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.”