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The sphere packing problem. (English) Zbl 0770.52009

The paper sketches a solution of the sphere packing problem in 3-space, i.e., that any packing of congruent spheres has a density exceeding that of the face-centred-cubic lattice packing, namely \(\pi/\sqrt{18}\).
The demonstrations of the author are based on Delaunay triangulations. Thus, his geometric approach is roughly dual to that of L. Fejes Tóth [cf. ‘Reguläre Figuren’, Akadémiai Kiadó, Budapest (1965; Zbl 0134.159) and ‘Lagerungen in der Ebene, auf der Kugel und im Raum’, Springer, Berlin (1972; Zbl 0229.52009)], which is based on Voronoi cells. (A comparison of both methods is given in the introduction of the paper.)
The results of the present article provide a sketch rather than a complete rigorous mathematical proof in a carefully controlled way. However, one may hope that the complicated numerical demonstrations of the author will give the needed encouragement to mathematicians to replace the corresponding steps by rigorous proofs. (E. g., the optimization problems used in this paper are of the same nature and complexity as the problem of minimizing the volume of a Voronoi cell).

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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[1] Conway, J. H.; Sloane, N. J.A., Sphere Packings, Lattices, and Groups (1988), Springer: Springer Berlin · Zbl 0634.52002
[2] Tóth, L. Fejes, Reguläre Figuren (1965), Akadémiai Kiadó: Akadémiai Kiadó Budapest · Zbl 0134.15901
[3] Tóth, L. Fejes, Lagerungen in der Ebene auf der Kugel und im Raum (1972), Springer: Springer Berlin · Zbl 0052.18401
[4] Habicht, W.; van der Waerden, B. L., Lagerung von Punkten auf der Kugel, Math. Ann., 123, 223-234 (1951) · Zbl 0043.35604
[5] Hales, T., Remarks on the density of sphere packings in three dimensions (1990), also: Combinatorica, to appear
[6] Hsiang, W.-Y., On the density of sphere packings in \(E^3\), I (1991), preprint
[7] Hsiang, W.-Y., On the density of sphere packings in \(E^3\), II (1991), preprint
[8] Lagrange, J.-L., Oeuvres, Vol. 7 (1857-1892), Gauthier-Villars: Gauthier-Villars Paris
[9] Muder, D., Voronoi polyhedra of 12 or fewer sides (1991), preprint
[10] Todhunter, I., Spherical Trigonometry (1859), MacMillan: MacMillan Cambridge
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