What is Fractal?What is Fractal?
• Not agreed upon the primary definitionNot agreed upon the primary definition• Self-similar objectSelf-similar object• Statistically scale-invariantStatistically scale-invariant• Fractal dimensionFractal dimension• Recursive algorithmic descriptionsRecursive algorithmic descriptions• latine word latine word fractusfractus = irregular/fragmented = irregular/fragmented• term Procedural Modeling is sometimes term Procedural Modeling is sometimes
misplaced with Fractalsmisplaced with Fractals
1883: Cantor Set1883: Cantor Set
• Cantor set in 1D: Cantor set in 1D: – Cantor DiscontinuumCantor Discontinuum– bounded uncontinuous bounded uncontinuous
uncountableuncountable setset
• 2D: Cantor Dust2D: Cantor DustGeorg Cantor
1890: Peano Curve1890: Peano Curve
• Space fillingSpace filling• Order lines Order lines curve curve
““remove squares until nothing remains”remove squares until nothing remains”
Analogy: Sierpinski CarpetAnalogy: Sierpinski Carpet
1918: Julia Set1918: Julia Set
• 11stst fractal in complex plane fractal in complex plane• Originally not intended to be visualizedOriginally not intended to be visualized
1926: Menger Sponge1926: Menger Sponge
• Contains every Contains every 1D object1D object
(inc. K(inc. K3,33,3, K, K55))
1975: History Breakthrough 1975: History Breakthrough
• Benoit Mandelbrot: Benoit Mandelbrot: Les objets fractals, forn, Les objets fractals, forn, hasard et dimension, 1hasard et dimension, 1975975
• Fractal definitionFractal definition• Legendary Mandelbrot SetLegendary Mandelbrot Set
2003: Fractals Nowadays2003: Fractals Nowadays
• Fractal image / sound compressionFractal image / sound compression• Fractal musicFractal music• Fractal antennasFractal antennas• ……
Knowledge SourcesKnowledge Sources
• B. Mandelbrot: B. Mandelbrot: The fractal geometry of natureThe fractal geometry of nature, , 19821982
• M. Barnsley: M. Barnsley: Fractals Everywhere, Fractals Everywhere, 19881988• Contemporary web sources:Contemporary web sources:
– http://math.fullerton.edu/mathews/c2003/FractalBib/Links/FractalBib_lnk_1.html
– Google yields over 1 000 000 results on “fractal”Google yields over 1 000 000 results on “fractal”
Coastal LengthCoastal Length
• Smaller the scale, longer the coastSmaller the scale, longer the coast• Where is the limit?Where is the limit?• USA shoreline at 30m details:USA shoreline at 30m details:
143 000 km!143 000 km!
Fractal DimensionFractal Dimension
• More definitionsMore definitions• Self-similarity dimensionSelf-similarity dimension
• N = number of transformationsN = number of transformations• rr = scaling coefficient = scaling coefficient
• Koch Curve exampleKoch Curve example• N = 4, N = 4, rr--1--1 = 3 = 3• Dimension = log 4 / log 3 = 1.26…Dimension = log 4 / log 3 = 1.26…
Fractal TaxonomyFractal Taxonomy
• Deterministic fractalsDeterministic fractalsa)a) Linear (IFS, L-systems,…)Linear (IFS, L-systems,…)
b)b) Non linear (Mandelbrot set, bifurcation diagrams,…)Non linear (Mandelbrot set, bifurcation diagrams,…)
• Stochastic fractalsStochastic fractals– Fractal Brovnian Motion (fBM)Fractal Brovnian Motion (fBM)– Diffusion Limited Aggregation (DLA)Diffusion Limited Aggregation (DLA)– L-SystemsL-Systems– ……
Example: Deterministic FractalExample: Deterministic Fractal
• Square: rotate, scale, copySquare: rotate, scale, copy
90%
10%
Contractive Contractive TransformationsTransformations
• Copy machine associationCopy machine association• Fractal – specified as a set of contractive Fractal – specified as a set of contractive
transformationstransformations• Attractor = fix point Attractor = fix point
Iterated Function SystemsIterated Function Systems
• IFS = set of contractive affine transformationsIFS = set of contractive affine transformations
• Iterated process:Iterated process:– First copy First copy
– Second copySecond copy
– AttractorAttractor
• Affine transformationAffine transformation
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IFS ComputationIFS Computation
• Deterministic:Deterministic:– Apply transformations to the object until infinitumApply transformations to the object until infinitum
• Stochastic (Chaos Game algorithm):Stochastic (Chaos Game algorithm):– Choose random transformation Choose random transformation ffii
– Transform a point using Transform a point using ffii
– Repeat until infinitumRepeat until infinitum
Lorenz AttractorLorenz Attractor
• Edward Norton Lorenz,Edward Norton Lorenz, 1963 1963• IFS made from weather forecastingIFS made from weather forecasting• Butterfly effect in dynamic systemButterfly effect in dynamic system
Midpoint DisplacementMidpoint Displacement
• Stochastic 1D fractalStochastic 1D fractal• Break the lineBreak the line• Shift its midpoint a littleShift its midpoint a little
Midpoint in 2DMidpoint in 2D
• Basic shape = triangle / squareBasic shape = triangle / square• Square: Diamond algorithmSquare: Diamond algorithm