# Prism

A **prism** is commonly defined as a polyhedron formed by extruding a "base" polygon into three dimensions, creating two copies of the base joined by rectangles. More generally, it may refer to the prism product (Cartesian product) of any *n*-polytope and a dyad (line segment), resulting in an (*n* + 1)-polytope. Three-dimensional prisms may be referred to more specifically as **polygonal prisms**.

Given a base polytope , the prism has twice the number of vertices of , and has facets comprising two copies of and the prisms of each of 's facets. For example, the hexagonal prism has two hexagons and six squares, the squares being the prisms of the base hexagon's edges. The dual of a prism is the bipyramid of the base's dual, at least abstractly.

## Specific cases[edit | edit source]

The prism product has many nice properties in the study of polytopes by symmetry:

- Starting with a point and repeatedly taking the prism operator produces the hypercubes, an infinite class of regular polytopes.
- The prisms of the regular polygons constitute an infinite class of uniform polyhedra. In general, the prism of any uniform polytope is also uniform.
- The prism of a vertex-transitive polytope is also vertex-transitive. The vertex figure of the prism is the pyramid of the vertex figure of the base.
- Similarly, scaliform and CRF polytopes will preserve these attributes in their prism.
- The prism of any orbiform polytope is a segmentotope.

If the base polytope has circumradius *r* and the height of the prism is *h*, the circumradius of the prism is given simply by √*r*^{2}+(*h*/2)^{2}, and its hypervolume is equal to *vh* where *v* is the hypervolume of the base.

If the prism's base is also a prism, then it can also be seen as a duoprism of a rectangle and the base of the base facet.

## External links[edit | edit source]

- FerrÃ©ol, Robert (2015). "Hyperprisme" (in French).
- Klitzing, Richard. "n/d-p".
- Wikipedia contributors. "Prism (geometry)".