A Million-Circle Fractal

The "doesn't halt" behavior of the algorithm is a central feature, but lacking any formal proof it may be difficult for people to convince themselves that it is true. One of the supporting cases is that of a very large fractal with a simple shape -- a circle. I have generated such a million-circle fractal with the following parameters:

c=1.30; N=1; 97.72% fill

The fractalization "frame" is 24x24 inches (or other units). Run time was 14.7 hours. How do you represent such a fractal? The biggest circle has radius 6.828 inches, while the smallest has radius .00086 inches, for a ratio of 7939:1. Such a large variation is extremely difficult to show graphically. Reasonably good resolution of the smallest circles would call for about 4000 pixels/inch, for a total image size of 96000 pixels on a side. Such a huge image is quite unwieldy to print or view.

One answer to this is a zoom approach. Paul Bourke has done this with the data listed here. Scroll down to see this image.

It was decided that scholars in mathematics, computer science, etc. could best be served by providing a text file with complete high-resolution (single precision) data. The file available here gives the image size and number of circles on the first line, followed by a million data lines, one for each circle. Each line gives the x position of the circle's center, the y position, the radius, and the cumulative number of trials needed for placement of that circle. The number of trials needed to place a given circle can be found by subtracting the cumulative trials for the previous data line. A total of 1 690 697 421 trials was needed.

Interested persons can download this file and read it into a variety of graphics and/or statistics programs to carry out their own further investigations of its properties.

The text file of the complete data can be inspected or downloaded -- click here. This is a very large file -- 40 Mbyte.

In the image below, the upper left part shows a complete view of all the data, with the circles shaded like hemispheres for easier grasp. In this image the smallest features are far too tiny to be seen. To the right is an approximate 10x expansion of the indicated region, and below this is another approximate 10x expansion of the expanded region. In this last view you can see the individual very small last-placed circles. It can be seen that even after a million circles have been placed (at almost 98% fill), there is still plenty of room for more circles of the size called for by the rules.