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

By: Wu-Yi Hsiang, Weiping Zhang (Editor)

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Hardcover | 1 December 2000

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424 Pages

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This work provides proof of Kepler's conjecture about optimal density, and establishes the least action principle, which states that the hexagonal dense packings in crystals are the geometric consequence of optimization of density.

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Non-Fiction Science Chemistry Crystallography Physics Materials & States of Matter Condensed Matter Physics including Liquid State & Solid State Physics Mathematics Geometry Differential & Riemannian Geometry Non-Euclidean Geometry Booktopia Publisher Services

World Scientific Publishing

Foreword
Acknowledgment
List of Symbols
Introduction	p. 1
Sphere Packings and the Sphere Packing Problem	p. 1
Kepler's Conjecture on Sphere Packings	p. 4
Density of Finite Packings without a Container and the Least Action Principle of Crystal Formation	p. 6
Locally Averaged Density and the Intrinsic Density of the Second Kind	p. 13
Main Theorems on Sphere Packings	p. 14
The Basics of Euclidean and Spherical Geometries and a New Proof of the Problem of Thirteen Spheres	p. 19
Vector Algebra and Basic Spherical Geometry	p. 20
Spherical Configurations, Area Estimates and a New Proof of the Impossibility of Thirteen Touching Neighbors	p. 61
Circle Packings and Sphere Packings	p. 83
The Problem of Circle Packings	p. 83
Sphere Packings and Crystal Formations, Kepler's Conjecture and a Least Action Principle of Crystal Formation	p. 89
Some Basic Ideas and Crucial Understanding which the Proofs of the Major Theorems are Based	p. 106
Geometry of Local Cells and Specific Volume Estimation Techniques for Local Cells	p. 123
Basic Geometry of Local Packings and Local Cells	p. 123
Technique of Volume Estimation of the Core Part	p. 138
Volume Estimation of a Rectilinear Slab	p. 164
Volume Estimation of Local Cells	p. 187
Estimates of Total Buckling Height	p. 201
The Correlation Between Buckling Heights and Area Estimates	p. 204
Area Estimates of Star Configurations	p. 215
Estimation of Total Buckling Height	p. 225
The Proof of the Dodecahedron Conjecture	p. 235
The Proof of Case 1: m = 12	p. 236
The Proof of Case 2: m [is equal to or less then] 11	p. 237
The Proof of Case 3: 13 [is equal to or less then] m [is equal to or less then] 15	p. 238
Geometry of Type I Configurations and Local Extensions	p. 239
Geometry of Type I Local Packings and Type I Configurations	p. 241
Geometry of Local Extension	p. 300
The Proof of Main Theorem I	p. 327
The Proof of Main Theorem I for Core Packings with at Least Thirteen Close Neighbors	p. 329
The Proof of Main Theorem I for Core Packings with at Most Twelve Close Neighbors	p. 332
Retrospects and Prospects	p. 383
Retrospects	p. 383
Prospects	p. 387
References	p. 397
Index	p. 401
Table of Contents provided by Blackwell. All Rights Reserved.

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

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