  This paper addresses two problems needed to support four-dimensional ($3d +
t$) spacetime numerical simulations. The first contribution is a general
algorithm for producing conforming spacetime meshes of moving geometries. Here,
the surface points of the geometry are embedded in a four-dimensional space as
the geometry moves in time. The geometry is first tessellated at prescribed
time steps and then these tessellations are connected in the parameter space of
each geometry entity to form tetrahedra. In contrast to previous work, this
approach allows the resolution of the geometry to be controlled at each time
step. The only restriction on the algorithm is the requirement that no
topological changes to the geometry are made (i.e. the hierarchical relations
between all geometry entities are maintained) as the geometry moves in time.
The validity of the final mesh topology is verified by ensuring the
tetrahedralizations represent a closed 3-manifold. For some analytic problems,
the $4d$ volume of the tetrahedralization is also verified. The second problem
addressed in this paper is the design of a system to interactively visualize
four-dimensional meshes, including tetrahedra (embedded in $4d$) and
pentatopes. Algorithms that either include or exclude a geometry shader are
described, and the efficiency of each approach is then compared. Overall, the
results suggest that visualizing tetrahedra (either those bounding the domain,
or extracted from a pentatopal mesh) using a geometry shader achieves the
highest frame rate, in the range of $20-30$ frames per second for meshes with
about $50$ million tetrahedra.
