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
Fractals Around Us
Fractals Inside Us
Fractal Flora
Fractal Weather
Artificial Fractal Shapes
Fractal Images
Fractal Patterns M. C. Escher: Smaller and Smaller
1883: Cantor Set Cantor set in 1D: 2D: Cantor Dust Cantor Discontinuum bounded uncontinuous uncountable set 2D: Cantor Dust Georg Cantor
1890: Peano Curve Space filling Order lines curve
1891: Hilbert Curve
1904: Koch Snowflake Helge von Koch
1916: Sierpinski Gasket
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 Stochastic 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%
Example: Deterministic Fractal
Example: Deterministic Fractal
Contractive Transformations Copy machine association Fractal – specified as a set of contractive transformations Attractor = fix point
Example: Sierpinski Gasket
Iterated Function Systems IFS = set of contractive affine transformations Iterated process: First copy Second copy Attractor Affine transformation ~
Sierpinsky Gasket IFS
Barnsley’s Fern IFS
Barnsley’s Fern
Reality Versus Fractal
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
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
Diamond Algorithm
Diamond Algorithm
Diamond Algorithm
Fractal Terrain
Diamond Algorithm Applications Terrains Landscapes Textures Clouds