Least Action Principle Of Crystal Formation Of Dense Packing Type And Kepler's Conjecture

Wu-yi Hsiang, Weiping Zhang

• Publisher: World Scientific
• Year: 2001
• Print ISBN: 9789810246709, 9789810246709
• eBook ISBN: 9789812384911
• Topic: Mathematics
• Subtopic: Differential Geometry
• Index: Mathematics

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.

Table of contents

• Contents
• Foreword
• Acknowledgment
• List of Symbols
• Chapter 1 Introduction
• Chapter 2 The Basics of Euclidean and Spherical Geometries and a New Proof of the Problem of Thirteen Spheres
• Chapter 3 Circle Packings and Sphere Packings
• Chapter 4 Geometry of Local Cells and Specific Volume Estimation Techniques for Local Cells
• Chapter 5 Estimates of Total Buckling Height
• Chapter 6 The Proof of the Dodecahedron Conjecture
• Chapter 7 Geometry of Type I Configurations and Local Extensions
• Chapter 8 The Proof of Main Theorem I
• Chapter 9 Retrospects and Prospects
• References
• Index