Finite Subdivision Rules

We introduce and study finite subdivision rules. A finite subdivision rule is a finite list of instructions which determines a subdivision of a given planar tiling. Given a finite subdivision rule and a planar tiling associated to it, we obtain an infinite sequence of tilings by recursively subdividing the given tiling. We wish to determine when this sequence of tilings is conformal in the sense of Cannon’s combinatorial Riemann mapping theorem. In this setting, it is proved that the two axioms of conformality can be repaced by a single axiom which is implied by either of them, and that it suffices to check conformality for finitely many test annuli. Theorems are given which show how to exploit symmetry, and many examples are computed. This paper is concerned with recursive subdivisions of planar complexes. As an introductory example, we present a finite subdivision rule in Figure 1. There are two kinds of edges and three kinds of tiles. A thin edge is subdivided into five subedges (three of these are thick and two are thin) and a thick edge is subdivided into three subedges (two of these are thick and one is thin). There are three kinds of tiles: a triangle with thin edges; a quadrilateral with a pair of opposite thin edges and a pair of opposite thick edges; and a pentagon with thick edges. Each tile is subdivided into subtiles of those three kinds, and these subdivisions restrict on the boundary arcs to the subdivisions defined for the edges. Because of this, one can recursively subdivide planar complexes made up of tiles of these three kinds. For example, Figure 2 shows the second and third subdivisions of the quadrilateral tile type. This figure was produced by Kenneth Stephenson’s computer program CirclePack. Our motivation for the above subdivision rule is illustrated in Figure 3. This figure, which was drawn from an image produced by Jeffrey Weeks’s computer program SnapPea, shows a right-angled dodecahedron D in the Klein model of hyperbolic 3-space. The geodesic planes through the twelve faces intersect the sphere at infinity in twelve thick (red) circles. The group G generated by the reflections in these twelve geodesic planes is a discrete subgroup of Isom(H), and D is a fundamental domain for the action of G on H. The images of D under elements of G form a tiling T of H. Define combinatorial balls B(n), n ≥ 0, recursively by B(0) = D and B(n+1) = star(B(n), T ) for n ≥ 0. For each n, the geodesic planes through the faces of ∂B(n) intersect the sphere in circles. The thin (blue) circles in Figure 3 are the circles at infinity corresponding to the faces of ∂B(1). For each n, let S(n) be the tiling of the 2-sphere determined by the circles coming from faces of ∂B(k) for k ≤ n. The tiles of S(n) are triangles, Date: September 15,1999. 1991 Mathematics Subject Classification. Primary 20F32, 52C20; Secondary 05B45.