In Euclidian space, one sphere can be touched by how many equal-sized spheres simultaneously? Intuitively, the answer is 12. Is there a (geometrical) proof of this?
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1Have a look a these pages: https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres, https://en.wikipedia.org/wiki/Sphere_packing – Thomas Aug 28 '15 at 09:12
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As size of touching sphere decreases, their number increases indefinitely. – Narasimham Aug 28 '15 at 09:21
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@Narasimham: It is true that "equal-sized" is, strictly speaking, ambiguous here: equal to each other, or equal to the original sphere? But it's obvious that the second interpretation is intended. – TonyK Aug 28 '15 at 09:33
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"Same" sized" may reduce the ambiguity by 50 percent.. :) – Narasimham Aug 28 '15 at 09:37
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4Very famous problem. Newton and Gregory argued about it – Newton said, 12, Gregory, 13. See https://plus.maths.org/content/newton-and-kissing-problem – Gerry Myerson Aug 28 '15 at 10:08
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So, Thomas, have you followed the links? – Gerry Myerson Aug 29 '15 at 22:38
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I have followed the links, Gerry. Now I am thinking about the more general kissing problem involving surrounding spheres that are equal in size but perhaps larger or smaller than the central one. The "shadow" method would tell us how many can "theoretically" be accommodated (but notice that it's not straightforward: if you halve the radius of the surrounding spheres, the number you can theoretically accommodate less-than-quadruples because these surrounding spheres also move closer to the center). But then we face the hard problem of how much of this theoretical space can actually be used.... – Thomas Pogge Aug 30 '15 at 05:28
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1While you are thinking about the more general problem, maybe you could post an answer to the particular problem in the question, based on what you have learned by following the links. Oh, and if you want to be sure I see a comment addressed to me, you have to write @Gerry in it somewhere. – Gerry Myerson Aug 30 '15 at 23:42