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On 2D packings of cubes in the torus

Authors: Andrew V. Reztsov and Ian H. Sloan
Journal: Proc. Amer. Math. Soc. 125 (1997), 17-26
MSC (1991): Primary 05B40; Secondary 11H31, 52C15, 65D32
MathSciNet review: 1401751
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Abstract: The 2D packings of cubes (i.e. squares) in the torus ${\mathcal {T}}^{2}={[0,1)}^{2}$ are considered. We obtain the exact expression $N_{2} (\lambda ) = \left \lfloor \lambda \lfloor \lambda \rfloor \right \rfloor $ for the quantity $N_{2} (\lambda )$, the maximal number of 2D cubes in a packing. (Here $1/\lambda $ is the length of sides of cubes, $\lambda \in {\bold {R}},\ \lambda >2$.) Corresponding best packings are constructed. Both rank 1 best lattice packings and rank 2 best lattice packings are given.

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Additional Information

Andrew V. Reztsov
Affiliation: Research Fellow, Division of Science and Technology, Tamaki Campus, The University of Auckland, Private Bag 92019, Auckland, New Zealand

Ian H. Sloan
Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, New South Wales, Australia

Received by editor(s): June 16, 1995
Communicated by: William W. Adams
Article copyright: © Copyright 1997 American Mathematical Society