Well, thank you, Sam, for those flattering comments. And I'm sorry you didn't mention about me being late for lectures and things like that. I want to talk about, well, what's you see on the screen that we've just that that comes a little bit later. But eventually I shall talk about the pattern which you see in the front of a building, the paving there. And I want to explain how that came about and what it's all about.
But before getting to that, I want to make some more general comments first about crystal symmetries and then about non crystal symmetries. So these things are, I assume, very familiar to people that you can have a plain pattern, which is it's has a translational symmetry. So you can slide this along in some way, and it's the same as it was before. And in addition to that, has some rotational symmetry.
Now there's a theorem, that very ancient theorem, I think, which the only rotational symmetries you can have will be two, four, three, four, four, five and six, four. And just to illustrate those different symmetries, we have them here. If you have a parallelogram and a pattern of them arranged like this, then about the centre of that, you're going to have a two fold symmetry rotation through 180 degrees since it's into itself.
Equilateral triangles again, take the centre of one of the triangles. Then you can rotate the whole pattern into itself 120 degrees. So it's three times near back to where you start for follow the very familiar square pattern and six fold hexagons. Now these are all very familiar. Why are they the only ones as well? There are various ways you can prove that. One of the simplest is as follows Let me just get the stuff working here.
Suppose you have some pattern which has both a rotational symmetry and a translational symmetry. And I suppose there discrete points which don't sort of crowd in infinitely infinitesimally close to each other, something like that. So we'd have a, a set of points on the plane. And I'm going to suppose that I found a point which, about which the pattern has a symmetry, which is one of these symmetry.
So n is the is an info symmetry. So if you rotate through 380, 360 degrees over n pattern, we're going to itself. So that's a point in the plane with that that property. And here's another one. And I'm going to choose those as close as possible. So let's find somewhere where I find two such symmetry points. If there's one, there's going to be another, because I've got a translational symmetry. So you move this along somewhere and there'll be another point about which it has rotational symmetry.
And we're going to find two which are as close as they can be. Now, I rotate this one about that one by the symmetry 360 over N and then that since the pattern is the same when you rotate through that, that also must be an end fold symmetry point. Then I look at this one and I rotate in the opposite direction. This one. And so that one must be an info symmetry point. And you see those are closer together, which contradicts these two being as close as possible.
With the only exceptions being, well, maybe they're not closer together, which would be the case if n equals two. Then of course, they're not an equal three. They're not either and equals four. They're exactly the same distance and equal six. You get away with it because they coincide. And so those are the only possibilities. If you have an equals five, you can see they're obviously closer here.
If n were larger than six, they'd cross over, but they'd still be closer together, be more like that picture. Okay, so that's the proof. Really? Very simple. Not much. You can understand that pretty reasonably, I hope. Now, what about this pattern? Well, it has a look as though it's both symmetrical in the sense you translate it and it goes into itself. And five fold rotational symmetry. All of these stars, pentagons and things like that. In fact, both those statements are almost true.
So this pattern could be extended to infinity. And I'll show you the way you can do that. Extend it to infinity. And we have the property that if you give me any percentage less than 100%, so 99.9%, then I would be able to find a way of sliding that pattern along itself, a translation symmetry, so that it goes into itself to 99.9%, that is to say, of the line segments. 99.9 of them would be in exactly the same place as before, and only 1% would be different.
And you imagine and I want it better than that. How about 99.99%? So I say, okay, yeah, I can do that too. And I find another translation and it will agree to 99.9% and also I should say rotation. I can find points about which it will rotate and the symmetry in the first case would have been 99%, 99% secondary, 99.9%. And you might say, well, I'm no, I'm at 99.999999. What have you like? There will be a translation and a rotation of this pattern.
Five fold rotation of symmetry and a translational symmetry to that precision. So it's almost in that sense, never quite exactly 100%, but anything slightly shorter? 100%. Yes, you can find find it. Well, now, you might go back to the original argument and say, well, what goes wrong with the argument? Let's take it. So we're looking at not exactly symmetric points, but we're looking at points which are has the symmetry to 99.9% to something. The thing is that if those are 99.9. Percent points.
These ones are likely to lose just a little bit of symmetry because this one's only 90.9. 99.9. That'll probably be 99.8 around. So you lose just slightly, slightly, they'll get closer. But you'll have lost just a little bit of the symmetry, the precision. The presentation will be slightly less. And you keep doing that. And of course, by the time you've got down to the size of the percentages here, you will have lost all all the precision.
