Roger Penrose - Forbidden crystal symmetry in mathematics and architecture

well thank you very much for that introduction I hope I can live up to it first of all let me in fact the main thing I want to talk about is something which is not quite finished namely well you see what you see in front of you up here is actually a drawing or whatever they do these days with computers I suppose of the new mathematics building that's the only thing which is in color which is not quite finished it's finished enough that I have an office there somewhere I can't really get into it there's only one chair and two desks and a whole lot of crates so it's not very hospitable yet but that's partly my fault because I collected too many things in my other office but right here will be an area tiles with a particular arrangement which is based on things that I've been doing so I want to explain that that's really the purpose of this talk I'll come to the detailed explanation of what's going on there towards the end of the talk I shall say lots of things about other architectural use of these tilings in other parts of the world but before doing that I want to explain the tilings themselves so let me well I need to go to the next picture I think where you can see it the front entrance is right here and you have to walk over my tiles in order to get in the building and it's not quite finished you might see some goings on back there where it's not finished there's some other than goings on back in front which aren't in the picture very conveniently where it's not finished either that's not finished clearly ah but so and that one in the middle isn't finished I think that's going to be the last tile which is placed in it's certain sense its central in the whole design but you may see some kinds of regularities as you look at it but not altogether obvious what these regularities are so I shall be explaining what that's all about but before doing that let me explain the general idea of what crystal symmetries are allowed and what aren't and so can I move to the visualizer please thank you I'm using this kind of old-fashioned technology because that's the only one I understand these are the crystallographic symmetries so we have point this well suppose or the other way we have them twofold symmetries a pattern regular pattern of parallelograms will have a twofold symmetry about the center of each parallelogram threefold if you have equilateral triangles then about the center of any triangle you will have the whole pattern has a threefold symmetry you rotate 120 degrees whole pattern goes back into itself fourfold core squares that's the most familiar and six-fold we have the familiar pattern of hexagons and it's a fear on the mathematical theorem that these symmetries are the only ones that you can have when I say that I mean in addition to translational symmetry so you have to have a B translational symmetry that means you slide the whole picture of parallel to itself in some directions and the pattern goes into itself so you want to have a rotational symmetry and together with that a translational symmetry and I want to give you a quick argument to show that that is the case these are really the only symmetries you can have this is a sort of standard theorem suppose you have a pattern of points or something something which isn't like a Fraxel so it's they're discrete and have sort of finite minimum distances between points and things that you're just you can imagine some pattern and that pattern has a translational symmetry so you could slide it along and it goes into itself but also rotational symmetry so you rotate through 360 degrees over N where n is such is your degree of symmetry and the pattern goes into itself so what I'm going to propose then that you have such points of symmetry so here's a point of info symmetry and there's another point of interest see if there's one there must be another because if the whole pattern slides along then this will go into another have to be another info symmetry point so it's gotta have more than one if it's got to have more than one there will be somewhere in the pattern two of them which are as close as possible so I'm going to choose those as long as it's not a fractal or something silly like that she's chose choose a pair of points which are as close as they can be and that's those ones up there now you see I'm going to rotate this one by 360 degrees over N into that point and this one in the other direction through 360 degrees over N into that point and these two will then be closer which contradicts this being the closest so that's a contradiction unless N equals two when they're not closer a lot further off when N equals three they're not closer further off when N equals six that's above them when they coincide so you get away with it they're in the case of sin equals four of course there's same distance as before so so that's okay but N equals five for example they're closer and anything larger than six they'll cross over like here and they'll certainly be closer so that tells you that the only symmetries you can crystal symmetries you can have are those ones given two three four and six okay straightforward okay well what about this pattern well it has a lot of regularity to it and it has a sort of fivefold Ness to it also in fact you can see various regions which are fivefold symmetry up to a point and which have translational symmetry up to a point in fact let me tell you that if you give me any percentage less than 100% so in 99.9 percent then I could slide this picture over itself so the pattern agrees to with what you had before to that percentage 99.9 percent of whatever it was and also has the rotational symmetry fivefold symmetry two ninety-nine point whatever it is and you might say well why doesn't the theorem work and the argument is well you go back to the proof here and you see if this point up here wasn't a perfect one you see suppose it was only ninety-nine point nine percent point and that's a ninety-nine point nine percent point then that's likely to lose just a little bit of accuracy it'll be nine nine point eight likely and that one's probably ninety-nine point eight so that although they're closer they're not quite so good so you lose a bit of the symmetry but each time you perform this argument here but the points may be closer so there's a trade-off between those two things and that's exactly what happens with this happen so anyway I'll just show you that I think it's worth pointing out a number of features of this pattern for example well actually have these rings here I'll talk about those later that's a 10-fold it's a regular decagon and every time you find one of those regular decagons there's always ten Pentagon's surrounding it that's always the case wherever you find one there is another one here sometimes you find them overlapping like this one in this one and you still get rings of ten Pentagon's they just happen to go through each other that's