Proprism

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 if all its product polytopes are convex.

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 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 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 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}⁴, a tesseract duo-prism {4,3,3}², and regular 8-cube {4,3,3,3,3,3,3}.

In geometry of 4 dimensions or higher, duoprism is a polytope resulting from the Cartesian product 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) 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.[2]

The Cartesian product of two polygons is the set of points:

${\displaystyle P_{1}\times P_{2}=\{(x,y,u,v)|(x,y)\in P_{1},(u,v)\in P_{2}\}}$

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 symbols, {3} × {3}, and is composed of 9 vertices.

The tesseract, can be constructed as the duoprism {4} × {4}, the product of two equal-size orthogonal squares, composed of 16 vertices. The 5-cube can be constructed as a duoprism {4} × {4,3}, the product of a square and cube, while the 6-cube can be constructed as the product of two cubes, {4,3} × {4,3}.

The prism {3} × {3} × {3} can be seen in orthogonal projection within a regular enneagon.

In geometry of 6 dimensions or higher, a triple product is a polytope resulting from the Cartesian product 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 (polygon) or higher.

The lowest-dimensional forms are 6-polytopes being the Cartesian product of three polygons. The smallest can be written as {3} × {3} × {3} in Schläfli symbols if they are regular, and contains 27 vertices. This is the product of three equilateral triangles and is a uniform polytope. The f-vectors can be computed by (3,3,1)³ = (27,81,108,81,36,9,1).

The 6-cube, can be constructed as a triple product {4} × {4} × {4}. The f-vectors can be computed by (4,4,1)³ = (64,192,240,160,60,12,1).