finding gold in the forest …a connection between fractal trees, topology, and the golden ratio

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