A $σ$-morphic convex protoset
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908469556150272 |
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| author | Džuklevski, Aleksa |
| author_facet | Džuklevski, Aleksa |
| contents | We say that a tile is $σ$-morphic if it tiles the plane in exactly $\aleph_0$ many noncongruent ways (up to an isometry). It is an unsolved problem of whether a $σ$-morphic tile exist in the plane. In this note we present a construction of a set of convex tiles that is $σ$-morphic. The result is interesting since all the constructions of $σ$-morphic sets of tiles that arise in the literature make use of bumps and nicks, which necessarily make the tiles non-convex. We construct our set by cleverly dividing the tiles of the set of tiles discovered by Schmitt into convex tiles so that they behave in the same manner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A $σ$-morphic convex protoset Džuklevski, Aleksa Combinatorics 52C20, 05B45, 52A37 We say that a tile is $σ$-morphic if it tiles the plane in exactly $\aleph_0$ many noncongruent ways (up to an isometry). It is an unsolved problem of whether a $σ$-morphic tile exist in the plane. In this note we present a construction of a set of convex tiles that is $σ$-morphic. The result is interesting since all the constructions of $σ$-morphic sets of tiles that arise in the literature make use of bumps and nicks, which necessarily make the tiles non-convex. We construct our set by cleverly dividing the tiles of the set of tiles discovered by Schmitt into convex tiles so that they behave in the same manner. |
| title | A $σ$-morphic convex protoset |
| topic | Combinatorics 52C20, 05B45, 52A37 |
| url | https://arxiv.org/abs/2507.20867 |