Rep-Tiles
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914065478057984 |
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| author | Blair, Ryan Cahn, Patricia Kjuchukova, Alexandra Schwartz, Hannah |
| author_facet | Blair, Ryan Cahn, Patricia Kjuchukova, Alexandra Schwartz, Hannah |
| contents | An $n$-dimensional rep-tile is a compact, connected submanifold of $\mathbb{R}^n$ with non-empty interior which can be decomposed into pairwise isometric rescaled copies of itself whose interiors are disjoint. We show that every smooth compact $n$-dimensional submanifold of $\mathbb{R}^n$ with connected boundary is topologically isotopic to a polycube that tiles the $n$-cube, and hence is topologically isotopic to a rep-tile. It follows that there is a rep-tile in the homotopy type of any finite CW complex. In addition to classifying rep-tiles in all dimensions up to isotopy, we also give new explicit constructions of rep-tiles, namely examples in the homotopy type of any finite bouquet of spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rep-Tiles Blair, Ryan Cahn, Patricia Kjuchukova, Alexandra Schwartz, Hannah Geometric Topology An $n$-dimensional rep-tile is a compact, connected submanifold of $\mathbb{R}^n$ with non-empty interior which can be decomposed into pairwise isometric rescaled copies of itself whose interiors are disjoint. We show that every smooth compact $n$-dimensional submanifold of $\mathbb{R}^n$ with connected boundary is topologically isotopic to a polycube that tiles the $n$-cube, and hence is topologically isotopic to a rep-tile. It follows that there is a rep-tile in the homotopy type of any finite CW complex. In addition to classifying rep-tiles in all dimensions up to isotopy, we also give new explicit constructions of rep-tiles, namely examples in the homotopy type of any finite bouquet of spheres. |
| title | Rep-Tiles |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2412.19986 |