# Proprism > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Proprism > Markdown URL: https://mediated.wiki/source/Proprism.md > Source: https://en.wikipedia.org/wiki/Proprism > Source revision: 1285820554 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) The [{3}×{3} duoprism](/source/3-3_duoprism) is a proprism as the product of two orthogonal triangles, having 9 squares between pairs of edges of 2 sets of 3 triangles, and 18 vertices, as seen in this skew orthogonal projection. In [geometry](/source/Geometry) of 4 dimensions or higher, a **proprism** is a [polytope](/source/Polytope) resulting from the [Cartesian product](/source/Cartesian_product) of two or more polytopes, each of two dimensions or higher. The term was coined by [John Horton Conway](/source/John_Horton_Conway) 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 polytopes](/source/Uniform_polytope).[1] ## Properties The number of [vertices](/source/Vertex_(geometry)) in a proprism is equal to the product of the number of vertices in all the polytopes in the product. The minimum [symmetry order](/source/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](/source/Convex_polytope) if all its product polytopes are convex. ## f-vectors An [f-vector](/source/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 (fA,**1**)*(fB,**1**), like [polynomial multiplication](/source/Polynomial_multiplication) polynomial coefficients. For example for product of a triangle, f=(3,3), and dion, f=(2) makes a [triangular prism](/source/Triangular_prism) with 6 vertices, 9 edges, and 5 faces: - fA(x) = (3,3,**1**) = 3 + 3x + x2 (triangle) - fB(x) = (2,**1**) = 2 + x (dion) - fA∨B(x) = fA(x) * fB(x) - = (3 + 3x + x2) * (2 + x) - = 6 + 9x + 5x2 + x3 - = (6,9,5,**1**) [Hypercube](/source/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](/source/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}. ## Double products or duoprisms Further information: [Duoprism](/source/Duoprism) In geometry of 4 dimensions or higher, **duoprism** is a [polytope](/source/Polytope) resulting from the [Cartesian product](/source/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](/source/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](/source/Four-dimensional_space) called these **double prisms**.[2] The [Cartesian product](/source/Cartesian_product) of two [polygons](/source/Polygon) is the [set](/source/Set_(mathematics)) of points: - P 1 × P 2 = { ( x , y , u , v ) | ( x , y ) ∈ P 1 , ( u , v ) ∈ P 2 } {\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](/source/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](/source/Schl%C3%A4fli_symbol), {3} × {3}, and is composed of 9 vertices. The [tesseract](/source/Tesseract), can be constructed as the duoprism {4} × {4}, the product of two equal-size orthogonal [squares](/source/Square), composed of 16 vertices. The [5-cube](/source/5-cube) can be constructed as a duoprism {4} × {4,3}, the product of a square and cube, while the [6-cube](/source/6-cube) can be constructed as the product of two cubes, {4,3} × {4,3}. ## Triple products 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](/source/Polytope) resulting from the [Cartesian product](/source/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](/source/Polygon)) or higher. The lowest-dimensional forms are [6-polytopes](/source/6-polytope) being the [Cartesian product](/source/Cartesian_product) of three [polygons](/source/Polygon). The smallest can be written as {3} × {3} × {3} in [Schläfli symbols](/source/Schl%C3%A4fli_symbol) if they are regular, and contains 27 vertices. This is the product of three [equilateral triangles](/source/Equilateral_triangle) and is a [uniform polytope](/source/Uniform_polytope). The f-vectors can be computed by (3,3,**1**)3 = (27,81,108,81,36,9,**1**). The [6-cube](/source/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**). ## References 1. **[^](#cite_ref-1)** [John H. Conway](/source/John_Horton_Conway), Heidi Burgiel, Chaim Goodman-Strauss, *The Symmetries of Things* 2008, [ISBN](/source/ISBN_(identifier)) [978-1-56881-220-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-56881-220-5) (Chapter 26, p. 391 "proprism") 1. **[^](#cite_ref-2)** *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](https://web.archive.org/web/20030121092141/http://etext.lib.virginia.edu/etcbin/toccer-new2?id=ManFour.sgm&images=images%2Fmodeng&data=%2Ftexts%2Fenglish%2Fmodeng%2Fparsed&tag=public&part=all)—contains a description of duoprisms (double prisms) and duocylinders (double cylinders). [Googlebook](https://books.google.com/books?id=Y7cEAAAAMAAJ&q=The+Fourth+Dimension+Simply+Explained) --- Adapted from the Wikipedia article [Proprism](https://en.wikipedia.org/wiki/Proprism) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Proprism?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.