On the densest lattice packing of centrally symmetric octahedra
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- by Stefan Chaładus PDF
- Math. Comp. 58 (1992), 341-345 Request permission
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 $(h({\mathbf {m}}) = \max |{m_i}|)$ of which satisfies \[ \begin {array}{*{20}{c}} {{\text {(i)}}} \\ {{\text {(ii)}}} \\ {{\text {(iii)}}} \\ \end {array} \quad \begin {array}{*{20}{c}} {h({\mathbf {m}}) < {{(\tfrac {4}{3}h({\mathbf {n}}))}^{1/3}}\quad {\text {if}}\;{n_4} \leq - 2{n_1} + {n_2} + {n_3},} \hfill \\ {h({\mathbf {m}}) < {{(\tfrac {{27}}{{19}}h({\mathbf {n}}))}^{1/3}}\quad {\text {in any case,}}} \hfill \\ {h({\mathbf {m}}) \leq h{{({\mathbf {n}})}^{1/3}}\quad {\text {if}}\;{n_4} \geq {n_1} + {n_2} + {n_3}.} \hfill \\ \end {array} \] The closing examples show that the above estimations cannot be improved.References
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S. Chaladus, On a decomposition of integer vectors. III, Discuss. Math. (to appear).
H. Minkowski, Gesammelte Abhandlungen, Vol. I, p. 354; Vol. II, pp. 1-42, Teubner, 1911.
- J. V. Whitworth, On the densest packing of sections of a cube, Ann. Mat. Pura Appl. (4) 27 (1948), 29–37. MR 30549, DOI 10.1007/BF02415557
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Math. Comp. 58 (1992), 341-345
- MSC: Primary 11H31; Secondary 52C07
- DOI: https://doi.org/10.1090/S0025-5718-1992-1094941-6
- MathSciNet review: 1094941