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In mathematics, a finite subdivision rule is a recursive way of dividing a
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ... or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric
fractals In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ... . 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. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of
hyperbolic manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
s.
Substitution tiling In geometry, a tile substitution is a method for constructing highly ordered Tessellation, tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational ...
s are a well-studied type of subdivision rule.

# Definition

A subdivision rule takes a
tiling Tiling may refer to: *The physical act of laying tile Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, Rock (geology), stone, metal, baked clay, or even glass. They are generally ...
of the plane by polygons and turns it into a new tiling by subdividing each polygon into smaller polygons. It is finite if there are only finitely many ways that every polygon can subdivide. Each way of subdividing a tile is called a tile type. Each tile type is represented by a label (usually a letter). Every tile type subdivides into smaller tile types. Each edge also gets subdivided according to finitely many edge types. Finite subdivision rules can only subdivide tilings that are made up of polygons labelled by tile types. Such tilings are called subdivision complexes for the subdivision rule. Given any subdivision complex for a subdivision rule, we can subdivide it over and over again to get a sequence of tilings. For instance, binary subdivision has one tile type and one edge type: Since the only tile type is a quadrilateral, binary subdivision can only subdivide tilings made up of quadrilaterals. This means that the only subdivision complexes are tilings by quadrilaterals. The tiling can be
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger song), "Regular" (Badfinger song) * Regular tunin ...
, but doesn't have to be: Here we start with a complex made of four quadrilaterals and subdivide it twice. All quadrilaterals are type A tiles.

# Examples of finite subdivision rules

Barycentric subdivision In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ... 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.J. W. Cannon, W. J. Floyd, W. R. Parry.
Finite subdivision rules
'. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196.
The
Penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any fin ... can be generated by a subdivision rule on a set of four tile types (the curved lines in the table below only help to show how the tiles fit together): Certain rational maps give rise to finite subdivision rules.J. W. Cannon, W. J. Floyd, W. R. Parry.
Constructing subdivision rules from rational maps
'. Conformal Geometry and Dynamics, vol. 11 (2007), pp. 128–136.
This includes most Lattès maps.J. W. Cannon, W. J. Floyd, W. R. Parry. ''Lattès maps and subdivision rules''. Conformal Geometry and Dynamics, vol. 14 (2010, pp. 113–140. Every prime, non-split alternating knot or link complement has a subdivision rule, with some tiles that do not subdivide, corresponding to the boundary of the link complement.B. Rushton.
Constructing subdivision rules from alternating links
'. Conform. Geom. Dyn. 14 (2010), 1–13.
The subdivision rules show what the night sky would look like to someone living in a
knot complement In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
; because the universe wraps around itself (i.e. is not
simply connected In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
), an observer would see the visible universe repeat itself in an infinite pattern. The subdivision rule describes that pattern. The subdivision rule looks different for different geometries. This is a subdivision rule for the
trefoil knot In knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in t ... , which is not a
hyperbolic knot Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they m ...
: And this is the subdivision rule for the
Borromean rings In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, which is hyperbolic: In each case, the subdivision rule would act on some tiling of a sphere (i.e. the night sky), but it is easier to just draw a small part of the night sky, corresponding to a single tile being repeatedly subdivided. This is what happens for the trefoil knot: And for the Borromean rings: # Subdivision rules in higher dimensions

Subdivision rules can easily be generalized to other dimensions. For instance,
barycentric subdivision In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ... is used in all dimensions. Also, binary subdivision can be generalized to other dimensions (where
hypercube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ... s get divided by every midplane), as in the proof of the
Heine–Borel theorem In real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fo ...
.

# Rigorous definition A finite subdivision rule $R$ consists of the following. 1. A finite 2-dimensional
CW complex A CW complex is a kind of a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantiti ...
$S_R$, called the subdivision complex, with a fixed cell structure such that $S_R$ is the union of its closed 2-cells. We assume that for each closed 2-cell $\tilde$ of $S_R$ there is a CW structure $s$ on a closed 2-disk such that $s$ has at least two vertices, the vertices and edges of $s$ are contained in $\partial s$, and the characteristic map $\psi_s:s\rightarrow S_R$ which maps onto $\tilde$ restricts to a homeomorphism onto each open cell. 2. A finite two dimensional CW complex $R\left(S_R\right)$, which is a subdivision of $S_R$. 3.A continuous cellular map $\phi_R:R\left(S_R\right)\rightarrow S_R$ called the subdivision map, whose restriction to every open cell is a homeomorphism onto an open cell. Each CW complex $s$ in the definition above (with its given characteristic map $\psi_s$) is called a tile type. An $R$-complex for a subdivision rule $R$ is a 2-dimensional CW complex $X$ which is the union of its closed 2-cells, together with a continuous cellular map $f:X\rightarrow S_R$ whose restriction to each open cell is a homeomorphism. We can subdivide $X$ into a complex $R\left(X\right)$ by requiring that the induced map $f:R\left(X\right)\rightarrow R\left(S_R\right)$ restricts to a homeomorphism onto each open cell. $R\left(X\right)$ is again an $R$-complex with map $\phi_R \circ f:R\left(X\right)\rightarrow S_R$. By repeating this process, we obtain a sequence of subdivided $R$-complexes $R^n\left(X\right)$ with maps $\phi_R^n\circ f:R^n\left(X\right)\rightarrow S_R$. Binary subdivision is one example: The subdivision complex can be created by gluing together the opposite edges of the square, making the subdivision complex $S_R$ into a torus. The subdivision map $\phi$ 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 $f:\mathbb^2\rightarrow R\left(S_R\right)$ given by the standard covering map. Under subdivision, each square in the plane gets subdivided into squares of one-fourth the size.

