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On 2D packings of cubes in the torus

Authors:	Andrew V Reztsov Ian H Sloan

Institution:	Research Fellow, Division of Science and Technology, Tamaki Campus, The University of Auckland, Private Bag 92019, Auckland, New Zealand ; School of Mathematics, University of New South Wales, Sydney 2052, New South Wales, Australia

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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