So you can see that it slips through the proof as well. Illustration is why when you give a proof of something, you really mean what you say. It's got to be the statement has to be exactly what you say. The if it's only APR, you may find that there are loopholes. And this is a good example of that. Now, let me tell you how this pattern is constructed. It's very simple in principle.
Let's take the regular Pentagon and what I'm going to do as I'm going to subdivide that Pentagon into six smaller ones, which almost fill it. So I can find out where this one is by joining those two points, if you like, and that tells you what that line is. So if I join none adjacent points, I can see where this central Pentagon is, and then I join those and I can find those lines and so on. So I find that there are six regular pentagons which almost fill that Pentagon.
Now, what I'm going to do is to blow this up to the scale so that this one in the middle is the now the same scale as that one. And then do it again. But up. Do it again. Okay. Now, if I. I think. Yes. Here we go. If I do that here, too, this picture. Then this Pentagon. That's the big Pentagon. That was that one. And that Pentagon. There was that one. I mean, that one say. And then I subdivide these ones and you see this little gap in the middle there, which is a bit of a nuisance.
Never mind. We're almost there. But what I'm going to do now is imagine that we've subdivided again. And then this one would be subdivided. So there'll be a little triangle pointing in their little triangle pointing out there, and I'll have a sort of spiky rhombus the next time around, and it'll look like that. So this is the this big Pentagon there is but almost half the picture here. But so that one there is now become this one and so on.
And there's my spiky rhombus in the middle. And I find that there's exactly room for another one of these pentagons down there. I still have two gaps. This well, I call that a pentacle. If the lines have gone through each others what people call a pentagram. But this is a pentacle. It goes in and out like that. Okay. It's a star shape. And the other one will I call it a justice camp? I think that's one way to describe it. There's the jester, if you like. Okay. Okay.
So we're going to keep doing that, blow it up again the next time the pentacle will get a little spikes like that. And the nice thing about it is that I can always just find room for pentagons. You see, if I take the pentacle and I put the spikes on the next the next stage where I subdivide all these, there's little spikes sticking out there. I have another pentacle in the middle. And three, five more just as caps.
And with the jester's cap here, the other way up in the mind is that just this cap? There I find three pentagons, one pentacle and three just as caps. And with the rhombus, one pentagon, one pentacle and one jester's cap. So I don't have any new shapes. So each time I can blow it up and subdivide the ones I had before. According to this pattern, there's only one little thing much where you might not, but it's more likely to worry you.
I think if you if you, you know, mathematicians like to worry about things like this when they don't need to. There's an ambiguity here. You see, this could have been done the other way up. And you say, well, okay, do it one way. No, I'm going to be I have a rule which makes precise which way I do that. And what is that rule? Well, I'm going to adopt the following rule.
Looks a bit funny here. I'm just telling you this, but it's not hard to see that wherever you find one of these spiky rhombus is that Pentagon either on one side or the other. Rhombus, I should say spiky rhombus, one side or the other. You will find this pattern and this pattern. You look over there and see where that one is. If it's not on this side, you'll find on the other side. And then there's a little rhombus in the middle.
And the argument is that whichever way we do, this is governed by where that Pentagon is. So you take a symmetry about the major transverse, what do you call it, diagonal of this rhombus here. And this thing flips over a mirror symmetry about that. And that tells you where that Pentagon goes. So this one is down there. That one's down. If it were up there, that won't be up. That's the correct rule. That one's wrong. Now, why didn't I say this is the correct rule and that one's wrong?
Well, there's a good reason for that. That is that next time when I subdivide the thing in the middle is going to be ambiguous depending on which way you look. Whereas if you take this rule is consistent all the way through, okay, well, that's what you do. And if you do that, you get a pattern like this. Well, let's do it. Okay. I'm going to start with a nice big painting in there, and then I'm going to subdivide that according to the rules. And I've just been giving you the volume up three.
Is that? Do I just shout or is there a way of doing it with. Oh, it's down there. Oh, another one. It's not going to interfere with the other one. Awful squawks and things like that. Good. If it squawks out, I'll hurriedly take one out of pocket. Can you hear me now? Good. Okay. So that's the Pentagon before I subdivide that. There we go. Just according to the rules I was giving you.
