Welcome to the World of Fractals and Chaos
What are Fractals?
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.
What is Chaos Theory?
Chaos theory studies how small changes in initial conditions can lead to vastly different outcomes, often called the "butterfly effect." These systems appear random but follow deterministic rules.
Explore the Visualizations
Use the navigation above to explore different fractals and chaotic systems. Each section includes interactive controls and educational content to help you understand these fascinating mathematical phenomena.
The Mandelbrot Set
The Mandelbrot set is one of the most famous fractals, discovered by Benoit Mandelbrot in 1980. It's generated by iterating the formula: z = z² + c
Each pixel represents a complex number c. The color indicates how quickly the iteration escapes to infinity (or if it remains bounded).
Click and drag to zoom into a region
Julia Sets
Julia sets are closely related to the Mandelbrot set. Each point in the Mandelbrot set corresponds to a unique Julia set. They use the same formula z = z² + c, but with a fixed c value.
Adjust the parameters below to explore different Julia sets!
Sierpinski Triangle
The Sierpinski Triangle is a fractal named after Polish mathematician Wacław Sierpiński. It demonstrates self-similarity - each smaller triangle is identical to the whole.
Watch as the triangle is constructed recursively, subdividing each triangle into smaller copies.
Koch Snowflake & L-Systems
The Koch Snowflake is created by repeatedly replacing each line segment with a specific pattern. It's an example of an L-System (Lindenmayer System).
Starting with a triangle, each iteration adds smaller triangular bumps to each edge, creating an infinitely complex boundary with finite area!
Note: Complexity grows exponentially. Advanced mode allows up to iteration 20 but may freeze your browser.
Lorenz Attractor
The Lorenz Attractor is a set of chaotic solutions to the Lorenz system, a simplified model of atmospheric convection developed by Edward Lorenz in 1963.
It demonstrates sensitive dependence on initial conditions - the hallmark of chaos theory. Two nearly identical starting points will diverge exponentially over time.
The equations: dx/dt = σ(y - x), dy/dt = x(ρ - z) - y, dz/dt = xy - βz