Quasiperiodic packing densities

Abstract

The problem of the packing density on quasiperiodic lattices is discussed in a systematic way using projection techniques. For compact domains a direct construction is presented using a Voronoi construction on a quasilattice in perpendicular space defined by the forbidden volume of the packed objects. A generalized inflation law, valid for arbitrary shapes of the acceptance domain using the properties of linear mappings of the hyperlattice on itself which commute with the symmetry group, is used to show that the packing densities and the whole structure of the projected quasilattice are periodic under scale transformations. We find the optimal compact acceptance domains and packing densities for several icosahedral problems. For the packing of spheres on the primitive lattice, an icosahedron and the truncated triacontahedron give equal densities but different quasilattices. For the packing of icosahedra one finds only the second lattice and a very high density. For the fcc and bcc lattices the maximum density acceptance domain is a triacontahedron and the densities are considerably lower. The results of Henley for including correlations to increase the density are reformulated in terms of a graph problem in perpendicular space. Including only the graphs equivalent to his, we find the same packing density for the two primitive and for the fcc lattice. It is shown that a generalization leads to an interesting and very complex problem in graph theory which we are unable to solve.