Discovery of the Aperiodic Monotile - Numberphile

first of all Craig before we start properly I have to express my extreme disappointment with you because you guys released this Einstein tile I went and made this great video we did all this recording and traveled around Scotland and we were just about to release it and then you went and found another one a better one well I guess we should have held off until you released the first video that would have been nice just a few weeks a few weeks buffer and then we could have done a whole and I've done it all again but we've had to do a bit of surgery on our first video and then we're going to make another one later you should give me a heads up and I'll know for next time I should have I should have let you know can I first before we start talking about all this excitement around these tiles just to find a little bit about you um who are you where like you know tell me where you are and where you're from and what kind of job you do like where where do you fit into the great jigsaw puzzle of uh tiling nice well uh my name is Craig Kaplan I am an associate professor in the school of computer science at the University of Waterloo which is one of the larger uh universities in Canada particularly known for math computer science engineering but of course great programs in all sorts of fields I I kind of got into tilings as a graduate student which was a long time ago now but I uh in in a computer Graphics course that was being taught there as a graduate student as a final project we did a program to let you draw escher-like Tilex edit the shapes of tiles and it was a lot of fun and after that the professor of the course said you know this this could be research this could be something to play around with and I my initial reaction was like come on that's not research right that's just playing around that's just having fun which if that if you ever say that to yourself you're on to something right so I got sucked in at that point but really A lot of my work was about applications of tiling Theory uh and computer Graphics in Art and Design so I kind of came into tyler3 Theory through the back door right I I was interested in applying what was already very well known and of course with the mathematical background I mean it was only natural I got sucked into the big open problems and they were just something to to ruminate over for many years I never expected I'd make much of a contribution to any of those open problems but I'm very lucky to find myself here today Penrose tiling that you can't that you guys have kind of knocked off its pedestal were you were you already super familiar with that was that was that like you know was that the celebrity tiling to you oh yeah yeah I mean the Penrose tailing is absolutely and maybe still is the celebrity the the A-list celebrity in the room I was familiar with its properties in fact uh even in my PhD thesis I developed some automated tools to let you draw escher-like tilings based on Penrose tiles um so modify the edges of Penrose tiles to make them look more like whatever shape you wanted uh way back in the day Penrose himself had done such a modification and produced the this hilarious drawing of I think they kind of look like chickens uh like two kinds of birds and I I emulated that in my work uh but yeah I mean you know I'm I'm very interested in the intersection of mathematics and art and going into you know the 70s 80s 90s what were the big emissaries of mathematical Arts of the world it was basically the mandrel broad set and Penrose tiles and uh you know I I am extremely grateful for the existence of Penrose tiles to uh promote uh the popularity of mathematics in general and tilings in particular to the broader World they they have they have done their job exceedingly well never mind being a brilliant and profound contribution to mathematics so professor obviously the Penrose tiling has this property that but it requires two tiles to do it and it was obvious the question was could it be done with one tile what was that kind of like the for Mars Last Theorem of tiling was is that was that the top thing was that the ultimate tiling Holy Grail my mind yes that like even 20 years ago back in those days that was the thing that I became fascinated by like it's such an a deep and wonderful question it's kind of the the fundamental question of tiling Theory or the what used to be the fundamental open question of Thailand the one that if I could pick one to solve it would be that one and you know how lucky to be here today as I said before yeah I mean it's a wonderful question there's only one way that I would say it's different from from oslow's theorem and this is something I've been emphasizing a lot which is unlike something like for mozilla's theorem I wouldn't say we had a clear notion of which way we wanted the answer to go from us Last Theorem I mean it took a long time to say that there were no Solutions Beyond n equals 2 etc etc uh but we we knew which way we thought the answer would turn out just as like P versus NP Ramon hypothesis they're open but we have a pretty good idea of what we think the answer ought to be the existence of an aperiodic monotile this is the kind of thing where the mathematician says right you could like you could try to prove it one way Monday Wednesday Friday and try to disprove it Tuesday Thursday Saturday and then take Sundays off did you have a gut feeling I was I was pretty much on the fence um I there's no obvious reason why such a shape should not exist right there's no clear evidence saying why you shouldn't be able to find one