A Density Method for Bounding Heesch Numbers in 𝔼3
Abstract
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.
Related
- 数的密度聚类理论:静力学核心 / Numbers as Density Clusters: The Static Core
- A density deficit for sums of three cube-full numbers
- The Heesch number in 𝔼3 is unbounded
- Parallel Searching Method for Verifying Finiteness of Prime Numbers Presented as Specific Linear Combinations of Factorials
- Density and separation for augmented Zarankiewicz numbers
BibTeX
@misc{dzuklevski2026density,
title = {{A Density Method for Bounding Heesch Numbers in $\mathbbE^3$}},
author = {Aleksa Džuklevski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02579},
eprint = {2610.02579},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02579},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Density Method for Bounding Heesch Numbers in 𝔼^3
AU - Aleksa Džuklevski
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02579
UR - https://arxiv.org/abs/2610.02579
AB - 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.
ER -