How to explore the Mandelbrot set
- Click anywhere to zoom in three times around that point; right-click or shift-click to zoom out. On a computer you can also zoom smoothly with the mouse wheel.
- Raise Detail as you zoom deeper: more iterations reveal finer structure near the edge of the set, at the cost of slower rendering.
- Try different colour palettes, jump to famous locations such as Seahorse Valley, or switch to Julia sets and the Burning Ship fractal. Copy the link to share the exact view, or download a PNG.
What the Mandelbrot set is
The Mandelbrot set, named after mathematician Benoit Mandelbrot who studied it in 1980, is the set of complex numbers c for which repeatedly applying z → z² + c, starting from zero, never escapes to infinity. Points inside the set are drawn black; points outside are coloured by how quickly they escape. The boundary between the two is infinitely detailed: however far you zoom, new spirals, tendrils and tiny copies of the whole set appear, a property called self-similarity.
Each point c of the Mandelbrot set also has its own Julia set, computed with the same formula but with c fixed and the starting point varying; points inside the Mandelbrot set give connected Julia sets, points outside give scattered "dust". The Burning Ship fractal uses absolute values in the formula and produces a shape like a burning ship. Rendering uses smooth colouring and runs entirely in your browser; very deep zooms eventually hit the limits of standard double-precision numbers and become blocky.
Is the Mandelbrot set infinite?
Its area is finite, about 1.5065, but its boundary is infinitely long and detailed.
Why does deep zooming become blurry or blocky?
Browsers use 64-bit floating-point numbers, which run out of precision around zoom levels of 10¹⁴.
What is the difference between Mandelbrot and Julia sets?
The Mandelbrot set varies c and starts at z = 0; a Julia set fixes c and varies the starting point z.