The restaurant had a wide variety of food
The restaurant offered an extensive selection of food. Penrose tiling is made of five tiles and they’re all different shapes and sizes
Big news and maths! There’s a new aperiodic monotile in town, so we’ve come to the town of New Tile to see what it’s all about.
So you know your regular floor tiles - how they, you know, they go over and over and cover the entire plane with a pattern? This new tile goes infinitely but with no repeats - happy new tile!
Okay, so what’s this all about? We need to break that word down first of all. We need to understand what tiling or tessellation is. So let’s look at probably the most simple shape that tiles - a triangle. When we tessellate triangles, it tends to be a kind of one up one down tooth situation going on, and every single triangle tessellates - even the scalene ones. And what that means is that they fit together beautifully, like you would see on a pavement or a wall - no gaps and no overlaps.
So those are the two main rules - they fit together, no gaps and no overlaps. So every single triangle tessellates, every single quadrilateral tessellates - so if we were to find a kind of new tiling that had a new property where it wasn’t periodic, it would have to at least be five sides.
Periodic - okay, periodic. The second part of this - if you can tessellate in such a way that there is translational symmetry - so the same section repeats over and over again - second color, this section here - I can pick that up and move it over and it looks exactly the same as this section down here - okay, so this is periodic because it has translational symmetry - it’s got a regular repeating pattern over and over again.
You can see how you would take even just a section of two of them, section of three of them - any arbitrary amount - and you can find another patch that looks exactly the same and you can pick the entire tessellation up.
So this is what we mean by global - the entire tessellation up, move it all over and maybe up one, up two, and it will look exactly the same, fit in exactly where it was before. For different shapes, presumably that repeatable bit can be more and more complicated - I imagine, yeah, it can. Um, so I think that’s called a primitive unit.
If you have an interesting combination that you can move around - hexagons looks quite cool - but you could do it with one hexagon as well - have fun with that!
Mathematicians have discovered a brand new kind of tile - it looks like a football shirt. Is this periodic? Yes, this is periodic and it’s a tiling. What we are aiming for is an aperiodic tiling and we have tons of them - actually they’ve existed for a long time. I guess the most famous one is the Penrose tiling.
Yeah, I really didn’t think this would go out of date, but it did. I am shocked by the news, but surprised and you know I’ve got more space. The Penrose tiling is was the the lowest amount of tiles that we could have which made this a periodicity happen - it forced a periodicity to happen. So the Penrose tiling is either made of kites and darts, or a kind of wide rhombus and a thinner rhombus. On their own, they definitely tile periodically, but with the Penrose tiling it’s a set of the two of them - so you need two tiles to make it happen, and there’s these - they’re called matching rules.
Sometimes you can do it with little notches and it forces you to tile them in such a way that there will never be a translational symmetry - so you pick the whole thing up, you move it around, it never looks the same. You might notice that there are some bits which look like they repeat. I do love that his name is Penrose - it’s got this kind of pentagonal rose motif that comes up.
How come if you can see repeats, we’re saying it’s non-repeating? It’s okay to have local repeats - those little bits that look the same - but if you were to pick up the whole tessellation, move it around and try and find somewhere else it fits, you’re not gonna find it. So Penrose needed to, and this was an open problem for a long time - could it even be done? Absolute mystery - mathematicians didn’t know if there could be an aperiodic t be in that particular shape yeah
Nicely done! Well done and particularly good at this, there’s one I actually do think it is part of being good at maths: recognizing the patterns. When David Smith discovered the hat, he also discovered another tile made by putting kites together called the turtle. Here’s the turtle when they were first discovered: this one is made out of eight kites joined together and this one is made of ten kites joined together. They were treated as separate entities, but they are in fact one in the same as part of a continuum. If you look at the angles, you get a right angle down here, a reflex three-quarter angle or inverted right angle over here. The angles are the same as we work our way around, and the lengths are different. All of the lengths are either one or root three apart from one of them which is two, but you can think of it as two ones joined together.
We have the hat and we have the turtle and they are part of a continuum of shapes. If you took all of the edge lengths and kind of pushed them down so that you made some of them smaller and some of them bigger depending on which way you go, you get two boundaries: a Chevron and a comet shape. There’s a middle one as well, right in the middle which looks like a mix between a hat and a turtle which also tiles periodically. The underlying grid for the Chevron and the underlying grid for the comet, although they’re both triangular, the ratios don’t match up, so the lengths don’t match and it means you can’t Rectify them. You can’t put the Chevron onto the comic grid and you couldn’t put the comet on the Chevron grid with the same area and have them line up with the lattice points.
All of the shapes in between the periodic points fall into this no man’s land of a periodicity. The reason the particular one was released to the world is because you can draw it really nicely on a grid of a dual grid of hexagons and triangles together. The dual of the Triangular tiling is a hexagon grid and vice versa. The hat is made out of eight of these if you see the kite that’s created by two Jewel grids together. It was sitting in front of our faces. The reason that it would have this aperiodic tiling is because things were constantly rotating at rational angles or something never to repeat, but it’s actually quite an orderly pattern. There’s only six orientations that it would be at, but it doesn’t even make use of all of the orientations that could be in that particular shape.