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