# Prism (EntityClass, 5)

### From Higher Dimensions Database

A **prism** is a shape which has been constructed by extrusion.

When forming a prism from a prism, the shape formed is symmetrical in the way that the two new dimensions cannot be told apart: the order does not matter. For this reason, the shape formed by forming a prism from a prism is called a *diprism*, and the shape formed by forming a prism from a diprism is called a *triprism*, and so on. Thus, a *k*-prism is the shape formed by extruding a base shape into a new dimension *k* times.

There is a simple relationship between the numbers of elements of a prism, and the numbers of elements of its base:

H_{n}(+A) = 2H_{n}(A) +H_{n-1}(A)

where *H*_{n}(X) is the number of *n*-dimensional elements of the shape X, A is the base shape and +A is the prism.

The first term in the RHS of this equation corresponds to the elements of the base that get duplicated and form the "ends" of the prism, and the second term corresponds to the elements that get extruded (and hence are also prisms) and form the "sides" of the prism.

Notice that the only difference from the equation for pyramids is the coefficient of the first term.