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Fractals

Santhosh Kumar 18/11/2023

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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Categories: Physics

Tags: 18.11.2023, Newsbits

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