Mathematical Games Hosted by Ed Pegg Jr. [Episode 4: The Hat and Other Tilings]

Hello, everyone. This is Ed Penk Jr. And we're going to do another episode of Mathematical Games today, but I'm going to wait a minute or so for people to come in. So it's been a rather exciting month in the world of recreational mathematics with all these new things coming out. And let me know when the numbers have stabilized. Can everyone hear me okay out there? All right, so Mathematical Games was a column by Martin Gardner and Scientific American. And for 25 years and many books, he talked about the world of recreational mathematics. One item he discussed quite a bit was tiling problems. And with this series, I'm hoping to do justice to some of the topics that Martin Gardner explored and further some of them. So for this, we're going to talk about the new discoveries in recreational mathematics that were just, just discovered dealing with tiling. And here's the COVID of scientific American from January 1977. This was a, an important issue for me. Yes, I'm an old person. I was a boy when I got this issue in the mail and this cover particularly fascinated me. The Penrose tiles when, when they were discovered. And I, I actually got to be friends with Roger Penrose later. But I still quite remember when I was 13 or so and got this in the mail and got, got to see this discovery for the first time. It amazed me that two tiles could be so complicated. And just a few weeks ago, we learned about the HAT tiling, which pretty much solved the problem for the fewest tiles needed for a aperiodic set. And it turns out that just one tile is needed, this shape that's been called the hat, but it's a rather complicated tiling. So we started exploring it pretty much the hour that it was announced on Archive. Let me see over here. And welcome Sabrina and everyone else. Okay, so to start off, we want to talk about what is periodicity. So it's well known that any triangle can tile the plane. So here's here we have a demonstration on any, any triangle can tile. And we've added this bit of periodicity, if we can remove it. But if we move this dot here, we basically shift the grid and we can overlay the triangles on top of each other. And if you pick the right values, you can overlay the triangles on top of each other. And this is what is known as, this is what is known as periodicity. When you can pick up a copy of the tiling and overlay it on itself, then it's a periodic tiling. For example, there. Similarly, all quadrilaterals can tile. We can Shift these around and by basically flipping each quadrilateral on its edge, no matter how distorted it is, you get an infinite tiling of the plane. And this, you can also pick up this tiling, move it around and put it on top of itself. So this is also periodic. And the vector displacement is basically from this point to this point is a translation vector for moving the tiling onto itself. And we can also do tilings with pentagons. And I've added a way of moving, moving the pentagons around. These are a little bit harder to control. But with, with practice, which I haven't done enough of, you can move the, the pentagons onto themselves. All of these tilings shown here are, are periodic tilings. And there are now known to be 15 different types of pentagons which can tile the plane. These are all the different variations that can tile the plane. Most of them were found by an amateur mathematician by the name of Marjorie Rice. She found for example, this one and three others. So she found more of the, more of the pentagon tilings than anyone else. A homemaker. And the last one that was found, not sure if this one works. Yeah, I messed when I added the, when I added the, the vectors, I messed up tiling number 15, so I'm not sure what I did there, but I'll need to fix this before I make this translation live. But each one of these tilings has a motif. And when you, when you repeat that motif, you get an infinite tiling of the plane with pentagons. And as a diversion we, we can also do other tilings, for example, with squares. Here's one way of tiling a big section with all the squares being different sizes. And the way it works is you have this strange Matrix here with 1, 2, 3, 4, 5, 6, 7 at the bottom and the numbers 86, 87, 88, 89, 90, 91. If you use this checkerboard pattern and fill in the numbers, 1 +98 is equal to 91 +8. On either side of each line. The sums are the same. And that allows, this matrix, allows you to do this tiling which you can expand any size, do you. For any square number, you can take one to a googleplex squared and make a gigantic tiling where all this different squares are, have different sizes. And you'll get this, this strange shape that was found by Adam Ponting. So let's look at some of these range stranger tilings that are known. These are some of the things that have been explored by recreational tilers. And we'll get to a person who's done quite a bit. This is the tiles with finite convex spectra. Let's see if I can find the next one. Interesting one. Many