Laminating Lattices with Symmetrical Glue

Veit Elser, Simon Gravel

Abstract: We use the automorphism group Aut(H), of holes in the lattice L 8=A 2 ⊕ A 2 ⊕ D 4, as the starting point in the construction of sphere packings in 10 and 12 dimensions. A second lattice, L 4=A 2 ⊕ 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.