  In 1961 Tullio Regge provided us with a beautiful lattice representation of
Einstein's geometric theory of gravity. This Regge Calculus (RC) is strikingly
different from the more usual finite difference and finite element
discretizations of gravity. In RC the fundamental principles of General
Relativity are applied directly to a tessellated spacetime geometry. In this
manuscript, and in the spirit of this conference, we reexamine the foundations
of RC and emphasize the central role that the Voronoi and Delaunay lattices
play in this discrete theory. In particular we describe, for the first time, a
geometric construction of the scalar curvature invariant at a vertex. This
derivation makes use of a new fundamental lattice cell built from elements
inherited from both the simplicial (Delaunay) spacetime and its circumcentric
dual (Voronoi) lattice. The orthogonality properties between these two lattices
yield an expression for the vertex-based scalar curvature which is strikingly
similar to the corresponding and more familiar hinge-based expression in RC
(deficit angle per unit Voronoi dual area). In particular, we show that the
scalar curvature is simply a vertex-based weighted average of deficits per
weighted average of dual areas. What is most striking to us is how naturally
spacetime is represented by Voronoi and Delaunay structures and that the laws
of gravity appear to be encoded locally on the lattice spacetime with less
complexity than in the continuum, yet the continuum is recovered by convergence
in mean. Perhaps these prominent features may enable us to transcend the
details of any particular discrete model gravitation and yield clues to help us
discover how we may begin to quantize this fundamental interaction.
