In Euclidean space, the densest sphere packings and their densities are only known in dimensions 1, 2 (Thue, Fejes Tóth), 3 (Hales), 8 (Viazovska), and 24 (Cohn et al.). However, several nontrivial lower and upper bounds for the density δ(d) of the densest packing in dimension d have been established. A simple "folklore" result states that δ(d) ≥ 1/2^d. In this talk we present the intuition and details of three other lower and upper bounds for δ(d): the Minkowski–Hlawka theorem for a lower bound, Blichfeldt's upper bound, and Rogers's upper bound. These results, among others, place δ(d) within a narrow strip of possible densities.
