Lattice Coverings of n-Space By Spheres

A lattice ∧ in euclidean n-space, En is a group of vectors under vector addition generated by n independent vectors, X1, X2 … , Xn , called a basis for the lattice. The absolute value of the n × n determinant the rows of which are the co-ordinates of a basis is called the determinant of the lattice and is denoted by d(∧). For any lattice ∧ there is a unique minimal positive number r such that, if spheres of radius r are placed with centres at all points of ∧, the entire space is covered. The density of this covering may be defined as (Jnrn)/(d(∧)) where Jn is the volume of the unit sphere in n-dimensional euclidean space. This density will be denoted by θn(∧). The density of the most efficient lattice covering of n-space by spheres, θn is the absolute minimum of θn(∧) considered as a function from the space of all lattices to the real numbers.