And then I subdivide that. No. Now, you see, there would have been this little ambiguity possibly here because there's that rhombus. And I look to see where should I put the Pentagon? Well, I look around and I say, there is that pattern of Pentagons which I want you you would be able to find. And I look at that, that one's over there has to be there. And therefore this one's down there. Okay. So that's satisfying the rules. And then next time, once more and I get the pattern that we had before.
Okay. So that tells you how it's built. It's got hierarchical construction that can whip those away. And just to confuse you a bit, I'm turning it upside down. Now, where was that original big Pentagon? It's not so easy to find, but I. Each time I had to figure it out all over again. The point about this is that it has a uniformity that's not so obviously hierarchical. It does have a hierarchical construction, but it's really something much more regular than that.
It has a highly irregular structure, and the hierarchical nature is not at all obvious. In fact, it has various properties. I point out a few of these. For example, you see here you have a regular deck again, ten sided figure. There's another one. And every time when you find one of these regular deck of guns, it's always subdivided in the same way. Three pentagons to rhombus and one pentacle.
That one just as cap. Okay. And every time you find one of these, it's always surrounded by a ring of ten pentagons. Yeah, there's another one. Always a ring of ten pentagons, wherever it is. Sometimes these things overlap. Like here you've got two of them overlapping. Another one there. Whenever that happens, you still find your ring of ten pentagons that just go quite happily through each other. And it has that very nice property.
Other things which are very evident to me just while I stand here, is that if I take any line in the picture, I can put a ruler against it. So any line in the picture, it will keep on going right across the picture with the same density of lines all the way up. So if you imagine a boat, you had a field of corn and there was one growing at each point vertex of this pattern. And you drive past in the car and you look at it. They all line up at some point and then it all line up again.
You say, Oh, that's a nice regular pattern. Well, it's one of these things. Maybe this you'd be lucky. Lucky to find a farmer who'd do that, but never mind. Anyway, I just wanted to indicate that there is a lot of structure. There's a lot of a lot of there are a lot of features which are not at all evident from the hierarchical construction. And for example, now we have this ten sided one in the middle here with its ring of Pentagons.
But in this case, you also find right the way around it a nice ring where you have rhombus, Pentagon, rhombus, Pentagon, rhombus, spending all the way around ten, ten, rhombus and pentagons alternating. And you find those all over the place. Okay.
Now, I want to say something else about these patterns, and that is there was a problem originally to do with tiling, playing with squares, with coloured edges, and you had different squares with different arrangements of colours and you had to match the colours. And the question that was raised by a Chinese-American mathematician called How one could you find a computer program which will answer yes or no given those sets of colourings finite set of the squares?
Yes or no? Can you tile the entire plain with these coloured squares? And he came to the conclusion that yes, there would be an algorithm for doing that, a computer program for doing that, if it were the case that for any set of coloured squares, if it did tile a plain, you could be sure that it would tile a plane in a periodic way. Well, his student, Robert Berger, thought about this problem and eventually proved that there was no computer program for solving the tiling problem of this kind.
And therefore, as part of his argument, in fact, he produced a set of, I think it was originally 2000, 20,426 different coloured squares. He then got it down to about 100 and then Rafi Robinson got it down to about six. And he had a way of making it into a jigsaw puzzle rather badly drawn here, but that's what it is. You can see the squares on this piece of paper, but that's not got the designs on them. You have to fit these together and the only way you can do it is known periodically.
So that was made it consistent that there was no computer program for solving the tiling problem with these coloured squares and the fact that there are shapes which are only known periodically. Well, I was visited in the Old Mathematical Institute by the American mathematician Simon Kitchen, and he was describing these things to me. And he said and Robinson was somebody like to get the number down as small as possible.
And so he told me there were six. And I said, Well, I think I can do it with five. And my five will look completely different from his. In fact, what I was actually doing was with six. Now, those six, you see, are actually the same basically as this pattern. However, the Pentagon's are divided into three classes, depending upon well, as with this one, we have the Pentagon surrounded by five of us here we have it only three. And here we have another one. Only two here.
So either two, three or five, and I'm going to make those different. And then I make it into a jigsaw puzzle where these are the that's the five. That's the two. The three and that's the two one. And just to show you what's going on, a couple of these so that. The five one is the is the orange one. And you can see what's goes on. Here we go. But you see, that's still six like Rafael Robinson six.
