Metrics on the face-centered cubic lattice

The face-centered cubic lattice and its analogues in dimensions 4 and 5 are known to solve the sphere-packing problem in those dimensions. This property makes these lattices important for image processing in situations where high angular resolution is required, or if a good distance approximation is required. Given that the approximation to Euclidean distance is better for these lattices than for the standard Cartesian lattice, it is necessary to provide a distance measure for the lattices. We show how to construct two different metrics for the face-centered cubic lattice, and how this metric can be extended to higher dimensions.