\fullkepler" i i 2004/3/13 page 1 i i A Proof of the Kepler Conjecture (DCG Version) Thomas C. Hales March 13, 2004 i i i i \fullkepler" i i 2004/3/13 page 2 i i 2 i i i i \fullkepler" i i 2004/3/13 page i i i Contents Preface vii I Historical Overview 1 1 Introduction 3 1.1 The face-centered cubic packing . 3 1.2 Early History, Hariot, and Kepler . 5 1.3 History . 9 1.4 The Literature . 10 1.4.1 Bounds . 10 1.4.2 Classes of packings . 11 1.4.3 Other convex bodies . 12 1.4.4 Strategies of proof . 12 2 Overview of the proof 15 2.1 Experiments with other Decompositions . 15 2.2 Contents of the Papers . 18 2.3 Complexity . 19 2.4 Computers . 20 2.5 Acknowledgments . 20 Bibliography 23 II Formulation 29 3 The Top Level Structure of the Proof 33 3.1 Statement of Theorems . 33 3.2 Basic Concepts in the Proof of the Kepler Conjecture . 37 3.3 Logical Skeleton of the Proof of the Kepler Conjecture . 39 3.4 Proofs of the Central Claims . 40 4 Construction of the Q-system 43 4.1 Description of the Q-system . 43 i i i i i \fullkepler" i i 2004/3/13 page ii i i ii Contents 4.2 Geometric Considerations . 46 4.3 Incidence Relations . 48 4.4 Overlap of Simplices . 52 5 V -cells 57 5.1 V -Cells . 57 5.2 Orientation . 61 5.3 Interaction of V -cells with the Q-system . 63 6 Decomposition Stars 69 6.1 Indexing Sets . 69 6.2 Cells attached to Decomposition Stars . 71 6.3 Colored Spaces . 73 7 Scoring [Ferguson, Hales] 77 7.1 De¯nitions . 78 7.2 Negligibility . 83 7.3 Fcc-compatibility . 84 7.4 Scores of Standard Clusters . 86 III Detailed Decomposition (Sphere Packings III) 91 8 Local Optimality 95 8.1 Results . 95 8.2 Rogers Simplices . 96 8.3 Bounds on Simplices . 97 8.4 Breaking Clusters into Pieces . 101 8.5 Proofs . 106 9 The S-system 111 9.1 Overview . 111 9.2 The set ±(v).............................112 9.3 Overlap . 119 9.4 The S-system de¯ned . 120 9.5 Disjointness . 121 9.6 Separation of simplices of type A . 122 9.7 Separation of simplices of type B . 123 9.8 Separation of simplices of type C . 123 9.9 Simplices of type C0 . 124 9.10 Scoring . 125 10 Bounds on the Score in Triangular and Quadrilateral Regions 127 10.1 The function ¿ ............................127 10.2 Types . 128 10.3 Limitations on Types . 130 10.4 Bounds on the Score in Quadrilateral Regions . 131 i i i i \fullkepler" i i 2004/3/13 page iii i i Contents iii 10.5 A Volume Formula . 134 IV Bounds (Sphere Packings IV) 139 11 Upright Quarters 143 11.1 Erasing Upright Quarters . 143 11.2 Contexts . 144 11.3 Three anchors . 145 11.4 Six anchors . 146 11.5 Anchored simplices . 146 11.6 Anchored simplices do not overlap . 147 11.7 Five anchors . 150 11.8 Four anchors . 151 11.9 Summary . 154 11.10 Some flat quarters . 155 12 Bounds in Exceptional Regions 159 12.1 The main theorem . 159 12.2 Nonagons . 161 12.3 Distinguished edge conditions . 161 12.4 Scoring subclusters . 162 12.5 Proof . 163 12.6 Preparation of the standard cluster . 164 12.7 Reduction to polygons . 165 12.8 Some deformations . 166 12.9 Truncated corner cells . 168 12.10 Formulas for Truncated corner cells . 169 12.11 Containment of Truncated corner cells . 170 12.12 Convexity . 173 12.13 Proof that Distances Remain at least 2 . 174 13 Convex Polygons 177 13.1 Deformations . 177 13.2 Truncated corner cells . 178 13.3 Analytic continuation . 179 13.4 Penalties . 180 13.5 Penalties and Bounds . 181 13.6 Penalties . 182 13.7 Constants . 184 13.8 Triangles . 185 13.9 Quadrilaterals . 186 13.10 Pentagons . 187 13.11 Hexagons and heptagons . 188 13.12 Loops . 189 i i i i \fullkepler" i i 2004/3/13 page iv i i iv Contents 14 Further Bounds in Exceptional Regions 193 14.1 Small dihedral angles . 193 14.2 A particular 4-circuit . 194 14.3 A particular 5-circuit . 196 V Pentahedral Prisms (Sphere Packings V) { Ferguson 203 VI Tame Graphs and Linear Programs (Sphere Packings VI) 207 15 Tame Graphs 211 15.1 Basic De¯nitions . 211 15.2 Weight Assignments . 213 15.3 Plane Graph Properties . 215 16 Classi¯cation of Tame Plane Graphs 217 16.1 Statement of the Theorem . 217 16.2 Basic De¯nitions . 217 16.3 A Finite State Machine . 219 16.4 Pruning Strategies . 219 17 Contravening Graphs 223 17.1 A Review of Earlier Results . 223 17.2 Contravening Plane Graphs de¯ned . 228 18 Contravention is tame 231 18.1 First Properties . 231 18.2 Computer Calculations and Their Consequences . 232 18.3 Linear Programs . 233 18.4 A Non-contravening Four-circuit . 236 18.5 Possible four-circuits . 237 19 Weight Assignments 239 19.1 Admissibility . 239 19.2 Proof that tri(v) > 2 . 240 19.3 Bounds when tri(v) 2 f3; 4g. 242 19.4 Weight Assignments for Aggregates . 245 20 Linear Program Estimates 247 20.1 Relaxation . 248 20.2 The Linear Programs . 249 20.3 Basic Linear Programs . 250 20.4 Error Analysis . 251 21 Elimination of Aggregates 253 21.1 Triangle and Quad Branching . 253 i i i i \fullkepler" i i 2004/3/13 page v i i Contents v 21.2 A pentagonal hull with n = 8 . 254 21.3 n = 8, hexagonal hull . 254 21.4 n = 7, pentagonal hull . 254 21.5 Type (p; q; r) = (5; 0; 1) . 256 21.6 Summary . 256.