Re: [math-fun] Kissing number (again).
Oops! I wrote:
Jud McCranie wrote:
Consider the new ball touching the center ball and is in contact with ball A.
You can't assume that. It might be in touch with just one of the previously-placed balls, and you can't even guarantee which one.
I'm sorry, I misread "center ball" as one of the ones you placed. You can indeed choose the new ball so it is touching the kissed ball and one previously placed ball A. But to do even that, you have to prove at least part of the theorem from my last message.
Swing it one way (staying in contact with ball A and the central ball) until it touches ball B. Swing it the other way until it touches ball C. Now would it be the case that only certain areas in that swing need to be considered? Would it be that, say, any position from 0 to 0.1x is essentially the same because it makes no essential difference in how the rest of the balls are placed, and 0.1x to 0.36x are essentially the same, etc? If so, that would be how it can be reduced to a finite number of cases.
You really do have to consider the whole arc. There is a way to make the number of cases finite, and that is to say, "The new ball G is at some unspecified place on this epsilon-arc." You can't say that it makes "no essential difference" where it is, but you can say the difference it makes is no greater than the diameter of the set of places it could be. In the future, when you're "swinging a newer ball around G," you will have to take account of not really knowing where you placed G, so the space of possible tangencies to G has positive area, and you have to say "some unspecified place in this epsilon-patch" instead of "along this epsilon-arc". And at the end, you won't usually have a packing, because you have to overpack to take account of the uncertainties. All you can say is that, if you end up with a sufficiently great amount of overpacking, that there couldn't have been a packing of that many balls in the first place. Dan
On Fri, 26 Sep 2003, Dan Hoey quoted and wrote:
I'm sorry, I misread "center ball" as one of the ones you placed. You can indeed choose the new ball so it is touching the kissed ball and one previously placed ball A. But to do even that, you have to prove at least part of the theorem from my last message.
Swing it one way (staying in contact with ball A and the central ball) until it touches ball B. Swing it the other way until it touches ball C. Now would it be the case that only certain areas in that swing need to be considered? Would it be that, say, any position from 0 to 0.1x is essentially the same because it makes no essential difference in how the rest of the balls are placed, and 0.1x to 0.36x are essentially the same, etc? If so, that would be how it can be reduced to a finite number of cases.
Why are you even talking about swinging just the one ball C? There might well be other balls D,E,F,... in the way.
You really do have to consider the whole arc. There is a way to make the number of cases finite, and that is to say, "The new ball G is at some unspecified place on this epsilon-arc." You can't say that it makes "no essential difference" where it is, but you can say the difference it makes is no greater than the diameter of the set of places it could be. In the future, when you're "swinging a newer ball around G," you will have to take account of not really knowing where you placed G, so the space of possible tangencies to G has positive area, and you have to say "some unspecified place in this epsilon-patch" instead of "along this epsilon-arc". And at the end, you won't usually have a packing, because you have to overpack to take account of the uncertainties. All you can say is that, if you end up with a sufficiently great amount of overpacking, that there couldn't have been a packing of that many balls in the first place.
Dan's "finite number of cases" is correct only in a rather useless sense. The trouble is, that there's no way to tell whether or not there's a solution in any one of his "cases". I have a much simpler way of making the number of cases finite, which is simply to consider the whole problem as just one case! Unfortunately, it suffers from the same defect. John Conway
participants (2)
-
Dan Hoey -
John Conway