Finite subdivision rule
In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals.
Cannon's conjecture 
Here, a geometric action is a cocompact, properly discontinuous action by isometries. This conjecture was partially solved by Grigori Perelman in his proof of the Geometrization conjecture, which states (in part) than any Gromov hyperbolic group that is a 3-manifold group must act geometrically on hyperbolic 3-space. However, it still remains to show that a Gromov hyperbolic group with a 2-sphere at infinity is a 3-manifold group.
Cannon and Swenson showed  that a hyperbolic group with a 2-sphere at infinity has an associated subdivision rule. If this subdivision rule is conformal in a certain sense, the group will be a 3-manifold group with the geometry of hyperbolic 3-space.
Rigorous definition 
A finite subdivision rule consists of the following.
1. A finite 2-dimensional CW complex , called the subdivision complex, with a fixed cell structure such that is the union of its closed 2-cells. We assume that for each closed 2-cell of there is a CW structure on a closed 2-disk such that has at least two vertices, the vertices and edges of are contained in , and the characteristic map which maps onto restricts to a homeomorphism onto each open cell.
2. A finite two dimensional CW complex , which is a subdivision of .
3.A continuous cellular map called the subdivision map, whose restriction to every open cell is a homeomorphism.
Each CW complex in the definition above (with its given characteristic map ) is called a tile type.
An -complex for a subdivision rule is a 2-dimensional CW complex which is the union of its closed 2-cells, together with a continuous cellular map whose restriction to each open cell is a homeomorphism. We can subdivide into a complex by requiring that the induced map restricts to a homeomorphism onto each open cell. is again an -complex with map . By repeating this process, we obtain a sequence of subdivided -complexes with maps .
Binary subdivision is one example:
The subdivision complex can be created by gluing together the opposite edges of the square, making the subdivision complex into a torus. The subdivision map is the doubling map on the torus, wrapping the meridian around itself twice and the longitude around itself twice. This is a four-fold covering map. The plane, tiled by squares, is a subdivision complex for this subdivision rule, with the structure map given by the standard covering map. Under subdivision, each square in the plane gets subdivided into squares of one-fourth the size.
Combinatorial Riemann Mapping Theorem 
Subdivision rules give a sequence of tilings of a surface, and tilings give an idea of distance, length, and area (by letting each tile have length and area 1). In the limit, the distances that come from these tilings may converge in some sense to an analytic structure on the surface. The Combinatorial Riemann Mapping Theorem gives necessary and sufficient conditions for this to occur.
Its statement needs some background. A tiling of a ring (i.e., a closed annulus) gives two invariants, and , called approximate moduli. These are similar to the classical modulus of a ring. They are defined by the use of weight functions. A weight function assigns a non-negative number called a weight to each tile of . Every path in can be given a length, defined to be the sum of the weights of all tiles in the path. Define the height of under to be the infimum of the length of all possible paths connecting the inner boundary of to the outer boundary. The circumference of under is the infimum of the length of all possible paths circling the ring (i.e. not nullhomotopic in R). The area of under is defined to be the sum of the squares of all weights in . Then define
Note that they are invariant under scaling of the metric.
A sequence of tilings is conformal () if mesh approaches 0 and:
- For each ring , the approximate moduli and , for all sufficiently large, lie in a single interval of the form ; and
- Given a point in the surface, a neighborhood of , and an integer , there is a ring in separating x from the complement of , such that for all large the approximate moduli of are all greater than .
Statement of theorem 
If a sequence of tilings of a surface is conformal () in the above sense, then there is a conformal structure on the surface and a constant depending only on in which the classical moduli and approximate moduli (from for sufficiently large) of any given annulus are -comparable, meaning that they lie in a single interval .
The Combinatorial Riemann Mapping Theorem implies that a group acts geometrically on if and only if it is Gromov hyperbolic, it has a sphere at infinity, and the natural subdivision rule on the sphere gives rise to a sequence of tilings that is conformal in the sense above. Thus, Cannon's conjecture would be true if all such subdivision rules were conformal.
Examples of finite subdivision rules 
Barycentric subdivision is an example of a subdivision rule with one edge type (that gets subdivided into two edges) and one tile type (a triangle that gets subdivided into 6 smaller triangles). Any triangulated surface is a barycentric subdivision complex.
Every prime, non-split alternating knot or link complement an associated canonical finite subdivision rules, with tiles that do not subdivide, corresponding to the boundary of the link complement. Two interesting examples are the trefoil knot complement's subdivision rule:
And the Borromean rings complements' subdivision rule:
In each case, we can the first tile type to be our subdivision complex . Then we get cell structures that are more and more refined, for the trefoil complement:
And for the Borromean rings complement:
There are also finite subdivision rules for all closed hyperbolic 3-manifolds created by gluing together right-angled hyperbolic polyhedra.
Applications to biology 
The ideas of combinatorial conformal geometry that underlie Cannon's proof of the "combinatorial Riemann mapping theorem", were applied by Cannon, Floyd and Parry (2000) to the study of large-scale growth patterns of biological organisms. Cannon, Floyd and Parry produced a mathematical growth model which demonstrated that some systems determined by simple finite subdivision rules can results in objects (in their example, a tree trunk) whose large-scale form oscillates wildly over time even though the local subdivision laws remain the same. Cannon, Floyd and Parry also applied their model to the analysis of the growth patterns of rat tissue. They suggested that the "negatively curved" (or non-euclidean) nature of microscopic growth patterns of biological organisms is one of the key reasons why large-scale organisms do not look like crystals or polyhedral shapes but in fact in many cases resemble self-similar fractals. In particular they suggested (see section 3.4 of ) that such "negatively curved" local structure is manifested in highly folded and highly connected nature of the brain and the lung tissue.
Applications in other areas 
- Brian Rushton. Alternating links and subdivision rules., Master's thesis, Brigham Young University, 2009. Introduction
- James W. Cannon. The combinatorial Riemann mapping theorem. Acta Mathematica 173 (1994), no. 2, pp. 155–234.
- Perelman, Grisha (11 November 2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159 [math.DG].
- Perelman, Grisha (10 March 2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109 [math.DG].
- Perelman, Grisha (17 July 2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245 [math.DG].
- J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension 3. Transactions of the American Mathematical Society 350 (1998), no. 2, pp. 809–849.
- J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196.
- J. W. Cannon, W. Floyd and W. Parry. Crystal growth, biological cell growth and geometry. Pattern Formation in Biology, Vision and Dynamics, pp. 65–82. World Scientific, 2000. ISBN 981-02-3792-8,ISBN 978-981-02-3792-9.
- J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 11 (2007), pp. 128–136.
- B. Rushton. Constructing subdivision rules from alternating links. Conform. Geom. Dyn. 14 (2010), 1–13.
- B. Rushton. Creating subdivision rules from alternating links. Alg. Geom. and Top. 12 (2012) 1961–1992
- P.J. Lu and et al. Decagonal and quasi-crystalline tilings in medieval islamic architecture. Science, 315:1106&ndash1110, 2007
- D. Zorin. Subdivisions on arbitrary meshes: algorithms and theory. Institute of Mathematical Sciences (Singapore) Lecture Notes Series. 2006.
- Bill Floyd's research page. This page contains most of the research papers by Cannon, Floyd and Parry on subdivision rules, as well as a gallery of subdivision rules.
- Brian Rushton's research page. This page contains a gallery of subdivision rules of 3-manifolds and n-cubes.