ON THE DENSEST LATTICE PACKING OF CENTRALLY SYMMETRICAL OCTAHEDRA

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 n = [n1, n2, n3, n4] (0 less-than-or-equal-to n1 less-than-or-equal-to n2 less-than-or-equal-to n3 less-than-or-equal-to n4, n4 > 0) contains a nonzero vector m is-an-element-of Z4, the height (h(m) = max\m(i)\) of which satisfies (i) h(m) < (4/3 h(n)) 1/3 if n4 less-than-or-equal-to -2n1 + n2 + n3, (ii) h(m) < (27/19 h(n))1/3 in any case, iii) h(m) less-than-or-equal-to h(n)1/3 if n4 greater-than-or-equal-to n1 + n2 + n3. The closing examples show that the above estimations cannot be improved.