A fractal is a mathematical shape that exhibits self-similarity at every scale. Fractals have fractional (non-integer) dimensions, unlike ordinary geometric shapes.

What is the Mandelbrot set?

The Mandelbrot set is the most famous fractal in mathematics, defined by iterating z = z² + c for complex numbers. Points that remain bounded belong to the set, while those that escape produce colourful patterns at the boundary.

For every point c in the complex plane, its Julia set is the set of initial values z₀ for which the same iteration z = z² + c remains bounded. If c is inside the Mandelbrot set, its Julia set is connected; otherwise it shatters into Fatou dust.

Where do fractals appear in nature?

Fractals appear in Romanesco broccoli, blood vessels, lungs (23 levels of branching), snowflakes, lightning, coastlines, and mountain ranges.

Home▸Drawing with Maths▸Mandelbrot Set Explorer - Interactive Fractal Zoom, Julia Sets and Complex Numbers

🧪 Mandelbrot Set Explorer - Interactive Fractal Zoom, Julia Sets and Complex Numbers

Explore the Mandelbrot set with infinite zoom, custom colour maps, and Julia set exploration. Discover self-similarity, fractal dimension, and the beauty of complex quadratic iteration - great for A-level and university maths.

Drawing with Maths2DModerate60 FPS

⚙ Under the hood

This simulation demonstrates the Mandelbrot set and related fractal geometry. Users can explore the set by zooming in and observing its intricate patterns, which are generated using complex numbers.

Mandelbrot SetJulia SetsComplex Numbers

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install