except for the historical fact that nobody had after looking for a long time the only thing that we can say for sure that we could have said for sure up until this year is that if no such tile exists it would be very hard to prove that fact at least that was my gut feeling because that's a statement about all possible shapes that none of them are aperiodic monotones whereas if an aperiodic monotone does exist at least you have a thing whose properties you're trying to prove so happily we live in the good timeline where there is a shape that we can talk about that has tell me the discovery stormy tell me your War Story how did it how did it first cross your desk what's your from your I know I know there are multiple people involved Tell Me Through Your Eyes it's a wild ride it's completely unlikely and should not be taken as a model for how scientific discovery ever happens it's completely ridiculous right right it started last November um David Smith who is not a mathematician he's uh let's call him a shape hobbyist as he likes to the term he likes to use um he emailed me out of the blue now you know he wasn't a complete stranger we we both knew who the other was we had interacted a little bit via email over the years and we knew each other from a small mailing list online of people interested in thailands uh but he emailed me out of the blue in November and uh was asking me about some of the work I had done recently on uh using software to measure computational tiling properties of shapes was he did it was he was he tipping his hand straight away was he was he saying I think I've got something or was he kind of feeling you out first uh almost straight away his first email was really just asking about my research um and he didn't specifically mention that but I think that was the day that he had initially found it and was wondering what the deal was but a few days later so you know his first if I remember correctly he first emailed me on November 17th and then on November 20th he emailed me again I I think I had I responded and emailed me again and said well here's the thing I'm stuck on it's this funny poly kite shape and that was the first drawing that I had seen in the Hat so he wasn't playing too close to the chest what what did you think when you first saw this shape in an email from David the first time you laid eyes on the shape did you did it did it resonate with you or did you think oh here's another here's another one that's not going to work or yeah well so I trusted David's Judgment at the outset and and I say that because I had seen some of the previous experiments that he had done on his own that he had either posted on his blog or shared on the mailing list and uh he knew what he was doing I mean that much was clear to me even if I was not trying to like evaluate it from a research perspective I could tell that he had a keen uh intuition for interesting shapes so I was willing to Grant him some benefit of the doubt but that being said my first contact with the hat I mean it's a really boring mundane shape there's really nothing that you could point at and say Ah that's why it's going to be interesting it's just kind of a random blob right so my first reaction upon seeing it was like well we've got to suspend judgment like David already said uh it has a h number of at least three which you know there's some technical details there although you with Edmund Harris made a wonderful video but about page numbers so that would be a good place to go look as a reminder of what that of what that's about so when he said it has a h number of at least three he was already communicating to me like there's something interesting going on here and he couldn't make a tile periodically that's the other half of what makes it interesting so I I couldn't see anything about the shape immediately that that you know keyed off my intuition but I was certainly willing to dive deeper and try to you know plug it into my software and see what came out of that how long after it was given to you to play with and start doing some analysis on did you realize this could be something special uh I think it was only a couple of days later I mean uh first of all David kept experimenting and sending me well by by the you know by three four days later he was sending me photographs of cut out paper hats that he had arranged on it on a table and you know he could see you could see the progress he was making trying to trying to make the shape tile of the plane and what was really interesting about his photographs was he had cut out uh he had a you know computer-controlled craft cutter he had cut out the reflected hats like the opposite-handedness hats in a different color paper than the unreflected has and so immediately you could see it as photographs the sparse distribution of reflected hats among the unreflected ones and that was a big clue that's like wait a second like he wouldn't do that unless he was somehow being forced to create those kinds of Arrangements his intuition would otherwise naturally have guided him towards something simply so the fact already that we were getting this weird sparse arrangement of reflected hats that was a tip-off to me and then on the other on the other side I had by then in fact adapted my software to be able to automatically compute large patches of hats by Brute Force automatically so I could walk out much farther computationally than David was able to achieve by hand and just that larger data set if we started to color it in like say color in the reflected hats in a different color maybe color other things by orientation it was looking