of these were discovered by recreational mathematicians and it's still an unsolved problem whether they found all the interesting cases where you can split up a, a convex shape into tiles that are identical. So it's still not known whether we've found all the interesting ones. This one is found by Mike Hirshhorn, a mathematician in Australia. These are all pentagons and they make this nice 18 gonna. And this is the record. You can split up a dodecagon into 48 different pieces. No one is sure whether this is the, the highest of the finite cases. And this particular tile can do a whole bunch. This is the, the record setter for fin finite convex spectra. And there's some cases that are infinite. You can keep on adding layers to this one to get more tiles. So when we're doing these explorations with tiles, it's often this sort of thing that's at the start. And we're hoping to find more tiles like, like this one that have amazing properties. Kind of a side note item here. Here we have a 9 by 11 grid and here we're moving, we move the tiles around and get a 10 by 10 grid. So basically these are the same shapes, but this one has one fewer tile than the one over here. And the, the way the trick works is you've got a, a square number four by four and you do three by five, you get one less, but that disappears as you get to larger sizes. And this is called the melting tile illusion. This is used by some puzzle makers to make peculiar puzzles where you've solved the problem, you fit it into the frame, but then the puzzle maker ends you one more tile to put in. And it seems impossible because you've already fit in the frame. But by shifting the orientation, you can get in that one extra tile by moving the pieces around. Let's see here. This, this concept of, of. Periodicity comes in with a lot of dissections. And at the demonstration site we have hundreds of these dissections. For example, here's a square in six pieces. And when we move the pieces around, we get a dodecagon with six pieces. Underlying the dodecagon is a periodic square tiling. Underlying the dodecagon or underlying the square is a periodic dodecagon tiling. Because the dodecagon and triangles will tile the plane. Virtually all of the dissections that we have on the site use this property of periodicity to Solve the problem and freeze. Solve this problem using periodicity. So that's a application of periodicity for a tiling problem. So what about non periodicity? Well, that's where you cannot move one tile. You can't lift up all the tiles and move them and get the same thing. On the infinite plane. If there's a defect somewhere, you're prevented from doing that movement action. There is no translation vector. And here's one way of tiling the plane with L trominos so that you cannot lift up everything and move it onto itself, because this is a non periodic tiling. However, it's easy with the ultramino to make a periodic tiling. You just basically make a two by three rectangle and repeat that. But it's also easy to use the ultramodo to make something which is not periodic by adding those defects. Here's another range tiling, the pinwheel tiling, where you have a 1, 2 domino split into two triangles. And at the lowest level you take that triangle and split it into five copies of itself. And since the since it's a non standard dissection of a triangle into copies of itself, as you repeat that action, things just get weirder and weirder. There's a building in Australia that uses the pinwheel tiling as its covering. Here's another non periodic dissection, the isopenta dissection. And things get stranger and stranger. Anybody recognize this? It's actually not the Penrose tiling. All of the tiles here are identical because it's a 3D item using rhombuses. This is what's known as the Wing era roof. However, it's very similar to the Penrose tiling. If you look at it from the top. Yes, it's, it's sort of a 3D Penrose tiling. So this is kind of a cheating way of doing the Einstein problem with a single tile. You just basically slip things into 3D and use this, this special property of the Penrose tiling. But that's kind of cheating. A lot of the non periodic items we've shown so far use triangles and quadrilaterals. But as we've shown, if we have triangles and quadrilaterals, you can always tile the plane with them. So can we get non periodic behavior from pentagons? Yes, and one way is to use the Hirshhorn medallion I showed earlier. You can basically spiral the pentagons around itself and you can keep going with those spirals and it covers the plane with a rotational translation. But there are many different periodic tilings with this particular pentagon. If you want to do non periodic behavior with a pentagon that's forced or a set of pentagons, you can't use one of these standard 15 pentagons to do your tiling because they will make periodic patterns as given. Ammon did several pentagon and hexagon pairs which had non periodic properties. So we have more non periodic things. We can use the PEEP and Tomino and we've shown the Eltromo. The