Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 36 (1995), No. 1, 145-148.

On the Radial Ratio in Dense Mixed Sphere Packings

Luis G. De Lucía

Auburn University, Department of Mathematics P.O. Box 82, 228 Parker Hall, Auburn, AL 36849, U.S.A. E-mail: deluclg@mail.auburn.edu

Abstract. Packings of E^{3} are constructed with balls of two radii, 1 and r < 1, with density higher than the sphere lattice packing density (δ_{L}^{3} = \frac{π}{\sqrt{18}}) for every r in the interval (0.482 …, 0.577 …].

Troughout this note, we will work in the Euclidean 3-space, E^{3}. A packing of E^{3} is a collection of compact and convex subsets of E^{3} with mutually disjoint interiors. The volume of a set K will be denoted by Vol(K), and the convex hull of a set of points S by Conv(S).

Intuitively, the density of a packing P is the proportion of space occupied by the elements of the packing. Let B(ρ) be the ball of radius ρ > 0 centered at the origin O. Let δ(P, B(ρ)) denote the ratio of the sum of volumes of those elements of P which are contained in B(ρ) to the volume of B(ρ). Then the density of the packing P is given by

δ(P) = lim sup_{ρ→∞} δ(P, B(ρ))

For a detailed discussion of densities of packings see [5].

For a fixed r ∈ (0, 1], a mixed sphere packing of E^{3}, is a packing consisting of balls of radius 1 and balls of radius r. A mixed sphere packing with balls of radius 1 and r will be denoted by P_{r}.

For each r ∈ (0, 1] we define the optimum density δ(r) as the supremum of δ(P_{r}) taken over all mixed sphere packings P_{r}. For completeness’ sake, we define

δ(0) = lim sup_{r→0} δ(r)

In the plane there are mixed circle packings with density higher than the circle packing density (δ^{2} = \frac{π}{\sqrt{12}}) for all r < 0.645 … [8]. Furthermore it is known that no such packing exists for r > 0.74299 [1]. However, much less is known about the analogous problem in space. Let δ^{3} denote the density of the densest packing of E^{3} with unit balls, and δ_{L}^{3} denote

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1995 Heldermann Verlag, Berlin

the density of the densest lattice packing with unit balls. It is known that δ_{L}^{3} = \frac{π}{\sqrt{18}}, and it is conjectured that δ^{3} = δ_{L}^{3}. Furthermore this conjecture may be strengthened to say that δ(r) = δ_{L}^{3} for values of r sufficiently close to 1, as suggested by the case in two dimensions.

For this reason it is of interest to find the values of r ∈ [0, 1] for which δ(r) > δ_{L}^{3}. It is easy to show that δ(r) > δ_{L}^{3} for r ∈ [0, 0.444 …). Namely, when r = 0, L. Fejes Tóth’s result in the plane [7] generalizes to give δ(0) = δ^{3}(2 − δ^{3}) > δ^{3} ≥ δ_{L}^{3}. For r ∈ (0, \sqrt{2} − 1], density higher than δ_{L}^{3} is achieved by placing the centers of the small balls in the “deep holes” (see [3]) of the densest lattice packing of balls of radius 1. Leaving all the centers fixed in the previous packing, we can increase the radius of the small balls and decrease the radius of the balls of radius 1 simultaneously so that they do not overlap. Then we notice that the density is larger than δ_{L}^{3} as long as the ratio of these radii remains smaller than 0.444 . . ..

Recently Vassallo and Wills [12] showed that δ(r) > δ_{L}^{3} for r ∈ [0.577 …, 0.623 …). Although to find mixed sphere packings P_{r} for r ≥ 0.623 … for which δ(P_{r}) > δ_{L}^{3} remains the main problem, Vassallo and Wills point out that it is not known wether this can be done in the “gap” 0.444 … ≤ r < \frac{\sqrt{3}}{3} = 0.577 . . ..

It is the aim of this note to narrow this gap by means of the following

Theorem. If r is in the interval ((\frac{1}{2}(\sqrt{\frac{3}{2}} − 1))^{\frac{1}{3}} , \frac{\sqrt{3}}{3}] = (0.482 …, 0.577 …] then there is a mixed sphere packing with balls of radius 1 and balls of radius r with density higher than δ_{L}^{3}.

Proof: For h ≥ 1, let x_{1} = (\frac{−1}{\sqrt{3}}, 1, h), x_{2} = (\frac{−1}{\sqrt{3}}, −1, h), x_{3} = (\frac{2}{\sqrt{3}}, 0, h), x_{4} = (\frac{−1}{\sqrt{3}}, 1, −h), x_{5} = (\frac{−1}{\sqrt{3}}, −1, −h), and x_{6} = (\frac{2}{\sqrt{3}}, 0, −h), and let T_{h} =Conv(\{x_{1}, …, x_{6}\}). For each i ∈ {1, 2, …, 6}, let B_{i} denote a sphere of radius 1 centered at the point x_{i}. Clearly the interiors of these spheres are pairwise disjoint. Let B_{7} denote a sphere of variable radius r centered at the origin O.

