Why is this a fractal?

A defining feature of fractals is complex self-similarity at different scales.  For the Martin fractal, this is evident as one watches what happens while it grows as points are added.  It starts out as a small square or diamond, then puts out four “tentacles,” fills in to become a square or diamond again, then puts out four more tentacles to repeat the process.  (Like some sort of alien life form taking over a planet.)

Below are five sets of images that display the growth of the fractal from 1,000 to 100,000,000 points.  Each set represents an increase in points by a factor of ten.  Successive sets zoom out to keep the images to a manageable size while showing the detail in the growing fractal; the largest image in each set becomes the smallest in the next.  (At the scale of the 100,000,000-point image, the 1,000-point image would only be a single pixel.)  Press animate beneath each image to watch the fractal grow.

animate   1,000 – 10,000 points

animate   10,000 – 100,000 points

animate   100,000 – 1,000,000 points

animate   1,000,000 – 10,000,000 points

animate  10,000,000 – 100,000,000 points

Some of the images seem to have their own vague pattern repetition.  For example, the tentacles of the 1,000,000-point image have something of a ball-and-string appearance:

The points are colored with an RGB-CMY rainbow color palette in the order in which they are generated:  red, yellow, green, cyan, blue, magenta.  The images thus tend to be red and yellow on the inside, and blue and magenta on the outside.  But there are exceptions; in the fourth set of images, you can watch the fractal “implode.”