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Finding Gold In The Forest …A Connection Between Fractal Trees, Topology, and The Golden Ratio

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Fractals First used by Benoit Mandelbrot to describe objects that are too irregular for classical geometry No fixed mathematical definition Typical characteristics: self-similarity, detail at arbitrary scales, simple recursive definition

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Fractal Dimension An important characteristic of a fractal The main tool for applications Self-similar fractals have a nice fractal dimension d given by N = (1/r) d where N is number of pieces, r is scaling factor, so d = ln N / ln r

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The Cantor Set Start with a unit interval, remove middle third interval, and continue to remove middle thirds of the subintervals Is self-similar and has a fractal dimension of ln 2/ ln 3

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Topology Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects Topology studies features of a space like connectivity or number of holes A topologist doesnt distinguish between a tea cup and a donut

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Homology Homology tries to distinguish between spaces by constructing algebraic and numerical invariants that reflect the connectivity of the spaces In general, the basic definitions are abstract and complicated For nice subsets of 2, the only non-trivial homology can be determined by counting holes

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Same Fractal Dimension, Different Topology

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Fractal Trees Compact, connected subsets that exhibit some kind of branching pattern There are different types of fractal trees Many natural systems can be modeled with fractal trees

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Rat Lung Model

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Retina Analysis

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Binary Fractal Trees Specified by four parameters: 2 branching angles 1 and 2,and two scaling ratios r 1 and r 2, denoted by T(r 1, r 2, 1, 2 ) Trunk (vertical line segment of unit length) splits into 2 branches, one with angle 1 with the trunk and length r 1, second with angle 2 and length r 2 Idea: each branch splits into 2 new branches following the same rule

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T(.5, 1, 240 º, 240 º ) First iteration of branching

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T(.5, 1, 240 º, 240 º ) Second iteration of branching

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T(.5, 1, 240 º, 240 º ) Third iteration of branching

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T(.5, 1, 240 º, 240 º )

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Symmetric Binary Fractal Trees T(r, ) denotes tree with scaling ratio r (some real number between 0 and 1) and branching angle (real-valued angle between 0 º and 180 º ) Trunk splits into 2 branches, each with length r, one to the right with angle and the other to the left with angle Level k approximation tree has k iterations of branching

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Some Algebra A symmetric binary tree can be seen as a representation of the free monoid with two generators Two generator maps m R and m L that act on compact subsets Addresses are finite or infinite strings with each element either R or L

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Examples T(.55, 40 º )

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Examples T(.6, 72 º )

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Examples T(.615, 115 º )

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Examples T(.52, 155 º )

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Self-Contact For a given branching angle, there is a unique scaling ratio such that the corresponding symmetric binary tree is self-contacting. We denote this ratio by r sc. This ratio can be determined for any symmetric binary tree. If r < r sc, then the tree is self-avoiding. If r > r sc, then the tree is self-overlapping.

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Overlapping Tree

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Self-Contacting Trees The branching angles 90° and 135° are considered to be topological critical points, one reason being that the corresponding self-contacting trees are the only ones that are space-filling All other self-contacting trees have infinitely many generators for the first homology group

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All self-avoiding trees are topologically equivalent

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Topology and Fractal Trees? At first, topology doesnt seem very useful for studying fractal trees- the topology is either trivial or too complicated Idea: study topological and geometrical aspects of a tree along with spaces derived from a tree What derived spaces?

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Closed ε-Neighbourhoods For a set X that is a subset of some metric space M with metric d, the closed ε- neighbourhood of X is X ε = { x | d(x, X) ε }

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Example

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Closed ε-Neighbourhoods of Trees The closed ε-neighbourhoods, as ε ranges over the non-negative real numbers, endow a tree with much additional interesting structure They are a function of r, θ, and ε What features do we study?

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Holes of Closed ε-Neighbourhoods Number Persistence Complexity Level Symmetry Location Type

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Persistence The range of ε-values that a hole class persists over.

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Levels The level of a subtree is related to the branch that forms its trunk Level k hole is related to level k subtree Every hole is the image of a level 0 hole

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Location Where are the holes?

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Critical Values Critical set of ε-values for (r,θ) based on persistence Critical values of r for a given θ, based on complexity Critical values of θ, based on location Different relations give different classifications of the trees that focus on different aspects

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Specific Trees It is possible for a closed ε- neighbourhood to have infinitely many holes for non-zero value of ε T(r sc, 67.5°)

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Specific Trees It is often not straightforward to determine exact critical ε-values for a given tree, but they are not always necessary- sometimes estimates are good enough T(r sc, 120°)

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What is the self-contacting scaling ratio for the branching angle 120°? It must satisfy 1-r sc -r sc 2 =0 Thus r sc = (-1 + 5)/2

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The Golden Rectangle

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The Golden Ratio The Golden Ratio Φ is the number such that 1/Φ = (Φ-1)/1 Thus Φ = (1 + 5)/2 1.618033988749… and 1/Φ = (-1 + 5)/2 = Φ - 1

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The Golden Ratio Many people, including the ancient Greeks and Egyptians, find Φ to be the most aesthetically pleasing ratio

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The Golden Ratio Φ can be considered the most irrational number because it has a continued fraction representation Φ = [1,1,1,…] Φ can be expressed as a nested radical

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The Golden Ratio Φ is related to the Fibonacci numbers F 1 = F 2 = 1 and F n = F n-2 + F n-1

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The Golden Trees Four self-contacting trees have scaling ratio 1/Φ Each of these trees possesses extra symmetry, they seem to line up The four angles are 60°, 108°, 120° and 144°

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Golden 60

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Golden 108

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Golden 120

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Golden 144

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