The lattice triple packing of spheres in Euclidean space

Abstract:

We say that a lattice $\Lambda$ in $n$-dimensional Euclidean space ${E_n}$ provides a $k$-fold packing for spheres of radius 1 if, when open spheres of radius 1 are centered at the points of $\Lambda$, no point of space lies in more than $k$ spheres. The multiple packing constant $\Delta _k^{(n)}$ is the smallest determinant of any lattice with this property. In the plane, the first three multiple packing constants $\Delta _2^{(2)},\Delta _3^{(2)}$, and $\Delta _4^{(2)}$ are known, due to the work of Blundon, Few, and Heppes. In ${E_3},\Delta _2^{(3)}$ is known, because of work by Few and Kanagasabapathy, but no other multiple packing constants are known. We show that $\Delta _3^{(3)} \leqslant 8\sqrt {38} /27$ and give evidence that $\Delta _3^{(3)} = 8\sqrt {38} /27$. We show, in fact, that a lattice with determinant $8\sqrt {38} /27$ gives a local minimum of the determinant among lattices providing a $3$-fold packing for the unit sphere in ${E_3}$.