*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 ). 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 the ratio of 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 |