Totally not getting what you want to say.The large sphere of 96 in circumference has a radius of 15.28.3789 is way too few for a 96 in (circumference) sphere.A large sphere with radius 9.67156 could hold 1074 ping pong balls.
Subfive wrote:
What about this: use volume of sphere equation...with radius of 9.67156. The radius is half of the number of ping balls that will fit in the diameter of a sphere of 96 in.
I get ~ 3789 ping balls fitting in the 96 in sphere.
rekrunner wrote:
If you insist on interpreting "~96" as it was written, why not attempt to solve the problem for a 96.00" sphere (merely 4 significant digits), then express your answer as an approximation ("~") with two significant digits?
you can do that, but it's harder, and the answer is no more precise than mine.
4700 +/- 400 is one significant digit
There is one s.f. in 400, but there are still two s.f's in 4300, 4400, ..., and 5100.
In a large sphere of exactly 20 ping pong balls in diameter, I am able to find 12,582,912 possible arrangements of maximum density sphere packing. Only 3 of those are HCP. FCC is 2 more.
That's interesting. How many of them are unique, i.e. not rotations, mirror images or chiral opposites of other ones?
Furthermore, all of these possibilities I can find are centered at the origin of the large sphere. I can also imagine innumerable offsets in every combination of the x,y,z axes. You can imagine now why I say my search was not exhaustive.
If you are interested, for every one of these arrangements, I can calculate how close the closest ping pong ball was to the large sphere, and how far the farthest one was.
Knowing those distances can spare you from having to consider any offsets from origin at all.
You could build all the patterns starting from the edge of the sphere and consider every unique possibility for which every PPB is within 1 PPB diameter of the edge. Then no translation can add space for another one. At most, moving it to another edge would repeat a pattern already considered from the original starting point.
You are much more interested in sphere stacking than I am. But offhand I wondered if there are helical stacking patterns. Turns out there are
http://en.wikipedia.org/wiki/Boerdijk%E2%80%93Coxeter_helixReducing it to two significant digits doesn't make finding a solution easier. The math is still the same. If anything, bringing undo attention to the uncertainty only serves as a distraction.
Regarding duplicate solutions, I can only imagine two scenarios. Each solution has one mirror along the "y-axis". Furthermore 6144 have a mirror along the "z-axis". I think the rest are unique.
Regarding offsets, you cannot eliminate them so easily. As you move a cluster of tightly packed ping pong balls, with respect to the larger sphere, in effect some ping pong balls will disappear on one side, while others can reappear on the other side. This does not always happen at the same rate. After evaluating all the possibilities, I ran another simulation with all of the possibilities offset by 1/8th of a ping pong ball (down). This resulted in a different layering providing a better solution that was better than my first. This solution is also not symmetric from top to bottom. More balls appeared on the top half, than were clipped on the bottom half.
Move it down by 1/20th of a ball -- 4647.
rekrunner wrote:
Regarding offsets, you cannot eliminate them so easily. As you move a cluster of tightly packed ping pong balls, with respect to the larger sphere, in effect some ping pong balls will disappear on one side, while others can reappear on the other side. This does not always happen at the same rate.
It wouldn't happen at all if you considered only arrangements where all the PPB's actually fit inside the sphere.
There is surely a maximum such that if every point on the edge of the sphere is within that distance of a PPB, no transposition will create room for another one. Construct, from a single PPB at a point on the edge of the sphere, an arrangement of PPB's until each point of the sphere is within that maximum distance from a PPB, and you'll have the maximum possible # of PPB's for that particular arrangement. Then evaluate the possible arrangements for that starting point.
There is no need to consider other starting points because any arrangement that can be constructed from elsewhere has an equivalent that can be constructed from your original starting point. That goes for starting from the origin too - any arrangement can be transposed so that at least one PPB touches the edge, making it equivalent to something you could construct from any point on the edge.
I've only ever considered arrangements with all ping pong balls inside the sphere.I consider offsets, because my initial set of 12.6 million maximum density arrangements, centered at the origin, does not capture all the possibilities. By moving these arrangements, by up to one half of a ping pong ball, in x, y, and z directions I find even more unique possibilities.By instructing us to "construct an arrangement" and then "evaluate the possible arrangements", all you've done is specify that same set of unique arrangements upfront. You haven't reduced the potential solution set.And since the maximum density arrangement seems to be only about 87% efficient, it's not given that a maximum density approach results in a maximal solution. Seems like, even if we found a maximal solution, we would have to rule out the possibility that spacing some balls less maximally could make room for more.
