  The main purpose of this article is to develop a novel refinement strategy
for four-dimensional hybrid meshes based on cubic pyramids. This optimal
refinement strategy subdivides a given cubic pyramid into a conforming set of
congruent cubic pyramids and invariant bipentatopes. The theoretical properties
of the refinement strategy are rigorously analyzed and evaluated. In addition,
a new class of fully symmetric quadrature rules with positive weights are
generated for the cubic pyramid. These rules are capable of exactly integrating
polynomials with degrees up to 12. Their effectiveness is successfully
demonstrated on polynomial and transcendental functions. Broadly speaking, the
refinement strategy and quadrature rules in this paper open new avenues for
four-dimensional hybrid meshing, and space-time finite element methods.
