Re-Tile Everything. And I mean everything. The entire universe, if you want to. Using a single tile, you can now do this an infinite number of ways.
Yes, the big mathematics news of the week is the discovery of the hitherto elusive “einstein”, or monotile (so THAT’S what the great Albert’s surname means!), which can tile the plane (2 dimensions) aperiodically, in other words, in ways which never repeat themselves.
Another great, Roger Penrose, spent decades on the search, narrowing it down to pairs of tiles (for example, “kites” and “darts”) which can tile aperiodically. But never did he, or anyone until now, find that single tile which would do the job. Its very existence was an open question. Its finder? One David Smith, 64, of East Yorkshire, UK, a “shape hobbyist,” as he calls himself.
I dabbled some years ago, trying a shape which was half a square (diagonally cut) joined to an equilateral triangle. Nope, it seems. If I had found a shape which seemed to tile aperiodically, how would I even prove it rigorously? I don’t have the mathematical education to do that. Anyway, now the thing has been found. Its simplest version is made of 8 identical sixth parts of a hexagon, and has 13 sides, as in the illustration. A rather technical preprint article on the find, yet to be peer reviewed, but getting plenty of attention, is here.
This is so much more interesting as a bathroom tile than, say, squares, triangles or hexagons (the only regular shapes which can tile the plane regularly). Simply because it allows infinite tiling variations to each one of us. Also, because this is a purely mathematical object, there’s no way to copyright it, just as there isn’t with, say, a square.
It’s early days yet, but I expect this shape to revolutionize the world of tessellation, or tiling, in general. Wallpaper, physical and computer. The aforementioned bathroom tiles, or ones for the kitchen or elsewhere. Floors. Yes, there’s an awkwardness at the edges which you don’t have with squares, because here you have to cut off pieces to make a straight edge, or live with an irregular one. Small price, I think. Textiles: clothing, curtains and drapes, tablecloths, carpets. It might be a craze which comes and goes, getting its 15 minutes of fame, or it might linger for years, decades, being infinite in possibilities. The tile itself is also, I must add, only one of a large set of such tiles, oh yes!
What we still don’t yet have, though it has not yet been proven impossible, is an aperiodic mono tile which tiles WITHOUT flipping or reflection. I might just dabble some more myself: this tile’s finder is much more like me than the serious mathematicians he enlisted to prove his tile’s aperiodicity. I would also have to do a similar thing if I ever found a suspect. Interestingly, as is sometimes the case between dimensions in mathematics, aperiodic monotiling was already proven in THREE dimensions quite a while before this second find. (In math, one can have as many dimensions as one’s work requires, literally without limit.) In 3D, the tiles are called quasicrystals.
My own tiling finds in 2D, 3D and so on have all been periodic, but at least they have the virtue of being countably infinite in form, as I detail here.
This was enough of a find to satisfy and delight me when I made it more than 20 years ago.
There is, however, more to discover in the worlds of tiling, as I hope to have shown here. Who will make the next cannonball splash?
BLOG by Tony Hanmer
Tony Hanmer has lived in Georgia since 1999, in Svaneti since 2007, and been a weekly writer and photographer for GT since early 2011. He runs the “Svaneti Renaissance” Facebook group, now with nearly 2000 members, at www.facebook.com/groups/SvanetiRenaissance/
He and his wife also run their own guest house in Etseri: www.facebook.com/hanmer.house.svaneti
