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Least Action Principle of Crystal Formation of Dense Packing Type and Kepler's Conjecture cover

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.
Free Access
FRONT MATTER
  • Pages:i–xxi

https://doi.org/10.1142/9789812384911_fmatter

No Access
Introduction
  • Pages:1–17

https://doi.org/10.1142/9789812384911_0001

No Access
The Basics of Euclidean and Spherical Geometries and a New Proof of the Problem of Thirteen Spheres
  • Pages:19–81

https://doi.org/10.1142/9789812384911_0002

No Access
Circle Packings and Sphere Packings
  • Pages:83–122

https://doi.org/10.1142/9789812384911_0003

No Access
Geometry of Local Cells and Specific Volume Estimation Techniques for Local Cells
  • Pages:123–200

https://doi.org/10.1142/9789812384911_0004

No Access
Estimates of Total Buckling Height
  • Pages:201–233

https://doi.org/10.1142/9789812384911_0005

No Access
The Proof of the Dodecahedron Conjecture
  • Pages:235–238

https://doi.org/10.1142/9789812384911_0006

No Access
Geometry of Type I Configurations and Local Extensions
  • Pages:239–326

https://doi.org/10.1142/9789812384911_0007

No Access
The Proof of Main Theorem I
  • Pages:327–382

https://doi.org/10.1142/9789812384911_0008

No Access
Retrospects and Prospects
  • Pages:383–396

https://doi.org/10.1142/9789812384911_0009

Free Access
BACK MATTER
  • Pages:397–402

https://doi.org/10.1142/9789812384911_bmatter

“The book presents an exposition of the ideas suggested by W Y Hsiang to prove this interesting and difficult conjecture …”
Mathematics Abstracts