450 likes | 661 Views
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.
E N D
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
Fractal Patterns M. C. Escher: Smaller and Smaller
Georg Cantor 1883: Cantor Set • Cantor set in 1D: • Cantor Discontinuum • bounded uncontinuous uncountableset • 2D: Cantor Dust
1890: Peano Curve • Space filling • Order lines curve
1904: Koch Snowflake Helge von Koch
Analogy: Sierpinski Carpet “remove squares until nothing remains”
1918: Julia Set • 1st fractal in complex plane • Originally not intended to be visualized
1926: Menger Sponge • Contains every 1D object (inc. K3,3, K5)
1975: History Breakthrough • Benoit Mandelbrot: Les objets fractals, forn, hasard et dimension, 1975 • Fractal definition • Legendary Mandelbrot Set
2003: Fractals Nowadays • Fractal image / sound compression • Fractal music • Fractal antennas • …
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”
Coastal Length • Smaller the scale, longer the coast • Where is the limit? • USA shoreline at 30m details: 143 000 km!
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…
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 • …
Example: Deterministic Fractal • Square: rotate, scale, copy 90% 10%
Contractive Transformations • Copy machine association • Fractal – specified as a set of contractive transformations • Attractor = fix point
Iterated Function Systems • IFS = set of contractive affine transformations • Iterated process: • First copy • Second copy • Attractor • Affine transformation ~
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
IFS examples Dragon Curve
Lorenz Attractor • Edward Norton Lorenz, 1963 • IFS made from weather forecasting • Butterfly effect in dynamic system
Midpoint Displacement • Stochastic 1D fractal • Break the line • Shift its midpoint a little
Midpoint in 2D • Basic shape = triangle / square • Square: Diamond algorithm
Diamond Algorithm Applications • Terrains • Landscapes • Textures • Clouds