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Finding Gold In The Forest
…A Connection Between Fractal Trees, Topology, and
The Golden Ratio
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
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
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
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 doesn’t distinguish between a tea cup and a donut
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
Same Fractal Dimension, Different Topology
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
Rat Lung Model
Retina Analysis
Binary Fractal Trees
• Specified by four parameters: 2 branching angles 1 and 2,and two scaling ratios r1 and r2, denoted by T(r1, r2, 1, 2)
• Trunk (vertical line segment of unit length) splits into 2 branches, one with angle 1 with the trunk and length r1, second with angle 2 and length r2
• Idea: each branch splits into 2 new branches following the same rule
T(.5, 1, 240º, 240º)
• First iteration of branching
T(.5, 1, 240º, 240º)
• Second iteration of branching
T(.5, 1, 240º, 240º)
• Third iteration of branching
T(.5, 1, 240º, 240º)
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 0º and 180º)
• 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
Some Algebra
• A symmetric binary tree can be seen as a representation of the free monoid with two generators
• Two generator maps mR and mL that act on compact subsets
• Addresses are finite or infinite strings with each element either R or L
Examples
• T(.55, 40º)
Examples
• T(.6, 72º)
Examples
• T(.615, 115º)
Examples
• T(.52, 155º)
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 rsc. This ratio can be determined for any symmetric binary tree.
If r < rsc, then the tree is self-avoiding.
If r > rsc, then the tree is self-overlapping.
Overlapping Tree
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
All self-avoiding trees are topologically equivalent
All self-avoiding trees are topologically equivalent
Topology and Fractal Trees?
• At first, topology doesn’t 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?
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
Xε= { x | d(x, X) ≤ ε }
Example
Example
Example
Example
Example
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?
Holes of Closed ε-Neighbourhoods
• Number• Persistence• Complexity• Level• Symmetry• Location• Type
Persistence
The range of ε-values that a hole class persists over.
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
Location
Where are the holes?
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
Specific Trees
• It is possible for a closed ε-neighbourhood to have infinitely many holes for non-zero value of ε
T(rsc, 67.5°)
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(rsc, 120°)
T(rsc, 120°)
• What is the self-contacting scaling ratio for the branching angle 120°?
• It must satisfy
1-rsc-rsc2=0
Thus
rsc= (-1 + √5)/2
The Golden Rectangle
The Golden Ratio
• The Golden Ratio Φ is the number such that
1/Φ = (Φ-1)/1Thus
Φ = (1 + √5)/2 ≈ 1.618033988749…and
1/Φ = (-1 + √5)/2 = Φ - 1
The Golden Ratio
Many people, including the ancient Greeks and Egyptians, find Φ to be the most aesthetically pleasing ratio
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
The Golden Ratio
• Φ is related to the Fibonacci numbers
F1 = F2 = 1
and
Fn = Fn-2 + Fn-1
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°
Golden 60
Golden 108
Golden 120
Golden 144