Let each sphere in a sphere packing expand uniformly until it touches its neighbors on flat faces. Call the resulting polyhedron the local cell. Then the local density is given by ρ congruent V_sphere/V_(local cell). When the local cell is a regular dodecahedron, then ρ_dodecahedron = (πsqrt(5 + sqrt(5)))/(15sqrt(10)(sqrt(5) - 2)) = 0.7547....

local cell | local density conjecture | sphere packing