But but yeah. So if you imagine the coloured ones are different and they have a matching rule according to those little knobs and matches which I just showed you then the point is that you force that kind of pattern, which I've been describing. It does have this hierarchical arrangement, but I'm not saying do it that way. I'm saying do it anyway you can where you make the notches and the knobs fit together. And the only way you can do it is in the sort of arrangement which I just showed you.
Now, the reason I said I thought I could do it with four. You see, if you look at these tiles, you see that there's this little funny star shape thing there and two of them on here. And there's only one little place where you have a a slot for them to go into. And that's one this one. So you cut this one out and glue it to that. Take another two versions of this, glue it on here and here, and you don't need that one.
So you could do it with five. So I went home and then I started thinking about this, cutting up and seeing whether I should improve on that. And I got it down to four and I was quite pleased with that. And then I got it down to two and I wasn't I wasn't so pleased. Now, why wasn't I so pleased? I think the reason that I wasn't so pleased was I thought, Oh, this is so easy. It must have been known to people. I think that's why I thought that.
But I do remember being disappointed when it came down to I'd rather like my set of four, but instead of two I originally hit on is shown in this picture here where we have. I well, it's cuts and dots. Here we have the cut and here we have the dart and those are the only two. And I have a rule where they have to they've got corners which are either coloured black or white, and you have to match the corners. You can do this with knobs and notches, if you like, make a jigsaw puzzle out of it.
And well, I'll show you one in a minute. And each one of the cuts you have to put lines on in the appropriate way, and each dart you do as well. And I've done that on this side, on the right hand side, and you will see that it brings out the pattern we just had before with the Pentagon's. It's just they assembled to make the Pentagon's and the the dotted ones the shaded ones make either the rhombus here or the just as cap here or the pentacle over there.
And that is the pattern we just had before. And you can see that if I do it right, then watch. Or which one I need that. Let's try that. And probably East Orange one. Let's try that one. Yeah, I think that's it. So you can see they're really the same kind of pattern. But the cuts and cuts are done in a way that you don't need so many different styles. That's all. Now, you can also modify them, as I say, to make a jigsaw puzzle type arrangements.
And here's one which is sort of influenced by Escher to some degree, where there are two birds, the big bird and the little bird. And the only way you can assemble them is a jigsaw puzzle, is in an arrangement like this which never repeats itself. And you can then go back and see how that ties in with the these ones. And I hope I can. It wasn't designed to fit this transparency, but I managed to find a place where they fit.
And there I think you can see the big birds, the cuts and the little birds are the dots. And that's all they are. But that makes a nice jigsaw puzzle. Okay. Now, you can also do two in another way, which are actually more commonly used. It turns out because they're just rhombus. And the colouring I've done here is you have to match those colours and it forces this non periodic arrangement which I've just been showing you. And what you get, I'll start off by relating it to the cats and dogs.
There you are. So each. Each dot is drawing, you have the same lines drawn on in each cut the same way and up with this pattern of promises. Now, this pattern has been used in all sorts of places. I'll show you some places where it has been used, but often just like this. And I always thought that's a bit disappointing because you don't know why the pattern is like that. You think, well, lots of ways of doing wrong business. You could just make them regular patterns.
So why do you do them like this? And it's you might say, well, that's what you're supposed to do, but that's not good enough, is it? But if you have the matching rules like here, then it does force that type of arrangement. I use a few things I should say about this. One is okay as it forces it, but are there more than one way of doing it? Is they want more than one way of doing it? Well, it depends. The answer is both. It's unique and there's infinitely many different ways of doing it.
In fact, true to the left, note the number of points that are on the on the real line two to the left note number of different ways of doing it, but they're all the same in a certain sense. If you were given two of these patterns and you wanted to see whether they were the same or different, and you could only examine a finite portion you would never tell because any finite portion in one of them will appear in the other one infinitely many times.
So as far as finite arrangements are concerned, they're identical. The only difference is our way to infinity. And that's quite a subtle thing. Well, let me not go into the subtleties of that, but it is the case that, strictly speaking, as far if you go all out to infinity, there, all the different ways of doing it.
