[Properties of] discrete spaces

Most work with surfaces done on computers—whether for computer graphics, computer-aided design, solving boundary value problems or otherwise—makes use of discrete approximations. Typically surfaces are represented by collections of patches—with a simple mesh of triangles often being used. The triangles are however normally specified not so much by their network of connections as by the explicit coordinates of their vertices. And while there are various triangulation methods that for example avoid triangles with small angles, no standard method yields networks analogous to the ones I consider in which all triangle edges are effectively the same length.

In pure mathematics a basic idea in topology has been to look for finite or discrete ways to capture essential features of continuous surfaces and spaces. And as an early part of this Henri Poincaré in the 1890s introduced the concept of approximating manifolds by cell complexes consisting of collections of generalized polyhedra. By the 1920s there was then extensive work on so-called combinatorial topology, in which spaces are thought of as being decomposed into abstract complexes consisting say of triangles, tetrahedra and higher-dimensional simplices. But while explicit coordinates and lengths are not usually discussed, it is still imagined that one knows more information than in the networks I consider: not only how vertices are connected by edges, but also how edges are arranged around faces, faces around volumes, and so on. And while in 2D and 3D it is possible to set up such an approximation to any manifold in this way, it turns out that at least in 5D and above it is not. Before the 1960s it had been hoped that in accordance with the Hauptvermutung of combinatorial topology it would be possible to tell whether a continuous mapping and thus topological equivalence exists between manifolds just by seeing whether subdivisions of simplicial complexes for them could be identical. But in the 1960s it was discovered that at least in 5D and above this will not always work. And largely as a result of this, there has tended to be less interest in ideas like simplicial complexes.

And indeed a crucial point for my discussion in the main text is that in formulating general relativity one actually does not appear to need all the structure of a simplicial complex. In fact, the only features of manifolds that ultimately seem relevant are ones that in appropriate limits are determined just from the connectivity of networks. The details of the limits are mathematically somewhat intricate (compare page 1030), but the basic approach is straightforward. One can find the volume of a sphere (geodesic ball) in a network just by counting the number of nodes out to a given network distance from a certain node. And from the limiting growth rate of this one can immediately get the Ricci scalar curvature—just as in the continuous case discussed above. To get the Ricci tensor one also needs a direction. But one can get this from a geodesic—which is in effect the analog of a straight line in the network. Note that unlike in a continuous space there is however usually no obvious way to continue a geodesic in a network. And in general, some—but not all—of the standard constructions used in continuous spaces can also immediately be used in networks. So for example it is straightforward to construct a triangle in a network: one just starts from a particular node, follows geodesics to two others, then joins these with a geodesic. But to extend the triangle into a parallelogram is not so easy—since there is no immediate notion of parallelism in the network. And this means that neither the Riemann tensor, nor a so-called Schild ladder for parallel transport, can readily be constructed.

Since the 1980s there has been increasing interest in formulating notions of continuous geometry for objects like Cayley graphs of groups—which are fundamentally discrete but have infinite limits analogous to continuous systems. (Compare page 938.)