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arXiv · 2408.02196

Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes

Chao Yang·Zhujun Zhang
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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.

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Chao Yang, Zhujun Zhang. 2024-08-05. Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes. https://doi.org/10.1090/proc/17186

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