# Burning Ship Fractal

> Zoom into the Burning Ship fractal, the Mandelbrot iteration with absolute values: find the small ship near the large one, then edit the equation.

The Burning Ship fractal comes from a small change to the Mandelbrot iteration: before each squaring, the real and imaginary parts of $z$ become their absolute values. So the map is $z \to (|\operatorname{Re} z| + i|\operatorname{Im} z|)^2 + c$, starting at zero. Michael Michelitsch and Otto E. Rössler described it in 1992. With real numbers $z = a + ib$ and $c = p + iq$, one step is $a \to a^2 - b^2 + p$ and $b \to |2ab| + q$, and the function $B$ of the document's prelude writes this as a loop that you can edit. Most pictures of this fractal are reflected so that the ship is upright, and this document plots $B(x, -y)$ for the same reason.

Zoom near $x = -1.75$, $y = 0.02$ to find a small copy of the ship, with tall masts.

$B$ gives the escape count of a point divided by 128, and 1 for a point that does not escape in 128 steps. To make the count continuous, the test for the escape is $|z|^2 > 65536$. This does not change the set, because a point with $|z| > 2$ always escapes. The count is then $n + 1 - \log_2(\ln(|z|^2) / \ln 65536)$. The loop stops when the point escapes, so only a point that does not escape does all 128 steps.

- [Open the interactive version](https://graph-paper.io/showcase/en/burning-ship)
- [Learn more on Wikipedia](https://en.wikipedia.org/wiki/Burning_Ship_fractal)