Thirteen spheres problem and Fejes Tóth Conjecture

In the Euclidean plane one can pack the unit circles in such a way that every circle touches the maximal six neighbors, and the corresponding hexagonal arrangement is unique up to congruence. It is natural to ask the similar questions for sphere packing in the physical space, and it turns out that even the maximality of touching neighbors for local arrangement already constitutes a challenging problem in classical solid geometry, namely, “the problem of the thirteen spheres”. It was originated from an unpublished recorded discussion between David Gregory and Isaac Newton in 1694, and the solution was not correctly settled to be twelve until 1953. We will give a comparative analysis on three different proofs on the impossibility of thirteen touching neighbors, and discuss the importance of area estimates in the three proofs of the impossibility. The hexagonal close packings were all originally mistaken to be the face-centered-cubic lattice packing by Kepler in 1611, and later clarified by Barlow in 1883. They are the only known examples in which every sphere touches the maximal twelve neighbors. It was conjectured by L. Fejes Tóth that hexagonal close packings should be characterized by this property of maximality of touching neighbors everywhere. In this thesis we will prove that it is the case, and in fact such a characterization can even be localized to the study of double-layer local structures