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Fractals

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Santhosh Kumar 18/11/2023
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Fractals mind map Context Physicists using fractal geometry to study quantum systems providing a unique perspective on the uncertainties of quantum physics. Definition Geometric shape Detailed structure at small scales Characteristics Fractal dimension Exceeds topological dimension Self-similarity Patterns repeat at smaller scales Types Exact self-similarity Identical at all scales Example: Koch snowflake Quasi self-similarity Approximate pattern repetition Example: Mandelbrot set Statistical self-similarity Stochastic pattern repetition Example: Coastline of Britain Qualitative self-similarity Time series patterns Multifractal scaling Multiple fractal dimensions Scaling Non-integer dimensional scaling Differentiability Nowhere differentiable Infinite detail at small scales Mathematical Background Measure theory Hausdorff–Besicovitch dimension Applications In nature Geometric patterns Natural processes In technology In art In architecture In law Chaos theory Geometric depictions of chaotic processes Generation Techniques Iterated function systems Fixed geometric replacement rules Strange attractors Chaotic systems L-systems String rewriting Origin of Term Coined by Benoît Mandelbrot Based on Latin "frāctus" Theoretical fractional dimensions

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