These movies have been constructed to help mathematicians understand the algorithm and the halting problem. They show circles fractalized within a circle with N=1 and various c values. Familiarity with the basic material in the Shier-Bourke paper is assumed.
You should think of the pattern as beginning with a black disc, which is the region to be filled. Each new circle is white, and it is surrounded by a red band whose width is the radius of the next-to-be-placed circle. The bounding circle also has such a red band extending inward. Thus at each stage the remaining black region(s) is the only place where the center of the next circle can go. One can use this formulation to re-state the halting problem: The algorithm never halts if it can be shown that the area of the black region(s) is always > 0.
The framing rate has deliberately been made rather slow so that you can look around a bit before the next circle appears. Each movie shows 25 circle placements.
The black area diminishes with addition of a new circle which takes up some of the space. At the same time, existing black regions grow because the radius of the next-to-be-placed circle is steadily falling according to the defining negative-exponent power law (the "area rule"), and new black regions appear. The total black area divided by the area of the bounding circle is the placement probability at the given stage of the algorithm. The total black area may go up or down from one placement to the next in a quite noisy way.
It should be kept in mind that the mathematics of the area rule dictates that in all of these movies the placed circles are "space-filling in the limit" if the algorithm does not halt.
c=1.1. This is a "low density" example. There are large connected black regions at all stages. It would seem intuitively obvious that the algorithm does not halt, but this is far from a formal proof. movie
c=1.2. It can be shown that if c > 1.20977... (N=1) the algorithm must halt if circle 0 is placed at the center of the bounding circle, i.e., the algorithm cannot be unconditionally nonhalting. It is conjectured that the algorithm is unconditionally nonhalting with N=1 for 1 < c < 1.20977... . movie
c=1.3. If circle 0 falls near the center of the boundary, the algorithm halts because there is no place for circle 1. Halting runs are rare. It is conjectured that the algorithm is conditionally nonhalting if placement of circle 1 is successful. movie
c=1.4. Halting events are more common. The total black area is much smaller than for lower c values. movie
c=1.45. Most runs halt; this one is a "survivor". This is close to the highest usable c value for N=1. It can be seen that the black area is quite small, and at many points in the movie there are only a very few tiny black regions. The boundary fills rapidly, and circle-to-circle spacings are small. Experience shows that if a run does not halt in the first few placements it does not halt at all. movie
