The dense packing of microscopic spheres (atoms) is the basic geometric arrangement in crystals of mono-atomic elements with weak covalent bonds, which achieves the optimal "known density" of B/O18. In 1611, Johannes Kepler had already "conjectured" that B/O18 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/O18 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 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.

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