Semicrosses and quadratic forms

In a series of papers, Galovich, Hamaker, Hickerson, Stein and Szabó investigated the tiling and packing of euclidean n-space by translates of certain star bodies, the semicross and the cross. If the translating vectors have integer co-ordinates and form a lattice, they reduced the geometric questions to splittings of finite abelian groups by the set {1, 2 … k} or {±1, ±2, …, ±k}. In this reduction groups of prime orders play a pivotal role. We will use the representation of primes by quadratic forms to construct such splittings.