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

2. Michael Biehl, Modelling and Simulation 2008/09 2 Fractal objects: - iterative construction of geometrical fractals - self-similarity and scale invariance Outline Fractal dimension: - conventional vs. fractal dimension - a working definition - the box-counting method Motivation: - examples of self-similarity

3. Michael Biehl, Modelling and Simulation 2008/09 3 Self-similarity in nature identical/similar structures repeat over a wide range of length scales

4. Michael Biehl, Modelling and Simulation 2008/09 4 Self-similarity in nature

5. Michael Biehl, Modelling and Simulation 2008/09 5 mosaic from the cathedral of Anagni / Italy Self-similarity in art

6. Michael Biehl, Modelling and Simulation 2008/09 6 an artificial, fractal landscape Self-similarity in computer graphics

7. Michael Biehl, Modelling and Simulation 2008/09 7 Self-similarity in physics Clusters of Pt atoms Diffusion limited aggregation

8. Michael Biehl, Modelling and Simulation 2008/09 8 Heart beat intervals Self-similar time series heart beat intervals time beat number medicine: further examples: economy (e.g. stock market) weather/climate seismic activity chaotic systems random walks

9. Michael Biehl, Modelling and Simulation 2008/09 9 Fractal objects: iterative construction ∙ initialization: one filled triangle The Sierpinsky construction remove an upside-down triangle from the center of every filled triangle ∙ iteration step: ( 1 ) ∙ repeat the step... ( 2 ) ( 3 )

10. Michael Biehl, Modelling and Simulation 2008/09 10 The fractal is defined in the mathematical limit of infinitely many iterations Fractal objects: iterative construction ( 8 )( ∞ )

11. Michael Biehl, Modelling and Simulation 2008/09 11 Fractal objects: properties (a) self-similarity ∙ exactly the same structures repeat all over the fractal zoom in and rescale

12. Michael Biehl, Modelling and Simulation 2008/09 12 Fractal objects: properties (a) self-similarity ∙ exactly the same structures repeat all over the fractal zoom in and rescale

13. Michael Biehl, Modelling and Simulation 2008/09 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. Michael Biehl, Modelling and Simulation 2008/09 14 Scale invariance 1m

15. Michael Biehl, Modelling and Simulation 2008/09 15 Fractal vs. integer dimension Embedding dimension d in a d-dimensional space: d numbers specify a point x y Dimension (of an object) D in a d-dimensional space, all objects have a dimension D ≤ d Example: d=2 D=1 D=2 D=0

16. Michael Biehl, Modelling and Simulation 2008/09 16 intuitive: length, area, volume rescale by a factor b length s Fractal vs. integer dimension b ·s b ·s b2·A b2·A area A

17. Michael Biehl, Modelling and Simulation 2008/09 17 intuitive: length, area, volume rescale by a factor b length s b2·A b2·A area A Fractal vs. integer dimension b1·s b1·s D

18. Michael Biehl, Modelling and Simulation 2008/09 18 working definition of dimension D: Fractal vs. integer dimension - object Q, embedded in a d-dimensional space - measure aspect A(Q), e.g. perimeter, area, volume,… A(Q) = A 1 in the original space A(Q) = A b after rescaling all d directions by b - compare results dimension D of aspect A(Q)

19. Michael Biehl, Modelling and Simulation 2008/09 19 Fractal vs. integer dimension b=2 aspect: black area “more than a line – less than an area”

20. Michael Biehl, Modelling and Simulation 2008/09 20 Fractal vs. integer dimension ∙ initialization: 3 lines forming a triangle another (famous) example: Koch islands ∙ iteration: replace every straight line by a, e.g. a spike first iteration:

21. Michael Biehl, Modelling and Simulation 2008/09 21 Koch island: Fractal vs. integer dimension

22. Michael Biehl, Modelling and Simulation 2008/09 22 Koch island: scale by factor b=3 length s length 4 s Fractal vs. integer dimension

23. Michael Biehl, Modelling and Simulation 2008/09 23 Summary ∙ qualitative properties of fractal objects: - self-similarity - scale invariance ∙ construction of example fractals: - the Sierpinsky construction - Koch islands ∙ quantitative characterization of fractals: - fractal dimension (vs. integer dimension) - working definition / measurement ∙ introduction: self-similar objects

24. Michael Biehl, Modelling and Simulation 2008/09 24 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 ? Problems

25. Michael Biehl, Modelling and Simulation 2008/09 25 Stochastic fractals repeating structures of equal statistical properties length scale ?

26. Michael Biehl, Modelling and Simulation 2008/09 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. Michael Biehl, Modelling and Simulation 2008/09 27 Measuring fractal dimension ∙ cover the object by boxes of size ∊ <∊><∊> ∙ count non-empty boxes ∙ repeat for many ∊ box-counting: resolution-dependent measurement

28. Michael Biehl, Modelling and Simulation 2008/09 28 Measuring fractal dimension ∙ cover the object by boxes of size ∊ ∙ count non-empty boxes ∙ repeat for many ∊ box-counting: resolution-dependent measurement ∙ consider the number n of non-empty boxes as a function of ∊ (in the limit ∊→0)

29. Michael Biehl, Modelling and Simulation 2008/09 29 n ~ ( 1/∊ ) D ( as ∊→0 ) obtain D from 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 D = log(n) /log(1/∊) Measuring fractal dimension area A

30. Michael Biehl, Modelling and Simulation 2008/09 30 Sierpinsky revisited suitable shape of boxes ?

31. Michael Biehl, Modelling and Simulation 2008/09 31 1 1 ∊n Sierpinsky revisited

32. Michael Biehl, Modelling and Simulation 2008/09 32 1 1 ∊n 1/2 3 Sierpinsky revisited

33. Michael Biehl, Modelling and Simulation 2008/09 33 1 1 ∊n 1/2 3 1/4 9 Sierpinsky revisited

34. Michael Biehl, Modelling and Simulation 2008/09 34 1 1 ∊n 1/2 3 1/4 9 1/8 27 k 0 1 2 3 1/∊ =2 k n =3 k k log(3) k log(2) D= Sierpinsky revisited n ~ (1/∊) D

35. Michael Biehl, Modelling and Simulation 2008/09 35 - Box-counting is only one method for estimating D, widely applicable, but costly to realize - important alternatives: Sandbox-method correlation functions Remarks / Outlook - 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 - in practice: linear regression ln(n) vs. ln(1/∊) for a range of box sizes

36. Michael Biehl, Modelling and Simulation 2008/09 36 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 Outlook