# Accuracy of Generated Fractals

Note: I refer to the Mandelbrot set in general as the M-set for short.

When I was writing the post on Rough Mandelbrot Sets I tried out some variations on the rough set. One variation was to measure the generated M-set against a previously calculated master M-set of high precision (100000 iterations of $z = z^2 + C$). In the image below the master M-set is in white and the generated M-sets are in green (increasing in accuracy):

Here instead of approximating with tiles I measured the accuracy of the generated sets against the master set by pixel count. Where $P = \{ \text{set of all pixels} \}$ the ratio of $P_{master} / P_{generated}$ produced something that threw me, the generated sets made sudden but periodic jumps in accuracy:

Looking at the data I saw the jumps were, very roughly, at multiples of 256. The size of the image being generated was 256 by 256 pixels so I changed it to N by N for N = {120, 360, 680} and the increment was still every ~256. So I’m not really sure why, it might be obvious, if you know tell me in the comments!

I am reminded of the images generated from Fractal Binary and other Complex Bases where large geometric entities can be represented on a plane by iteration through a number system. I’d really like to know what the Mandelbrot Number System is…

Below is a table of the jumps and their iteration index:

 Iterations Accuracy measure 255 256 0.241929 0.397073 510 511 0.395135 0.510806 765 766 0.510157 0.579283 1020 1021 0.578861 0.644919 1275 1276 0.644919 0.679819 1530 1531 0.679696 0.718911