Modelling and Simulation 2008

1. Modelling and Simulation 2008 A brief introduction to self-similar fractals

2. Outline Motivation: - examples of self-similarity Fractal objects: - iterative construction of geometrical fractals - self-similarity and scale invariance Fractal dimension: - conventional vs. fractal dimension - a working definition - the box-counting method

3. Self-similarity in nature identical/similar structures repeat over a wide range of length scales

4. Self-similarity in nature

5. Self-similarity in art mosaic from the cathedral of Anagni / Italy

6. Self-similarity in computer graphics an artificial, fractal landscape

7. Self-similarity in physics Diffusion limited aggregation Clusters of Pt atoms

8. Self-similar time series medicine: further examples: heart beat intervals economy (e.g. stock market) weather/climate seismic activity chaotic systems random walks Heart beat intervals time beat number

9. Fractal objects: iterative construction The Sierpinsky construction ∙initialization: one filled triangle ∙ iteration step: remove an upside-down triangle from the center of every filled triangle ∙repeat the step ... ( 3 ) ( 2 ) ( 1 )

10. Fractal objects: iterative construction The fractal is defined in the mathematical limit of infinitely many iterations ( ∞ ) ( 8 )

11. Fractal objects: properties (a) self-similarity ∙ exactly the same structures repeat all over the fractal zoom in and rescale

12. Fractal objects: properties (a) self-similarity ∙exactly the same structures repeat all over the fractal zoom in and rescale

13. Fractal objects: properties (b) scale invariance: ∙there is no typical … … size of objects … length scale Sierpinsky: contains triangles of all possible sizes apart from “practical” limitations: - size of the entire object - finite number of iterations (“resolution”)

14. Scale invariance 1m

15. D=1 D=2 Fractal vs. integer dimension Embedding dimensiond in a d-dimensional space: d numbers specify a point y D=0 Example: d=2 Dimension (of an object) D x in a d-dimensional space, all objects have a dimension D ≤ d

16. area A b·s b2·A Fractal vs. integer dimension intuitive: length, area, volume rescale by a factor b length s

17. b2·A b1·s Fractal vs. integer dimension intuitive: length, area, volume rescale by a factor b length s D area A

18. dimension D of aspect A(Q) Fractal vs. integer dimension working definition of dimension D: • object Q, embedded in a d-dimensional space • measure aspect A(Q), e.g. perimeter, area, volume,… • compare results A(Q) = A1 in the original space A(Q) = Ab after rescaling all d directions by b

19. b=2 Fractal vs. integer dimension aspect: black area “more than a line – less than an area”

20. Fractal vs. integer dimension another (famous) example: Koch islands ∙ initialization: 3 lines forming a triangle ∙iteration: replace every straight line by a, e.g. a spike first iteration:

21. Fractal vs. integer dimension Koch island:

22. Fractal vs. integer dimension Koch island: length s scale by factor b=3 length 4 s

23. Summary ∙ introduction: self-similar objects ∙ construction of example fractals: - the Sierpinsky construction - Koch islands ∙ qualitative properties of fractal objects: - self-similarity - scale invariance ∙ quantitative characterization of fractals: - fractal dimension (vs. integer dimension) - working definition / measurement

24. Problems Problems with the working definition • we measure, e.g.,the black area in the Sierpinsky • fractal, only to conclude that it has no area !? • implicitly we make use of the construction scheme, • what about “observed” fractals like the following ?

25. Stochastic fractals repeating structures of equal statistical properties length scale ?

26. Measuring fractal dimension Box-counting: resolution-dependent measurement of D ∙ cover the object by boxes of size ∊ ∙ count non-empty boxes ∙ repeat for many ∊ < ∊ >

27. Measuring fractal dimension box-counting: resolution-dependent measurement ∙ cover the object by boxes of size ∊ ∙ count non-empty boxes ∙ repeat for many ∊ <∊>

28. Measuring fractal dimension box-counting: resolution-dependent measurement ∙ cover the object by boxes of size ∊ ∙ count non-empty boxes ∙ repeat for many ∊ ∙ consider the number n of non-empty boxes as a function of ∊ (in the limit ∊→0)

29. Measuring fractal dimension n ~ ( 1/∊ ) D ( as ∊→0 ) obtain D from D = log(n) /log(1/∊) integer dimensional objects? as the grid gets finer (∊→0), the shape is more accurately approximated and we obtain n → A/∊2 i.e. D=2 area A

30. Sierpinsky revisited suitable shape of boxes ?

31. ∊ n 1 1 Sierpinsky revisited

32. ∊ n 1 1 Sierpinsky revisited 1/2 3

33. ∊ n 1 1 Sierpinsky revisited 1/2 3 1/4 9

34. ∊ n k 0 1 1 1 2 3 1/∊ =2 k n =3 k k log(3) k log(2) D= Sierpinsky revisited n ~ (1/∊)D 1/2 3 1/4 9 1/8 27

35. Remarks / Outlook in practice: linear regression ln(n) vs. ln(1/∊) for a range of box sizes - • Box-counting is only one method for estimating D, • widely applicable, but costly to realize • important alternatives: Sandbox-method • correlation functions • in deterministic self-similar fractals, all these • methods yield the same D • for “real world fractals”, results can differ significantly • further topics: self-affine fractals, multi-fractals

36. Outlook • Diffusion Limited Aggregation • simple, random growth process • model of various real world processes • yields self-similar aggregates with 1 < D <2 • quantitative study in terms of fractal dimension