all so that's a general feature another feature for example which is perhaps a little more obvious from where I what depends where you're sitting if you're sitting up the edge it's probably fairly obvious that you see these lines line up here I have wherever you find a line in there in the picture and you put your ruler along that you just find other lines lying in that and the density of them just doesn't fall off it doesn't matter where you do it so the pattern has a lot of regularity about it and this regularity well it's not completely obvious from the way I'm going to tell you how it was constructed first of all it's constructed from something very simple here we have a regular pen again subdivided into six smaller ones I think it's easy if I just do it down here regular pentagon six smaller ones with a few little gaps here now what I'm going to do is blow this up too so that the smaller Pentagon's are the same size as the original one and then subdivide each of those blow it up subdivide blown up subdivide now if I do that it doesn't quite work let me just imagine I now have blown it up so this this was a bigger Pentagon sort of out here and this was subdivided then I've start dividing again and you notice there was a little gap here and that little gap joins up another little gap so we have a little rhombus shape hole in the pattern okay now I'm going to do it again and when I do it again if I subdivide this this look there's going to be a little gap there so this rhombus will grow spikes and it will look like that now when it does that I find there's just room to put another Pentagon inside there okay and then the next step this will grow spikes and you get something which looks like what you see I should point out that the shapes I have the original Pentagon this star now which I'm calling a Penta core that's a pentacle and this thing I'm trolling the jesters cap well it's more like a justice cap this way up so they're just as caps tentacles and rhombuses we had the little rhombuses before now when you grow spikes for each of these shapes which you go at the next stage of hierarchy you can always find that you can fill out the gaps with shapes that you had before so you can blow up fill it up with these shapes blow it out fill it up and that will end up with something like this well it will if you've been just slightly careful slightly careful about this little subtlety which I have to mention it's interesting because I discovered after with having done this because a Japanese man had done exactly the same thing except that you made the wrong choice here and that doesn't lead to what you get jae-won to say now I put a tick on this and across on this so here is the rhombus with its spikes and I put a Pentagon there that's remember you see this that's the rhombus with its spikes and I can put a Pentagon either up here or down there there's a choice now what I want to do is to force one of those two choices and if you look at this spiky rhombus here does that go down or the top well what you do is you look one side or the other and I'm just telling you this but it does happen that you always find a pattern of Pentagon's like this except this one maybe it bottom or at the top you then reflect in the middle rhombus so that one goes to there if that's on the bottom this one if this is on the bottom that has to be on the bottom if this one run the top that didn't be on the top if you make the choice the Japanese man took this one then you find that the next stage it goes wrong whereas this one keeps going forever and so the pattern with the subdividing blowing up subdividing blowing up will just get bigger and bigger and bigger and cover as much of the plane as you choose in fact will cover the entire plane okay now I want to show that sort of backwards now suppose we have a big Pentagon here I hope most of that Pentagon is on the screen there's another there's another one sitting up here and another one sitting up here and one sitting there and sitting here and so on but there's a big Pentagon and what I want to do to that big Pentagon is subdivide in the way I've been describing here we go and then I subdivide that again there yeah and then I subdivide that again and you get the pattern I just showed you okay now where was that big Pentagon now you see there's something interesting about this that the pattern has a greater uniformity than you might have thought from the hierarchical construction in fact you know I have to find I can probably find it but I won't even try it has greater uniformity than the hierarchical organization seems to suggest but yet their hierarchy is hidden in that all the time now sometimes people point out that this kind of thing was in ancient Islamic art and there's an article here I'll show you and there's a lot of very interesting ancient Islamic art where you see regions of ten five fold and ten fold symmetry fascinating stuff but I haven't seen any indication of the kind of very kind of strict structure that scones the hierarchical arrangement in that strict sense or anything like that nevertheless they're fascinating but it's not so clear what the deep connection is with the things I've just been showing you however if you go a little bit more recently than these ancient Islamic things namely 216 19 then we find in the works of Johannes Kepler these famous astronomer in a book he wrote called harmonic and Mundi and in that book you will find these fascinating pictures now I should say that I my father owned a copy of this book and I had seen this picture I'd seen it although it wasn't in my mind when I started doing this except somehow I was no doubt somehow influenced to think that Pentagon's unlost a dead loss you see what he's done fascinating things with Pentagon's and things with Pentagon's but this design in particular I want to point out and as I later discovered to my surprise here we have that picture drawn bigger that's the very same pictures in Kepler including a little line there which I don't quite know why he put it there but he did and now here is my Pentagon pattern now I've put some marks here if I can find exactly where you will see that the Kepler pattern exactly fits this including this little line there so what was he doing I don't know I suspect he was probably thinking I mean they didn't even know about atoms and so on and crystals and goodness knows I suspect that he was interested in maybe some kind of atomic arrangement and perhaps injury knew about crystals and things and he perhaps wondered where the biological things because you often see fivefold symmetry and so on in biology that perhaps that was something underlying that I have no idea and I have no idea how he intended to continue this pattern there is people have tried other continuations which I don't believe I think his continuation was probably much more like what I was doing here I wouldn't be at all surprised but it is fascinating that he did in do indeed do has had this