One absolute value transforms the Mandelbrot formula into a flaming wreck of infinite detail.
In 1992, Michael Michelitsch and Otto Rössler published a paper asking a simple question: what happens if you take the Mandelbrot iteration and force the real and imaginary parts of to always be positive? The answer turned out to look like a burning ship.
The Mandelbrot iteration is . The Burning Ship iteration is almost identical — except before squaring, both components of are made positive using the absolute value:
Those two absolute value signs are the entire difference between a smooth symmetric Mandelbrot set and the jagged, asymmetric landscape of the Burning Ship. By folding the complex plane onto itself at both axes before squaring, the iteration produces sharp creases and angular structures that look nothing like the Mandelbrot's organic curves.
When you render the full fractal (zoom out to see the whole set), the main body resembles a sailing ship hull viewed from the side. At the top, a vertical mast-like antenna extends upward, and flickering flame-like tendrils surround the structure.
Zoom in and you'll find the self-similar property of fractals in action: smaller copies of the ship appear along the hull and mast. These “mini-ships” are shrunken, rotated, and sometimes distorted echoes of the main body — a signature of the quadratic iteration structure underneath.
One oddity: the standard rendering convention shows the Burning Ship upside-downcompared to most other fractals. That's because the absolute value folds the complex plane into its upper half, making the interesting structures appear in the lower-left quadrant. Most renderers (and FractalSet) flip the y-axis so that the ship shape appears right-side up and readable.
The same folding trick can be applied in other ways. Using only (just the real part) gives the Buffalo fractal. Setting the absolute value differently gives the Perpendicular Mandelbrot. The entire family of fractals produced by such modifications is called Multibrot variants.
Rendering tip
The Burning Ship renders best at higher iteration counts (400+) because many boundary points take a long time to escape. Try the lava or Mandelbrot palette to make the flame-like tendrils visible.
