Similarity isometries of shifted lattices and point packings

A lattice $$\Gamma$$ Γ (of rank and dimension d) is a discrete subset of ℝd that is the $$\mathbb{Z}$$ Z -span of d linearly independent vectors v1, . . . ,vd $$\in$$ ∈ ℝd over ℝ. The set {v1, . . . ,vd} is called a basis for $$\Gamma$$ Γ , and $$\Gamma$$ Γ = $$\mathbb{Z}$$ Z v1 ⊕···⊕ $$\mathbb{Z}$$ Z vd.