I recently posted a preprint together with Alexei Andreanov in which we enumerate all the locally optimal 2-periodic sphere packings (also known as double-lattice packings) in dimensions up to $d=5$. This enumeration includes lattices, since they can be represented as 2-periodic arrangements by using a sublattice of index 2 as the unit cell. For the optimal packing density (among 2-periodic packings), we found that in $d=3$, there is one non-lattice that achieves it (the hexagonal close packing), in $d=4$ there are no non-lattices that achieve it, and for $d=5$ there are two. The highest locally optimal non-lattice double-lattice density in $d=4$ is the intriguing $\delta = 2/\sqrt{144+64\sqrt{5}} \approx 0.118$ (compared to $\delta = 1/8 = 0.125$ for the $D_4$ lattice).

I recently posted a preprint together with Alexei Andreanov in which we enumerate all the locally optimal 2-periodic sphere packings (also known as double-lattice packings) in dimensions up to $d=5$. This enumeration includes lattices, since they can be represented as 2-periodic arrangements by using a sublattice of index 2 as the unit cell. For the optimal packing density (among 2-periodic packings), we found that in $d=3$, there is one non-lattice that achieves it (the hexagonal close packing), in $d=4$ there are no non-lattices that achieve it, and for $d=5$ there are two. The highest locally optimal non-lattice double-lattice density in $d=4$ is the intriguing $\delta = 1/(2 \sqrt{9+4\sqrt{5}}) \approx 0.118$ (compared to $\delta = 1/8 = 0.125$ for the $D_4$ lattice).

I recently posted a preprint together with Alexei Andreanov in which we enumerate all the locally optimal 2-periodic sphere packings (also known as double-lattice packings) in dimensions up to $d=5$. This enumeration includes lattices, since they can be represented as 2-periodic arrangements by using a sublattice of index 2 as the unit cell. For the optimal packing density (among 2-periodic packings), we found that in $d=3$, there is one non-lattice that achieves it (the hexagonal close packing), in $d=4$ there are no non-lattices that achieve it, and for $d=5$ there are two. The highest non-lattice double-lattice density in $d=4$ is the intriguing $\delta = 2/\sqrt{144+64\sqrt{5}} \approx 0.118$ (compared to $\delta = 1/8 = 0.125$ for the $D_4$ lattice).

I recently posted a preprint together with Alexei Andreanov in which we enumerate all the locally optimal 2-periodic sphere packings (also known as double-lattice packings) in dimensions up to $d=5$. This enumeration includes lattices, since they can be represented as 2-periodic arrangements by using a sublattice of index 2 as the unit cell. For the optimal packing density (among 2-periodic packings), we found that in $d=3$, there is one non-lattice that achieves it (the hexagonal close packing), in $d=4$ there are no non-lattices that achieve it, and for $d=5$ there are two. The highest locally optimal non-lattice double-lattice density in $d=4$ is the intriguing $\delta = 2/\sqrt{144+64\sqrt{5}} \approx 0.118$ (compared to $\delta = 1/8 = 0.125$ for the $D_4$ lattice).