Random close packing (RCP) is the densest configuration that can be achieved by randomly packing equal spheres, with packing fraction φ_RCP ≈ 0.64 in 3D (≈ 0.82 in 2D for discs). Ordered packings do better: hexagonal close packing achieves φ = π/(2√3) ≈ 0.9069 in 2D. The Kepler conjecture (proved by Hales 2005 via computer) states face-centered cubic and hexagonal close packing achieve φ = π/(3√2) ≈ 0.7405 in 3D — the maximum. The gap between RCP and the crystal reflects an amorphous-to-crystal transition that doesn't occur spontaneously in monodisperse hard-sphere systems at intermediate rates. Polydispersity (size variation) frustrates crystallization, lowering φ_max. Colors show local coordination number — hexagons glow teal (6-fold), defects are orange/red.