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Nankai Tracts in Mathematics - Vol. 3
LEAST ACTION PRINCIPLE OF CRYSTAL FORMATION OF DENSE PACKING TYPE AND KEPLER'S CONJECTURE
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 p/Ö18. In 1611, Johannes Kepler had already "conjectured" that p/Ö18 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 p/Ö18 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.
"The book presents an exposition of the ideas suggested by W Y Hsiang to prove this interesting and difficult conjecture ..."
| 424pp |
Pub. date: Dec 2001 |
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