covering disks with smaller disks - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:23:11Z http://mathoverflow.net/feeds/question/97939 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97939/covering-disks-with-smaller-disks covering disks with smaller disks Oliver Roche-Newton 2012-05-25T13:16:05Z 2012-05-25T17:29:51Z <p>How many disks with radius 1/2 are needed to cover a disk with radius 1? It certainly cannot be done with less than 5 small disks, and some non-rigorous drawings of mine suggest it can be done with 7 small disks. Can it be done with less? </p> <p>Slightly more generally, can anybody point me in the direction of any work on covering bounded bodies with disks? Am i right in thinking that covering disks with smaller disks is the most inefficient kind of covering?</p> http://mathoverflow.net/questions/97939/covering-disks-with-smaller-disks/97943#97943 Answer by Joel Reyes Noche for covering disks with smaller disks Joel Reyes Noche 2012-05-25T13:42:18Z 2012-05-25T13:42:18Z <p>You'll need seven disks with $r=0.5$ to cover a disk with $r=1$. See <a href="http://mathworld.wolfram.com/DiskCoveringProblem.html" rel="nofollow">http://mathworld.wolfram.com/DiskCoveringProblem.html</a>.</p> http://mathoverflow.net/questions/97939/covering-disks-with-smaller-disks/97944#97944 Answer by Igor Rivin for covering disks with smaller disks Igor Rivin 2012-05-25T13:44:58Z 2012-05-25T13:44:58Z <p>A more general version of this question is discussed (and answered) in <a href="http://www.cs.uwm.edu/faculty/ad/disks.pdf" rel="nofollow">this paper by Dumitrescu and Jiang</a></p> http://mathoverflow.net/questions/97939/covering-disks-with-smaller-disks/97945#97945 Answer by Joseph O'Rourke for covering disks with smaller disks Joseph O'Rourke 2012-05-25T13:50:06Z 2012-05-25T13:50:06Z <p>Just to help think about the problem, here is the natural cover by seven $\frac{1}{2}$ disks: <br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img src="http://cs.smith.edu/~orourke/MathOverflow/HexCover.jpg" alt="Disk cover"><br /></p> http://mathoverflow.net/questions/97939/covering-disks-with-smaller-disks/97961#97961 Answer by Zsbán Ambrus for covering disks with smaller disks Zsbán Ambrus 2012-05-25T17:29:51Z 2012-05-25T17:29:51Z <p><a href="http://www2.stetson.edu/~efriedma/circovcir/" rel="nofollow">Erich Friedman's packing center claims</a> that you can't cover with 6 disks, and that this was proved by Károly Bezdek in 1979. If you want a more exact reference, ask Erich Friedman in email.</p>