by Wu-Yi Hsiang (University of California, Berkeley, USA & Hong Kong University of Science & Technology, Hong Kong)

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 Thirteen 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