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Register a new userMaximally Dense Disc Packings on the Plane
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Suppose one has a collection of disks of various sizes with disjoint interiors, a packing, in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book textit{Regular Figures} (MR0165423), Laszlo Fejes Toth found a series of packings that were his best guess for the maximum density for any $1> q > 0.2$. Meanwhile Gerd Blind in (MR0275291,MR0377702) proved that for $1ge q > 0.72$, the most dense packing possible is $pi/sqrt{12}$, which is when all the disks are the same size. In (MR0165423), the upper bound of the ratio $q$ such that the density of his packings greater than $pi/sqrt{12}$ that Fejes Toth found was $0.6457072159..$. Here we improve that upper bound to $0.6585340820..$. Our new packings are based on a perturbation of a triangulated packing that have three distinct sizes of disks, found by Fernique, Hashemi, and Sizova, (MR4292755), which is something of a surprise.
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