Mandelbrot approximations in G-Force

A few years back, when I was doing FlowFields for G-Force based on chaos theory and fractals, I did some Julia set ones but concluded that the Mandelbrot set was impossible. However, I recently realized that Nth-order approximations to Mandelbrot were doable for any finite N. Last night I got 1st through 6th order ones working, and they look quite a bit different from what I expected.

My initial expectation was that the Nth-order approximation would fit inside the Nth nested bounding curve. This image from Peitgen et al., Chaos and Fractals: New Frontiers of Science. Therefore, I expected that the approximations might also be nested.

Here’s a slightly different view of the “lemniscates” from mathworld.wolfram.com. Note that although these were nominally computed from the exact same formula, they are not identical to the ones in the previous picture.

For a baseline, here’s an image of my Julia Seahorse FlowField. Julia sets are doable because the “motion” of each pixel is defined by a global constant (the complex number C that “indexes” that set). If you know X, Y, and C, you know everything you need to know. But for the Mandelbrot set, pixel motion needs X, Y, and the coordinates of where that pixel originally started. Every pixel needs to know its ancestral home. There’s no way to code that in G-Force, so I concluded that Mandelbrot cannot be done.

Here’s the 1st-order approximation. It clearly has a fractal boundary. It seems to have both up-down and left-right mirror symmetry, which matches wolfram.com‘s first bounding curve, but not Peitgen’s. The formula is srcZ = Z^2 + Z. Each pixel takes one correct step, but then forgets where it came from, so each subsequent step is “wrong”.

The 2nd-order approximation begins to look vaguely like the final set. The formula is srcZ = (Z^2 + Z)^2 + Z, which represents two steps before the pixels forget. (The white lines are the waveshape used to “excite” the FlowField, and should be ignored.)

Here’s the 3rd-order approximation, srcZ = ((Z^2 + Z)^2 + Z)^2 + Z. Surprisingly, this appears to no longer be a connected set, since there is a substantial gap between the left bulb and the main body.

The 4th-order approximation

The 5th-order approximation.

The 6th-order approximation.