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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
3.
Fractal Dimension
3.1.
Self-Similarity Dimension
3.1.1.
Definition and Formula
3.1.2.
Calculation Methods
3.1.3.
Examples with Simple Fractals
3.1.4.
Limitations and Assumptions
3.2.
Box-Counting Dimension
3.2.1.
Box-Counting Algorithm
3.2.2.
Grid-Based Measurements
3.2.3.
Practical Implementation
3.2.4.
Error Analysis and Convergence
3.3.
Hausdorff Dimension
3.3.1.
Mathematical Definition
3.3.2.
Hausdorff Measure
3.3.3.
Calculation Techniques
3.3.4.
Relationship to Other Dimensions
3.4.
Other Dimension Measures
3.4.1.
Correlation Dimension
3.4.2.
Information Dimension
3.4.3.
Capacity Dimension
3.4.4.
Packing Dimension
3.5.
Multifractal Analysis
3.5.1.
Concept of Multiple Dimensions
3.5.2.
Singularity Spectrum
3.5.3.
Multifractal Measures
3.5.4.
Generalized Dimensions
3.5.5.
Applications in Complex Systems
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4. Classical Fractals