Fractals

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