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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.23301 |
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| _version_ | 1866915820546818048 |
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| author | Dobbelaere, Bert Kagey, Peter Thomas, Drake Vindas-Meléndez, Andrés R. |
| author_facet | Dobbelaere, Bert Kagey, Peter Thomas, Drake Vindas-Meléndez, Andrés R. |
| contents | In areas as diverse as contemporary art, play structures, climbing equipment, and modular construction toys, we see the presence of building block-like polyhedral complexes, which are generalizations of the pieces in the game Tetris. We give an algorithm for counting the number of $n$-celled structures on polygonal and polyhedral cells of certain periodic two- and three-dimensional tilings; moreover, we count these structures up to translations, rotations, and reflections of the tiling. We describe this algorithm with respect to structures in the snub square tiling, provide numerical data related to existing three-dimensional art and structures, and suggest puzzles based on these constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23301 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Building with Blocks: Enumerating Polyforms on Tilings Dobbelaere, Bert Kagey, Peter Thomas, Drake Vindas-Meléndez, Andrés R. Combinatorics 05B45, 05B50, 52C22 In areas as diverse as contemporary art, play structures, climbing equipment, and modular construction toys, we see the presence of building block-like polyhedral complexes, which are generalizations of the pieces in the game Tetris. We give an algorithm for counting the number of $n$-celled structures on polygonal and polyhedral cells of certain periodic two- and three-dimensional tilings; moreover, we count these structures up to translations, rotations, and reflections of the tiling. We describe this algorithm with respect to structures in the snub square tiling, provide numerical data related to existing three-dimensional art and structures, and suggest puzzles based on these constructions. |
| title | Building with Blocks: Enumerating Polyforms on Tilings |
| topic | Combinatorics 05B45, 05B50, 52C22 |
| url | https://arxiv.org/abs/2602.23301 |