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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 p/v18. In 1611, Johannes Kepler had already ""conjectured"" that p/v18 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/v18 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

Kepler's conjecture. --- Sphere packings. --- Crystallography, Mathematical. --- Crystallography --- Crystallometry --- Mathematical crystallography --- Crystals --- Lattice theory --- Packing of spheres --- Packings of spheres --- Combinatorial packing and covering --- Conjecture, Kepler's --- Kepler conjecture --- Sphere packings --- Mathematics --- Mathematical models

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This book should be accessible for graduate or advanced undergraduate students both in mathematics and computer science. We have designed this book especially

Algebraic logic. --- Monotonic functions. --- Fixed point theory. --- Lattice theory. --- Machine theory. --- Abstract automata --- Abstract machines --- Automata --- Mathematical machine theory --- Algorithms --- Logic, Symbolic and mathematical --- Recursive functions --- Robotics --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Fixed point theorems (Topology) --- Nonlinear operators --- Coincidence theory (Mathematics) --- Functions, Monotonic --- Functions of real variables