Mandelbrot & Julia set boundaries: D_f ≈ 2 (Shishikura 1998)
D_f = ? | Computing...
The Mandelbrot set boundary has Hausdorff dimension exactly 2 (Shishikura 1998) — the maximum for a planar curve.
Box-counting: cover the boundary with boxes of size ε, count N(ε). Then D_f = lim_{ε→0} log N(ε) / log(1/ε).
Different Julia sets (parameter c) span dimensions from 1 (quasi-circle) approaching 2 (c near Misiurewicz points).
The log-log plot shows the scaling; slope = fractal dimension.