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In 1961 Tullio Regge provided us with a beautiful lattice representation ofEinstein's geometric theory of gravity. This Regge Calculus (RC) is strikinglydifferent from the more usual finite difference and finite elementdiscretizations of gravity. In RC the fundamental principles of GeneralRelativity are applied directly to a tessellated spacetime geometry. In thismanuscript, and in the spirit of this conference, we reexamine the foundationsof RC and emphasize the central role that the Voronoi and Delaunay latticesplay in this discrete theory. In particular we describe, for the first time, ageometric construction of the scalar curvature invariant at a vertex. Thisderivation makes use of a new fundamental lattice cell built from elementsinherited from both the simplicial (Delaunay) spacetime and its circumcentricdual (Voronoi) lattice. The orthogonality properties between these two latticesyield an expression for the vertex-based scalar curvature which is strikinglysimilar to the corresponding and more familiar hinge-based expression in RC(deficit angle per unit Voronoi dual area). In particular, we show that thescalar curvature is simply a vertex-based weighted average of deficits perweighted average of dual areas. What is most striking to us is how naturallyspacetime is represented by Voronoi and Delaunay structures and that the lawsof gravity appear to be encoded locally on the lattice spacetime with lesscomplexity than in the continuum, yet the continuum is recovered by convergencein mean. Perhaps these prominent features may enable us to transcend thedetails of any particular discrete model gravitation and yield clues to help usdiscover how we may begin to quantize this fundamental interaction.

A Discrete Representation of Einstein's Geometric Theory of Gravitation: The Fundamental Role of Dual Tessellations in Regge Calculus