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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
Mathematical Foundations
Complex Numbers and the Complex Plane
Review of Complex Numbers
Real and Imaginary Parts
Complex Conjugates
Modulus and Argument
Arithmetic Operations
Addition and Subtraction
Multiplication and Division
Powers and Roots
Geometric Interpretation
Argand Diagram
Polar Form
Exponential Form
Iteration and Dynamical Systems
Iterated Functions
Function Composition
Recursive Sequences
Convergence and Divergence
Orbits and Trajectories
Definition of Orbits
Periodic Orbits
Chaotic Orbits
Fixed Points and Stability
Attracting Fixed Points
Repelling Fixed Points
Neutral Fixed Points
Stability Analysis
Attractors and Basins
Definition of Attractors
Basins of Attraction
Repelling Sets
Basin Boundaries
Metric Spaces and Topology
Basic Set Theory
Open and Closed Sets
Neighborhoods
Compactness
Distance Metrics
Euclidean Distance
Hausdorff Distance
Applications in Fractal Geometry
Continuity and Limits
Continuous Functions
Uniform Convergence
Dimension Theory
Classical Dimensions
Topological Dimension
Euclidean Dimension
Fractal Dimension Concepts
Motivation for Non-Integer Dimensions
Relationship to Classical Dimensions
Dimension as a Measure of Complexity
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1. Introduction to Fractals
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3. Fractal Dimension