Burning Ship Fractal

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.