FractalsFractals

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

Fractals Around UsFractals Around Us

Fractals Inside UsFractals Inside Us

Fractal FloraFractal Flora

Fractal WeatherFractal Weather

Artificial Fractal ShapesArtificial Fractal Shapes

Fractal ImagesFractal Images

Fractal PatternsFractal Patterns

M. C. Escher: Smaller and SmallerM. C. Escher: Smaller and Smaller

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

1891: Hilbert Curve1891: Hilbert Curve

1904: Koch Snowflake1904: Koch Snowflake

Helge von KochHelge von Koch

1916: Sierpinski Gasket1916: Sierpinski Gasket

““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%

Example: Deterministic FractalExample: Deterministic Fractal

Example: Deterministic FractalExample: Deterministic Fractal

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

Example: Sierpinski GasketExample: Sierpinski Gasket

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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Barnsley’s Fern IFSBarnsley’s Fern IFS

Barnsley’s FernBarnsley’s Fern

Reality Versus FractalReality Versus Fractal

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

IFS examplesIFS examples

Dragon CurveDragon Curve

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

Diamond AlgorithmDiamond Algorithm

Diamond AlgorithmDiamond Algorithm

Diamond AlgorithmDiamond Algorithm

Diamond AlgorithmDiamond Algorithm

Fractal TerrainFractal Terrain

Diamond Algorithm ApplicationsDiamond Algorithm Applications

• TerrainsTerrains• LandscapesLandscapes• TexturesTextures• CloudsClouds