Eschermatics - Roger Penrose

[Music] I would like to welcome you all to the mathematics Institute my name is Alan go really and I'm in charge of the public lecture series for the Institute before we start the event let me remind you that as usual the green site indicate the emergency exit I also want to acknowledge the sponsor that you can see there XTX market the dispense of the whole oxford mathematics lecture series XX market is a leading quantitative driven electronic market maker with offices in London Singapore and New York and we're very grateful for the support now on to the main event as you can see or as you know what you can feel in the air to that tonight is a very special event it's actually co-organized with the clay mathematics Institute and Oxford mathematics as such it will be a little bit different from the usual format the public lecture today will be followed by the presentation of the clay Award for dissemination of mathematical knowledge I call it Kadam for sure so you will have the distinct honor to be part of the event and the presentation the award ceremony for the Kadam and since we want it to be directly after the talk we will not take a question today we will go directly into the award ceremony Nick Woodhouse which is who is the former head of our department and the current president of the clay mathematics Institute will give us a full description of the awards and of professor Penn roses achievements so I will be very brief in my introduction for tonight's lecture I will just say the following when you work in mathematics and physics one side of becoming famous is that you name may be associated with a theory say oh the theory yes that that person made it now you realize that Roger is in a completely different category when you see the partial list of theories that has been named that either eyes name or the naturally associated with him for instance you'll find the twister theory the notion of spin work cosmic censorship vile curvature path Aziz Schrodinger Newton equation Andromeda paradox conformal cyclic cosmology just to name a few no it gets even better when you name is used for a famous phenomena or theorem as you probably guess people don't usually name things after themself my some have tried but it usually doesn't work so for that to happen for a name to be given to you name to be given to something spectacular first of all you have to done something quite remarkable but you also must have the admiration and the respect of entire community and again here the list is quite long and I'm just naming a few people in mathematics and physics know about Penrose inequalities Penrose interpretation of quantum mechanics the more penrose pseudoinverse the Newman Penrose formalism Penrose diagram and of course the Penrose working singularity theorems just to give you an idea but things get even better and transcends science when you name associated with things for the public at large beyond physics and mathematics and here against we have Penrose tiling that you might have seen coming in here tonight the Penrose cube that you can buy online the Penrose stairs the Penrose triangle and of course these last things are closely associated with this contact and influence on the artist MC Escher and is the topic of tonight lecture so without further ado please help me welcome Roger and let's hear directly from him about the study well thank you very much yeah well it's a great pleasure and honor for me to be able to address this very distinguished audience here and I should explain the title first of all it's a book which I'm supposed to be writing and this is a way to get me started on this book I keep telling my publishers that I'm just hungry way getting through and it'll be nearly finished this is to start in a certain sense because I had to organize my thoughts well I had done that much already but I had to organize my thoughts particularly I should explain the idea is to use Asia pictures as an introduction to certain various areas of mathematics so it would be so that it would be a book for the general public and you would introduce the idea this by looking at s your pictures I found it quite difficult in a way because one of the problems is it's not as though sticking this actually illustrates this this one illustrates that this illustrates the other thing you find that the same picture illustrates about three things all at once and then to try and put this into some kind of organization I found extremely difficult at least is something I can start with assure I think when he was in Italy he did a lot of prints lots of pictures in which he okay there were sometimes a little exaggerated but very accurate pictures of scenery and so on buildings and I think it was when he visited Alhambra in Spain and he realized all these incredible Moorish patterns which he tried to understand what was going on there were different kinds of arrangements and then he learnt from a mathematician I think it was well actually I'm not quite sure it was I did not say but he got this picture which shows the 17 different crystalline crystal groups if you like cemetery groups of the plane where you have discrete motions and there are seven seventeen different ones illustrated by these pictures you see here well asha then developed these pictures in in his own unique way using tessellations of various creatures and so on I should explain you see who did this by in this one you can see on the outside you've got the ordinary squares or something and then he will modify them in terms of creatures this is the sort of thing you have there are symmetries either you have a translation which is just a motion without any kind of rotation you can have glide planes where you translate an oil and you also reflect the same time you can have simple rotations that's the one down here remember all the other ones are yes a reflection that's the other one so all these different motions combined together to give you the various seventeen different groups that you can have and each one he would illustrate in his own unique way he was able to economise a little bit although this one ought to be in color I think that I just have a black and white version so you have to imagine they're all different colors and the two of these different groups are illustrated with the same picture if you take the colors into consideration then you have one of them and if you don't then you have the other one so so you can do two at once that's the sort of the economy that he liked I should just show these things rather quickly because I have an awful