great interest in these things he was interested in there's the thing called the Kepler problem which is packing spheres as closely as you can in three-dimensional space and this problem was unsolved for a long time and only fairly recently was it sort with the aid of a computer I may say and it turned out Kepler was right and what he suggested you also see there are other symmetries displayed like these and in fact you find these things in quasi crystals also I wasn't going to show you too many of these other ones it just take too long I just shown you showing you here 1201 this is by some people a gala and and galen and and listen and it's a design listen is an experimental person and he'd actually spotted twelvefold symmetries in certain apparently crystalline materials and he showed me the diffraction pattern it's got a curious story showing the diffraction pattern and i saw this pattern of little spots where you get diffraction electrons really bounce off in certain directions and where have I seen that pattern before I couldn't think I've seen it before from I've seen it before and I realize it's right here it is this little little spikes are on the corners of this little pattern in Kepler what was he doing I haven't the foggiest but it's fascinating to see how how much there was in that little picture set of pictures of Kepler here's another one with that 12 or 12 fold symmetry the tri-fold ones are rather nice there are also eight fold ones which Robert MN and someone else produce they're never found them quite so attractive I think that the five one well five fold ones are nice and the twelve old ones are also particularly nice so but I'm not showing any more version four in so I don't think they've been used in architectures far enough okay so that's the basic idea now there's another bit to the story which is somewhat different which has to do with well that doesn't do much sometimes in some of these machines you put that up and you see only what I've covered up and not this depends on the the way the lighting works this was a set a set of tile shaped it was I won't go into the whole story it was quite a long interesting story to do with mathematical logic and it the course of the story an American mathematician called Robert burger discovered a set of tooth 20400 26 shapes which would only tile the plain tile the entire plane but only in a way which never repeated itself and he needed to do that to show that the certain thing that is actually to show that the tiling problem is not computable that was actually what is trying to do but that's not part of my talk here but Raphael Robinson who's another American mathematician managed to reduce this number from twenty thousand four hundred six to just to six with C and so this was pretty impressive it had gone down his steps from some other people but this was his achievement to do it with six and I was talking to an American another American mathematician Simon Cochin and he told me that raffle Robertson was some do you like to get the numbers done small as possible and I thought well six I see I know I can do it with five well you see these are the shapes that you get in in the tiling pattern I'm showing you here you see you'll get there the Pentagon's the the rhombuses the justice caps and the Pentacles but if you want to make a tiling problem which forces this arrangement you can do it by putting little knobs and notches on the pieces and that forces them to fit into that arrangement that's I won't try to prove that but that's true but you need three different versions of the Pentagon's depending upon whether there's five other Pentagon's next to it only three or only two and that that's the difference between the three kinds of Pentagon's but the thing is you'll notice that this Pentagon here has that funny little thing there on the bottom and this thing has that funny little thing and this is two of them so all you need to do is to take that one glue it on there and glue it twice on there and you've only got five pieces so I knew I could do it with five then I started fiddling around from it and I realized you could do it with two and so that's the two and the way it works well I had another version first which I'll give it to you you have to match the colors you can do it with little knobs and not just like this as well but if you just match the colors and here we have a version which is pouring the uncovered but you can see underneath is that tiling pattern well I'm going to show how these others are related to the Pentagon pattern here they're all part of the same story to make sure and keep these things in sight and let's see something about the story but first of all let me mention the other one which came first which is kites and darts these are the kites and the darts down here and at the top I hope you can see you put the kites you mark the kites and the dots each each dart is marked the same way in each Keiser's mark the same way and they fit together to form the tiling which I just showed you I think it would be better if I show you their kites and darts assembled and here we have a rhombus the the original not rhombus one the Pentagon's so there we are I think that's it and if you look carefully you'll see that every kite is marked the same way and every dart is marked the same way so this pattern of kites and darts is equivalent to this pattern it produces this but only with only two shapes so that's how you can do it with two of course I haven't put the notches and the knobs on it let me do that by putting whether actually I put knobs on here with black and white corners colors so I can do that here I just noticed that their opposite way from in that other picture of the distal matter but here we have the kites and darts I'm now going to mark the corners and if you match those corners that the only way of putting them together is one of these non periodic arrangements I should perhaps and make a point about this does that mean that the only way of covering our plane that there's only one way of doing it yes and no is the answer to that strictly speaking mathematically the answer is no that is to say there are many ways of continuing to infinity however and a certain finite sense they're all same that is to say if you find any two of these complete tilings of the plane with the same shapes that is and i take one of them and i take a region no matter how big that's finite i can find that in the other one so you can't tell ever which one you're in it doesn't know how big the region is you can always find it in the other one infinitely many times okay that's not so hard to prove actually but curious okay you can also do it by making a jigsaw puzzle instead of the notches and Rob's jigsaw puzzle okay here's one that's a sort of ester ization I would say I'm not sure this will match on that I didn't try to get them to me oh that's not too bad they might do but I haven't I haven't actually tried that there are about the same you