Sphinx is another one that you can make non periodic tilings with. But a using the rhombus that it makes, you can also make periodic tilings. So one thing that people often look for is they'll find a non periodic tiling and then they'll. They'll see if it's forced at all. And if, but if it's not forced then it doesn't qualify. We can. There are a lot of reptiles. We've got a few hundred different tilings on the site. Many of them are fractals. So if you go to demonstrations wolfman.com you can see many of these different items. And Since I have 50 different programs in one notebook, some of them are clashing with each other. That is why some of these are pink. But this is one fractal that you can divide into two copies of itself and you can repeat this as many times as you want. Here are some, some fractal tilings of order 5. This whole thing is a fractal and it divides into five copies of its. And these are. Use the Chaos game to define the pattern. So one, one thing people often look for was is there a fractal tiling that entail the plane and forces non periodic behavior. But unfortunately many of these weird tiles also lend themselves to periodic behavior when they're all the same size. And at the substitution tiling site I've got dozens and dozens of different substitution tilings that are known. Unfortunately all of these have under underlying them either periodic behavior or they're. They're too complicated to. Or they have more than one tile involved. So what's been, what's been needed in this, this problem that's, that's been open for, for many years is whether there's an aperiodic tiling with a single tile. And there's two parts to that. First, your tiling set. In this case, one tile has to make a non periodic tiling. And second, and this is the harder part, arbitrary arbitrarily large periodic regions or patches are impossible with that tile. And that's getting both of those things at the Same time has been an open problem for 30 years now. Penrose got it down to two tiles. You've sort of working on the, on the work of Kepler though. Kepler developed a set of monsters with non periodic behavior. The first one with the aperiodic tiling system was Robinson. And this is a demonstration that shows the Robinson tiles. The various steps in this are somewhat strange, but this is one way of building up a Robinson tiling programmatically. And this was the minimal set for a while and then they started finding things like the chair tiling. And we have a demonstration on that by Bradley. And I think he shows that this Rule 686 is equivalent to the chair tiling. And there's also the Wang tiles. This set of tiles at the bottom here is able to tile the plane with edge matching. However, it doesn't use a substitution tiling. It uses something stranger. But this is the, the minimal known set of Wang tiles with four colors that, that works. So let's get back to the hat. Here's the shape here, rather simple. And it was this person and David Smith who found it. He describes himself as a shape enthusiast. And pictured in front of him there is one of the two aperiodic tiles that he discovered. And he found this in November while using this program from JAPS Puzzle page. And I'll show that to you here, Jeps Puzzle page. This is a useful program for looking at really weird sets. In this case dividing a hexagon into 12 p. And with this one particular tile he couldn't get a solution for it, but he could get, he could fill up larger regions, but he couldn't get it to do anything in a, in a, in a satisfying way. Wasn't working right with his sense of how shapes worked. So he contacted a friend of his and also a friend of mine, Craig Kaplan, who developed this particular app. Yes, we're going to be talking about an app for a bit. You can go to the site here, this app called Good Fences and basically all you're doing is taking a shape and surrounding it with copies of itself. And that's what's called a Corona. So here we have the L Tromino and we can surround it with four copies of itself. We can completely surround it. So this was Craig's research into what's known as the heesch problem for what tiles can you surround copy? Can you surround itself with copies? And this is an unsolved problem as to how far you can go so that you can surround the tile with copies of itself, say three, four Five times, but no more. And here's a demonstration that I made with Craig called the Snake. I didn't name it, Craig did. When Craig came up with some results, in fact, I'll show the results are here. The Polyform tiling page@Polyamo.org UK he's got all sorts of different results on tiling at this page. And when he made an update, I happened to notice that he had this shape and this shape and this shape and this shape and this shape, all in order. Three heath tilings. And he said, oh yeah, it looks like you found something. We've now proven that there are an infinite number of Heath three tiles just, just because we're, we're basically adding another chevron on to the snake. As we make the snake longer, we, we basically