lot to show you there's another one and these groups are illustrated if you remember what those numbers and letters were so on then you can see which one it is I'm not going to go through that in detail because I have too many other things to say but I'm just giving you examples of the way and we she Illustrated these different symmetry groups either with these birds and fish have to look carefully to see which they are all we have in this case dogs and you like the way that make too much difference because this involves a glide plane you see so you have to turn it over to get the symmetry okay upside down doesn't make too much differences particularly those ones and here we see another one then the group scissor Sara illustrated that bottom I shall run through this rather quickly because I've got far more transparencies to show than I normally would have in a lecture like this okay thank you quite see what was happening there ever never mind they can get very elaborate for example with this one you see the the animal of the creatures I have to see what they are they're beetles or something of some sort and their arms organs a lot can complicate his way but he was good at doing that and of course you can different groups you have to look carefully to see which group it is is you here have a key at the bottom so you can tell anyway that that gives you a a good way to start because he does illustrate these different symmetry groups and one can go and talk about what groups are on motions of the plane and that sort of thing this one this last picture here has a certain history to it which I should explain I once met Asia I've come back to that story ester how I happen to get in touch with him but I did meet him once and I came with a certain set of puzzle pieces which in the shape of the ones that at the top at the very top I had a bunch of wooden pieces and they were the shape that we have here just and the thing was you have to assemble them set they cover the whole plane it was quite interesting because you have to have 12 different orientations 6 one way up and 6 the other way up and they all have to be incorporated to cover the plane and then you can I had to exceed it and wrote me a letter and said well he didn't quite see whence the principle was and so I showed him this pattern here basically have rhombuses and these little lines mark how you might ever this this one has to match the next one and you have to have a modification of the shapes which so that one agrees with the other and this this is what it was this work shape was designed so that a lot of false tries as well but that's there basically and then I explained how it was done and I suggested some possible animal or I think it was a bird of some kind but rather crude and I said I'm sure you can do a lot better than that and he did produce the picture which I just showed you previously this is actually the only example he had of what you could call an on ice non I so he drew tiling you see it's a tiling of one shape but the different instances of that shape may not be on equal footing with each other so although I think the dark ones and the light ones I'm not sure if it is Illustrated that way but some of them have a different relation to the pattern as a whole so you see two one and another one and if you move that one to the other one the pattern doesn't follow it so you have that's what's called a non Ising he'd rule tiling it's quite interesting historically because in the famous speech that Hilbert when he introduced this his famous problems at the turn of the nineteenth twentieth century he one of the problems he made was if you have a three-dimensional shape is it possible that you have a shape which will tell only non-high sahih drily i'm not sure if that word was used at that time but he that's what he was trying to do I think he thought that plane shapes are completely trivial and so there was nothing interesting you could do with plane shapes but then one of his colleagues for students I think he should I think it was who produced an example which showed that even in the case of plane shapes this conjecture was incorrect and that you did have shapes which would only tile the plane that would tile the plane but only in a way watch which was non I so he drew so that was an example but anyway the picture that usher made was I believe the last was certainly the last watercolor he ever made which partly due to my slowness and responding to his letters but never mind about that anyway you can do other things with similar sorts of shapes let me just give you an example here very similar to some of the ones which did have periodic shapes but this isn't it's not obviously periodic it may be it extends in some such way they're not all the same size I'm not even quite sure what's involved in the arrangement here of course you could do more less symmetrical things still by just having any other animals and things like that there's no symmetry at all and of course many of these examples were things like that what about three dimensions well of course that's a bit hard to illustrate but he certainly did things in three dimensions here's an example of just cubical arrangements but this is what I wanted to show here is a a sketch of this picture that's another one there too so three-dimensional things were things that with interest interest in and here we have another three-dimensional picture this fish you can see that that's again it's good illustrator symmetry this is just translation so it's not so exciting but it's as a picture it's impressive of course the only one I know of which which illustrates a tiling of three dimensions is this one here you have these flat worms or whatever they are swimming around and the tiling that you see in the background they're swimming through it I mean just parts it of course is the tiling of regular tetrahedra and ringgit and octahedra and they fit together to form a three-dimensional lattice tiling a three space so you see that he was interested in three dimensions but harder to draw pictures of bursting so I don't know that he did so many of us who certainly didn't do so many of them you also explored with distorting geometry in interested ways and here you have an example where the building you go up and it twists around and you find a copy of the building up there but the geometry is twist around in some strange way and this is the sketch for it I think so you see examples of that rather like these creatures which he invented because they crawl along like a caterpillar or something and then they roll up and