see that if I concentrate on on that bird there it's a yeah that one and that they won't go all the way around but yes there we are these ones fit but I'd have to find a better place to match them but as I said any finite region which will be in tips etc etc but if you want to know how to do it you do it this way they're really cuts and darts disguise and then there's a hierarchy which you can infer I won't go into that here but just to show you those sort of things you can do now here is a rhombus pattern when people use it they tend to do it without any adornments I prefer it when you see where you are the trouble with this if you just take it's not as a jigsaw puzzle there zillions of ways of doing of course because you can't are with either one without using the other one and in many ways of doing that so you have to have some rule about how you fit them together and the rule could just be well it's that pattern but that's not very economical you could say the rule is that they have to match the stripes which I showed you before and so on there's also a close relationship between the kites and darks and the rhombus one you see that many of the lines are in common between the two okay let's go back to the rhombus pattern again and there memorize that pattern and here is a rather finer one I hope you can see the rhombuses all right I'm afraid there's some places where it's got a little worn but that is indeed what it is of course you see I did these things were by hand but of course now the computer people take over and they can make much more impressive pictures than I ever could so that's a computer picture I'm going to show you another computer picture you can go even find an end now that is rhombus pattern I don't think you can quite see the rhombuses but what you probably can see is Mauri patterns now it's quite interesting because you can get one of them in the middle so it takes a little skill to do this and I may have lost it because I yeah you have to move the thing at right angles so the way where you want to move it so it's tricky I'm trying to get that spot in the middle now you see various lines going across those lines are the places where the patterns differ where they agree are the spaces in between now there's another spot let's try that one I'm not sure I'm getting any better at it but I'll try I think that the lines are further well I'm going to cheat now I'm going to use some little guide marks at the edge now I'm not sure that I get it accurately enough it's quite hard to do ah yes you may see there yeah it agrees everywhere except along those lines there and if I move it far enough away I can make the patch where degrees as bigger as big as you like I think I might just try one more here whether I can actually find the right marks I'm not sure I'm up with the edge here to try and find it let's hope that spreads it yep it agrees everywhere except that one line across the middle except I'm not find it you can do it this way I think but then that's I don't know let me just match that match the things at the edge of you legend can you see it that's it there we are that's it huh okay that's the demonstration okay now let me make a point I used to give lectures on these things a long time ago and in the 70s and people would say well doesn't that mean there's a sort of area of crystallography opening up and I used to say well yes in principle that means you could imagine such things but I could couldn't imagine how nature would do it now there's a reason for saying that and that is that the assembly can't be in the way you would imagine a crystal to be made because this sort of classic crystal assembly is you have a little the Matins come along and sort of sit in a little Ridge one by one and then they come on another layer and another like that that's a local assembly now you see the Assembly is never local and I knew this because suppose we have that pattern these are cats and dogs and that is correctly assembled you could continue that to infinity if you put a car kite there you can still continue it to infinity if I take that kind off I'm sorry that's a dot if I put it take that dot off and put a kite here you can still continue that to infinity but if I put a dot there and a kite there it goes wrong just about there so it's a non-local feature and it seems puzzling how you could get crystalline type substances to grow if it is this kind of tiling so I was sort of a bit skeptical that maybe you could have such a thing well I think I want to move now over to the Samora of the PowerPoint images if we could do that please thank you this is a design for a green for a poster which was in the mathematical Institute where we're moving to this other building and I don't think it's been put up in the new building yet but the idea was to have some assembly of these tilings these are all tilings that were given to me by people who manufactured individual tiles the ones in the middle were given by mathematician called Ron Graham who I think he just wanted to play with these ideas and and he put little knobs on them so that you could fit them together only you're right well you're not supposed to turn them over otherwise you can do it the wrong way too these are some of his too so you could see the cuts and the darts this kite and dart pattern is also I have five different colors and the coloring of the pattern is uniquely determined by the timing I won't go into what it is but it makes that the pattern is is absolutely the pattern of colors is absolutely fixed by the by the timing pattern with certain rules about it you may be able to see things sort of jumping out at you when you look at the pattern here we have a rhombus version and here we have the rules about the hierarchy how they work I would rather to go into that detail here we have some amusement with I showed the birds to you those are the birds these are really the birds kites and darts like they're not here you can see them as individual birds the other side were not the other side the alternative way of coloring it brings back the original exit pentagon tiling you notice there's a little foreign creature in the middle there's a dog there if you put one dog in this pattern the entire pattern run out to infinity is completely unique that's exactly one way of doing it that quite curious it's a little bit hard to see what's going on with that pattern down in the corner here we have some actual materials this is an actual quasi crystal i think they now call these things crystals instead of quasi crystals I'm not quite sure notion of crystal has extended to include these things but you see this beautiful regular dodecahedron which is certainly not allowed in ordinary crystallography and here we have the diffraction pattern these things were originally discovered by chessmen by looking at the diffraction patterns and you if you look carefully you see that there are Pentagon's and things in the diffraction pattern itself I can me oh I