Tiling with notched cubes

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Tiling with notched cubes

In 1966, Golomb showed that any polyomino which tiles a rectangle also tiles a larger copy of itself. Although there is no compelling reason to expect the converse to be true, no counterexamples are known. In 3 dimensions, the analogous result is that any polycube that tiles a box also tiles a larger copy of itself. In this note, we exhibit a polycube (a ‘notched cube’) that tiles a larger copy...

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Lattice Tilings by Cubes: Whole, Notched and Extended

We discuss some problems of lattice tiling via Harmonic Analysis methods. We consider lattice tilings of R by the unit cube in relation to the Minkowski Conjecture (now a theorem of Hajós) and give a new equivalent form of Hajós’s theorem. We also consider “notched cubes” (a cube from which a reactangle has been removed from one of the corners) and show that they admit lattice tilings. This has...

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An Efficient Strategy for Tiling Multidimensional OLAP Data Cubes

Computing aggregates over selected categories of multidimensional discrete data (MDD) cubes is the core operation of many on-line analytical processing (OLAP) systems. In order to support efficient computations of these aggregates in a multidimensional OLAP (MOLAP) system, a careful design of the database storage architecture must be undertaken. In particular, tiling (i.e., subdivision of an MD...

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ژورنال

عنوان ژورنال: Discrete Mathematics

سال: 2000

ISSN: 0012-365X

DOI: 10.1016/s0012-365x(99)00310-6