Undecidability of Translational Tiling with 2 Polycubes
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915448007688192 |
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| author | Kim, Yoonhu |
| author_facet | Kim, Yoonhu |
| contents | In this paper, we prove that it is undecidable whether a set of two polycubes can tile $\mathbb{Z}^3$ by translation. The proof involves a new technique that allows us to simulate two disconnected polycubes with two connected polycubes. By expanding this technique to higher dimensions, we also prove that a set of disconnected tiles in $\mathbb{Z}^n$ can be simulated by the same number of connected tiles in $\mathbb{Z}^n$ for $n \geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11725 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Undecidability of Translational Tiling with 2 Polycubes Kim, Yoonhu Combinatorics 52C22 (Primary) 68Q17 (Secondary) In this paper, we prove that it is undecidable whether a set of two polycubes can tile $\mathbb{Z}^3$ by translation. The proof involves a new technique that allows us to simulate two disconnected polycubes with two connected polycubes. By expanding this technique to higher dimensions, we also prove that a set of disconnected tiles in $\mathbb{Z}^n$ can be simulated by the same number of connected tiles in $\mathbb{Z}^n$ for $n \geq 3$. |
| title | Undecidability of Translational Tiling with 2 Polycubes |
| topic | Combinatorics 52C22 (Primary) 68Q17 (Secondary) |
| url | https://arxiv.org/abs/2508.11725 |