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PreprintCC BY-NC-SA 4.0arXiv

Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles

  • Teruhisa Sugimoto
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Abstract

Using a convex pentagonal monotile belonging to the Type 5 family, we investigate the relationships among the hat tile, turtle tile, and Tile$(1, 1)$. By applying Sugimoto's Perspective and Amfirifma's Perspective, we obtain four types of concave polygons, AH-tile, BH-tile, AT-tile, and BT-tile, each having Heesch number 1 under the conditions considered. We show that these polygons correspond to clusters used to generate the non-periodic tilings 𝒯h and 𝒯s. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to ASPmr{A−tile, B−tile}.

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Cite

BibTeXRIS
BibTeX
@misc{sugimoto2026tile,
  title = {{Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles}},
  author = {Teruhisa Sugimoto},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09603},
  eprint = {2609.09603},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09603},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles
AU  - Teruhisa Sugimoto
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09603
UR  - https://arxiv.org/abs/2609.09603
AB  - Using a convex pentagonal monotile belonging to the Type 5 family, we investigate the relationships among the hat tile, turtle tile, and Tile$(1, 1)$. By applying Sugimoto's Perspective and Amfirifma's Perspective, we obtain four types of concave polygons, AH-tile, BH-tile, AT-tile, and BT-tile, each having Heesch number 1 under the conditions considered. We show that these polygons correspond to clusters used to generate the non-periodic tilings 𝒯_h and 𝒯_s. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to ASPmr{A−tile, B−tile}.
ER  -