# Burning Ship

Complex · w = f(z, t)

`w = (abs(re(z)) + i · abs(im(z)))² + c`

[Open in the app](https://www.wavelace.com/app#p=50) · [This page](https://www.wavelace.com/presets/burning-ship)

### What it draws

`w = (|Re z| + i · |Im z|)² + c` is the Mandelbrot recipe with one change: before each squaring, both parts of `z` are made positive. The square is then `x² − y² + 2|x · y| · i`, so only the imaginary part is touched. Each point of the square is fed in twice, as the starting `z` and as the constant `c`, and `Iterations` sets how many steps a surviving orbit has to take. Starting at `z = c` rather than at zero changes nothing, since the first step from zero is `c` either way.

### What the fold does

Taking absolute values is not a complex-differentiable operation, and that is the whole point. A holomorphic map is conformal, so the Mandelbrot boundary curls into smooth spirals and buds. Folding the plane along both axes at every step breaks that. What appears instead are straight edges, sharp creases and plumes that look like smoke and rigging.

On the real axis the fold does nothing at all: an orbit that starts real stays real, and `|x|² = x²`. Along that line the set is exactly the Mandelbrot interval from `−2` to `0.25`. Off the axis the fold pushes every step into the upper half plane, so the picture has no symmetry between the two imaginary directions. The ship itself, the little hull with an antenna that names the whole thing, sits near `−1.78` with a small negative imaginary part, close to the left edge of this square. Most reproductions draw the imaginary axis pointing downward, which is what turns it right way up.

### History

Michael Michelitsch and Otto E. Rössler published the map and its name in 1992, in a paper on the burning ship and its quasi-Julia sets. It belongs to a small family of non-holomorphic variations on the quadratic map, alongside the tricorn, which conjugates `z` instead of folding it.

### Try

- `Top`, then look along the left edge for the hull and its antenna.
- Raise `Iterations` toward 300: the rigging fills in and the smoke thins out into fine threads.
- Type `z² + c` for the unfolded map, and the creases give way to smooth spirals and buds.
- Fold a cube instead: `(|Re z| + i · |Im z|)³ + c`, written as `(abs(re(z)) + i · abs(im(z)))^3 + c`.
- Raise `Span of z` to 4: the whole thing sits in a wide plain of escape terraces.

### Read more

- [Burning Ship fractal](https://en.wikipedia.org/wiki/Burning_Ship_fractal)
- [Mandelbrot set](https://en.wikipedia.org/wiki/Mandelbrot_set)
- [Tricorn (mathematics)](https://en.wikipedia.org/wiki/Tricorn_(mathematics))
- [Holomorphic function](https://en.wikipedia.org/wiki/Holomorphic_function)