It depends on where you start. In the hierarchies, there are hierarchical, hierarchical in the same way that I started, but you could sort of start in different places, and that makes a difference, such as infinity. But as far as financial arrangements are concerned, they're all the same. Okay. Now. There are these are all the five fold ones or ten fold. It depends whether you you can call them either five fold or ten fold, depending on your rules.
I'm not going to make a distinction there, but there are others. And here we have a 12 fold, one that's rather nice. One made by Galen Dennison. He's Swiss. I'm not sure it was Swiss. These were there was a competition, actually, for the I thought the ones that they produce were the nicest ones. These are 12. So that's one of them. And they haven't got matching rows, but you can make matching rules.
It's a little bit more complicated how to do it. You can't do it with just I think there are three different shapes here. But you can you have to have another shape to make to force the matching rules. It's really a four tile thing. Here's another version of the same thing. Really rather nice. There are also the eight fold ones which Robert Mann and somebody could beat to produce. Very rapid momentum is very rapid.
Martin Gardner had an article on these things and he announced he was going to produce these something that he hadn't ever shown before. And I don't think he said very much about what they were like. And Robert Manne, this rediscovered the The Rumpus ones, which is very remarkable I thought in less than a month, I suppose. Okay, now what do I want to say?
Well, sometimes people say, well, you find these things in Islamic patterns, Islamic buildings, and you certainly find a lot of extraordinary looking patterns in Islamic buildings with all sorts of symmetries. Yes, it's nice example here. And you see this part of the wall. I'm not sure I've counted what these ones. So that's ten fold, I think, here and so on. But they're all limited regions and there's no rule about how you might go out to infinity.
Or if you see a nice pattern of these things, it's almost always the case that there's a evident symmetry to it, that it just repeats itself. And so just having interesting symmetries, I mean, the symmetries don't extend to the whole pattern. They just local symmetries. So you can have these local symmetries which are not crystallographic, but the pattern as a whole is a crystallographic thing. One of the more remarkable examples was pointed out by a beach map. And this. No, I said that's wrong.
That's it. Yes. And Paul Steinhart and a Chinese chap. I've gotten his name now. But anyway, they did try and relate it to the tilings that I had. I think there are some relationships, but. But there's no evidence that there was any kind of a rule enforcement periodicity. In fact, the examples like the one up there you can see actually is is completely periodic. So and it's less obvious here what's going on.
And they have a certain hierarchical aspect. You see, they're subdivided, but not in a way that repeats the patterns on the big scale. So it's not clear whether that may be somewhere in some building. A such an example will actually be found of these non periodic things. But I want to show you something else which is a bit different somewhat later, but not that much later.
1619. These are pictures done under the instruction of Johannes Kepler, the famous astronomer who discovered the Kepler in orbits, ellipses and so on. But he also fooled around with all sorts of things and he like polyhedron non none and how you fill space with different shapes and so on like that. But he had a page in one of his, his books, harmonica mundi volume two 1619 in which there are all these different, very curious symmetries.
Now, my father owned a copy of this book, and I had seen this page some years before finding these patterns, which I've just been showing you. But I wasn't thinking about them at the time. But let me show you something. I'm going to concentrate now on this person called a one there. Let's make it a bit bigger. There we are. And now I want to show you something. This is made of pentagons, regular pentagons put together. And then these little holes in them which are decaying in all shapes.
And these stars, these pentacles. Now, I'm going to try and find the right spot here, which is maybe a little tricky. Here we are. And look, it fits. Exactly. Not only does it fit exactly, but in Kepler's picture, he drew an extra little line there for some reason. And that line also fits. Exactly. Now. What was he doing? I don't know. I really don't know. I was giving a lecture like this somewhere, and somebody in the audience at the end said, yes.
There is a letter that Kepler wrote to somebody and he explained what he was doing in his pictures. I said, Oh, that's very interesting. Can you find out and let me know? Well, I heard nothing more from her. There's times I've got people to look at Kepler's letters and things, and the best they could find were letters that he had written to the person who actually did the drawing. So that was a he had a an artist actually to draw these things for him, but were strictly under his instruction.
So they were Kepler's pictures, no doubt. My guess is that he was probably interested in he certainly interested in face space filling with with regular shapes and so on and crystals. But people didn't even know really about atoms in those days. But it was one possibility and I imagine that the idea of crystals being atoms arranged in a regular way was certainly a strong possibility.
