Finite subdivision rules

Abstract

Abstract. We introduce and study finite subdivision rules. A finite subdivision rule R consists of a finite 2-dimensional CW complex S R , a subdivision R(S R ) of S R , and a continuous cellular map ϕ R : R(S R ) → S R whose restriction to each open cell is a homeomorphism. If R is a finite subdivision rule, X is a 2-dimensional CW complex, and f : X → S R is a continuous cellular map whose restriction to each open cell is a homeomorphism, then we can recursively subdivide X to obtain an infinite sequence of tilings. 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 replaced 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. IntroductionThis 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 [16].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 [18], 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 3 ), and D is a fundamental domain for the action of G on H 3 .

Constructing subdivision rules from polyhedra with identifications

“…We then explain how to convert these replacement rules into finite subdivision rules in a large number of cases. As an example of our method, the tilings in Figures 1 and 2 correspond to the same orbifold that Cannon, Floyd and Parry studied in [3]. The tiling in their paper is different from the one we obtain, but strongly related.…”

“…One approach, adopted by Cannon, Floyd, and Parry, among others, is to study subdivision rules [3]. Subdivision rules are recursive formulas for refining coverings or tilings of a surface.…”

“…These proved difficult to study, so Cannon, Floyd and Parry proposed a simpler class of subdivision rules: finite subdivision rules [3]. A finite subdivision rule is a way of dividing a 2-dimensional CW-complex (usually the sphere) into a labeled tiling or cell structure, together with a finite recursive formula for dividing each tile into smaller tiles.…”

“…The most famous examples include barycentric subdivision and hexagonal refinement, where a triangle is chopped up into smaller triangles, which are chopped up into smaller tiles, etc. To remove all ambiguity, we give a concrete definition of a finite subdivision rule, taken from [3]. (1) A finite 2-dimensional CW-complex S R , called the subdivision complex, with a fixed cell structure such that S R is the union of its closed 2-cells.…”

Auto-Generation System Based on Fractal Geometry for Batik Pattern Design

“…Finite subdivision rules-use a recursive topological algorithm for refining tiling [19] and are similar to the process of cell division [20].…”