really weird I mean like again this is a this is not a theorem this is a statement I need tuition an intuitive statement that my program would not spontaneously create a composition of tiles that's more complicated than it strictly needs to be so the fact that the computer was coming up with these crazy non-repetitive arrangement of tiles it's another big clue and that sort of that set off the investigation so as early as like a week after he first emailed me David uttered the phrase Einstein problem in an email he said could this maybe be a solution to the Einstein problem wouldn't that be a thing and and I was willing to agree with him at that point like yeah we have to we have to figure this out there's something going on here I know it's like a mathematical breakthrough but because it's so visual it really conjures up images of me like Howard Carter first glimpsing into Tutankhamun's tomb and seeing a beautiful thing and thinking oh my God have have I found what I think I found after all this dreaming it is yeah well I mean that's that's the the visceral excitement the Wonder I mean that's the the moment I I always think I don't know if this is too obscure reference because we're talking 30 years at this point but I always think of the movie contact I don't know if you if you remember contact the moment when Jody Foster First turns the dishes in the vla and hears the message for the first time and her eyes like Snap open and she just gapes right she can't believe what she's hearing I mean just to we should all be so lucky as to have that feeling once in a lifetime yeah one moment can you put your finger on a moment or was it too iterative or was there a moment when you thought it this is it this is we have we actually have got it we found her it was a little bit more gradual than that but yeah I'd say even early on like late November I was starting to feel that feeling of like this I think this is it I mean there was a lot more work to do and we absolutely should get on to the rest of the story because I don't want to leave out my other collaborators who were who were absolutely crucial who did things I could never have done um but you know even in late November there was that feeling that high of like I think this is it like we're gonna have we're not done we you know this is very far from a proof of a mathematical theorem but we're gonna get it this is the shape well as we move on to your collaborators and how the process continues this is probably a good time to ask what did you do about like secrecy and things like that like famously Andrew Wiles when he cracked from our Last Theorem was very secretive about it no one even knew he was working on it until he had announced the proof were you trying to put a lid on it were you saying to David hey keep quiet about this I think I think we've got the big one or were you like you know broadcasting to everyone and it was all hands to the pump we we were not broadcasting the news far and wide that's for sure yeah I think we were trying to keep it a little bit to ourselves and eventually the four of us so I think even in the second part of November like late November I was telling David maybe we should not spread this around too much at this point interestingly uh David had already emailed a couple of people before me uh asking them for help and it just so happened that they were unavailable for different reasons so I was a squeaker for me I got I got pulled in at just the right moment and and happened to have the right tools to help with the analysis so those people are like the people who turned down the Harry Potter books yeah so look maybe it's a little bit like that yeah yeah but uh but yeah we we were trying to keep it to ourselves a little bit so I was telling people I think I'm on the trail of a major Discovery and then you know talking to colleagues like maybe I would say well it's related to aperiodic tilings but I wouldn't elaborate beyond that and that was all the way up into March when we finally announced the paper and we met the other people and who else you had to go to to stick the landing right good so uh from through the end of December I got David's Indulgence to let me work on problem with the feeling simply that once we roped in other hardcore mathematicians they probably solve it immediately and then I wouldn't get to do anything else so I it was really completely selfish it's like please give me a chance to keep playing I think I can make a little bit more progress just between the two of us and David was nice enough to oblige and I promise like first thing after New Year's day we'll we'll email a few other people and get the ball rolling so indeed yeah so early January we reached out to Hein Goodman Strauss who is you know a long time researcher in tiling Theory and you know is well known in that Community made some key contributions to our understanding of aperiodic tilings he he's one of the few people other than Penrose who came up with a two-shape aperiodic tiling or I should say a an aperiodic tile set of size two the trilobite and crab and you can I mean it's easy to find online which in some sense are two shapes that with the appropriate matching conditions recapitulate the structure of What's called the chair tiling and so hayam got in and you know didn't raise any objections right from the get-go he's like yeah this is looking really interesting and then a week or two after that uh we also reached out to Joseph Myers now Joseph is also another interesting person uh in terms of his his past and his current work he's not uh an academic he has a PHD in combinatorics from