Locally dense finite lattice packings of spheres
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We consider finite packings of unit-balls in Euclidean 3-spaceE3 where the centres of the balls are the lattice points of a lattice polyhedronP of a given latticeL3⊃E3. In particular we show that the facets ofP induced by densest sublattices ofL3 are not too close to the next parallel layers of centres of balls. We further show that the Dirichlet-Voronoi-cells are comparatively small in this direction. The paper was stimulated by the fact that real crystals in general grow slowly in the directions normal to these dense facets.
The results support, to some extent, the hypothesis that real crystals grow preferably such that they need little volume, i.e that they are locally dense.
Mathematics subject classification numbers, 1980/1985Primary 52A43 52A45
Key words and phrasesLattice packings
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- U.Betke, P.Gritzmann, J. M.Wills, Slices of L. Fejes Tóth's sausage conjecture,Mathematika 29 (1982) 194–201.MR 84m: 52017Google Scholar
- A. H.Boerdijk, Some remarks concerning close-packing of equal spheres,Philips Res. Rep. 7 (1952) 303–313.MR 14310 Google Scholar
- P.Gritzmann, J. M.Wills, Finite packing and covering,Stud. Sci. Math. Hung. 21 (1986) 149–162.MR 88h: 52022Google Scholar
- P. M.Gruber, G. G.Lekkerkerker,Geometry of numbers; North-Holland, Amsterdam, 1987.MR 88j: 11034Google Scholar
- W.Kleber,Einführung in die Kristallographie; VEB Verlag Technik Berlin, 1973.Google Scholar
- J. M.Wills, On the density of finite packings,Acta. Math. Hung. 46 (1985) 205–210.MR 87e: 52025Google Scholar