Pyramid product

The pyramid product is an operation that can be applied on any two polytopes. It is considered to be an extension of the traditional sense of a pyramid, which refers to a polytope formed by adding a vertex perpendicular to the base polytope's hyperplane. The pyramid product of two polytopes A and B can be called an A-B disphenoid and its dimension is equivalent to the sum of the dimensions of A and B plus one. Hence, the coordinates of A and B lie on enitrely different hyperplanes, but with the addition of another dimension which determines its height. For example, the coordinates of a triangular-square disphenoid with height h can be given as (assuming both bases have edge length 1):


 * (0, $\sqrt{3}$/3, 0, 0, h/2),
 * (±1/2, -$\sqrt{3}$/6, 0, 0, h/2),
 * (0, 0, ±1/2, ±1/2, -h/2).

Pyramid products are closely related to tegum products, but are differentiated mainly by the existence of an additional dimension. Generally, a pyramid product is isogonal if both bases are isogonal and congruent to each other, isotopic if both bases are isotopic and congruent to each other, or noble if both bases are noble and congruent to each other. There are exceptions though, such as the regular pentachoron, which can be constructed as the pyramid product of a dyad and a triangle.

Like the prism product and tegum product, the pyramid product is both commutative and associative.