Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866916804292509696 |
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| author | Yang, Chao Zhang, Zhujun |
| author_facet | Yang, Chao Zhang, Zhujun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02196 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes Yang, Chao Zhang, Zhujun Combinatorics Computational Complexity Metric Geometry 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. |
| title | Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes |
| topic | Combinatorics Computational Complexity Metric Geometry |
| url | https://arxiv.org/abs/2408.02196 |