Mandelbrot Set Visualizer
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Understanding the Mandelbrot Set
What is the Mandelbrot Set?
The Mandelbrot set is the set of complex numbers c for which the function f(z) = z² + c does not diverge when iterated from z = 0.
Key Concepts
- Iterative function: z_{n+1} = z_n² + c
- Initial condition: z₀ = 0
- Bounded sequence: |z_n| ≤ 2 for all n
- Complex plane mapping
Mathematical Properties
Topological Properties
Connected set
Simply connected
Compact set
Self-similar structure
Geometric Properties
Symmetry about real axis
Cardioid main bulb
Period bulbs
Fractal boundary
Complex Dynamics
Julia sets relation
Parameter space
Periodic orbits
Critical points
Advanced Topics
Fractal Dimension
Hausdorff dimension ≈ 2
Box-counting dimension
Self-similarity dimension
Numerical Methods
Escape time algorithm
Distance estimation
Boundary tracing
Generalizations
Multibrot sets
Higher dimensions
Parameter spaces