Given two unit-diameter disks tangent to a given line and to each other, determining a region bounded by two circular arcs and a line segment, is the Ford disk packing of that region the unique packing that covers as much total area as possible, among all ways of packing the region with disks tangent to the line?

As usual, disks in a packing must have disjoint interiors. For background on Ford disk packings, see http://en.wikipedia.org/wiki/Ford_circle . Note that I am not asking about Apollonian packings; in my packings, all disks must be tangent to the bounding line segment.

I would also be interested in links to existing literature on other packing problems of a similar nature, where the, um, tiles in the packing (is there a more apt generic word than "tiles"?) are required to be tangent to one of the sides of the region being packed.


There is an affirmative answer to a related question in which we view these disks as horodisks in the upper half-plane model of the hyperbolic plane. In this setting, it is natural to extend the Ford disk packing by applying (integer) horizontal translation and to add one more horodisk to the packing, namely, the shifted upper half plane ${\rm Im} \ z \geq 1$. We superimpose a dissection of the plane into ideal triangles whose edges are geodesics joining the points at infinity of mutually tangent horodisks. Each triangle in this hyperbolic Delaunay triangulation has some finite (hyperbolic) area $A$, and its overlap with the three disks that intersect it has area $3A/\pi$. Laszlo Fejes-Toth, in his 1953 article "Kreisausfullungen der hyperbolischen Ebene" (Acta Mathematica Hungarica 4(1-2), 103-110), showed that $3/\pi$ was the largest fraction of an ideal triangle that can be covered by disjoint horodisks.

If one applies the ideas of Bowen and Radin (see for instance their article "Densest packing of equal spheres in hyperbolic space", https://www.ma.utexas.edu/users/radin/papers/hyper.pdf), one can make rigorous sense of the assertion that these horodisks, taken together, occupy exactly $3/\pi$ of the hyperbolic plane, and that no horodisk packing can beat this bound.

I do not understand Toth's article to the extent of being able to assert that Ford packings are the only packings that achieve Toth's bound, but I expect that this is true.

Thanks to Henry Cohn for pointing out this variant of my question.


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