What is the Mandelbrot set?

The Mandelbrot set is the set of all complex numbers c for which the iteration z → z² + c, starting from z₀ = 0, remains bounded (does not escape to infinity). Points inside the set are coloured black; points outside are coloured according to how many iterations they take to escape beyond a bailout radius — this is called the escape-time algorithm. The boundary of the Mandelbrot set is infinitely detailed: zooming in reveals self-similar spirals, miniature copies of the whole set, seahorse valleys, and elephant valleys, all at every scale of magnification.

What is smooth colouring and why does it look better than integer iteration count?

Standard escape-time colouring assigns each pixel the integer number of iterations before escape. This creates visible 'bands' of equal colour. Smooth colouring (also called normalised iteration count) uses the formula μ = N + 1 − log(log|z_N|) / log(2), where |z_N| is the modulus of the complex number at the time of escape. This non-integer value interpolates smoothly between integer escape counts, eliminating the banding and producing smooth gradient transitions between colour regions. The formula is derived from the renormalisation map of the Mandelbrot iteration.

What is a Julia set, and how is it related to the Mandelbrot set?

For each complex number c, the filled Julia set is the set of starting points z₀ for which the iteration z → z² + c remains bounded. The Mandelbrot set is essentially a catalogue of all Julia sets: if c is inside the Mandelbrot set, its Julia set is connected (one piece); if c is outside, the Julia set is a 'dust' of disconnected points (a Cantor set). In Julia mode in this simulator, clicking the canvas sets the parameter c, and the display shows the Julia set for that specific c. Points near the Mandelbrot boundary produce the most intricate Julia sets.

Why does the Mandelbrot set have infinite self-similar detail?

The Mandelbrot set is a fractal with Hausdorff dimension 2 — its boundary is so intricate that it locally fills the plane. The self-similarity arises because the iteration z → z² + c is a polynomial map whose critical point (z = 0) has the same dynamical structure at every scale. This manifests as miniature copies of the entire Mandelbrot set embedded throughout the boundary region at all scales of magnification. Furthermore, the Mandelbrot set is connected (proved by Douady and Hubbard, 1982) and simply connected — there are no isolated islands.

Home▸Drawing with Maths▸Mandelbrot Set Explorer — Interactive Fractal Zoom & Julia Sets

🧪 Mandelbrot Set Explorer — Interactive Fractal Zoom & Julia Sets

Explore the Mandelbrot set with smooth-colouring pixel renderer. Zoom into seahorse valleys, spiral whorls, and mini Mandelbrots. Toggle Julia set mode, switch colour palettes, and fly to famous locations. Free browser-based fractal explorer.

Drawing with Maths2DModerate60 FPS

⚙ Under the hood

This simulation allows users to explore the Mandelbrot set and related Julia sets through interactive fractal zoom. By zooming in on the boundary of the Mandelbrot set, they can observe infinitely complex patterns and appreciate the beauty of mathematical fractals.

Mandelbrot SetJulia SetsZooming

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