Finding Gold In The Forest

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

2. 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

3. 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

4. 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

5. 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

6. 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

7. Same Fractal Dimension, Different Topology

8. 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

10. Rat Lung Model

11. Retina Analysis

12. 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

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

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

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

16. T(.5, 1, 240º, 240º)

17. 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

18. 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

19. Examples • T(.55, 40º)

20. Examples • T(.6, 72º)

21. Examples • T(.615, 115º)

22. Examples • T(.52, 155º)

23. 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.

24. Overlapping Tree

25. 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

26. All self-avoiding trees are topologically equivalent

27. All self-avoiding trees are topologically equivalent

28. 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?

29. 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 }

30. Example

31. Example

32. Example

33. Example

34. Example

35. 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?

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

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

38. 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

39. Location Where are the holes?

40. 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

41. 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°)

42. 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°)

43. 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

44. The Golden Rectangle

45. 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

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

48. 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

49. The Golden Ratio • Φ is related to the Fibonacci numbers F1 = F2 = 1 and Fn = Fn-2 + Fn-1

50. 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°