4.2 Hyperbolic Space

Among all possible rules, the formation of geometrical shapes of the kind we have just been discussing is very rare. Slightly more common is the type of behavior that we see in a rule like:

Essentially the same behavior also occurs in a mixed-arity rule with a single relation on the left-hand side:

We can think of the structure that is produced as being like a binary tree of triangles:

The same structure can be produced from an Apollonian circle packing (e.g. [18][1: p985]):

If each triangle is required to have the same area, the structure can be rendered in 2D as:

If we tried to render this with every triangle roughly the same size, then even in 3D the best we could do would be to have something that crinkles up more and more at the edge, like an idealized lettuce leaf:

But just as we can think of the grids we discussed before as being regularly laid out in ordinary 2D or 3D space, so now we can think of the object we have here as being regularly laid out in a hyperbolic space [19][20] of constant negative curvature.

In particular, the object corresponds to an infinite-order triangular tiling of the hyperbolic plane (with Schläfli symbol {3,∞}). There are a variety of ways to visualize the hyperbolic plane. One example is the Poincaré disk model in which hyperbolic-space straight lines are rendered as arcs of circles orthogonal to the boundary:

(The particular graph here happens to be the Farey graph [21].)