get the same tiling pattern. So he'd called it the peg Snake. It wasn't my idea, but it was this problem, this he's problem that David Smith back here thought he might have found a good solution for. So we took the. Our Craig and David used the HE software that he developed for this site and if you look here, they've now extended it to Polykites somewhere here. Polyimon, Polyhexes, Polyaminos, Polykites. These are the results that they just added a few weeks ago, these initial results. They were looking for coronas for the hat, but they managed to get up to level, level, what is it? 1, 2, 3, 4, 5. They got to level 5 and when they were working on level 6, they knew they had something because either this was a level 6 heath tiling, which would be a record setter, or they found something else, but it looked like you could keep this tiling going and that's what they found out. So first they looked for all the coronas that could be made and they found that there was 188 different Coronas, Level 2 Coronas for the hat. And here they all are here because they were. Look, they thought they'd solved the heat problem to some level. This is also a Wolfram demonstration here. But they managed to get well beyond that and it looked like you could make an infinite tiling that was very strange. So they kept looking at it and then they tried to figure out what was going on with this tiling. And here's, here's David Smith's website where he talks about the problem, The various methods and methodologies. This shape here is one that came out later and we'll get to that. So they, they found all these triangular rearrangements and they Couldn't quite figure out what was going on. And it was later that they found this second tiling which also seemed to use the same underlying pattern that was different and it also was not periodic in any way. They. They couldn't force a periodic behavior out of it. So they brought in Heim Goodman Strauss over here, this is Craig Kaplan. And they still couldn't figure out. So they brought in Joseph Samuel Myers and that's when they discovered that it was an actual whole family of tiles. And here we have how they can modify any of these will tile the plane and not be periodic anyway. Except for this one you can do periodic behavior with. It's pretty simple to see how. And also this one at the other end, Tile 10 you can do periodic behavior with. And it's. It's easy to see how this one in. This in the middle here is also periodic, but it's not obvious how you can do that. However, and the. In the paper on Arxiv, they show how this tiling can be tiled periodically. However, none of the other variations will tile periodically. So you can take any of these and tile the plane with them. One thing I hope to do eventually is to do this with coronas to show how those work. However, of the 180 Coronas, only 63 of them actually work in an infinite tiling. So we'll need to throw out some of them to get those to work. But hopefully I'll get that done soon. So once they had all these, all these things pieced together, they published the paper 20-03-2023. And you can go to the archive link and see it and as I'll toss the link over here into the chat. And let's see. So they mentioned that this is an. What's known as an Einstein. It's in German, means one stone. And the paper goes into exquisite detail for what's going on with this tiling. It's a substitution tiling similar to what I did with the pinwheel pinwheel tiling. They're basically taking a shape and making substitutions, but there's four of them, they're all meta substitutions and they're all weird. So you have to kind of consider that each of these is actually built from pieces of hats and each time you make a substitution you're changing the angle more. So it took a while for them to figure out what was going on with this. It's almost simpler to figure out what's going on when you treat this as a fractal. And this is something Yoshiaki Araki showed. So within one day we had a pretty good idea of what was going on. One of the first things I wanted to do is look at a big patch and see how everything worked with orientations. So each color here represents a hat with the same orientation. And you can see that there's these strange bands set up here. For example, this one here, this one here, this one here. And it turns out that the spacing between these bands is in the golden ratio. And also there's this yellow band here, here, here. If you measure the distance between the yellows, you also get the golden ratio. There's also this cyan, these cyan bands going another way, same thing. And you get the golden ratio again. It turns out that the ratio between the flip hats and all the other hats is one over the golden ratio to the fourth, which is somewhat amazing. And at this link. We have, we have all sorts of different items for, for these tilings. So one of the first tilings I came up with, well, there's an underlying hexagonal grid under all of these. What happens if we, if we consider the, the shape of a Y, that's one way of dividing