The Burning Ship Fractal

The Burning Ship fractal consists of points, c, in the complex plane that obey the following rule

  1. Start with the complex number z = a + ib where a and b are real numbers.
  2. Generate a new complex number, z', by taking the absolute value of both the real and imaginary parts before squaring the resulting complex number, and adding the result to c. z' = (|Re(z)| + |Im(z)|*i)2 + c.
  3. Repeat steps 1 and 2. If the complex number z goes toward infinity, then the starting point, c, is not a member of the Mandelbrot set. All numbers that remain bounded are members of the set.
It can be shown that if the magnitude of z is greater than 2 , then z will approach infinity. The code assumes that the number c is in the set if |z| > 2 after 256 iterations. In order to show how rapidly a number fails the test, we color the pixel corresponding to the number of iterations.