Sphere packing

Sphere packing studies arrangements of non-overlapping spheres that maximize the fraction of space they occupy. Problems vary by dimension, region, and rules for the arrangement.

Packing density: The fraction of a region's volume occupied by packed objects. It is the quantity sphere-packing problems seek to maximize.
Euclidean space: A space governed by Euclidean distance and geometry. The classical sphere-packing problem is usually posed in Euclidean space.
Thomas Harriot: An English mathematician and astronomer who studied close packing of cannonballs around 1600. His correspondence helped introduce the cannonball stacking problem to European mathematics.
Sphere packing in materials science: The use of sphere-arrangement models to study how particles occupy space in materials. Idealized sphere packings provide benchmarks for granular and particulate structures.
Lattice sphere packing: A sphere packing whose centers form a lattice. It restricts the search to periodic arrangements, unlike unrestricted packings.
Kepler conjecture: The theorem that the densest packing of equal spheres in three-dimensional Euclidean space has density π/√18. It settles the central three-dimensional problem for congruent spheres.
Lattice: A discrete additive subgroup generated by linearly independent vectors in a vector space. Lattice points can serve as centers for periodic sphere arrangements.
Johannes Kepler: A German mathematician and astronomer who formulated the conjecture on densest sphere packing in 1611. His conjecture identified the close-packed density later proved by Hales.
Crystallography: The study of crystals and their atomic or molecular structures. Close-packed sphere models describe common arrangements of atoms in crystals.
Non-lattice sphere packing: A sphere packing whose centers do not form a lattice. It allows arrangements unavailable to lattice-based optimization.

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