Hyperbolic networks

Any surface that always has positive curvature must eventually close up to form something like a sphere. But a surface that has negative curvature (and no holes) must in some sense be infinite—more like cases (c) and (d) on page 412. Yet even in such a case one can always define coordinates that nominally allow the surface to be drawn in a finite way—and the Poincaré disk model used in the pictures below is the standard way of doing this. In ordinary flat space, regular polygons with more than 6 sides can never form a tessellation. But in a space with negative curvature this is possible for polygons with arbitrarily many sides—and the networks that result have been much studied as Cayley graphs of Fuchsian groups. One feature of these networks is that the number of nodes reached in them by following r connections always grows like 2^r. But if one intersperses hexagons in the networks (as in the main text) then one finds that for small r the number of nodes just grows like r²—as one would expect for something like a 2D surface. But if one tries to look at growth rates on scales that are not small compared to characteristic lengths associated with curvature then one again sees exponential growth—just as in the case of a uniform tessellation without hexagons.