Interactive Lab / The Mandelbrot Set

Escape-Time Discovered 1978 – 1980 Dimension 2 (boundary)

The Mandelbrot Set

M={ c:sup⁡n∣zn∣<∞,  z0=0,  zn+1=zn2+c }M = \{\, c : \sup_n |z_n| < \infty,\; z_0=0,\; z_{n+1}=z_n^2+c \,\}
★Cool Fact
The boundary has Hausdorff dimension exactly 2 (Shishikura, 1991) despite the set having finite area — possible only because it’s a fractal. Every visible bulb sprouts infinitely many smaller ones.

Rendering…

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