Re: [math-fun] Packing problem
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 ----- 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?
Dan, Rigorous posts don't get no props.
You definitely get props for the pun!
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).
I like the simplicity of your definition, but the elastic cord I have in mind would consist of circular arcs (where the cord is tangent to a roll) alternating with straight segments (where the cord isn't tangent to a roll), whereas your model seems to be that cord is circular, so I think we're talking about different idealizations. Jim Propp
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: https://youtu.be/8byNCTBrgC4 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
----- 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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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
----- 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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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
-----
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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According to the paper (Gravitational instabilities in binary granular materials; Christopher P. McLaren, Thomas M. Kovar, Alexander Penn, Christoph R. Müller, and Christopher M. Boyce; www.pnas.org/cgi/doi/10.1073/pnas.1820820116; unfortunately paywalled): When a “droplet” of heavy particles is surrounded by a sea of light particles (Fig. 4C) the droplet does not fall straight to the bottom as in immiscible fluids, but rather it immediately splits into two daughter droplets which fall on an angle through the light particles and undergo subsequent binary-splitting events. The observed binary splitting of granular droplets exhibits striking similarity to the bifurcations of falling dye droplets in miscible fluids (35–39). However, the physical mechanisms be- hind these two phenomena are different. When a droplet of dye is placed in a miscible fluid, it forms a vortex ring. This ring becomes unstable and fragments into smaller droplets which undergo the same instability and may fragment again (37). The RT instability is suspected to cause the fragmentation of the vortex ring (38, 39). The mechanism underlying the granular binary splitting reported here can be explained as follows: Gas channels around the granular droplet because of the lower per- meability of the heavy particles. The lack of gas flow to suspend the light particles just below the granular droplet causes a region below the droplet to solidify, or become stationary, due to force chains under the weight of the droplet. The solidification of par- ticles below the droplet is confirmed experimentally using particle image velocimetry (40) in Fig. 6. The droplet cannot fall through these solidified particles, but it still can descend through the sus- pended particles, and thus it splits and falls at an angle through the suspended light particles just above the solidified particles. At a certain point, a solidified region forms just below the daughter droplets for the same reason of gas channeling, and thus they undergo a subsequent splitting event. Thus, the mechanism of gas channeling and solidification of particles makes this branching phenomenon distinct from that observed in conventional miscible fluids. So the difference between the two cases (bubble of light particles starting at bottom, bubble of heavy particles starting at top) is differing flow of the injected gas around the two populations. (In case it's not clear, the particles only move because they vibrate the cell containing them *and* blow air through it.) On Wed, Apr 24, 2019 at 9:37 PM <bradklee@gmail.com> wrote:
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
-----
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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So the difference between the two cases (bubble of light particles starting at bottom, bubble of heavy particles starting at top) is differing flow of the injected gas around the two populations.
This is purely geometrical. A spherical obstruction or a wedge will have relatively low drag, and usually mostly laminar flow, but a *concave* wall (droplet is poor word choice in my opinion, and misleading) can be expected to cause a node in the streamline pattern, which then causes scattering. This mechanism has nothing to do with mass difference. Possibly rough reconnection behind a droplet explains why some of the particles get bounced out and stranded alone (how sad). --Brad
participants (5)
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Brad Klee -
bradklee＠gmail.com
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Dan Asimov -
James Propp -
Tom Duff