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Article #131Re: A Conjecture About Packings of Balls
From: Schramm Oded
Date: Fri, 13 Nov 1992 09:59
Date: Fri, 13 Nov 1992 09:59
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In article <9211082352.AA24164@zaphod.uchicago.edu> greg@math.uchicago.edu (Greg Kuperberg) writes: >In article <9211060833.AA05802@wisdom.weizmann.ac.il> you write: >>For a finite ball packing P in R^3 [with balls round but not >>necessarily congruent], let >>a(P)=2(number of tangencies)/(number of balls) >... >>Conjecture: A = sup a(P) = 12. >... >>I think I can improve the naive argument above to get something >>better than A<=24, but I don't expect to reach A=12. > >Your expectation is true and your conjecture is false. :-) ... >which yields a better kind of grout. If you do this once you get a >packing with average kissing number 7152/581 > 12.309, and if you >iterate, the supremum is (I think) 3486/283 > 12.318. ... Thanks G. Kuperberg. Without much sweat I can show that A <= 8 + 4 Sqrt[3] = 14.9282... Unless there is some miraculously ingenious idea, I expect that better and better estimates will become very difficult. Anyway, the actual value of A is not as interesting as the near-optimal packings. Do you think, G. K., that your construction gives the true A? Oded Schramm schramm@wisdom.weizmann.ac.il
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arc1-ca.pugleaf.net!sparky!uunet!zaphod.mps.ohio-state.edu!moe.ksu.ksu.edu!ux1.cso.uiuc.edu!news.cso.uiuc.edu!danReferences:
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Newsgroups: sci.math.research Path: arc1-ca.pugleaf.net!sparky!uunet!zaphod.mps.ohio-state.edu!moe.ksu.ksu.edu!ux1.cso.uiuc.edu!news.cso.uiuc.edu!dan From: schramm@wisdom.weizmann.ac.il (Schramm Oded) Subject: Re: A Conjecture About Packings of Balls References: <9211082352.AA24164@zaphod.uchicago.edu> Message-ID: <9211130959.AA18671@wisdom.weizmann.ac.il> Originator: dan@symcom.math.uiuc.edu Summary: The average number of contacts is <= 8 + 4*3^(1/2) Sender: Daniel Grayson <dan@math.uiuc.edu> X-Submissions-To: sci-math-research@uiuc.edu Organization: Weizmann Institute of Science X-Administrivia-To: sci-math-research-request@uiuc.edu Approved: Daniel Grayson <dan@math.uiuc.edu> Date: Fri, 13 Nov 1992 09:59:41 GMT Lines: 28