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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. - PowerPoint PPT Presentation
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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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Sierpinsky Gasket IFSSierpinsky Gasket IFS
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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