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Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes
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abstract
This paper focuses on the undecidability of translational tiling of $n$-dimensional space $\mathbb{Z}^n$ with a set of $k$ tiles. It is known that tiling $\mathbb{Z}^2$ with translated copies with a set of $8$ tiles is undecidable. Greenfeld and Tao gave strong evidence in a series of works that for sufficiently large dimension $n$, the translational tiling problem for $\mathbb{Z}^n$ might be undecidable for just one tile. This paper shows the undecidability of translational tiling of $\mathbb{Z}^3$ with a set of $6$ tiles.
Forward citations
Cited by 2 Pith papers
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Undecidability of Translational Tiling of the Plane with Orthogonally Convex Polyominoes
Translational tiling of the plane with a set of seven orthogonally convex polyominoes is undecidable.
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Undecidability of Translational Tiling with Three Tiles
Deciding translational tiling of Z^4 by three connected polyhypercubes is undecidable, shown by reduction from Wang's domino problem.
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