Hsiang Wu-yi Hsiang 
Least Action Principle Of Crystal Formation Of Dense Packing Type And Kepler’s Conjecture [PDF ebook] 

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/ In 1611, Johannes Kepler had already "conjectured" that B/ 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/ 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.