Explore It Online
The Burning Ship fractal is a variation of the Mandelbrot set whose jagged towers resemble ships and flames. A small change to the formula produces a landscape of repeating ship-like shapes.
Launch the interactive visualizer above to zoom into the boundary, pan across the formations, and change the color palette. Explore with keyboard or touch controls, or start a guided animation. The formula below explains how the shapes are generated.
The Formula
The Burning Ship uses the iteration z → (|Re(z)| + i|Im(z)|)² + c. It's the Mandelbrot formula with one change: before squaring, both the real and imaginary parts are replaced with their absolute values.
That single modification breaks the Mandelbrot set's rotational symmetry. Instead of the smooth, rounded cardioid, you get jagged upward-pointing structures that look like ships with masts and flames rising above them, which is where the name comes from.
Discovery
Michael Michelitsch and Otto E. Rössler described it in 1992. The absolute value operation acts as a fold: negative values get reflected to positive before the squaring step. This fold breaks the symmetry vertically rather than radially, and the resulting shapes have a roughness the Mandelbrot set completely lacks.
Zoom in, though, and you'll find smaller copies of the main ship embedded throughout the boundary. These mini-ships are surprisingly well-ordered compared to the chaos around them.
What Makes It Interesting
Mathematically, the Burning Ship is one of the simplest examples of how a non-analytic function (the absolute value) changes fractal structure. Analytic functions preserve angles and produce smooth boundaries. The absolute value doesn't. It creates corners, folds, and the sharp edges visible everywhere in this fractal.
The boundary regions are also noticeably more chaotic than the Mandelbrot's. Where the Mandelbrot boundary has a kind of organized complexity (spirals, seahorses, satellite copies), the Burning Ship's boundaries look turbulent and irregular.
To see the difference for yourself, compare with a Mandelbrot deep zoom. That explorer renders only the standard Mandelbrot set, not the Burning Ship, so open it alongside this visualizer and compare how the two boundaries behave as you zoom in.
Exploring It
Guided tour through the main ship formations. It zooms into the larger structures and their surrounding detail.
Slow undulating movement through the fractal, panning across the landscape.
Full manual control with keyboard or touch. Navigate, zoom, and find your own mini-ships.