Complex dynamics
Burning Ship fractal
Michael Michelitsch and Otto Rössler introduced the Burning Ship in 1992. Take the Mandelbrot iteration but absolute-value the real and imaginary parts of z before squaring. The result loses complex-analytic structure and gains ship-like, antenna-like, even skyline-like features.
At a glance
| Designer | Michelitsch & Rössler, 1992 |
|---|---|
| Iteration | zn+1 = (|Re zn| + i|Im zn|)2 + c |
| Symmetry | About the real axis only |
| Holomorphic? | No (absolute value is not differentiable in the complex sense) |
The eponymous ship
Around c ≈ -1.76 + 0.04 i the iteration boundary forms a striking shape resembling a vessel emitting smoke; this gave the fractal its evocative name.
Why not holomorphic?
Taking absolute values of Re z and Im zis not a holomorphic operation, so most theorems about complex dynamics (Sullivan’s no-wandering-domain, Fatou-Julia connectivity) do not directly apply. The Burning Ship is a non-holomorphic escape-time fractal, related to but distinct from the Mandelbrot family.
References
- Michelitsch, M. & Rössler, O., “The “burning ship” and its quasi-Julia sets,” Computers & Graphics, 1992.
- Mandelbrot set · Tricorn
Try it
Run an interactive playground at /tools/burning-ship.