# Pyramid (EntityClass, 5)

### From Higher Dimensions Database

A **pyramid** is a shape which has been constructed by tapering. All three-dimensional pyramids are self-dual, but this does not hold in higher dimensions (prime example: cubic pyramid).

When forming a pyramid from a pyramid, 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 pyramid from a pyramid is called a *dipyramid*, and the shape formed by forming a pyramid from a dipyramid is called a *tripyramid*, and so on. Thus, a *k*-pyramid is the shape formed by tapering a base shape to a point in a new dimension *k* times.

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

H_{n}(&A) =H_{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 pyramid.

The first term in the RHS of this equation corresponds to the elements of the base that form the "base" of the pyramid, and the second term corresponds to the elements that get tapered (and hence are also pyramids) and form the "sides" of the pyramid.

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