Tool

Mandelbrot Explorer

The Mandelbrot set is the set of complex numbers c for which the iteration z ← z² + c starting at z = 0stays bounded. Black points are inside the set, colored points escape to infinity, the color encodes how fast.

Click to zoom in · Shift+click to zoom out · Drag to pan0 ms

Max iterations: 200

Zoom: 1.00×

Log scale; 1× = default view.

Re(c): -0.700000

Im(c): 0.000000

Famous landmarks:

Iteration rule

z_{n+1} = z_n² + c   with   z_0 = 0
c ∈ M  ⟺  the orbit stays bounded  ⟺  |z_n| ≤ 2 for all n
escape test: stop when |z_n| > 2

Each pixel is the parameter c. Color is the smoothed escape time n + 1 − log log |z_n| / log 2.

What to try

Zoom into the boundary between black and color, that is where the self-similar structure lives.
Hunt for mini-Mandelbrots, scaled copies of the full set appear inside the boundary at every zoom level.
Try the “seahorse valley” around (-0.75, 0.1) or the “elephant valley” around (0.275, 0).
Push iterations higher when the image looks chunky at deep zoom.

FAQ