The lab
Mandelbrot set
A point c belongs to the set if iterating z → z² + c from z = 0 never escapes to infinity. The boundary between those that do and those that don't is infinitely complex — self-similar at every scale, never repeating, arising entirely from two lines of algebra. This is what order and chaos look like from the edge.
zₙ₊₁ = zₙ² + c · c ∈ ℂ · zoom to explore
256
smooth escape
click to zoom in · right-click or double-click to zoom out · drag to pan
interesting locations
Benoit Mandelbrot published "How Long Is the Coast of Britain?" in 1967, then spent the next two decades arguing that geometry should describe nature's actual roughness — not Euclid's idealized smoothness. This set, named for him posthumously, is the canonical example: a two-dimensional slice through a four-dimensional parameter space, where the boundary has Hausdorff dimension exactly 2. It is simultaneously bounded and infinitely wrinkled.
Drag to explore. The detail never runs out.