For 0 < r ≤ \sqrt{\frac{7}{3}} − 1 and h = 1, we have that B_{7} ⊂ T_{1} and B_{7} does not overlap any of the other spheres. Consider the intersection of T_{1} with the seven spheres, and observe that translates of T_{1} and its reflection −T_{1} tile E^{3} in a facet-to-facet manner. This tiling produces a periodic mixed sphere packing P_{r}.

Since twelve copies of T_{1} are placed around a vertex, and there are spheres of radius 1 at each of the six vertices of T_{1}, we have that Vol(T_{1} ∩ (B_{1} ∪ … ∪ B_{6})) = \frac{1}{2}(\frac{4π}{3} ). Also, B_{7} ⊂ T_{1} implies Vol(T_{1} ∩ B_{7}) =Vol(B_{7}) = \frac{4π}{3} r^{3}. As well, Vol(T_{1}) = 2\sqrt{3}. Thus we have

δ(P_{r}) = \frac{\frac{1}{2}(\frac{4π}{3} ) + \frac{4π}{3} r^{3}}{2\sqrt{3}}

Easy calculations then show that δ(P_{r}) > δ_{L}^{3} for (\frac{1}{2}(\sqrt{\frac{3}{2}} − 1))^{\frac{1}{3}} < r ≤ \sqrt{\frac{7}{3}} − 1.

For \sqrt{\frac{7}{3}} − 1 ≤ r ≤ \frac{\sqrt{3}}{3} and h = ((r + 1)^{2} − \frac{4}{3})^{\frac{1}{2}} ≥ 1, we have that the sphere B_{7} lies in T_{h} and it does not overlap any of the other spheres. Therefore, using a tiling with translates of T_{h} and −T_{h}, spheres of radii 1 and r can be arranged in a periodic packing P_{r} of E^{3}.

Now we only need to add the observation that Vol(T_{h}) = 2h\sqrt{3} = 2\sqrt{3(r + 1)^{2} − 4}. Hence we have

δ(P_{r}) = \frac{\frac{1}{2}(\frac{4π}{3} ) + \frac{4π}{3} r^{3}}{2\sqrt{3(r + 1)^{2} − 4}}

In finding the values of r for which δ(P_{r}) > δ_{L}^{3} we arrive at the inequality

8r^{6} + 8r^{3} − 3r^{2} − 6r + 3 > 0

Let f(x) = 8x^{6} + 8x^{3} − 3x^{2} − 6x + 3. Observe that the line segment for \frac{1}{2} ≤ x < \frac{3}{4} tangent to f at the point (\frac{1}{2}, \frac{3}{8}) lies above the x-axis. Since f′′(x) > 0 in this interval, the latter inequality is true for all values of r in the interval [\frac{1}{2}, \frac{3}{4}).

In conclusion, δ(P_{r}) > δ_{L}^{3}, whenever \sqrt{\frac{7}{3}} − 1 ≤ r ≤ \frac{\sqrt{3}}{3}, which completes the proof of the theorem.

Thus the gap of 0.444 … ≤ r < 0.577 … has been narrowed to 0.444 … ≤ r ≤ 0.482 … where it is still unknown wether δ(r) > δ_{L}^{3} or not.

References

[1] G. Blind: Über Unterdeckungen der Ebene durch Kreise, Journal reine angew. Math. 236 (1969), 145–173.

[2] D.W. Boyd: The Osculatory Packing of the Three Dimensional Sphere, Canad. J. Math. 25 (1973), 303–322.

[3] J.H. Conway and N.J.A. Sloane: Sphere Packings, Lattices and Groups, Springer-Verlag, New York etc. (1988).

[4] G. Fejes Tóth: Packing and Covering with Incongruent Circles, Studia Sci. Math. Hung. 19 (1984), 49–53.

[5] G. Fejes Tóth and W. Kuperberg: Packing and Covering with Convex Sets, Handbook of Convex Geometry (Amsterdam) B ed. P.M. Gruber and J.M. Wills, North-Holland, Amsterdam (1993).

[6] L. Fejes Tóth: Regular Figures, International Series of Monographs in Pure and Applied Mathematics, vol. 48 Pergamon Press Oxford etc. (1964).

[7] L. Fejes Tóth: Lagerungen in der Ebene, auf der Kugel und im Raum, Springer, Berlin (1972).

[8] L. Fejes Tóth: Some Density-Bounds for Packing and Covering with Incongruent Circles, Studia Sci. Math. Hung. 15 (1980), 63–70.

[9] L. Few: A Mixed Packing Problem, Mathematika 7 (1963), 229–242. [10 P.M. Gruber and C.G. Lekkerkerker: Geometry of Numbers, Mathematical Library,

vol. 37 North- Holland, Amsterdam etc. (1987).

[11] T.L. Saaty and J.M. Alexander: Optimization and the Geometry of Numbers: Packing and Covering, SIAM Review 17 (1975), 475–519.

148 L.G. De Lucía: On the Radial Ratio in Dense Mixed Sphere Packings

[12] S. Vassallo and J.M. Wills: On Mixed Sphere Packings, Contributions to Algebra and Geometry 35 (1994), 67–71.

Received January 6, 1995.