Hi Jim, Ahh yes, the older, much older, question, of yin and yan, and “why don’t the two behave exactly as one?” Surely I don’t know, but I can at least try out a guess. HYPOTHESIS: It’s mostly in the initial conditions. The bifurcation experiment starts with a flat wavefront sheet whereas the surfacing experiment starts with a curved wavefront bubble. What would happen with a wedge, I wonder? Or even with a single particle? But I am skeptical that the bifurcation really is a fractal. Actually the leading effect looks more like reflection in a waveguide to my eyes. If you pay careful attention, some of the smaller bodies in the bifurcation experiment seem to fall mostly straight down. As in the rising experiment they leave stranded dots behind them. This seems to support the hypothesis. The stranded particles intrigue me most of all. How do they relate to the leavings of a puffer in Conway’s game of life? Cheers, Brad
On Apr 24, 2019, at 9:19 PM, James Propp <jamespropp@gmail.com> wrote:
That's a really fun video! I especially liked the fractal bifurcation at the end.
I don't understand the asymmetry between heavy and light particles. Why do upward-falling bunches of light particles stick together, while downward-falling bunches of heavy particles split and re-split?
Jim
On Tue, Apr 23, 2019 at 11:34 PM Brad Klee <bradklee@gmail.com> wrote:
Hi Dan,
No, I think the diameter condition is imprecise. The principle energy contribution due to elastic stretching should depend on an arc-length integral along the cord.
With your example of the empty rosette, collapsing one of the outer tubes to the center obviously lessens the cord's arclength, so the configuration is globally instable despite resilience to infinitesimal perturbations. The 1-1-epsilon "shish kebab" configuration is also globally instable. Work from local minimum to global minimum then depends on normal forces and friction coefficients as well as dynamical changes to the elastic binding energy.
If you think stability theory leads to fun math, you may be right. I happened to watch a really nice experiment video today:
The theory behind this could certainly draw from the genre of packing problems, but would also need to emphasize the importance of work due to external forces. In each of the experiments, gravitational and vibrational forces move the ensemble toward the global minimum of the gravitational potential energy.
Per the original statement, gravitational and contact forces will certainly affect dynamics. A fall-out event requires gravity and some amount of moving or jostling the ensemble. Thus, stability to small perturbations may not be a strong enough requirement.
--Brad
On Tue, Apr 23, 2019 at 6:44 PM Dan Asimov <dasimov@earthlink.net> wrote:
Rigorous posts don't get no props.
My previous attemptedly rigorous post on the subject was just to say, I guess opaquely, that it seems to capture the essence of a stable arrangement — like one forced by an elastic collar — to merely require:
any sufficiently small perturbation of the arrangement (modify the centers but not the radii) cannot decrease the *diameter* of the arrangement (= the diameter of the smallest disk containing all disks of the arrangement).
It seems clear that a rosette of 7 equal disks, but with the central one removed, is stable in this sense (despite the hole).
—Dan
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On 4/23/19, Allan Wechsler <acwacw@gmail.com> wrote: I don't mind any particular model, but I think I just don't understand the no-slip condition. Can you rigorize it?
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