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Fractals

Fractals. 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.

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Fractals

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  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. Georg Cantor 1883: Cantor Set • Cantor set in 1D: • Cantor Discontinuum • bounded uncontinuous uncountableset • 2D: Cantor Dust

  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: • http://math.fullerton.edu/mathews/c2003/FractalBib/Links/FractalBib_lnk_1.html • Google yields over 1 000 000 results on “fractal”

  21. Coastal Length • Smaller the scale, longer the coast • Where is the limit? • USA shoreline at 30m details: 143 000 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 • 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: • 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

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