Tiling a Rectangle with Polyominoes
Copyright policy )Tiling a Rectangle with Polyominoes
A polycube in dimension $d$ is a finite union of unit $d$-cubes whose vertices are on knots of the lattice $\mathbb{Z}^d$. We show that, for each family of polycubes $E$, there exists a finite set $F$ of bricks (parallelepiped rectangles) such that the bricks which can be tiled by $E$ are exactly the bricks which can be tiled by $F$. Consequently, if we know the set $F$, then we have an algorithm to decide in polynomial time if a brick is tilable or not by the tiles of $E$.
polyomino, [info.info-dm] computer science [cs]/discrete mathematics [cs.dm], [NLIN.NLIN-CG] Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG], [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], Polyomino, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], tiling, QA1-939, [NLIN.NLIN-CG]Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG], Tiling, Mathematics, [math.math-co] mathematics [math]/combinatorics [math.co], [nlin.nlin-cg] nonlinear sciences [physics]/cellular automata and lattice gases [nlin.cg]
polyomino, [info.info-dm] computer science [cs]/discrete mathematics [cs.dm], [NLIN.NLIN-CG] Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG], [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], Polyomino, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], tiling, QA1-939, [NLIN.NLIN-CG]Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG], Tiling, Mathematics, [math.math-co] mathematics [math]/combinatorics [math.co], [nlin.nlin-cg] nonlinear sciences [physics]/cellular automata and lattice gases [nlin.cg]
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