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Mathematics
Fractals
1. Introduction to Fractals
2. Mathematical Foundations
3. Fractal Dimension
4. Classical Fractals
5. Iterated Function Systems
6. Escape-Time Fractals
7. Random and Stochastic Fractals
8. Fractals in Nature
9. Applications of Fractal Geometry
10. Connections to Related Fields
11. Computational Methods and Tools
12. Advanced Topics and Current Research
Fractal Dimension
Self-Similarity Dimension
Definition and Formula
Calculation Methods
Examples with Simple Fractals
Limitations and Assumptions
Box-Counting Dimension
Box-Counting Algorithm
Grid-Based Measurements
Practical Implementation
Error Analysis and Convergence
Hausdorff Dimension
Mathematical Definition
Hausdorff Measure
Calculation Techniques
Relationship to Other Dimensions
Other Dimension Measures
Correlation Dimension
Information Dimension
Capacity Dimension
Packing Dimension
Multifractal Analysis
Concept of Multiple Dimensions
Singularity Spectrum
Multifractal Measures
Generalized Dimensions
Applications in Complex Systems
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4. Classical Fractals