they can when they go downhill they can roll up steps I think they do it is they go out the door you can see them anyway he rather like these creatures and made use of them in various ways but the the the geometry of the space is distorted in interesting ways too so anyway that's other things in three dimensions he liked playing with polyhedra and also combinations with polyhedra such as here you have the two tetrahedra and there's a straightforward dodecahedron I think and that's according he played with I'll give you another example here which is a I have to see what it is cuz I'm not trying notice you can probably tell me qu you can see it better for me you're sitting than I can anyway these are two polyhedra interlocking which here I think it's a micro Sejal dodecahedron into inter locking and preserving the same symmetry as each other and here you have I think three octahedra and lizard creatures and ferrous things involved so he's got other ones dotted around like that so he was quite keen on that idea I'm showing interpenetrating polyhedra and the symmetry this is not the symmetry of the octahedron you have when you finished its you see you can I should have worked out what the symmetry was before I gave this talk but here we have again in re-entry I always like this one because you can see the planets of some sort and the two planets are both tetrahedra and they interlock and they don't seem that no evening anything about each other because one of these are all dinosaurs and things in the others are some kind of civilized community and they interlock in that so I going through this rather quickly because you really have to look at this pitch pictures in details to appreciate all the intricate the tremendous artistry he has in detail which is extraordinary and it needs an examination oh that one seems to be stuck to the bottom here and that one too here we have this is just elated that's the still it's related dodecahedron do these pentagrams and passing through each other and it has the same symmetry it's the icosahedron but it's more interesting in that the faces go through each other and the the poly the polygons also go through each other as well and I think this is illustrating sort of various kinds of rubbish run here and you have this perfect shape in the middle one of the things he liked doing this exploring the contrast between the image of a three-dimensional object and the three-dimensional object and here you see this one looks as though it's a three-dimensional object and then one at the bottom is flat but nevertheless it's the pitch a picture of a three-dimensional object I mean the spheres this one here is I guess a hemisphere or something but the top you see a picture so he's playing around with the picture of the object and the object itself and he does much more elaborate things of that nature striking one is this where you see in the background these - I don't know humanoids or something we seem to be forming this interlocking pattern and then they sort of get a bit fed up and walk out of the picture and make friends with each other so that's one example you has other examples of the same thing and this I've always liked that one you see the tiling here of these lizard creatures or whatever they were and that could continue as a plane tiling but then they decide to come to life and they walk around on top of these oh he drove other objects then after their having had a stroll they get back into the into the two-dimensional picture it's a paradox of course but he loved that kind of thing there we have a much more elaborate version they're the same kind of situation also mirrors involved so you can glide symmetry to you see because it's reflected you have a reflection in the symmetry with their black and white and somehow and they go around and you just have to follow it right it's very very complicated to what's going on because they go through and they're also reflected and you see the reflections merging with the original creatures and here's another case where he illustrates two kinds of paradox at the same time I mean one of them is this contrast between the picture of something and the thing itself here you have a picture of a hand it gets more and more realistic until you think it's a real hand and that's drawing a picture and then of course it does the same thing there but there's also a kind of paradox in that the thing is thing being drawn by this hand and therefore in some sense this is a it's a product of this thing and then this is itself a product of that so you have something which isn't it's a bit like a Russell paradox where something is the set which has itself as a member when you see that kind of thing being illustrated in that picture I I had this opportunity to visit that Cheshire as I mentioned before and when I was there he had this long table I was expecting when I visited him that he would have a very extraordinary house with staircases going out of the window that sort of thing and I was rather disappointed to find it was very neat in the organized house with a lovely picture window the view outside and a very neat and organized place and he had this long table I don't know about as long as yeah and as bad as why does that he was sitting at one end and he had two piles of prints and he said we'll look this size this one I don't have many left but the other side I have quite a few left you could choose one Wow and so I brought the pile over and I started going through it and it was really difficult choice like what some of the favorite ones he had already gone they were in the other pile well a lot of things and it was difficult to make the choice and eventually I picked out this one no he was very pleased when I picked that one because he said people don't usually understand that one so he thought since I seem to understand it I thought at least I thought I did I was very flattered that he that he was pleased with my choice I should explain this a little bit here we have a fish and you see as you can move around here the scales of the fish become a tessellation of other fish and as you go around if I'm going which way around it is and you find that one of these fish become this one here and then it's scales become bigger and bigger and bigger and then one of its scales and put found become the original fish so it's a bit like that the two hands drawing each other they didn't know somewhat more subtle way and so it takes a little while to figure out exactly what's going on here but again you have this sort of Russell paradox idea you see the scales or a subset