Mandelbrot Set
The canonical escape-time fractal: the set of complex numbers c for which z² + c remains bounded.
The Iteration
For each point c in the complex plane, start at z = 0 and repeatedly apply:
z_{n+1} = z_n² + c
If the magnitude of z stays bounded forever, c belongs to the Mandelbrot set. If it escapes — grows without bound — it does not.
The coordinate plane before any computation. Every pixel is a candidate. The real axis runs left to right; the imaginary axis runs top to bottom.
Escape-Time Coloring
We iterate up to a maximum count (typically 256). If |z| > 2 at iteration k, the point has escaped. The escape time determines the colour.
The black region is the Mandelbrot set itself — all points that never escape. The boundary is already recognisable: a main cardioid with circular bulbs attached.
The Boundary
The true complexity lives at the boundary. Zooming in reveals infinite self-similar detail: the set contains infinitely many miniature copies of itself, connected by intricate filaments.
Smooth escape-time colouring maps the iteration count to a continuous value, eliminating the harsh bands. The gradient reveals the rate at which points escape — fast escapes are bright, slow escapes are dark.