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March 22

Aperiodic monotiling

http://newscientist.com/article/2365363-mathematicians-discover-shape-that-can-tile-a-wall-and-never-repeat reports the first solution to the open problem of aperiodic monotiling. I'm an absolute newbie in subject but why is the following case not considered an aperiodic monotiling?

“

Each cell of an infinite rectangular grid can be labelled (x, y) for all integer x and y, and triangulated in 2 possible ways as in the diagram.

If, say, a + b (where a is the abs(x)th digit of $π$ , and b is the abs(y)th digit of e) is even, triangulate it one way, and if odd, triangulate the other way.

Other than reflections about the axes (if that makes it periodic, just use different transcendental numbers for each quadrant), there shouldn't be any repetition, and every tile is a rectangle split diagonally.

”

Thanks,
cmɢʟee⎆τaʟκ 13:37, 22 March 2023 (UTC)[reply]

In the usual understanding of the notion of "tile" in the context of a tiling of the plane, a (non-coloured) tile is a geometric shape, which is a planar area determined by its boundary. So diagonals do not count. Classifying square tiles into a number of distinguishable types by the orientation of some marking, such as a diagonal, is equivalent to giving them a distinguishing colour. So the proposed method does not qualify as a monotiling – there are two tile types. Moreover, while the tiling resulting from crossing π with e is non-periodic, if these numbers are normal, it contains arbitrarily large patches in which all tiles are identical, so while it is non-periodic, it is not aperiodic. --Lambiam 14:50, 22 March 2023 (UTC)[reply]

You can read the original paper here: arXiv:2303.10798. --Lambiam 14:55, 22 March 2023 (UTC)[reply]

@Cmglee:: Creating an aperiodic tiling from one tile is not difficult. As you point out, you can make an aperiodic tiling from a triangle or a rectangle tile. The hard open problem is or was whether there's a tile shape, with certain restrictions on the shape, that can tile the plane but *only* in an aperiodic way. A triangle or rectangle tile certainly can't help here, because those shapes can also tile the plane in a periodic way.

As Lambiam points out, see the details in this very fresh preprint. David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, “An aperiodic monotile”, (2023-03-20) arXiv:2303.10798 .

– b_jonas 16:12, 22 March 2023 (UTC)[reply]

Thanks for the great explanation. So aperiodicity being the *only* tiling is the key. Cheers, cmɢʟee⎆τaʟκ 04:11, 23 March 2023 (UTC)[reply]