The dense packing of microscopic spheres (i.e. atoms) is the basic geometric arrangement in crystals of mono-atomic elements with weak covalent bonds, which achieves the optimal "known density" of B/ 18. In 1611, Johannes Kepler had already "conjectured" that B/ 18 should be the optimal "density" of sphere packings. Thus, the central problems in the study of sphere packings are the proof of Kepler's conjecture that B/ 18 is the optimal density, and the establishing of the least action principle that the hexagonal dense packings in crystals are the geometric consequence of optimization of density. This important book provides a self-contained proof of both, using vector algebra and spherical geometry as the main techniques and in the tradition of classical geometry.
The basics of Euclidean and spherical geometries and a new proof of the problem of 13 spheres; circle packings and sphere packings; geometry of local cells and specific volume estimation techniques for local cells; estimates of total buckling height; the proof of the dodecahedron conjecture; geometry of type I configurations and local extensions; the proof of main theorem I; retrospects and prospects.