Fractal Guide
What is the Burning Ship Fractal?
The Burning Ship is one of the most visually striking variants of the Mandelbrot set, characterized by an inverted ship-like silhouette engulfed in flames.
Contents
1. Definition of the Burning Ship
The Burning Ship fractal is defined by z(n+1) = (|Re(z(n))| + i|Im(z(n))|)ยฒ + c. The key difference from the Mandelbrot set is taking the absolute value of the real and imaginary parts of z at each iteration.
This small change โ taking absolute values โ creates a shape entirely different from the Mandelbrot set: asymmetric and ship-like. It was first reported by Michael Michelitsch and Otto E. Rรถssler in 1992.
2. Differences from the Mandelbrot Set
The Mandelbrot set is symmetric about the real axis, but the Burning Ship is asymmetric. This is because the absolute value operation breaks the complex conjugate symmetry.
The Burning Ship's boundary appears 'rougher,' with flame-like jagged structures throughout. Zooming in reveals self-similarity โ small Burning Ships appearing recursively.
3. Mathematical Properties
The absolute value operation is non-differentiable, making the Burning Ship's boundary mathematically more complex than the Mandelbrot set. Rather than a smooth holomorphic function, it involves a folding map.
This folding structure creates the Burning Ship's characteristic 'jagged' appearance and unique fractal details. Near the boundary, miniature Mandelbrot-like structures have also been discovered.
4. Rendering Characteristics
The rendering algorithm is nearly identical to the Mandelbrot set, with the addition of taking absolute values of the real and imaginary parts at each iteration. This small difference produces dramatically different results.
The Burning Ship is often displayed upside-down. In Tool Palette's Fractal Gallery, you can zoom in by clicking to explore the fine structural details.
Explore the Burning Ship
Interactively explore the Burning Ship in the Fractal Gallery.
Go to Fractal Gallery