Cambridge uh so he's highly qualified and he has an excellent background for exactly these sorts of problems and he works as a software developer so he's he's not an active researcher he doesn't he doesn't publish papers actively but I've known of him for at least 20 25 years because of this fantastic work he did calculating or I should say Computing the tiling theoretic properties of polyaminos particularly or I should say Paulie forms polyaminos poly hexes and polyimens so he had systematically searched through spaces of these simple polyforms looking for how they tile the plane or don't with inevitably the agenda of trying to find an aperiodic monotone so he could very well have been the discoverer of the hat if only he had looked in the poly kite box early on instead of focusing all his Cycles on polyhex's polygons polyomons but because of this background he was perfectly situated to help in the analysis of his polycut like he had spent all of these years preparing himself to prove the a periodicity of a monotile if one should come along and so when we presented when we wrote them in and said here it is well it took almost no time at all for him to finish the proof by the end of December I was able to show that the Hat tiled the plane like which is half of a proof of a periodicity you need to show that there are tidelines then you need to show that all of those tilings are non-periodic and I mean we we emailed Joseph on I think it was January 17th and that was his first exposure to this problem and and this shape and on the 25th he emailed back and said okay I think I've got it like which I mean it's both a testament to Joseph's Amazing skill and maybe also a comment on the fact that you know once you know that the shape exists proving that it has the properties you want it to have maybe that's not such an Arcane process when you finally went public and there was this all this great publicity and it was a it was it was a huge deal I remember it was all it was certainly all over my timelines but but initially there was this issue that a certain percentage of the tiles had to be flipped reflected so you could argue it wasn't one tile it was too it was some people said well that's not one tile that's two different tasks did that did that irku was that like a stone in your shoe that it wasn't quite as perfect as it could be I I wouldn't say that it's undermined my feeling of having solved this open problem like it it didn't demolish the success of you know the existence of an aperiodic and and you know we could litigate this all we want in some sense right this is a this is a problem of definitions it's not a flaw in a theorem where you can say well this step I think you know you've ignored this case this corner case or something it's really a decision about what the correct definition is of a tiling by a single shape and it's clear to our planet yeah I mean well sure right like so you I mean we could debate it all we want but nobody's ever going to be right or wrong because the question of what the correct definition for something is well that's a social process right it's not it's not really a question of mathematical correctness so but at the same time the stone in my shoe was oh yeah you're right that is also a really interesting problem that we should work on the question of whether there exists a chiral aperiodic monoton so um I wasn't so much disappointed as excited about their still being a non-trivial open problem that demanded to be investigated well that one didn't take long you guys came out with the with the solution to that one sooner it was crazy I mean and I you know I said this is a problem that really ought to be investigated I am not saying that we then went off and started investigating it no way I mean we were just as everyone else was saying we were thinking wow yeah that is a good problem and who knows when we will understand enough about a periodicity to find the answer to that one see it's my it's even harder and then as fate would have it right we put the paper out I think it was March 20th maybe I think it was 20th or 21st right and then um people started putting all sorts of wonderful visual experiments online they started playing around like you said the response was incredible I mean just wonderful to see and in particular um yoshiaki araki who's a well-known figure uh who plays around with tilings and creates wonderful visual designs and explores the visual properties of tilings um he had started posting some of the pictures that he had created including some based on what we call tile one one the shape where all the edges have the same length in the Continuum between the hat and the turtle and David just saw something there that you know you know that that like you know it it got playing around in the back of his head something looked a little bit strange there and he started exploring well what happens if I try to tile the plane using tile one one but I don't let myself use Reflections and I was like five days later so it was a complete stroke of dumb luck there none of the authors of that paper you know none of the authors of the first paper had any clue that we were headed in the direction towards a chiral aperiodic monotile I would never have expected the answer to just shake out of the work we had already done but there it was which is not to say that uh the answer was in the original paper too just the shape right then we had to reopen the whole Playbook and kind of redo all the steps that we had done previously just based on this new shape proving it's a periodicity but I mean I was astounded I I didn't I didn't