Julia Set Variation Plotter
Understanding Julia Sets
What are Julia Sets?
Julia sets are mathematical sets of points in the complex plane that exhibit intricate boundary behaviors under iteration of complex functions.
Basic Formula: f(z) = z² + c
where:
- z is a complex number (point in plane)
- c is a complex parameter
- Set contains points where iteration remains bounded
- Boundary points create fractal patterns
- Different c values create different sets
- Connected vs. Disconnected sets based on c
Mathematical Properties
- Topological Properties
- Connectedness Theorem
- Hausdorff Dimension
- Local Connectivity
- Critical Points
- Basin Boundaries
- Periodic Orbits
- Dynamic Properties
- Escape Time Algorithm
- Periodic Points
- Attracting Cycles
- Repelling Cycles
- Critical Orbits
- Parameter Space
Variations and Extensions
Higher Powers
z^n + c variations
Rational Functions
Complex rational maps
Exponential Maps
e^z + c patterns
Tricorn Sets
Anti-holomorphic maps
Applications and Connections
- Complex Analysis
- Holomorphic Dynamics
- Conformal Mapping
- Potential Theory
- Riemann Surfaces
- Physical Applications
- Chaos Theory
- Pattern Formation
- Quantum Mechanics
- Fluid Dynamics