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On the densest lattice packing of centrally symmetric octahedra

Author: Stefan Chaładus
Journal: Math. Comp. 58 (1992), 341-345
MSC: Primary 11H31; Secondary 52C07
MathSciNet review: 1094941
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Abstract: The main purpose of this paper is the calculation of the critical determinant, and therefore the packing constant, for any centrally symmetric octahedron. The results are obtained partially by a numerical computation that is not rigorous. As an application, we prove that the lattice of integer vectors perpendicular to any integer vector $ {\mathbf{n}} = [{n_1},{n_2},{n_3},{n_4}]\;(0 \leq {n_1} \leq {n_2} \leq {n_3} \leq {n_4},{n_4} > 0)$ contains a nonzero vector $ {\mathbf{m}} \in {{\mathbf{Z}}^4}$, the height $ (h({\mathbf{m}}) = \max \vert{m_i}\vert)$ of which satisfies

\begin{displaymath}\begin{array}{*{20}{c}} {{\text{(i)}}} \\ {{\text{(ii)}}} \\ ... ...if}}\;{n_4} \geq {n_1} + {n_2} + {n_3}.} \hfill \\ \end{array} \end{displaymath}

The closing examples show that the above estimations cannot be improved.

References [Enhancements On Off] (What's this?)

  • [1] S. Chaladus, On a decomposition of integer vectors. III, Discuss. Math. (to appear).
  • [2] H. Minkowski, Gesammelte Abhandlungen, Vol. I, p. 354; Vol. II, pp. 1-42, Teubner, 1911.
  • [3] J. V. Whitworth, On the densest packing of sections of a cube, Ann. Mat. Pura Appl. (4) 27 (1948), 29–37. MR 0030549

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Article copyright: © Copyright 1992 American Mathematical Society