up a hexagon into three gems. And there's also an upside down Y. So I just figured all the hexagons that were split into three parts, I would color either red or blue. And this is what, this is the pattern that came out of that. And this is on the link I showed earlier. Then I said, well, let's just do everything. It's either a Y or a partial or an anti Y, an upside down Y or a partial. And you get these strange triangles popping out. This is, this was a very strange pattern. This is one of the, the first things that seem to go viral. In addition to the other hat items. Let's see here. And Doug Duggan Hammock ask is pitangital geometry related somehow to the hatani? I don't know the answer to that. However it does, I think it appears with the triality item that'll get to at the end. So Yoshiaki Iraqi started making all sorts of different animals out of the hat. And there's also a, an early, an early bit of contention whether it should be called the hat or the T shirt because when you have it in one orientation it sort of looks like a hat, but in the other orientation it looks even more like a T shirt. So for about a week it was fighting back and forth, but hat eventually won. It's, it's being called the hat. And we Also had some people trying to break it. Here's the closest attempt I saw to breaking this as a problem. All of these shapes in here are hats and it's periodic except for the little tiny white spots in certain places. That's where the, that's where it breaks down. So you can, you can pretty much cover 99.99% of the plane with hats infinitely and it's periodic, but there's a little tiny piece that doesn't work every so often. So this tiling does not work because it has those flaws. But it, but it shows how, how close the hat is to almost being periodic but not quite. And that little, that little tiny bit is enough to force it into being on is enough to force it into being aperiodic. So I just did a whole bunch of things making large patches with hats. But Brad Clee, my, my colleague here at Wolfram also did quite a bit. Here's one way of tiling the polygon and he found that there was a, an underlying stellar automata of sorts underneath and he called it the hat triality. And here's, he's building up the different varied various colored dots involved with this failure automata pattern. So here's a way of coloring the, a good portion of the plane red, blue or white based on how the stellar automata expands. And you can go directly from, from this stellar automaton over to a hat tiling. And with that he shows here's a, a huge patch of the tiles with, with this technique using sailor automaton, he's skipping the substitution tiling and going right to cellular automata to do the, the hat tiling. Nan says 99.9. Yeah, that, that probably is an overstatement, But we now have two methods forward for generating these tilings. One is cellular automaton based method and the other is a substitution tiling based method. So the combination of this family and the tiling itself may give a simple proof that this is non periodic. And the proof goes like this. I think this is brilliant. This hasn't been accepted yet, but I think it's a brilliant method. Suppose that the large, these large tilings infinitely are periodic. Then there's a minimal periodic patch with a translation vector. The Polyimon shapes. Tile 01, that's this one over here. The simple chevrons and tile 10 which is the Pymans over here would each have translation vectors. The area of this one is 4 if you scale it and the area of this one over here is 8 if you, if you scale it. The same way for this translation, this mythical translation vector to work, one of them needs to have a square root of two term. However, that vector does not exist on the triangular grid. Therefore there is no periodic tiling with this particular sequence here. And I think that's kind of a brilliant little proof using using the fact that you have this family to to prove your case. Of course, the paper also has several other different proofs involved, so they're not relying on this one, but I think it's a valid proof that they have, And that's all I had prepared. So I'll be glad to take questions on any of this if anybody would like to post questions in the chat. Let's see here. One thing I'm still hoping to do is. One more I still hope to add to this particular program here because there's quite a few substitution tiling systems that I haven't covered yet, and I hope to get back to those sometime soon. So if anybody has interesting substitution tiling systems that they'd like to share with me, it's it. Feel free to contact me@edpwolfam.com let's see, Technic Beam Are you sharing that new yes, this will be posted on Community sometime soon, probably later today, either today or tomorrow, and there will be a link to it from this page to the community page where this is posted. I might need to do some a little bit more cleanup because there still seems to be a little bit of clashing left. I might need to take out one page, but I'll give links to demonstration