## Least Action Principle of Crystal Formation of Dense Packing Type and Kepler's ConjectureThe 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/OeU18. In 1611, Johannes Kepler had already conjectured that p/OeU18 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/OeU18 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 | 19 |

Circle Packings and Sphere Packings | 83 |

Geometry of Local Cells and Specific Vol | 123 |

Estimates of Total Buckling Height | 201 |

The Proof of the Dodecahedron | 235 |

configurations | 242 |

The Proof of Main Theorem I | 327 |

Retrospects and Prospects | 383 |

397 | |

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### Common terms and phrases

already amount arrangement associated basic becomes big-hole boundary buckling effect cells centered central angles circle close neighbors cluster cocircular collective completes configuration Conjecture consider considerably consists contains contribution core packing corresponding crystal defined definition diagonal edge-excess edge-lengths edges equal exactly example extension faces fact finite five formula four function geometric given global density hand Hence hexagonal indicated in Figure kind larger least least equal Lemma length local cells local packing locally averaged density lower bound minimal Moreover namely natural optimal pair points possible problem proof prove quadrilateral radial reduced regular remaining Remark resp respectively separation side side-lengths simple small deformation smaller space sphere packings spherical spherical configuration St(A star structure suffices Suppose Theorem thirteen touching neighbors triangles triangular twelve type I configuration unit vector vertices volume estimation