Mandelbrot Set — Deep Zoom

The Mandelbrot set: c ∈ ℂ such that z_n+1 = z_n² + c does not diverge from z₀=0. Smooth coloring uses the escape time + log of the magnitude: μ = n − log₂(log|z|) for continuous gradients. The boundary has Hausdorff dimension 2 (Shishikura, 1998). The Julia set for each c is either connected (c ∈ M) or totally disconnected dust (c ∉ M). Interesting points: main cardioid |c − ¼|² + Re(c − ¼) < ¼, period-2 bulb |c+1| < ¼.