can I can move it there is a bigger picture of that very beautiful run become very beautiful dodecahedron a regular dodecahedron and with very nice edges and corners and things like that and it is believed that the atomic arrangements are of the kind that you've seen here there are lots of versions of these atomic arrangements that you can have but that's just okay I don't know in this particular case what arrangement this is it only three-dimensional version of course there were three dimensional I think Robert Aumann was the first person to produce a three-dimensional version of the pentagonal tilings but then barish mathematicians discovered very sophisticated ways of getting generating these things by taking lattices and high dimension and slicing them and projecting them I don't want to go into all that here let me see what the next is okay now let's see some architecture this was the first use I know of of any of these tilings this was done quite soon after I produced them by Japanese architect I was really quite impressed with him because this thing was there are all sorts of things in its kites and darts it's good little hard to see at first because they're decorated by certain arcs which was suggested by John Conway a petition to produce nice patterns but here you have two darts two two kites and one dart and there's a kite there's a dart sorry there two kites five kites and so on but the architect noticed when they put this thing up that there was a mistake in it and you insisted they took it down and corrected the mistake so I was impressed with that that he cared about that carefully enough but it is it's a very intriguing and interesting pattern I was quite flattered to see this thing produced and there's a younger version of me standing there now this story hole in in Melbourne in Australia I they wanted me to be there at the opening I wasn't able to go I've visited later on it's the most extraordinary place I'm not sure what I think of it you can go inside and you find more of these things all over the place so it is based on the on this rhombus tiling and quite correctly done I think um how but it's elaborated and all sorts of curious ways so it's quite a an interesting building but let's move on now here we have this must be in in the Australian huh I have a crib here which tells me no this is the one in the in the Science Centre I think so too two of these tilings in in Perth the Science Center I think this one's the Science Center that's right and it's it's straightforward rhombus tilings and you see them edge on and there they're two different the fat and thing rhombus is there it's a bit hard to see them in this picture but you get some feeling for the extent of them and that's the shot taken from above yes this is the the Science Center in a place near Perth in Western Australia very good impressive I think they're very nicely done now this is the kites and darts this is in my old college st. John's College in Cambridge and I think Peter Goddard who was the master at the time sort of like Big Pun of the idea because this is the entrance to the Penrose building if nothing to do with any a member of my family as far as I know but it happens to be somebody called Penrose who designed it and the piece of wood in the middle of the picture here is the door it's the entrance to the library and this is a sort of circular pattern on the floor and the door here swings round there's a pillar up the middle there and and it swings around so you can't really see the whole pattern all at once but you can swing the door around and see different parts of it here we have just seeing the pattern it looks like so it's straightforward kites and darts looks very nice now this is Helsinki and I shown this and I'm a little puzzled by it because it took me a little while to realize what it was at first sight it looks like something else but when you look carefully you see these triangles and that shape they make a dart so the darts are broken down let's see if I can find one here here there's a dart that's the nearest one there's a dart that it's cut in two places to make smaller tiles I suppose they didn't like the they like their tile shapes to be convex I don't know if that's what it was but that the the kites are are complete kites it was a really quite big area tiled in this way I don't think I've ever actually seen it but that's in Helsinki in Finland okay let's let's move on this is in Stoneybrook Simon Center I think it's called it's an institute and on the ground you'll see here so we're going to go here first here we have the straightforward rhombuses no adornments just rhombuses and here we have a view looks quite nice nicely done again but the two kinds are both the same so you have to know what you're doing to see what the pattern is all about but you see regions of fivefold symmetry such as up here and so on in that region of fivefold symmetry extends to somewhere beyond it to so to do that for a while and then something else comes along yes this now is the chemistry department floor in the University of Western Australia and this is a really big area I went up high to look down and see if I could see any mistakes and I didn't spot any which was fortunate I think it's probably correct yes I had quite a good look at it and I think it's correctly tiles but they do find sometimes mistakes in these things and there is the you get some feeling for the size of it because somebody standing there on top of it now the the two tiles are not distinguished in the nature but only in their shape and size now that's a different one these are kites and darts that's Carleton University and this is kites and darts nicely colored nicely arranged all right there how good shot of them I'm not quite sure what that color coding is doing but it's interesting looks nice now this is a place in India Allahabad where they have a big complex of buildings and it's supposed to be this cope the arrangements of the buildings and the designs of the building is all supposed to be based on a big version of one of these tilings and I wanted to get an aerial view of this complex of buildings and I want Google Google map I had a look and I could just about see while some of those we had some vague relation to the path and I suspect they didn't finish or they tried to do it one way and then did it a different way I have no idea but it was hard to see the new patterns in the large being constructed according one of these starlings but maybe they are on the other hand you see in the building here is a kite and dart pattern which is seems to be pretty accurately done building itself is a 10 sided building and that sort of reflects the 10 sidedness that you've seen is patent in the middle so this looks quite nicely done as far as I can see I couldn't give good terribly good pictures of these things to see what was going on there now here's something else everything I've shown you up to this point is real I want to show you a