Bad Wigins wrote:
rekrunner wrote:Regarding offsets, you cannot eliminate them so easily. As you move a cluster of tightly packed ping pong balls, with respect to the larger sphere, in effect some ping pong balls will disappear on one side, while others can reappear on the other side. This does not always happen at the same rate.
It wouldn't happen at all if you considered only arrangements where all the PPB's actually fit inside the sphere.
There is surely a maximum such that if every point on the edge of the sphere is within that distance of a PPB, no transposition will create room for another one. Construct, from a single PPB at a point on the edge of the sphere, an arrangement of PPB's until each point of the sphere is within that maximum distance from a PPB, and you'll have the maximum possible # of PPB's for that particular arrangement. Then evaluate the possible arrangements for that starting point.
There is no need to consider other starting points because any arrangement that can be constructed from elsewhere has an equivalent that can be constructed from your original starting point. That goes for starting from the origin too - any arrangement can be transposed so that at least one PPB touches the edge, making it equivalent to something you could construct from any point on the edge.
"^^ Your an idiot, thats just the circumference, you need the volume."
-these blatant displays of pure irony always make me laugh.
rekrunner wrote:
By instructing us to "construct an arrangement" and then "evaluate the possible arrangements", all you've done is specify that same set of unique arrangements upfront. You haven't reduced the potential solution set.
No, but it should make it easier to methodically find all the solutions and identify duplicates.
And since the maximum density arrangement seems to be only about 87% efficient, it's not given that a maximum density approach results in a maximal solution. Seems like, even if we found a maximal solution, we would have to rule out the possibility that spacing some balls less maximally could make room for more.
Partly why I'm not so interested in finding a solution for regular stacking. Why bother finding all the solutions if you already know they're all wrong?
One reason any stacking arrangement is inefficient is the last outer PPB-radius of a 15cm-radius sphere contains so much of its volume:
(15^3 - 14.5^3)/15^3 = .097
close to 10% of the volume of the sphere, and of course the region in which any regular arrangement breaks down leaving a lot of extra space. Figuring out how to move the inner PPB's around to fit more in that edge region is a whole nuther ball of wax from sphere stacking.
Maybe it would be better start with the outer PPB's in a sort of geodesic sphere and build smaller geodesic spheres inward toward the center.
Bad Wigins wrote:
(15^3 - 14.5^3)/15^3 = .097
close to 10% of the volume of the sphere
Bad Wigins meant:
(15^3 - 13^3)/15^3 = .35
about 35% of the volume of the sphere
I forgot a PPB is 4cm in diameter, not 1cm.
I'm looking all over XKCD and can't find the answer.
Well the PPB is actually 4.9634" in circumference (0.79" radius), but I get the point. It's clear that the spherical boundary of the larger sphere results in some necessary inefficiency, not more than ~14-15%. That's what makes this problem complex, compared to cannonball (or shot-put, if you want) stacking.Regarding "it should make it easier to methodically find all the solutions" I'm interested to see a better practical description of this method to find all the (potential) solutions, with equations in some coordinates. With regular sphere packing, it's easy to derive equations for the coordinates of the individual ping pong balls, and then consider x,y,z offsets. It gets harder for irregular deviations, even if it allows us to eliminate considering offsets.I didn't say we "know" all maximal density solutions are wrong. I said we "don't know" they are right. I'm not sure how to show it, but seems like you would have to.I also thought about your "geodesic" outside-in sphere packing approach, but couldn't figure out simple equations to describe it.
Bad Wigins wrote:
No, but it should make it easier to methodically find all the solutions and identify duplicates.
...
Partly why I'm not so interested in finding a solution for regular stacking. Why bother finding all the solutions if you already know they're all wrong?
One reason any stacking arrangement is inefficient is the last outer PPB-radius of a 15cm-radius sphere contains so much of its volume:
(15^3 - 14.5^3)/15^3 = .097
close to 10% of the volume of the sphere, and of course the region in which any regular arrangement breaks down leaving a lot of extra space. Figuring out how to move the inner PPB's around to fit more in that edge region is a whole nuther ball of wax from sphere stacking.
Maybe it would be better start with the outer PPB's in a sort of geodesic sphere and build smaller geodesic spheres inward toward the center.
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