With the exception of the hyperpyramidal numbers (which include the pyramidal numbers and the centered pyramidal numbers, i.e. the (centered polygons) pyramidal numbers, as 3-dimensional hyperpyramidal numbers,) all the figurate numbers considered are regular polytope numbers corresponding to convex regular polytopes in a d-dimensional Euclidean space in , d ≥ 0.
Among the hyperpyramidal numbers, the d-dimensional square hyperpyramidal numbers, although not regular polytopes, are of particular interest since they are building blocks for the construction of the hyperoctahedral numbers (orthoplex numbers), which are regular polytopes. For example, the nth tetrahedron is the nth square dipyramid, i.e. it is the adjunction of the nth square pyramid to the (n-1)th square pyramid joined at their square bases.
Otherwise, considering nonconvex regular (e.g. stellated) polytopic numbers or considering nonregular (e.g. Archimedean solids) polytopic numbers would open the door to a humongous number of possibilities...