, and 5 faces: :f_A(x) = (3,3,1) = 3 + 3x + x² (triangle) :f_B(x) = (2,1) = 2 + x (dion) :f_A∨B(x) = f_A(x) * f_B(x) ::= (3 + 3x + x²) * (2 + x) :: = 6 + 9x + 5x² + x³ :: = (6,9,5,1)

Hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

f-vectors can be computed as Cartesian products of ''n'' dions, ⁿ. Each has f=(2), extended to f=(2,1). For example, an

8-cube In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces. It is represented b ...

will have extended f-vector power product: f=(2,1)⁸ = (4,4,1)⁴ = (16,32,24,8,1)² = (256,1024,1792,1792,1120,448,112,16,1). If equal lengths, this doubling represents ⁸, a square tetra-prism ⁴, a tesseract duo-prism ², and regular 8-cube .

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 made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

) 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 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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. Descriptions The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons. In the case of 3-3 duopr ...

, 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 List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

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

tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...

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

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s, composed of 16 vertices. The

5-cube In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts ...

can be constructed as a duoprism × , the product of a square and cube, while the

6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It ...

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-polytope In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. Definition A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. ...

s 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 An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

s and is a

. The f-vectors can be computed by (3,3,1)³ = (27,81,108,81,36,9,1). The

, can be constructed as a triple product × × . The f-vectors can be computed by (4,4,1)³ = (64,192,240,160,60,12,1).

References

{{reflist Multi-dimensional geometry Polytopes