Categorical Tiling Theory: Constructing Directed Planar Tilings via Edge Reversal
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911142549389312 |
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| author | DiLeo, Catherine Sessoms, Preston Shapiro, Brandon T. |
| author_facet | DiLeo, Catherine Sessoms, Preston Shapiro, Brandon T. |
| contents | Tilings of the plane resemble the simplicial and other complexes from algebraic topology, but have not been studied from this perspective. We construct finite categories corresponding to polygons with labeled directed edges, and introduce the problem of modeling tilings of the Euclidean or hyperbolic plane as presheaves over such a category. Combinatorially, this amounts to choosing an ``alignment'' for a tiling: a direction for every edge and consistent labels for the edges of each polygonal tile. We show that for a fixed tiling, given a single alignment we can characterize every other alignment of the same tiling by comparison of the edge directions. We then construct a ``reflective'' alignment for any tiling with an even number of polygons at each vertex, and from this generate a large family of alignments with elegant symmetry properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Categorical Tiling Theory: Constructing Directed Planar Tilings via Edge Reversal DiLeo, Catherine Sessoms, Preston Shapiro, Brandon T. Category Theory Combinatorics Geometric Topology Metric Geometry 18F99 (Primary) 52C20, 18F20, 05C10, 51F15, 51M10 (Secondary) Tilings of the plane resemble the simplicial and other complexes from algebraic topology, but have not been studied from this perspective. We construct finite categories corresponding to polygons with labeled directed edges, and introduce the problem of modeling tilings of the Euclidean or hyperbolic plane as presheaves over such a category. Combinatorially, this amounts to choosing an ``alignment'' for a tiling: a direction for every edge and consistent labels for the edges of each polygonal tile. We show that for a fixed tiling, given a single alignment we can characterize every other alignment of the same tiling by comparison of the edge directions. We then construct a ``reflective'' alignment for any tiling with an even number of polygons at each vertex, and from this generate a large family of alignments with elegant symmetry properties. |
| title | Categorical Tiling Theory: Constructing Directed Planar Tilings via Edge Reversal |
| topic | Category Theory Combinatorics Geometric Topology Metric Geometry 18F99 (Primary) 52C20, 18F20, 05C10, 51F15, 51M10 (Secondary) |
| url | https://arxiv.org/abs/2509.06363 |