Still, I have no answer to many of the very interesting questions posed here. But I can answer this one:
Bram would probably respond: "OK, and how about a 'master' version, with concentric circles".
As far as jumbling goes, I can confidently say that concentric circles will make no difference whatsoever. All we are adding is the orbits of the points on a smaller circle. If the larger circle is at the jumbling threshold, we already know that for smaller R there is no jumbling. And the additional moves (inner vs outer circle) won't really add anything. We are still exploring, for each point in the orbit, all possible moves of rotation by the given angle about each disk center.
So... turns out I am giving a talk on jumbling circle puzzles this weekend. I have a lot of material, and only 15-20 minutes to fill, so I should be fine, but I am kind of stressing out. I would like to have some movies, zooming in and/or changing R, which will be very computationally intensive to generate (but I may try), and I would like to have answers to some of the remaining questions. In particular I'm still unsure about the significance of finding quasicrystals and aperiodic tilings here. It may be that the Penrose tiles for N = 5 are a special case... I am going to try to generate diffractograms for some > critical images for other N. That should disclose quasicrystalline structure definitely. But, again I may not have time for this.
Oh, I do have one new thing to mention here. I talked about two ways of exploring critical R values when we are allowed to vary N and R between the disks: (1) keep R1 = R2, then for any given N1, N2, there is again a critical R we can find, and (2) set R1 = minimum critical R for any R2, and R2 at the minimum critical R for that R1. In both cases, we have a clear definition of a jumbling transition, again generating a discrete family of fractals, two (both conditions) for each N1, N2.
But in general, for any given R1 which admits jumbling at all, there will be a minimum R2 which jumbles. So really we have, for any N1, N2, a 2d plot of what the jumbling transition looks like. Sort of like a pressure-temperature phase diagram in physical chemistry.
Creating such a diagram here is very tedious -- resolving the jumbling transition can take a lot of time and careful adjusting of search parameters. But I have started to fill in one such diagram, for N1 = 3, N2 = 5:
R1 is horizontal, R2 is vertical. Each plotted point is a critical (R1, R2) pair, generating a fractal image. Stepping through the images in quick succession, it looks like seeing successive slices in a 3d structure. The interesting parts are the sharp transitions. I don't totally understand them, but it seems to be that different points on the circle hit the fractal transition at different R. But generally, when you are just past a critical R, the generating point quickly starts to fill the space, and smear out, obscuring any fractal generated by another point. Which suggests there are potentially lots of other fractals hiding in here that I won't find by the methods I've used so far. A little more work here should clarify this important point.
The points define a curve: on one side (left/below) we have non-jumbling, discrete behavior; on the curve we have fractals; and above/right of the curve we have quasicrystalline structure.
Oh, and one more thing. I was discussing this recently with another computer scientist at a Gardner Celebration of Mind event, and he pointed me to the wonderful book Indra's Pearls, by Mumford, Series, and Wright:
I have been reading this for the past few days. This is kind of exciting, because the situations they analyze are sort of similar, but not the same as what we are doing. If our results here were known, I think they would likely have been mentioned in this book. We explore the orbits generated by turning two intersecting disks; they explore the orbits generated by two MÃ¶bius transformations of a particular kind. Actually I had already considered generalizing our problem beyond rotating circles to using general MÃ¶bius transformations! However, they use continuous transformations, of the entire plane; we use discontinuous transformations, and it is the points on the circle boundaries that are interesting. We can also imagine using MÃ¶bius transformations that "slice" along some curves, the points on one side moving, the points on the other not moving; that is what I had been wondering about. What we have now is in fact a special case of this more general problem.