And I suspect that Kepler was also interested in living things and maybe plants and so on might have had made use of some other symmetries, such as the ones in his pictures here. I have no idea. That's pure speculation. There was a time when people were just finding examples of what we call quasicrystals, which seem to have a ten fold, five fold and ten fold symmetries, and then they seem to find some eight fold ones.
And then I was visiting Switzerland. This chap, Nissen, did one of the pictures I showed you earlier, the 12th of one, he and a colleague. This these ones. And the reason he was interested in this was that he claimed to have an alloy which produced diffraction patterns which seem to suggest there were 12 fold symmetry. And nobody seemed to believe him when he showed me these things. And I thought, well, it looks fairly persuasive.
They weren't quite uniform over the whole picture. And then he showed me the data points from the diffraction pattern. These are electrons you shine electrons on and they scatter out at particular angles. And then you see the shapes of the the little points of light that you get where the electrons hit. And this tells you something of the symmetry. And he showed me this with a 12 fold symmetry, little points.
I looked at this and I started joining them up with the squares of triangles and like that. Now I've seen that picture before somewhere. And where on earth have I seen that before? And then it dawned on me. Yeah. It's that one. If you put the points of the diffraction pattern on the corners of this arrangement here, if it is a bit bigger, that's the diffraction pattern. Well, I published some other points scattered about, but it was it was that particular pattern.
Again, I have no idea what Kepler was doing. I'm sure he didn't know anything about diffraction patterns and those, but nevertheless, he had some deep insights in other ways, too. So who knows? I'll just leave you. That's a bit of a conundrum. Okay. Let's do something a little different here. If I can get it to work, which is not necessarily the case here. We have I'm afraid it's got it's an old slide and you'll see the got rather messed up.
I must get a new one. But these are composites, which you can probably just about see. But there there's a nice big pattern of the rhombus. I can do it with that, can't I? Yeah. I don't need the transparencies. Okay. So that promises. What's that like? Well, that's wrong. As is, too. But I think I can't even see them very well. I can see those blobs there. But that little rhombus, this point about this is you can put others on top and get nice diffraction patterns.
I have to worry about that spot, because that's just the light, isn't it? That's. Probably do. Never mind. Don't worry about that. Um. I don't know whether I can do this very well here. But you can find a little place where. Oh. That's good. See if I can get that spot in the middle now. And a nice one. If you see those lines, the lines are where they disagree and where you don't see a line. That's where they agree. Now, this is where she'll probably fail.
But let me try it. I've marked a spot where. Let's just see. That's not it. I thought I knocked it. To see if. Okay. If you get it right, which I. Which is hard to do, I'm afraid. I think it's best if a twisted a bit more. That's confusing that on the one. It's not working. Let's try and match my marks. That might help. No. It's not working very well. There should be just oh, yes, there's a line right down the middle and that's it matches everywhere except along that line. Uh. Not very good.
I think it's best that you do. You can probably see the line extending from here to here. That line is where it disagrees. Three degrees everywhere else. You see, that's one of these 99.9 things, if I got it right. I think I shouldn't wait too much time on that. It ought to be just about there. I think you just about it. Actually glue them together and then it. But of course, the defect isn't. I think that's the best I'm going to be able to do.
I'm sorry about it. I think one trouble also is that these machines. Well, I shouldn't blame it on the machine. It's not quite. Did you get it today or. So I took so long. I think perhaps I'll use this other machine. Oh, I have to get it going, don't I? I know how to move the picture, but do I turn it? How do I turn it on? Is it there? What is that square? No, it's not it. I'm very stupid at these things. Should I be pressing something back front over there?
Of course. Perhaps it was there all the time, was it? Yes. Well, you can see this actually hasn't been put up in the building yet. This is a nice poster design in our old building. I think it's being stored somewhere away, isn't it? Probably. Anyway, it's made up out of little pieces, plastic pieces that were given to me by a mathematician, Ron Graham. And some of them were built by. Made by some other people. Michael. What? His company and.
The big one. The middle is the wrong one, and they're basically cats and dogs with the matching rules done by little knobs and things.
And it's also coloured in a very specific way, which the colouring is unique up to permutation of the colours, and it produces some nice things if you sort of squint your eye and look it up on the top left, we see a nice, straightforward, got dots on the top right, rhombus, one bottom left we see the modification which gives you the birds and the relationship to the Pentagon's as well. So that's a little bit baffling. Right in the middle there is a a dog and the dog is not.
