Aperiodic Monotile with hole in it.

Question

Here is A partial tiling of the plane in the style of Heesch tiling, but with the new Monotile discovered and encircling a hexagonal hole.

Call such a combination a quasi-Heesch tiling, where you use a single tile to build a tiling in encircling coronae, but where the central shape no longer needs to be the same as the tiles. Here we have a hexagon and the newly discovered aperiodic monotile. My question is, since this is already exhibiting a q-Heesch number of 8, which is 2 greater than the record holding Heesch tile, and since the Monotile tiles the plane anyway, does this imply that the new aperiodic monotile permits tilings of the plane with hexagonal holes in it? is it possible to know if you are building a region that will eventually be forced to contain such a hole? Does the 6-fold symmetry of this q-Heesch tiling imply rather that it must fail at some point beyond the 8 coronae featured since such a symmetry is not present in the full perfect tiling?

Not sure if this is what you want, but the central $6$ tiles plus the hexagonal hole form a shape that itself tiles the plane periodically. — nickgard, Apr 07 '23 at 19:43
@nickgard it does? i tried to find a way to tile with the hexagonal meta tiles but couldnt work it. picture? — Locke Demosthenes, Apr 08 '23 at 20:31

nickgard · Answer 1 · 2023-04-08T21:06:45.973

4

One part of the question asks whether the new aperiodic monotile permits tilings of the plane with hexagonal holes in it. The following image is a cropped version of the central part of the full image from the question, showing mainly the six monotiles that surround the central hexagonal hole:

This central shape tiles the plane periodically, which is equivalent to a partial tiling of the plane using the monotile but leaving hexagonal holes:

edited Apr 08 '23 at 21:06

answered Apr 08 '23 at 21:00

nickgard

4,116

this is excellent, looking at the next coronae out, it appears the larger hexagonal metatile may also tile the plane. to me this implies that we could find larger and larger such encircled holes, and build a tiling from them that now has arbitrary distances between the holes. indeed if the original hole has infinite Heesch number, then presumably there will be a tiling with 1 hole, then every possible gap distance for two holes, and so on. this means that while any periodic tiling must include holes, we could form a periodic tiling that is asymptotically complete in the plane – Locke Demosthenes Apr 09 '23 at 10:34

Aperiodic Monotile with hole in it.

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