believe it at first I I think I I literally said by email to David like if this turns out to be a non-reflective Einstein I'll eat my hat well you know I guess I've had to do so at this point since it's all come out and there's been some swirl of excitement what what's has there been anything about it that has surprised you or a crazy thing someone who's called you or some article you've seen or what like of all the things that have happened since what's been your favorite oh uh there's been I mean a lot of great kind of fan art I guess that I've loved seeing online uh textile Arts are natural so people have uh made quilts and crochet and knitted hats all sorts of great I mean there's obviously graphic art is a natural choice and people have made great esheresque interpretations of the Hat the turtle the Specter and so on um Yoshi yoshiakiraki whom I just mentioned uh he was experimenting as well with um tiling a strip with Hats periodically you can't fill the plane with Hats periodically that's the whole point it's an aperiodic monotonial but you can actually tile a strip of any finite width in a rep in a way that repeats along that strip so he had printed out like transparent tape with Hats like a big roll of tape and he had applied it around a beer glass and then somehow we found like tessellation beer like a brand of beer called tessellation that's the whole package I I really enjoyed seeing that I think my favorite artistic interpretation of the Hat was uh John Paul wheatley's soccer ball the football so a lot of people had noticed that you can put six hats around a a space in the shape of a regular hexagon and you could tile the plane with those configurations of six hats plus hexagon and John Paul just realized well let's change that hexagon into a pentagon and now you get these pentagonal panels that can wrap around to in a kind of dodecahedral configuration and poof you have a football and it's gorgeous I hope to see it someday I feel like the Hat came along and blew everyone's mind and it became this celebrity shape really quickly and the Specter because of the nature of the timeline of its Discovery even though it's kind of even though I feel like it's kind of a bigger deal it hasn't taken on the mythical status of the Hat do you do you have a favor of those two shapes one that's a bit closer to your heart um I think I think probably the hat is closer to my heart for kind of the same reasons that you've already said I mean the the the plateau of discovery that the Hat achieved is greater than the like marginal extra step up to the Specter they're they're all fascinating and they're all gonna I think give us lots of interesting ideas for the future the Specter does some interesting things that the Hat cannot in particular you have you can modify its edges to do anything you want and whereas with the hat uh the edges have to be straight lines you you actually have no degrees of freedom if you want all the tiles to have exactly the same shape so the Specter in some sense is maybe going to pay off more artistic dividends but yeah the Hat will will probably always live on as relatively speaking a bigger Discovery in my personal history are you going to do anything crazy like tile your bathroom with them or anything I don't know I it's yeah I had uh maybe a backsplash I'm I'm much more hoping I'm holding out hope we're uh we're gonna break ground soon on my campus here at the University of Waterloo on a new Math building and so that would be the place where I can put my hopes that we're gonna have somewhere in that building where the the Hat the Specter the turtle any number of these can be on display gotta happen now you've got to get in the rear if they don't do that that's crazy yeah yeah I've I've I I have the dean has expressed excitement about that possibility and that's about all I need to hope for for now but yeah let's make this happen yeah I can't imagine what it must feel like to be a tile guy and have the number one problem in your field the Mount Everest of your thing and you along with your collaborators be the ones who crack up I can't imagine what that feels like I just it must just be the best feeling in the world or is it is it deflating is it like now what like it's like it's like something it's like something special has been taken from you in a way now yeah I I've told people a few times like it's there's a certain existential feeling that comes from like knowing in the moment that you're probably doing the most important thing you will ever do right like which it's gonna I sure take this as a challenge I will try to best myself absolutely but let's be honest right this could very well be the most important thing I ever do so that is a little bit of a melancholic feeling that being said um yeah the feeling is incredible as as I I hope anybody who has already tuned in to your podcast your your YouTube channel and so on you know hopefully part of the reason they do that is because they can imagine the joy of scientific discovery of mathematical Discovery and I'm here to tell you that yeah it is every bit as incredible as you would hope it feels like and it yeah it does and uh you know all I can do is try to revisit that feeling uh when I need it and ideally to try to recreate it in the future with something else there's a new tile 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 they cover the entire plane with a pattern this new tile goes infinitely but with no repeats happy new tile

Channel: Numberphile

Views: 195,271

Keywords: numberphile

Id: _ZS3Oqg1AX0

Length: 31min 4sec (1864 seconds)

Published: Mon Jun 26 2023