project this is a a project for a thing called the Transbay cent transit area which is a railway Center in San Francisco now you don't don't associate railways with the United States but it's I think that's the idea is that railways should become more important and this big Center it was a big project and it was advertised and this particular company seemed to have won the with the desert well net with this particular design now I want to say show you some pictures of it just to give you some feeling of the scale this is inside the trains will be underneath and this is just to give you some picture all these things I say are not real they're simulations but the intention is that this will become reality in some time I forget they gave me some kind of a date whether I'm supposed to believe business I'm not quite sure but it was I think it was a 20-17 which doesn't sound sound all that distant in the future I'm not sure that the tilings on the ground or anything to do with these tilings but to us that's giving you a picture of the scale this also even more on the top of this Transit Center will be a park so this is in the day this is the same park at night of course as I say not well none of this is real it's or real in the future I suppose that's the idea but it's a simulation but it looks wonderful you have this wonderful park in the middle of San Francisco set up above the ground level which is down here and down there and huge buildings which extend up and here it is at night so that you see now this is what the sort of size on top of this is that Park then you see that area that we just looked at before with the people walking around and then there is a skin being here which wobbles around and underneath that would be the railway station that would be the last thing to get me off I suppose the railways is considered well I suppose that's a big project to get the railways you've got to put them right across the states but you will see if you look carefully at the cover that their views now it's not flat which is it presents curious problems here you see more clearly they have these panels and they have the rhombus tiling on the panels now the reason they did this apparently is that if they chose a regular tiling then when they joined the panels together you saw that they didn't match and it was looked dreadful so they thought it would be good idea to use these tiles where you wouldn't know what was happening anyway and so you wouldn't spot I they didn't match along the edges of course and then they said well this is a nice seamless construction and the architects came across and visited me in Oxford normal there's a very delightful people and I said well it doesn't quite match does it I said well yes we know that but thought it doesn't matter too much so I said well there are two things you can do here now you see it's not flat that was their problem and so you have to have a curved surface intrinsically curved surface and you want to have a tiling which actually works on a curved surface and so they thought they fiddle it in this way so I said well actually you could make it so that it matches well you see they have these seams here between them okay those aren't supposed to match because the things might expand and contract and then you won't have a gap here however all these other ones are supposed to match so I say well you can do it if you don't mind slight changes in the size so you can have a conformal map which means you you locally they are the right shape but the sizes can vary just slightly and they said we'll do it so that's what they're going to do I think okay now the last thing here I want to show you coming back to where this is real now this is real assistant tilings that we have wadham college my college and these are straightforward rhombuses but you see I insisted on having some something which tells you that you've whether you match them correctly so you have these little they're two style shapes the big one the fat one and the thin one and every fat one smart same way and everything one is about the same way and you've got to match them and then you get these non-periodic packets well I've been to her they were laying these things I've been to a play with my wife and we came home London so they're doing these tiles it when I see how they're getting on there pretty well done it I wouldn't have a look it looked very nice had a somewhat uneasy feeling so I went up higher on the honor a little higher level I looked down on it I had even more of an uneasy feeling I could quite place what it was and then I looked right over up the edge the builders had put an extra tile and they could see it would fit another one would fit just at the edge it would match perfectly and it would fit but if you put that tile in somewhere in the middle of the lawn the thing would go wrong it wouldn't go and we weren't having that so they had to pull it out even though the lawn of course is a lawn and you don't see this pattern anyway this goes it gives you one version of how to mark tiles but I thought since we're having this new building this wonderful new it really isn't impressive new building I'm not saying and may we're going to have one of these tilings that there and I thought well well let's have something different so I'm going to show you something different let's go back to this original tiling go back to this I'd point it out that we did have in this pattern lots of these nice little deck guns and then rings the Pentagon's around it now suppose you take those decagons well let's do it a little bigger I think idea yeah here we go that's it it's here it's transparent and it's just a color okay can't see it on the table that's it here we have it that'sthat's the next version up one version okay it's the same thing you've got these little here we have the little decagons and there's a big deck again you see and where we have the little beggins remember we had this ring of Pentagon's you also sometimes get a ring a big bigger ring appendages here we have you know Pentagon rhombus spending and rhombus big and rhombus all the way around and that's the sort of feature you get now when you go to the bigger version of Pentagon's in the hierarchy this you see a bigger deck in here and then you hear my ring of Pentagon's around it okay so we get these rings and I thought would be nice to have those rings featuring I'm going to take this one off make it clearer so I've got to mark all those rings and I'm going to put have to do it the right way up this thing up that's it those rings just slightly further out for a reason which well let's say it makes it look a little nicer what I want to do but there's a mathematical reason to okay those are the rings I want undone I just drawn these rings in where you have them just outside the decagon and going around the pentagon's okay now the trouble is that if I try to mark those on these tiles they're not all marked the same way so I've got to have a few more I've