Least action principle of crystal formation of dense packing type and Kepler's conjecture (Q2782044)

The least action principle of crystal formation is a philosophical belief that nature chooses best among all possibilities. Crystals are often naturally created hexagonal close packings of atoms with density \(\pi/ \sqrt{18}\), thus the author considers that this density should be optimal. Kepler's conjecture states that the density of densest packing of equal spheres in 3-dimensional Euclidean space is \(\pi/ \sqrt{18}\).NEWLINENEWLINENEWLINEThere is a decade long history of claims for the proof of Kepler's conjecture independently by \textit{W. Y. Hsiang} [Int. J. Math. 4, No. 5, 739-831 (1993; Zbl 0844.52017), Math. Intell. 17, No. 1, 35-42 (1995; Zbl 0844.52019)], and \textit{T. Hales} [Math. Intell. 16, No. 3, 47-58 (1994; Zbl 0844.52018), in: Proc. intern. congr. math., ICM 2002. Vol. III, 795-804 (2002; Zbl 1012.52031) Discrete Comput. Geom. 17, No. 1, 1-51 (1997; Zbl 0883.52012), ibid. 18, No. 2, 135-149 (1997; Zbl 0883.52013)]. They both follow the local density approach established by L. Fejes Tóth in 1953. According to this method, the problem is reformulated as an optimization of a nonlinear function over a compact set in a finite-dimensional Euclidean space. Proving the conjecture and verifying suggested proofs turned out to be problematic with this approach, as multitude of subproblems have to be solved and the optimizations remain hard.NEWLINENEWLINENEWLINEThe book presents an exposition of the ideas suggested by W. Y. Hsiang to prove this interesting and difficult conjecture, claiming again that his proof is complete.