The problem: These web pages are devoted to the following packing question in two dimensions. What is the densest packing of the plane using discs of radius 1 and r? We are allowed to use any number of each. The above links lead to the following: Other problems: In the problem we consider we do not fix the relative density of the two sizes of discs. An interesting question which we do not consider is what is the densest packing if we add the constraint that the ratio of the number of discs of radius 1 to the number of discs of radius r must be equal to some fixed number. This question was studied extensively by Likos and Henley. [LH93] Another interesting question we do not consider is what is the densest packing if we are allowed to use discs with any radius between r and 1.

Goal: The focus of these web pages is not on rigorous results . (There are few for the question we consider.) Our goal is to present the best packings we know for different ranges of r. Certain special values of r admit particularly good packings and the density is expected to have a local maximum at these values of r. Away from these values of r we have considered various perturbations of the special packings and present the ones that are best. The result is a plot of the density as a function of r.

I do not claim that these are original results. Most (if not all) of the packings here can be found elsewhere, but I have not seen a comprehensive summary of what is known. Hence the web pages. I am sure there are some values of r for which there are better packing that I do not know about. I would appreciate finding out about such packings and will incorporate them into the web pages.

Compact packings: A packing is said to be compact if every disc D in the packing is surrounded by a sequence of discs D1,D2,...,DN each of which is tangent to D and such that each disc in the sequence is tangent to the discs before and after it in the sequence. (D1 and DN must also be tangent.) Kennedy [Ken04] has proved that if we use discs with radius 1 and r<1, then there are only nine value of r for which compact packings exist. (r=0.637...,0.545...,0.533...,0.414...,0.386..., 0.349...,0.280...,0.154...,0.101...) Seven of these compact packings appear in Fejes Toth's book Regular Figures. The packing with r=0.386... appears in Likos and Henley. [LH93] I have not found the compact packing for r=0.545... in the literature, but surely someone has found this packing before. All of these compact packings can be accessed through the link Packings . Clicking on the value of r will display the packing and information about it.

Near compact packings: There are some highly symmetric packings which are not compact but are particularily efficient and probably give a local maximum for the density. Pictures are given for r=0.216..., 0.183..., 0.0820..., 0.0717..., 0.0571.... There are two symmetric packings for r=0.619..., 0.233... which are shown but have a packing density lower that than of the triangular packing. All of these packings can be accessed just as the compact packings by following the Packings link.

Perturbed packings: In between the values of r listed above we consider perturbing the packings for the above special values of r. For a given compact or near-compact packing we can consider perturbations for r slightly above or slightly below the special value. We have searched for possible perturbations by linearizing around the compact or near-compact packing and solving the resulting linear programming problem. Often there are several perturbations that give the same density at first order. We compute the exact density to determine the best one. These best packings can be seen by clicking on the links "above" and "below". We caution the reader that to make the difference between the perturbed packing and the original packing visible we must change r by an amount that is often larger than the distance to the next special value of r. So the packings shown for illustration purposes may in fact correspond to a value of r for which they are not optimal.

Small values of r: For extremely small values of r, one can put the large dics in the triangular arrangement and fill in the holes with patches of triangular arrangements of the small discs. As r goes to zero, the resulting density will approach D+(1-D)D= 0.991332331, where D is the single species packing density (D=pi/sqrt(12)). However, determining the optimal packing for small values of r is an infinitely complicated problem. Given a small value of r there is some number of small discs, n(r), that can fit into the holes formed by the large discs. We expect that at the values of r where n(r) increases, the density will have a local maximum. For r slightly smaller, the optimal packing should be to keep the large discs on the triangular lattice and so the small discs will "rattle" in the hole. For r slightly greater than the value where the number jumps, we expect the optimal configuration is to perturb the triangular lattice for the large discs slightly to allow n(r) small discs to fit into the hole. This is all speculation. The problem of determining the values of r at which n(r) increases and the corresponding configuration is quite non-trivial. It is similar to the question of determining how many discs of given radius one can fit in a unit square. Our list of packings includes packings with 1,3, 4, 6 and 10 small discs in the hole. These correspond to symmetric packings. We have not attempted to find the values of r for which n(r) takes on other values. Thus for r below 0.1, we plot points only for the values of r corresponding to n=4,6 and 10.

Discussion: The resulting plot of the density as a function of r shows the following features. The compact and near-compact packing are local maxima and the cusp at these local maxima is sharper than one might have expected. In the plot the density is equal to the single species packing density for 0.645707215916564<= r <=1. This has been proved for r in the interval [0.742,1]. In the plot there is one more interval of r in which the best know density is just the single species density. We emphasize that this plot only gives the best known packing density as a function of r. Pictures of a few packings are given for r less than 0.1, but the plot of the density only shows values of r greater than 0.1. As r goes to zero the density converges to 0.991... and we expect the density to have infinitely many local maxima and minima as we discussed above. We do not know of any better packings for r>0.1. We know there are no other compact packings, but there could be another "near-compact" packing like that for r=0.216 which we have not found. All of these best known packings are periodic, but there is no proof that the optimal packing must be periodic.

Author: Tom Kennedy, Mathematics Department, University of Arizona, email: tgk@math.arizona.edu Comments are welcome. In particular, I would be very interested in any packings that do better than those given here.

Acknowledgement: This work was supported by the National Science Foundation (DMS-0201566, DMS-0501168). Disclaimer