The Mandelbrot Set

The Mandelbrot set is the set of complex numbers c for which the iteration $z_{n+1} = z_n^2 + c$ (starting from $z_0 = 0$) remains bounded — i.e. does not escape to infinity. Points inside the set are coloured black; points outside are coloured according to how quickly they escape.

Smooth (continuous) colouring uses the fractional iteration count $\mu = n - \log_2(\log_2|z_n|)$ which removes band artifacts and produces gradients.

How to Explore

• Left-click anywhere to zoom in on that point.
• Right-click to zoom out.
• Scroll wheel zooms in/out at the cursor position.
• Drag to pan the view.
• Use the Preset dropdown to jump to famous locations.
• Crank up Max iterations for more detail at high zoom.

Did You Know?

The Mandelbrot set boundary is a fractal: it has infinite perimeter and self-similar structure at every scale of magnification. Zooming in by a factor of 10²⁰⁰ still reveals new structures. A related family, Julia sets, are obtained by fixing c and varying $z_0$; every point in the Mandelbrot set corresponds to a connected Julia set.