[2305.17743] A chiral aperiodic monotile
source link: https://arxiv.org/abs/2305.17743
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[Submitted on 28 May 2023]
A chiral aperiodic monotile
The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.
| Comments: | 23 pages, 12 figures |
| Subjects: | Combinatorics (math.CO); Discrete Mathematics (cs.DM); Metric Geometry (math.MG) |
| MSC classes: | 05B45, 52C20 (Primary) 05B50 (Secondary) |
| ACM classes: | F.2.2; G.2.1 |
| Cite as: | arXiv:2305.17743 [math.CO] |
| (or arXiv:2305.17743v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2305.17743 |
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