Hover over the Mandelbrot set to see the corresponding Julia set update in real time
Move mouse over Mandelbrot (left) to update Julia set (right)
The Mandelbrot set M = {c ∈ ℂ : z_{n+1} = z²+c with z₀=0 does not diverge}. Each point c in M has a corresponding Julia set J_c = {z : the orbit of z under f_c stays bounded}. The duality: c ∈ M ⟺ J_c is connected. When c is outside M, J_c is a Cantor dust (totally disconnected). The boundary of M has Hausdorff dimension 2 (Shishikura 1998) — it is as "thick" as a 2D region despite being a 1D curve. Both sets are self-similar at all scales.