pentagon tilings
I’ve been doing my best to learn more about the new convex pentagon tiling which was recently discovered by David Von Derau, Casey Mann, and Jennifer McLoud-Mann at UW Bothell recently and I think I’ll just blog about it.
So for one thing I’m super-curious how on earth anyone found out about this before the actual paper got written... But also, it’s fun to try and work out what the hell the researchers were actually trying to do, using only the confused stories of all of those poor news reporters, trying to meet their deadline.
Like, were they sitting in the Pentagon Room at UW Bothell, where all the pentagons are stored, trying to fit them together? Clearly not... but also, that’s essentially what they were probably asking a computer to do: a brute force search over “all” pentagons to see which ones can tile the plane, in some sense.
Or at least, sort of, because here’s the thing. I don’t know how to tell whether a pentagon can tile the plane except by... you know... trying to tile the plane with it! So we want to know about the tiles, but the naively obvious thing to do is to generate tilings. Which is tricky!
It’s pretty easy to classify what all the different convex pentagons could be: there’s 5 angles, all between 0 and 180; their supplements add up to 360, and the sides are constrained in a certain way. This data all gets summed up in what’s called a configuration space. The configuration space for pentagons is pretty straightforward, I imagine. I think it’s going to be a thing called a convex polytope, or maybe some kind of “orbitope” if you want to ignore symmetries. Whatever. We understand what those spaces look like and there’s lots known about them.
But the configuration space for pentagonal tilings? I have no idea what that looks like and I suspect neither does anyone else. This explains all of those poor reporters trying to say things like “there’s infinitely many possibilities”... what I now think they really mean is that the configuration space is cut up into potentially infinitely many pieces, according to what I’ll call the isometry number and the edge matching statistics.
I just made both of those terms up, and I’ll have to explain what I mean later. Loosely, the isometry number tells you something about how symmetric the tiling is, and the edge statistics say something about how the tiles fit together. The current state of the art is, as far as I can understand, to try to look for tilings that using only a few different sorts of tiles, but to start with small isometry number and trivial edge matching statistics and work your way up.
There’s a third post I’ll have to make, and it’s this. In an interview, Mcloud-Mann said that their computer search was generating a lot of spurious data (stuff which didn’t correspond to actually tilings)... How to make sense of that? It seems like they perhaps came up with a different searching strategy than what I’ve outlined above! I wonder what they were doing? I’m very curious. If I were to hazard a guess, I’d say that they were searching through the configuration space of pentagonal tilings, but using some sort of combinatorial guesswork to see how the tiles might fit together locally, without trying to construct the whole tiling. If done right, this would fit what we know: it’d be more tractable than trying to exhaust the configuration space, but it would definitely generate a whole bunch of false positives.
Very interesting!