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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.
Contents:
- 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
Readership: Researchers in classical geometry and solid state physics.
