  The main purpose of this article is to facilitate the implementation of
space-time finite element methods in four-dimensional space. In order to
develop a finite element method in this setting, it is necessary to create a
numerical foundation, or equivalently a numerical infrastructure. This
foundation should include a collection of suitable elements (usually
hypercubes, simplices, or closely related polytopes), numerical interpolation
procedures (usually orthonormal polynomial bases), and numerical integration
procedures (usually quadrature rules). It is well known that each of these
areas has yet to be fully explored, and in the present article, we attempt to
directly address this issue. We begin by developing a concrete, sequential
procedure for constructing generic four-dimensional elements (4-polytopes).
Thereafter, we review the key numerical properties of several canonical
elements: the tesseract, tetrahedral prism, and pentatope. Here, we provide
explicit expressions for orthonormal polynomial bases on these elements. Next,
we construct fully symmetric quadrature rules with positive weights that are
capable of exactly integrating high-degree polynomials, e.g. up to degree 17 on
the tesseract. Finally, the quadrature rules are successfully tested using a
set of canonical numerical experiments on polynomial and transcendental
functions.
