Introduction

When Regge formulated the first discrete version of general relativity in 1961, one of his motivations was to set up a numerical scheme for solving Einstein's equations for general systems without a large amount of symmetry. The hope was that the formulation would also provide ways of representing complicated topologies and of visualising the resulting geometries. Regge calculus, as it has come to be known, has not only been used in large scale numerical calculations in classical general relativity but has also provided a basis for attempts at formulating a theory of Quantum Gravity.

The central idea in Regge calculus is to consider spaces with curvature concentrated on codimension-two subspaces, rather than with continuously distributed curvature. This is achieved by constructing spaces from flat blocks glued together on matching faces. The standard example in two dimensions is a geodesic dome, where a network of flat triangles approximates part of a sphere. The curvature resides at the vertices, and the deficit angle, given by 2π minus the sum of the vertex angles of the triangles at that point, gives a measure of it. In general dimension n, flat n-simplices meet on flat (n − 1)-dimensional faces and the curvature is concentrated on the (n − 2)-dimensional subsimplices or hinges. The deficit angle at a hinge is given by 2π minus the sum of the dihedral angles of the simplices meeting at that hinge.