Fractal Generator

The Mandelbrot set is the set of complex numbers c for which the sequence z(n+1) = z(n)² + c, starting at z(0) = 0, stays bounded (never grows past a magnitude of 2) no matter how many iterations you run. Points that stay bounded are colored black; points that escape are colored by how quickly they escape.

A Julia set uses the same formula, z(n+1) = z(n)² + c, but fixes c as a constant and varies the starting point z(0) across the plane instead. Every point on the Mandelbrot set corresponds to a connected Julia set, and every point outside it corresponds to a Julia set that is totally disconnected ("fractal dust").

The Burning Ship fractal takes the absolute value of the real and imaginary parts of z before squaring at each step — z(n+1) = (|Re(z(n))| + i|Im(z(n))|)² + c — which produces a ship-like silhouette instead of the Mandelbrot set's smooth cardioid and bulbs.

This tool uses standard double-precision floating point, which stays numerically accurate to roughly 10-12 significant digits — enough to zoom in several million times before rounding artifacts appear. Increase the iteration count as you zoom in further to keep fine boundary detail sharp.

Black regions are points where the escape-time algorithm never exceeded the bailout radius within the maximum iteration count — meaning the point is inside (or numerically indistinguishable from inside) the fractal set itself.