It's cheating. He's a different shape from the others. But if you have one dog in there, the tiling is completely unique, right out to infinity. So you don't have this business of up to the left, not that unique. And the one thing on the right with the diffraction pattern is an actual quasicrystal. This is an actual material produced by people in Japan and now you can produce ones that you can actually see.
I mean, it's probably only about that big, but you can see it quite clearly and it's a regular dodecahedron which is not allowed for a crystal. So it's quite interesting that you can have that. And there's a bigger picture of the actual quasi crystal. I think they're called crystals now. So I don't know. I think they have so many properties of crystals that people prefer to call them all crystals.
I'm not sure whether I like that or not. Never mind. No part of the fact that's upside down, which maybe shouldn't matter. It's a hit. I hope not at all upside down. That is the first time these things were ever used in architectural design. That was anywhere. Well, I hope the others are not like that. An architectural design on a building.
This was the Tokyo Metropolitan University and the architect there wanted to use clocks dart so you could probably see them a little bit hidden with having all sort of other marks on top of them, but cutting dots. And when the thing was assembled, the architect looked at it very carefully and he found that it made a mistake. And had they they hadn't take it out and put it back correctly, which impressed me a lot.
It's completely right. This, as far as I can tell, it's supposed to have some connection with the shape of Tokyo as well, which I don't know much about, but I think it was rather very nice. Nicely done. Okay. That's right. Way up. This is a rather gaudy looking building in Melbourne. I think it is Australia, in Melbourne where they've used quite enough all over the front and all over and inside too.
And it's it's rather an extraordinary place. It's one of the first buildings I know which made use of these things. And you can see up on the top, right. I think that's got the stripes on them and I was showing you, but it seemed to be pretty accurately done. They've got the stripes on the other ones too. Yeah. Which you have to match. Now there's a much more sedate version on the ground down here.
You see, this is just straightforward robberies. This is Stony Brook in Long Island, New York, and that's the mathematics building. And they wanted to use a tiling of this kind. But it's just straightforward promises. As you'll find a lot of see this against dreadful promises. This is in in Perth, in Australia and I think this is the Cosmology Centre or something. It's a very nice popular science place in, in Perth and they wanted to use the tiling from the starlings on the floor.
Perfectly nicely done. I think the biggest area I've ever seen is also in Perth. It is in their chemistry building and again it's just straight run business. And here we have cut and. This is. I think. In the United States. Carleton. Carleton University somewhere. Yep. Which is a nice done touch and ask without also bringing out some pattern in the colouring which is very attractive. This is the other place. St John's College in Cambridge.
And this is the entrance to the building was called the Penrose building. And I say it's nothing to do with me whatsoever that some architect who designed buildings at St John's College in the 19th century and it was converted to become the library building and for the entrance they wanted to put this door swings round. You see, the door is now open so you can go either side when it's closed that you can only see half the pattern and the door swings round and it's cuts.
And that's perfectly correctly done. And here we have what we call a judgement. This is a pattern which for it's just in front of the the. In front of the student bar and it's usually full filled with lots of cigarette stubs and beer stains and so on. But I think it's been reasonably scrubbed down, but not terribly perfectly, and it is matching rules.
So I was with my wife where they were laying this thing, and I went to a play at the Playhouse, I think, and we came back and they were just about finishing the laying down. And I thought, well then have a look and see what they've done. And so I looked at it and all right, I thought, but a little bit disturbing. And so I went up on the higher level to look down on it. And I kept thinking, there's something a bit disturbing about that.
And then I realised right at the edge one of the workmen had seen you could put another tile in which would fit alright, but it would fit alright there. But if you kept on going somewhere in the middle of the lawn, you'd find you get stuck. So I had to have it pull it out, I'm afraid. And what's there now is correct. But. But there is this danger because it is a non-local business. Let me move on to. Yes. This is actually a three dimensional version. This is, I hope, going to appear on the wall.
The tiling out front of our building was financed by my Andre Stern very generously, and he wanted having a new building made, and he wanted to have something in there a bit different. So I suggested this three dimensional version. It's just the rhombus, but you see that they're all the same size and shape. The rhombus is just the angle that you see that makes the either the fat one or the thin one. And it makes this undulating terrain, which is quite nice.
