Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866908307663355904 |
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| author | Yang, Chao Zhang, Zhujun |
| author_facet | Yang, Chao Zhang, Zhujun |
| contents | Recently, Greenfeld and Tao disprove the conjecture that translational tilings of a single tile can always be periodic [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension $n$ is part of the input, the translational tiling for subsets of $\mathbb{Z}^n$ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper shows that translational tiling of the $3$-dimensional space with a set of $5$ polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from $3$-dimensional space to $4$-dimensional space, we manage to show that translational tiling of the $4$-dimensional space with a set of $4$ tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the $n$-dimensional space with a monotile, for some fixed $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00846 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes Yang, Chao Zhang, Zhujun Combinatorics Computational Complexity Metric Geometry Recently, Greenfeld and Tao disprove the conjecture that translational tilings of a single tile can always be periodic [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension $n$ is part of the input, the translational tiling for subsets of $\mathbb{Z}^n$ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper shows that translational tiling of the $3$-dimensional space with a set of $5$ polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from $3$-dimensional space to $4$-dimensional space, we manage to show that translational tiling of the $4$-dimensional space with a set of $4$ tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the $n$-dimensional space with a monotile, for some fixed $n$. |
| title | Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes |
| topic | Combinatorics Computational Complexity Metric Geometry |
| url | https://arxiv.org/abs/2409.00846 |