movie The movie is in .mov format; about 1.6 Mbyte.
A movie of the algorithm running can be helpful in understanding how it works. This movie comes from joint work between myself and Paul Bourke, with my data and Paul's movie-making.
This is a simple circle fractal. In the movie Paul has drawn the circles with edge shading so that they appear as bumps coming out of the plane of the screen. The color is log-periodic, i.e., the color changes in proportion to the logarithm of the circle radius. The color runs through one complete period, with the first and last colors the same. In such a scheme same-size circles have the same color.
One object was to demonstrate the space-filling property. To this end a c value close to the upper limit was used, resulting in a final image with 94% fill. A total of 41,842,579 trials were made to place 1001 circles, for an average of about 42,000 trials per placement. The achievement of high percentage fills for a small number of circles requires such long searches. c=1.48, N=3, fractal D=1.35. No halting was seen in many runs.
The arrangement of circles has many "holes" with three nearby circles. When one of these is filled it eliminates one hole and creates three new ones. A "just in time" principle can be seen -- a hole is filled almost as soon as the next-to-be-placed circle is small enough to fit in. This can be explained in terms of the dimensionless gasket width described in the Shier-Bourke paper in Computer Graphics Forum: paper
The few largest circles occupy a very large fraction of the area. The largest circle has 5450 times larger area than the smallest (74 times larger diameter). This is typical of geometric fractals, and of fractal distributions found in nature (e.g., the Gutenberg-Richter law for earthquake energy release).