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PreprintCC BY-SA 4.0HAL

Crystallization of the undertile

  • Arnaud Chéritat
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Abstract

We classify all periodic tilings with a monotile known as R1 and called here the undertile, which underlies several aperiodic monotiles such as the Taylor-Socolar one. The classification is done using deflation, through a study of fracture lines between zones where deflation can be performed. A full classification of fracture patterns is given, that also applies to nonperiodic tilings with the undertile. This can be used to reprove nonperiodicity of some existing monotiles based on the undertile.

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Cite

BibTeXRIS
BibTeX
@misc{cheritat2026crystallization,
  title = {{Crystallization of the undertile}},
  author = {Arnaud Chéritat},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20663},
  eprint = {2610.20663},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05802394v1},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Crystallization of the undertile
AU  - Arnaud Chéritat
PY  - 2026
DA  - 2026/10/14
PB  - HAL
DO  - 10.48550/arxiv.2610.20663
UR  - https://hal.science/hal-05802394v1
AB  - We classify all periodic tilings with a monotile known as R1 and called here the undertile, which underlies several aperiodic monotiles such as the Taylor-Socolar one. The classification is done using deflation, through a study of fracture lines between zones where deflation can be performed. A full classification of fracture patterns is given, that also applies to nonperiodic tilings with the undertile. This can be used to reprove nonperiodicity of some existing monotiles based on the undertile.
ER  -