  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
Regular Figures (MR0165423), L\'aszl\'o Fejes T\'oth 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 $1 \ge q >
0.72$, the most dense packing possible is $\pi/\sqrt{12}$, which is when all
the disks are the same size. In Regular Figures, the upper bound of the ratio
$q$ such that the density of his packings is greater than $\pi/\sqrt{12}$ that
Fejes T\'oth 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 has three distinct sizes of disks, found by Fernique,
Hashemi, and Sizova (MR4292755), which is something of a surprise.
