Sphere packing and covering bounds

Use Euclidean balls of one fixed radius r in real d-space. Their interiors are disjoint exactly when every pair of distinct centers is at least 2r apart. Tangency is allowed, and boundaries have zero d-dimensional volume. For a full-rank lattice Λ = BZᵈ, the volume of a fundamental parallelepiped is the absolute determinant of B, independent of the particular lattice basis chosen. There is one center per fundamental cell.

Write κ_d for the volume of the unit Euclidean ball. The density of a lattice packing is therefore the volume κ_d rᵈ of one radius-r ball divided by the fundamental-cell volume. It cannot exceed one because packing interiors do not overlap. This volume bound is elementary and often very weak. It concerns a periodic infinite arrangement, not the fraction of a finite plotting window filled by clipped disks. For arbitrary nonperiodic arrangements, density requires an asymptotic definition; we do not replace that analysis with a screenshot.

The covering radius μ(Λ) is the supremum, over all points in space, of distance to the nearest lattice point. Balls of radius R centered at the lattice cover the space exactly when R is at least μ. By periodicity it suffices to inspect one fundamental cell, although the nearest center may lie in a neighboring cell. For a rectangular planar lattice, the farthest points are rectangle centers, giving half the rectangle diagonal.

The covering density counts ball volume per cell and can exceed one because it counts overlaps with multiplicity. It must be at least one for a full covering; average multiplicity below one would leave uncovered volume. The shortest vector only detects when neighboring balls touch. It does not identify the deepest holes, so the packing radius and covering radius must not be interchanged. A stretched rectangular lattice can have a modest shortest-vector length and a very large covering radius.

In the plane, a square lattice with nearest-center spacing two packs unit disks at density π/4. A triangular lattice with basis (2, 0) and (1, √3) has the same minimum spacing but cell area 2√3, giving density π/(2√3), approximately 0.9069. This comparison proves the triangular construction is denser than that square construction. The fact that this planar density is globally optimal is a separate classical theorem, not proved merely by comparing two examples.

Higher dimensions introduce increasingly complex lattices and nonlattice periodic arrangements. The determinant controls volume per center, but a short vector can violate separation even when the determinant looks favorable. Covering has its own competing arrangements and bounds. In an engineering application, first decide whether the task is collision avoidance, guaranteed spatial service, or another metric entirely. Changing the norm, allowing unequal radii, or constraining centers to a finite container defines a different problem and changes the relevant bounds.