# Quasi-isometry properties Subdivision rules can be used to study the quasi-isometry properties of certain spaces. Given a subdivision rule $R$ and subdivision complex $X$, we can construct a Graph (discrete mathematics), graph called the history graph that records the action of the subdivision rule. The graph consists of the dual graphs of every stage $R^n\left(X\right)$, together with edges connecting each tile in $R^n\left(X\right)$ with its subdivisions in $R^\left(X\right)$. The quasi-isometry properties of the history graph can be studied using subdivision rules. For instance, the history graph is quasi-isometric to hyperbolic space exactly when the subdivision rule is conformal, as described in the combinatorial Riemann mapping theorem.

# Applications

Islamic Girih tiles in Islamic architecture are self-similar tilings that can be modeled with finite subdivision rules. In 2007, Peter Lu, Peter J. Lu of Harvard University and Professor Paul Steinhardt, Paul J. Steinhardt of Princeton University published a paper in the journal ''Science'' suggesting that girih tilings possessed properties consistent with self-similar fractal quasicrystalline tilings such as
Penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any fin ... s (presentation 1974, predecessor works starting in about 1964) predating them by five centuries.
Subdivision surfaces in computer graphics use subdivision rules to refine a surface to any given level of precision. These subdivision surfaces (such as the Catmull-Clark subdivision surface) take a polygon mesh (the kind used in 3D animated movies) and refines it to a mesh with more polygons by adding and shifting points according to different recursive formulas.D. Zorin.
Subdivisions on arbitrary meshes: algorithms and theory
'. Institute of Mathematical Sciences (Singapore) Lecture Notes Series. 2006.
Although many points get shifted in this process, each new mesh is combinatorially a subdivision of the old mesh (meaning that for every edge and vertex of the old mesh, you can identify a corresponding edge and vertex in the new one, plus several more edges and vertices). Subdivision rules were applied by Cannon, Floyd and Parry (2000) to the study of large-scale growth patterns of biological organisms.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. , .
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 that such "negatively curved" local structure is manifested in highly folded and highly connected nature of the brain and the lung tissue.

# Cannon's conjecture

James W. Cannon, Cannon, William Floyd (mathematician), Floyd, and Walter Parry, Parry first studied finite subdivision rules in an attempt to prove the following conjecture: Cannon's conjecture: Every Mikhail Leonidovich Gromov, Gromov hyperbolic group with a 2-sphere at infinity Geometric group action, acts geometrically on hyperbolic space, hyperbolic 3-space.James W. Cannon
''The combinatorial Riemann mapping theorem''.
Acta Mathematica 173 (1994), no. 2, pp. 155–234.
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 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. 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.

# 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 Riemann surface, 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 $T$ of a ring $R$ (i.e., a closed annulus) gives two invariants, $M_ \left(R,T\right)$ and $m_ \left(R,T\right)$, called Extremal length#Discrete extremal length, approximate moduli. These are similar to the classical Extremal length#extremal distance in an annulus, modulus of a ring. They are defined by the use of weight functions. A weight function $\rho$ assigns a non-negative number called a weight to each tile of $T$. Every path in $R$ can be given a length, defined to be the sum of the weights of all tiles in the path. Define the height $H\left(\rho\right)$ of $R$ under $\rho$ to be the infimum of the length of all possible paths connecting the inner boundary of $R$ to the outer boundary. The circumference $C\left(\rho\right)$ of $R$ under $\rho$ is the infimum of the length of all possible paths circling the ring (i.e. not nullhomotopic in R). The area$A\left(\rho\right)$ of $R$ under $\rho$ is defined to be the sum of the squares of all weights in $R$. Then define : $M_ \left(R,T\right)=\sup \frac,$ : $m_ \left(R,T\right)=\inf \frac.$ Note that they are invariant under scaling of the metric. A sequence $T_1,T_2,\ldots$ of tilings is conformal ($K$) if mesh approaches 0 and: # For each ring $R$, the approximate moduli $M_\left(R,T_i\right)$ and $m_\left(R,T_i\right)$, for all $i$ sufficiently large, lie in a single interval of the form $\left[r,Kr\right]$; and # Given a point $x$ in the surface, a neighborhood $N$ of $x$, and an integer $I$, there is a ring $R$ in $N\smallsetminus\$ separating ''x'' from the complement of $N$, such that for all large $i$ the approximate moduli of $R$ are all greater than $I$.

## Statement of theorem

If a sequence $T_1,T_2,\ldots$ of tilings of a surface is conformal ($K$) in the above sense, then there is a conformal structure on the surface and a constant $K\text{'}$ depending only on $K$ in which the classical moduli and approximate moduli (from $T_i$ for $i$ sufficiently large) of any given annulus are $K\text{'}$-comparable, meaning that they lie in a single interval $\left[r,K\text{'}r\right]$.

## Consequences

The Combinatorial Riemann Mapping Theorem implies that a group $G$ acts geometrically on $\mathbb^3$ 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.