Fractals
Fractals are complex geometric structures generated from simple mathematical rules, often through recursion or iterative processes. A key property of fractals is self-similarity, meaning their patterns repeat at different scales, creating infinite levels of detail. Famous examples include the Mandelbrot set, Julia sets, fractal trees, and space-filling curves like the Hilbert curve. Fractals appear both in mathematics and in nature, modeling phenomena such as coastlines, clouds, plants, and river networks. Their combination of simplicity and complexity makes them a powerful tool for visualization, simulation, and artistic expression.
Algorithm explanation
Fractals are generated through recursive or iterative processes. For example, a fractal tree is built by drawing a branch and recursively calling the function to create smaller sub-branches at different angles. The Mandelbrot set is generated by iterating the function z = z^2 + c for each point in the complex plane and checking whether the sequence diverges. The number of iterations before divergence determines the pixel color.
Concepts used
- Recursion
- Complex Numbers
- Iteration
- Mathematical Visualization
- Self-Similarity
- Geometry
- Canvas Rendering
Complexity
O(n * m * k), where n x m is the screen resolution and k is the number of iterations per pixelO(1) for iterative fractals (Mandelbrot) or O(d) for recursive fractals, where d is the recursion depthReal-world usage
- Procedural terrain and landscape generation in games
- Fractal-based image compression
- Modeling natural phenomena (trees, rivers, clouds)
- Digital art and mathematical visualizations
- Scientific simulations and complex graphics