Laminating lattices with symmetrical glue

TL;DR

This paper introduces a novel method for constructing dense sphere packings in 10 and 12 dimensions using automorphism groups of specific lattices, leading to new lattice and non-lattice packings with notable densities.

Contribution

It presents an innovative approach to lattice construction via symmetry groups, resulting in alternative 12-dimensional laminated lattices and dense 10-dimensional packings.

Findings

Constructed a 12D laminated lattice with kissing number 648.

Discovered a dense, aperiodic 10D packing with center density 1/32.

Linked automorphism groups to new packing arrangements.

Abstract

We use the automorphism group $Aut(H)$, of holes in the lattice $L_8=A_2\oplus A_2\oplus D_4$, as the starting point in the construction of sphere packings in 10 and 12 dimensions. A second lattice, $L_4=A_2\oplus A_2$, enters the construction because a subgroup of $Aut(L_4)$ is isomorphic to $Aut(H)$. The lattices $L_8$ and $L_4$, when glued together through this relationship, provide an alternative construction of the laminated lattice in twelve dimensions with kissing number 648. More interestingly, the action of $Aut(H)$ on $L_4$ defines a pair of invariant planes through which dense, non-lattice packings in 10 dimensions can be constructed. The most symmetric of these is aperiodic with center density 1/32. These constructions were prompted by an unexpected arrangement of 378 kissing spheres discovered by a search algorithm.