pages if any of those are removed. Let's see. Do you think a tiling will ever be discovered that doesn't use mirroring? That's an open question. I'll be very interested to see if that happens with the polyforms. We didn't think that anything would happen with the poly kites, poly poly drafters and so on, but obviously the polykites or polygems can do something. So there's now a renewed interest in looking at polyform tilings. It seems that the polyhex, which is hexagons, Polyimon, which are plane triangles, and polyominos, which are based on squares. It's pretty certain that those are not going to lead anywhere in a way that's discoverable, and the site that I pointed out from Frank Kaplan has really good information for those four particular polyforms. However, there's there are a whole bunch of other underlying things like polychiros, which are based on the chiropattern poly drafters which are based on the 30, 60, 90 triangle and so on which have not been well explored yet even to say order four or five where there's still a reasonable number of cases to look at. In fact the. The polygems which are a gem is considered one third of the hexagon. This hat shape is at order for polygem and there's I think less than 50 of them. But no one had looked at at how those tiled. So this. This. This probably could have been discovered 10 years ago if somebody had done a systematic search. It was. It was one of the things on my list that I never got to. Let's see. Discussion of periodic scripts using the hat. Yes the. Where is it here? This has a. A periodic strip of hats and it looks like it has a width of 14 or so. And if you. If you move the. The defects further apart it's possible to. To make a wider strip. But I'm pretty sure that those. Those strips of given thickness will always have problems. It would be interesting to. To find out or so this is a. This is a displacement of this much. It would be interesting to find out what the maximal thickness is for a given displacement for a strip for a periodic strip. Open problems or unsolved problems in the field of mathematical tiling I talked about the spectra problem earlier. What is the maximum number of pieces you can divide a. A. A convex shape into the. The record for a finite number is 48 and I showed the solution for that earlier. But. But really it's unknown how many different shapes have have just order eight, for example. So that. That is a wide open area. Other substitution systems that are unknown quite classifying all of them is an unsolved problem. Let's see. And if you go to three dimensions it's kind of wide open. What are the substitution tilings in 3D? That is a really open problem. I hardly know of any that are. That are known at the moment. So if you go to the third dimension you've. There's a whole bunch that is. That is unknown. I'm. I'm pretty sure there are tiling sets in 3D which do aperiodic behavior but they're. They're really hard to define. There's. So that would be one thing I would at is 3D tilings basically start with a shape or two shapes and divide those. Those two three shapes so that you're only using that subset of shapes to do it. Similar to what we did earlier with triangles, the isopenta. There are known dissections of that sort in the space of icosahedral symmetry. But whether there's something else is unknown. Let me see here. I wonder if I can find it quickly. Let's see. I'm looking for. I'm searching the demonstration site at the moment. One thing I think might come out of this using the plastic constant. If this site will come back. Sometimes. The site. Oops. Why is it taking up forever now? There we go. This is one way of the power clique polyhedra. This is one way of building up K19. You've got 19 different points and every two set of points is a power of the plastic constant distance from each other. I'm pretty sure that this particular network leads to a new dissection of 3D space using just two shapes. But I haven't figured out how to do it yet. I'm pretty sure that there's a really nice discovery with this particular set in the plastic constant. It just hasn't been found yet. And yes, this would basically be a 3D quasicrystal. I've looked at a whole bunch of different algebraic spaces and the plastic constant is by far the most promising. Number two is the golden space based on the golden ratio. But that one's much more. More explored in terms of. Well, pretty much all of the regular polyhedra are based on it. It's a well explored area. So I don't think that's a promising field to look at. It might be, but I'm certain that the plastic constant space has not been really explored at all. Especially in 3D. So if anybody wants to to explore that, that's probably a really good place to look at. Yes. The. The Schmidt Conway Dancer by Prism. I think we even have a demonstration for that. Let's see here. So I will. I will try to get that notebook up sometime soon today. And many thanks for attending my talk. I will hopefully this time next month have something new to show. So many thanks everyone. Sa.