Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems—the pictures of Chaos.

Polar Coordinate Fractals

A comprehensive exploration of fractals generated using polar coordinates, where separate transformations are applied to distance and angle. This approach creates stunning and diverse visual patterns that differ from traditional Cartesian coordinate fractals.

• Polar Coordinate Fractals - Series Index - Complete series on polar coordinate fractals with interactive GLSL visualizations

Series includes:

• Uzumaki Spiral - Polar Coordinates - The "Most Illegal Uzumaki in Mathematics" in polar coordinates
• Polar Julia Set - Classic Julia set reimagined in polar coordinates
• Polar Mandelbrot Variant - Mandelbrot-like fractal using polar iteration
• Spiral Fractal - Pure spiral patterns (Archimedean and logarithmic)
• Multi-Lobed Fractal - Symmetric multi-lobed patterns
• Distance-Based Fractal - Power function distance transformations
• Angle-Dominant Fractal - Polynomial angle transformations
• Hybrid Polar Fractal - Complex combined transformations

Other Fractal Content

• Animated L-Systems and Tree Fractals - Fractal tree generation using L-systems
• Mandelbrot Set - Classic fractal in Cartesian coordinates

Mathematical Concepts

Fractals demonstrate several important mathematical concepts:

• Self-similarity: Patterns repeat at different scales
• Iteration: Simple rules applied repeatedly create complex structures
• Chaos theory: Small changes in initial conditions lead to dramatically different outcomes
• Complex numbers: Many fractals are generated in the complex plane
• Polar coordinates: Alternative coordinate system offering new fractal patterns

Interactive Exploration

All fractal series include interactive GLSL Toybox demos that allow you to:

• Adjust parameters in real-time
• Explore different iteration counts
• Modify color schemes and visualization parameters
• Experiment with different fractal patterns

Related Topics

• Math & Learning - Mathematical learning resources
• Coding Projects - Interactive visualizations and procedural art
• Introduction to GLSL and Shaders - Learn shader programming for fractal visualization