got a few more arcs to make this now I'm going to mark this the same way around yes that's it I put these two together what do we see now well we see a sort of fattened out version of this Pentagon tiling etc remember we've got Pentagon's with a lock funny shapes there are Pentagon's there's a Pentagon there we have a funny shape one there's another Pentagon and here we have a justice cap you see but sort of squashed funny way and here we have the rhombuses and here we have the pentacle so it's the same pattern as before but sort of swelled out in funny ways okay now I need to fill that out just a bit more to make it well I think it's best if I put the rhombus pattern on there which is here then I can't find remember it's um underlies this pattern and you see that every rhombus has the same design pattern on it so if you marked the rhombuses in that way so that the thin ones just have two little circular arcs and the fat ones have two circular arcs across each other intersect it's the same everywhere you just need two kinds of tiles marked in this way and then you will see this pattern of circular lines coming out at you and I thought it would be rather nice in fact there was something a bit fortuitous about it because either you see the pattern in such a way that the rhombuses are quite obvious I didn't want that I wanted it so you didn't see it rhombuses very well and you saw that pattern which I rather liked and it's a bit surprising that you get that quite subtle pattern coming out from just two basic shapes and I done this what they have these are made out of granite I should say the tiles and the arcs are inlaid stainless steel so it should look rather impressive when it's finished now they had some samples of this in front of the old mathematics building like being in Edinburgh and I kept came back late on the train and I have a look at these things but wasn't very late was quite light and I looked at the tiles and the tower to Charles look completely different fat and the thin rhombuses look quite different I didn't want that and the circular arcs didn't fit together they looked horrible well the reason they didn't fit together if it was a sunny day and they'd be they'd been what I call combed and that means that the light hits them in a certain direction and then it shines at you but if you're not in that particular direction then they look dark so that if the tiles are oriented differently then the arcs will of dark tile are a dark arc will join onto a light arc and you lose the pattern completely so I thought that looks dreadful as you as you look at the you just see the rhombus instead of the pattern well they said the solution to the combing problem is don't comb it so they do what's called polishing it instead and that's not directional so that should work I hope the other thing is the fact that the tiles look completely different I came back a couple of days later and looked at them again and then they looked exactly the same I thought what that's going on all these different tiles have they gone back to China to get new Granite's from no it occurred to me the difference was the first day it had been wet have been raining and the surface texture and the type of granite is slightly different and when it's red wet they look quite different when it's dry they look the same so this is wonderful it gives you a new perspective on the wet day the pout you see one pattern on a dry day you look at it and see a different pattern so I thought that's quite nice okay let's get back to the to the to the PowerPoint please well this is actually when the architects were trying to work out the the design it is just showing the pattern that's all this is doing here you see the arcs are in red they said don't worry it won't be red it will be in stainless steel and the tiles look quite different don't worry they won't look that different but it is quite useful to see this pattern because you can see this doesn't you can't see it so well but there is a sort of central point here and there we have a circle you see that's one of the many circles you've got circle illumise and just outside that you see this bigger floral arrangement now you see lots of those all over the place there's right round the edge you see there it's therefore all the way around there's one there's one there's one there's one so we wanted to have this in the center and this so that the tiles are such a size that you could actually see these other ones there are other ones right further into there's another one there so there's lots of these bigger rings oh thank you very much yeah I use the green one it works here especially on red here so here we see the that floral arrangement then we have bigger ones like that one here I thought it might be quite nice because mathematicians children could come and and walk along the the patterns you see if they get back to where they started some of them do some take a mints to the little of off the edge some way I think there's a nice theorem waiting to tell whether how many different kinds of patterns that are before they close up so it's nice to look at and that's the idea okay perhaps we could go back now oh no sorry I don't go back to any area so this is the latest not quite the latest but almost the latest they've done a little bit not quite finished in top pop there and things not quite finished there and some area here which is I finished which you can't see I think it's quite yes now this is a wet passage pest I think that's been wet yes because you see the difference in the tiles there the other one I think was probably quite dry see so there you can't very easily see the difference in the time this one shows the stainless steel it does seem to show up very nice nothing so this is that the white is just some cover that they put over top of the famous tomb they don't want to take them off until I think till the official opening and then you will see the actual stainless steel and how they join now so I think it should look pretty impressive at the last month I know here we have the view that's the entrance the main actress the building it's a stunning building and they say and it's all partly the design was geared so that the Radcliffe Observatory which is a very wonderful building which is sort of revealed night was all hidden by all sorts of other buildings previously and not only can you sort of see it when you get further back you could see the top much more clearly on the other side of the building you could see it very clearly and from the common room you see they designed it so that the building is in two halves here with a connecting room which is entirely transparent so that you sort of could see through it but it's a really a wonderful common room in there where you could sit down you have a beautiful view of the Radcliffe Observatory at the back and and you can see the tiling from the other side so let's know soon anyway I think that's the end of it thank you very much you you

