## Sphere PackingsSphere Packings is one of the most attractive and challenging subjects in mathematics. Almost 4 centuries ago, Kepler studied the densities of sphere packings and made his famous conjecture. In the course of centuries, many exciting results have been obtained, ingenious methods created, related challenging problems proposed, and many surprising connections with othe subjects found. Thus, though some of its original problems are still open, sphere packings has been developed into an important discipline. This book tries to give a full account of this fascinating subject, especially its local aspects, discrete aspects and its proof methods. |

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### Contents

I | 1 |

II | 7 |

III | 10 |

IV | 13 |

V | 18 |

VI | 23 |

VII | 25 |

VIII | 31 |

XXIX | 107 |

XXX | 111 |

XXXI | 122 |

XXXII | 124 |

XXXIII | 127 |

XXXIV | 132 |

XXXV | 139 |

XXXVI | 141 |

### Common terms and phrases

Assertion assume bd(S binary Blichfeldt blocking number C.L. Siegel caps cell complex centrally symmetric convex Clearly codeword convenience convex body corresponding deduce deﬁne definite deﬁnite quadratic form denote densest lattice det(A diﬀerent dihedral angle Dirichlet-Voronoi cells dis(Q equality holds Euclidean distance facets Fejes T´oth Figure ﬁrst function geodesic h(Sn Hamming distance Hence integer isometric isometry Jacobi polynomial k(Sn Kepler’s conjecture Kissing Numbers Korkin L-S-M reduced lattice packing lattice sphere packing Lemma light rays starting limsup linear linear code lower bound mathematics Minkowski Minkowski-Hlawka theorem n-dimensional convex body nonoverlapping Numbers of Spheres obtain parallelepiped polynomial polytope positive deﬁnite quadratic positive number problem Proof of Theorem r(Sn Reed-Muller code Remark result Rogers routine argument routine computation yields S(Sn set of points side length spherical simplex symmetric convex body tiling unit spheres upper bound vectors vertex vertices write Zong