# Burning Ship

The Mandelbrot set plots the fate of \[(x + i y) \to (x + i y)^2 + (a + i b).\]

The Burning Ship plots the fate of \[(x + i y) \to (|x| + i |y|)^2 + (a + i b).\]

The absolute values mean the Burning Ship formula is not analytic. It is continuous, but its derivatives are not continuous everywhere.

# 1 At The Helm Of The Burning Ship

Conference paper.

Claude Heiland-Allen, “At The Helm Of The Burning Ship”, EVA London 2019.


The Burning Ship fractal is a non-analytic variation of the Mandelbrot set, formed by taking absolute values in the recurrence. Iterating its Jacobian can identify the period of attracting orbits; Newton’s root-finding method locates their mini-ships. Size estimates tell how deep to zoom to find the mini-ship or its embedded quasi-Julia set. Pre-periodic Misiurewicz points with repelling dynamics are located by Newton’s method. Stretched regions are automatically unskewed by the Jacobian, which is also good for colouring images using distance estimation. Perturbation techniques cheapen deep zooming. The mathematics can be generalised to other fractal formulas. Some artistic zooming techniques and domain colouring methods are also described.


Burning Ship. Dynamical systems. Fractal art. Numerical algorithms. Perturbation theory.

The paper is available as open access from At The Helm Of The Burning Ship paper (scienceopen.com). A local mirror is here: At The Helm Of The Burning Ship paper (local mirror). The presentation slides are here: At The Helm Of The Burning Ship slides.