Metraipolis › archive › dc_ea12e57ebdf6

A Density Method for Bounding Heesch Numbers in $\mathbb{E}^3$ — arXiv:2610.02579

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.02579
License (as archived): CC0-1.0
Archived: 2026-10-06T08:24:45.646Z
SHA-256 of the archived text: 403dd0d9d87e875f15e4a51ebdf1c006a3da8074d22dd10e38a58956997db8d9
Stable URL of this page: https://metraipolis.com/documento/dc_ea12e57ebdf6

How to cite

A Density Method for Bounding Heesch Numbers in $\mathbb{E}^3$ — arXiv:2610.02579 — arXiv (en), https://arxiv.org/abs/2610.02579 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ea12e57ebdf6 (Metraipolis archive dc_ea12e57ebdf6, sha256 403dd0d9d87e875f15e4a51ebdf1c006a3da8074d22dd10e38a58956997db8d9)

BibTeX

@misc{dcea12e57ebdf6,
  title = {A Density Method for Bounding Heesch Numbers in $mathbbE^3$ — arXiv:2610.02579},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.02579},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ea12e57ebdf6; sha256 403dd0d9d87e875f15e4a51ebdf1c006a3da8074d22dd10e38a58956997db8d9},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.02579. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In this paper, we provide a novel approach to limiting Heesch numbers of a specific family of tiles. In particular, we show how a certain family of three--dimensional prisms has a finite Heesch number. In so doing, we initiate a structural study of ratio $q$ tilings by two prototiles, in which, loosely speaking, one tile appears $q$ times more often than the other. The method works as follows: we take planar intersections with coronas around the prism in two perpendicular directions and compare the frequencies of the resulting prototiles in those patches. If every tiling by these prototiles has limiting ratio at least $q >1$, then sufficiently large coronas of the prism would require incompatible tile counts in the two directions. We also provide some computational results and exhibit $6$ infinite families of notched prisms that have finite Heesch numbers greater or equal to $2$. Finally, we provide some negative computational results about the non--existence of certain prisms with given Heesch numbers.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-10-05 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.02579 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-06T08:18:48.292Z dalla pagina https://arxiv.org/catchup/math.CO/2026-10-05?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).