1. Fractals

2. What is Fractal? Not agreed upon the primary definition
Self-similar object
Statistically scale-invariant
Fractal dimension
Recursive algorithmic descriptions
latine word fractus = irregular/fragmented
term Procedural Modeling is sometimes misplaced with Fractals

3. Fractals Around Us

4. Fractals Inside Us

5. Fractal Flora

6. Fractal Weather

7. Artificial Fractal Shapes

8. Fractal Images

9. Fractal Patterns
M. C. Escher: Smaller and Smaller

10. 1883: Cantor Set Cantor set in 1D: 2D: Cantor Dust Cantor Discontinuum
bounded uncontinuous uncountable set
2D: Cantor Dust
Georg Cantor

11. 1890: Peano Curve
Space filling
Order lines  curve

12. 1891: Hilbert Curve

13. 1904: Koch Snowflake
Helge von Koch

14. 1916: Sierpinski Gasket

15. Analogy: Sierpinski Carpet
“remove squares until nothing remains”

16. 1918: Julia Set 1st fractal in complex plane
Originally not intended to be visualized

17. 1926: Menger Sponge
Contains every 1D object
(inc. K3,3, K5)

18. 1975: History Breakthrough
Benoit Mandelbrot: Les objets fractals, forn, hasard et dimension, 1975
Fractal definition
Legendary Mandelbrot Set

19. 2003: Fractals Nowadays Fractal image / sound compression
Fractal music
Fractal antennas …

20. Knowledge Sources B. Mandelbrot: The fractal geometry of nature, 1982
M. Barnsley: Fractals Everywhere, 1988
Contemporary web sources:
Google yields over results on “fractal”

21. Coastal Length Smaller the scale, longer the coast Where is the limit?
USA shoreline at 30m details:
km!

22. Fractal Dimension More definitions Self-similarity dimension
N = number of transformations
r = scaling coefficient
Koch Curve example
N = 4, r--1 = 3
Dimension = log 4 / log 3 = 1.26…

23. Fractal Taxonomy Deterministic fractals Stochastic fractals
Linear (IFS, L-systems,…)
Non linear (Mandelbrot set, bifurcation diagrams,…)
Stochastic fractals
Fractal Brovnian Motion (fBM)
Diffusion Limited Aggregation (DLA)
L-Systems …

24. Example: Deterministic Fractal
Square: rotate, scale, copy
90%
10%

25. Example: Deterministic Fractal

26. Example: Deterministic Fractal

27. Contractive Transformations
Copy machine association
Fractal – specified as a set of contractive transformations
Attractor = fix point

28. Example: Sierpinski Gasket

29. Iterated Function Systems
IFS = set of contractive affine transformations
Iterated process:
First copy
Second copy
Attractor
Affine transformation ~

30. Sierpinsky Gasket IFS

31. Barnsley’s Fern IFS

32. Barnsley’s Fern

33. Reality Versus Fractal

34. IFS Computation Deterministic: Stochastic (Chaos Game algorithm):
Apply transformations to the object until infinitum
Stochastic (Chaos Game algorithm):
Choose random transformation fi
Transform a point using fi
Repeat until infinitum

35. IFS examples
Dragon Curve

36. Lorenz Attractor Edward Norton Lorenz, 1963
IFS made from weather forecasting
Butterfly effect in dynamic system

37. Midpoint Displacement
Stochastic 1D fractal
Break the line
Shift its midpoint a little

38. Midpoint in 2D Basic shape = triangle / square
Square: Diamond algorithm

39. Diamond Algorithm

40. Diamond Algorithm

41. Diamond Algorithm

42. Diamond Algorithm

43. Fractal Terrain

44. Diamond Algorithm Applications
Terrains
Landscapes
Textures
Clouds