Usually disk packing problems require that no two disks of the packing intersect.
Does anybody know if the problem has been studied when disks may intersect but they are not allowed to contain the center of any other disk?
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6$\begingroup$ In a collection of closed disks of radius $r$, disks do not contain the centre of any other disk iff the disks of radius $r/2$ with the same centres are disjoint. $\endgroup$– Robert IsraelCommented Oct 29, 2012 at 15:46
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$\begingroup$ In my experience, disk-packing problems often allow two disks to share one point of tangency, and many of the recent computer generated results on disk packing take this into account. Gerhard "Ask Me About Circle Packing" Paseman, 2012.10.29 $\endgroup$– Gerhard PasemanCommented Oct 29, 2012 at 15:54
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$\begingroup$ What kind of questions are you interested in? $\endgroup$– Igor RivinCommented Oct 29, 2012 at 17:02
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$\begingroup$ @Igor Rivin I was just curious. Since packing problems received huge attention it amazed me that this version was not explicitly studied. But I guess Robert Israel's comment explains why. $\endgroup$– marcCommented Nov 5, 2012 at 6:47
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2 Answers
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Yes. Here is an example: Rigidity of infinite disk patterns by Zheng-Xu He.
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There is a subject called "circle packing". There are many exciting results, including the one mentioned in the previous answer. The subject originates from a Thurston conference talk. The discs bounded by circles are sometimes allowed to intersect. See, for example the book of Ken Stephenson, Introduction to circle packing, MR2131318.
answered Oct 29, 2012 at 22:13