Aperiodic monotiles: from geometry to groups
<|reference_start|>Aperiodic monotiles: from geometry to groups: In 2023, two striking, nearly simultaneous, mathematical discoveries have excited their respective communities, one by Greenfeld and Tao, the other (the Hat tile) by Smith, Myers, Kaplan and Goodman-Strauss, which can both be summed up as the following: there exists a single tile that tiles, but not periodically (sometimes dubbed the einstein problem). The two settings and the tools are quite different (as emphasized by their almost disjoint bibliographies): one in euclidean geometry, the other in group theory. Both are highly nontrivial: in the first case, one allows complex shapes; in the second one, also the space to tile may be complex. We propose here a framework that embeds both of these problems. We illustrate our setting by transforming the Hat tile into a new aperiodic group monotile, and we describe its symmetries.<|reference_end|>
@article{coulbois2024aperiodic,
    title={Aperiodic monotiles: from geometry to groups},
    author={Thierry Coulbois (I2M), Anah'i Gajardo (UdeC), Pierre Guillon (I2M),
  Victor Lutfalla (I2M)},
    journal={arXiv preprint arXiv:2409.15880},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.15880},
    primaryClass={cs.DM math.CO}
}