And here we have our building. So let me try and say something about this now, because it is the rhombus tiling. But it's got more to it. And I want to describe. I'm afraid I shot this up on that. There is the rhombus tiling. But you see more to it there. There's some patterns there. Now, you may remember that when I was showing the pattern of pentagons. So, um, let's take the bigger version here. You have places where there are these regular decorations.
And each time you have regular again, it's surrounded by a ring of ten pentagons. So I thought maybe you could enhance these rings somewhat by having a circle going around them. So there you are. Now, that's, of course, doesn't do the same thing to all the Pentagon's. So let's do something else. I'm going to complete these things a little bit. If I do that, you can see that it's really a sort of fattened up version of the the.
Here's the Pentagon fattened up and the and the justice kept fattened up. And then the Pentagon has done funny things at the expense of those. But you can see the relationship to these here. If there's the the pentacle. There's the pentacle. There is here and there. So if I add those extra lines, we sort of retrieved a curvilinear version of the one that we had before, and it would be nice to put those marks on some actual tilings and then tilings easiest ones to make the rhombus ones.
But if you do that, then you find that the rhombus. Some of them are different. You see that rhombus has a different arrangement in the other. So we have to add a few more lines. There we go. And if you do that, then the fat rhombus has everything the same two lines going across each other and the thin rhombus was two lines across. The pattern itself is now just that. And what we have first, which is this second. Let me add a few more lines then we have that arrangement.
And so I was rather hoping that that would be the major feature when you see the tiles and it was suggested that the tiles should have stainless steel arcs. I thought that sounded really nice. And I remember coming back, I think I was at a conference in Edinburgh or somewhere up north, and there were a set of I think six tiles in front of the building. And I looked at them and there seemed to be two things wrong with them. One was that the standing still didn't seem to join on.
I mean, they joined, but they but some were bright and some were dark. And the reason was that they were apparently was called combed and the combing has a different grain to it. So they said, oh, well, we can fix that by polishing instead of combing it. But the other problem was that the tiles looked very different. One was much darker than the other, and that wasn't what I wanted. I want them to look fairly similar to the pattern of the arcs is what you saw.
And then a few days later, I came back and they looked tiles looked the same. The two different rhombus look the same as each other looks much better. Maybe they've got some different tiles, but it wasn't that at all. The reason was the first day I'd seen them it had been raining and they were wet. And the second day it hadn't. It was nice and dry. And so we thought, well, that's nice because then you get two different kinds of pattern.
When it's dry, you see the mainly the arcs and when it's wet you can bring out the other pattern. So I thought, that's rather nice. Anyway, here we have the building as a whole. Well, from that angle. And here we have a sort of shot in front of the main entrance at the back there. And you can see these rings, the that's the ring of Pentagons as it was now changed into a circle. And you have several circles and you have sort of bigger ones going round.
Not sure whether you can see it very well here. Hmm. Maybe the other picture is better for that. Yeah. Here we are. Yeah. So you can see. Let's just check. I think if you take the circle just up there and then a little bit further out, you'll find a sort of floral pattern of a ten sided shape. So. I thought it might be amusing just to see, you know, how if you follow the lines of, well, there's a I can give you a conjecture.
I don't know if I want to say it to conjecture or not because I haven't really thought about it much. But you see these patterns always join up in some way. But do they form closed loops where you can see some of them? You feel the circle. Okay, that's a closed loop. You sometimes see double circles. I'm not sure if I can find one to like that. That's another shape where you can see the big curvilinear decoration and you also find the curvilinear Pentagon in the right places,
which is something I haven't quite expected. But you see them, too. And so. You get close friendships. But if they cross each other. Do you ever get a close shave? Are those four the only ones you ever get? If you find anything which doesn't belong to him versus just just sort of by inspection. The only ones I've found, I've never been able to close and they go right off the picture.
So the conjecture might be that those are the only closed loops that may well be false, because I haven't any idea really this much, because one of them is the shape of the pilings. If you watch it, they are okay. I think that's one of the few other things I can say, but perhaps people have questions and then to leave it at that. Oh, I just. Yeah, I'm sure that there's one more picture. No, that's just the beginning, isn't it? I thought there was one more picture. That's what lost the moment.
Now, it just showed what they look like when they're wet. But. And this is after this? Mm hmm. No, that's right. At the beginning, I got lost, number one. Thank you very much.