An amusement

This being notice of a tiling phenomenon – called an einstein - turned up by a tiling hobbyist from up north – one David Smith. The subject of two pieces in the New York Times (and no doubt elsewhere), reference 1 and reference 2, and one learned paper, reference 3.

Tilings have interested people for a long time, interest which has spread out beyond the simple plane to all kinds of exotic spaces. But here we are interested in small sets of tiles which can be used to tile a plane – but only in a non-periodic way, that is say not like most wallpapers, curtains or floor tilings. So, the square will tile the plane, but only in a periodic way. Ditto the equilateral triangle. Ditto the hexagon. While the circle will not tile the plane at all. 

This einstein is a bit more complicated than that, but not hugely complicated. It is called a polykite, that is to say it is made up of four pairs of identical kites. Which, when tiled, have the irritating property of occupying key points of a hexagonal tiling. Hiding in plain sight, as it were.

But going back to the 1960’s, some mathematicians wondered whether there were sets of tiles which could tile the plane, but only non-periodically. Never in a periodic or regular way, never repeating themselves; not like squares or equilateral triangles at all. After which some other mathematicians turned up some large sets of tiles which fitted this bill, which had this property. And then in the 1970’s, the chap who became Sir Roger Penrose took an interest and found a set of just two simple tiles with this property. The hunt was properly on. Several more such sets of two tiles were grubbed out of the ether. But was there a single tile with this property?

This had to wait for around half a century for Smith to come up with a promising candidate, which he turned over to a more mathematically minded acquaintance for consideration. Resulting in the first three of the references below.

From reference 3, I learn that considerable effort has been poured into proving theorems about tilings in all kinds of spaces, not just planes. We have Wang tiles. There is something called the Heesch number of a shape – which one might wrongly think had to be 0 or infinity. One can shift complexity from the shape to matching rules for that shape, rules which go beyond just saying that the matched tiles have to fit together, rules which might assign colours to the edges of shapes. One can assemble shapes into hierarchies. One can use computers to do the grunt work.

Effort which has sometimes turned up statements which one can prove cannot be proved, undecidable in the jargon. A concept which existed in the days when I knew about such things, but which were not, as I far as I was then aware, used in practical mathematics, as here.

But there is still work to be done. Smith allows his einstein to be reflected as well as rotated to make up a tiling. So now the hunt is on for an einstein which does not involve reflections.

While I rather liked the quilting and the ceramic tiling based on this einstein which are illustrated in reference 1. Would a non-periodic floor tiling provide visual interest or would it, after a while anyway, just irritate?

PS 1: I wonder if it is the case that any crystal can be usefully thought of as a periodic tiling by the molecule involved or by the atoms making up that molecule?

PS 2: I can claim a very junior connection to reference 3, in that it was supported in part by a Senior Rouse-Ball Studentship, while I, while at school, once took a Rouse-Ball prize. Presumably the same chap, but his entry in Wikipedia tells me nothing of what the link with my school might have been. He does not appear to have attended it himself. The rather longer biography at reference 5 does not help either. But I do still, occasionally, consult the prize book.

References

Reference 1: What Can You Do With an Einstein: Earlier this year, mathematicians discovered a unique shape. Now do-it-your-selfers have found ingenious ways to put it to use – Siobhan Roberts, New York Times – December 2023.

Reference 2: Elusive ‘Einstein’ Solves a Longstanding Math Problem: And it all began with a hobbyist “messing about and experimenting with shapes” – Siobhan Roberts, New York Times – March 2023.

Reference 3: An aperiodic monotile – David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss – March 2023. The copy I have is a pre-print, but I dare say the result stood review.

Reference 4: https://momath.org/. The National Museum of Mathematics of New York. A place which cropped up along the way.

Reference 5: https://mathshistory.st-andrews.ac.uk/Biographies/Ball/. 

Reference 6: Tilings and Patterns: an introduction - Branko Grünbaum , G. C. Shephard – 1989. A book about tilings and patterns which I acquired when I was thinking about the tiling of the visible surface of material objects in support of their projection into consciousness - for which purpose it was very much overkill. It only devotes a few of its 450 pages to periodic tilings and a quick scan today did not turn up anything like the present problem.