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

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.