Channel: undefined

Views: 513,990

Rating: 4.8154764 out of 5

Keywords: Science, Ri, Royal Institution, Sir Roger Penrose, Symmetry, Tessalation, Tiling, Architecture, Design, Maths, Mathematics Education (Website Category), Education, Forbidden Symmetry, Penrose tiles, Roger Penrose (Author), Repeating patterns

Id: th3YMEamzmw

Length: 58min 13sec (3493 seconds)

Published: Thu Jan 16 2014

Reddit Comments

[u/Kowzorz]

Thank you for sharing this. This was thoroughly entertaining.

"Does that mean there's only one way to tile a plane? Yes and no is the answer to that. Strictly speaking, mathematically the answer is no. That is to say: there are many ways of continuing to infinity. However in a certain finite sense, they're all the same. That is to say, if you find any two of these complete tilings of the plane, with the same shapes, that is, and I take one of them and I take a region no matter how big that is finite, I can find that in the other one. So you can't tell ever which one you're in. " t=23m

I'm reminded of normal numbers such as pi and the ever loved in this sub phi.

I wonder how a crystal grown in manners such as this video's designs would behave.

The moire pattern bit was neat because the pattern generated while he was not directly aligned with the radial lines a pattern similar (I wonder if it's geometrically similar) to the patterns doing the moire. He didn't seem to mention it during the demonstration.

I wonder how this tiling would behave in hyperbolic geometry.