It is surprisingly easy to generate a fractal image. A schematic of the Mandelbrot Set is as follows. Start with the equation $z_{n+1} = z_n^2 + c$, where $c$ is a complex number. Your “image” is a 2d array whose rows and columns correspond to the real and imaginary parts of $c$. Set $z_0 = 0$ and repeatedly apply the equation. If the value converges in a set number of iterations, set the corresponding entry in your array to 1, else 0.


Array/figure size in pixels


Bitmapped image, running time


Color the pixel by the number of iterations it took to converge. Try variants of the equation (see Julia sets). And note that the computation of each pixel is independent of every other: we can parallelize this! We visit this in another post. There are many ways to optimize the sequential implementation — try these out!


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