of the fish but yet the fish scales become a fish and then it's scales are one of the subsets one of the elements at least of this set of little fish if you like becomes the original one source you have a Russell paradox idea involving sets which are members of themselves or members of other sets which are members of themselves it's that sort of thing it's very much like a perhaps better known picture the picture gallery where you see there's a boy the right way around yes I think so the boy over here somewhere who's in a picture galleries you have to cover up the rest of it really to see that he's the picture gallery and he's looking at the picture and then you follow the picture around and you see eventually the boy is in in the picture so he's again it's a sort of Russell paradox kind of thing where the picture itself but it illustrates something else which was actually explored more by a mathematician Lenstra who is a dutch mathematician in Leiden and he and a colleague decided you see in the middle of the picture yes I didn't quite know what did well I don't know if that was it or what he didn't quite know what to do with it he decides at least that was a place to put his signature so he gave up on trying to fill that thing in the middle he sort of filled it up in the middle with the other picture but it's done in a way which you don't quite see there's anything strange about it but did I suppose this is a sort of singularity in the middle but here he gave up on the singularity and decided just to have a hole but what let's join his colleague decided to do was to find a transformation of this picture which would fit inside that hole and then of course there's a hole inside that picture and so the whole picture goes into that as well and he made a very nice film of this where you sort of zoom in on it and you see the pictures inside itself all the time and it gives you a very eerie feeling it's just going on and on and on with a slight rotation time but to do this and I think Asha also used this sort of thing when in it they use a bit of complex analysis here because it's a conformal picture and you use the conformal this is of course where one can talk about the mathematics in developing the idea complex analysis and how the mapping of the complex plane holomorphic mapping to another part of a complex plane you always have these things which are these things that which are conformal so that angles are preserved sizes are not preserved oh I better start that again yes yeah I tend to knock my microphones off by waving my arms around but so this is the sort of mathematics kind of lead into because it relates to the conformal geometry that is relates closely to certainly to the one dimensional complex and now Isis which is complex you have two real parameters to describe your complex number and so that's the normal complex plane and then when you transform with a complex analytic function the or a differentiable complex differentiable function then the mapping is one which is conformal and the Dutch mathematicians don't make great deal of use of this in order to make this picture map into itself you have to do something about making this square outside into a circle to make it fit but they managed to do this very well okay well conformal transformations are also interesting as you made use of them here it's a bit like we tessellations were playing we had before but as you work your way down you see the sizes change and so it really isn't quite a Euclidean motion but with emotion where you squash it down so it's a one of these conformal motions which relate in to complex analysis and that would be very nice area to discuss certainly in more detail one of the more famous of Ashes pictures is this one here and here we see one of his circle limits it's quite curious because it's very I find it very useful to use this picture to illustrate the different cosmologies in cosmology you usually make an assumption that the universe is spatially isotropic and homogeneous and these things the limited number of geometries you can use basically characterized by the curvature the curvature can be positive there can be more complicated topologies but there are basically the three types the positive curve with zero curvature and the negative curvature and Yesha was very ingenious in showing that he could use more or less the same shapes illustrate each of these three if you look carefully you see the number of The Devil's feets cousin coming together as three here was it where it's only two there and there different numbers of things that the the wings and so on and so you can get these without changing the shapes very much you can get these three different geometries Illustrated this geometry illustrates there is things which I often use it in lectures on cosmology as I said before it's administration of the what's called hyperbolic geometry that's seeing Euclidean geometry that's the geometry of the sphere hyperbolic geometry is negative curvature and it's often actually did several of these that she called circle limits he learnt about it from coxeter who was at the same meeting of the International Congress of mathematicians in 1950 I can't remember exactly 53 54 something where I went to this and my second year as a graduate student and this is where I first became acquainted with Ash's work yes sir I didn't at that time have the these examples here that he showed you you didn't have that then but he learned from coxeter who was attending this meeting and coxeter explained to Escher he said what would be interesting if you could illustrate the hyperbolic plane this is the plane that people usually mathematicians usually refer to as the Poincare a disc I always worry about that because it was initially due to Beltrami and I guess people don't seem to remember it was Beltrami she had all these different transformations he had the client representation they poincaré half-plane the park or a disc before and these other people did I'm not quite sure about rebound because riemann already had the half-plane apparently but I don't know whether he realized it was a non Euclidean geometry I'm not sure but anyway this is this was the first one that su made it's actually quite useful to use that one because then you can see how the you see these are the straight lines are either straight lines diameters or else you have arcs and circles which meet the boundary orthogonal E and the act since it's conformed all the angles are correctly represented then you can see how you can have lines which never meet each other even I mean these would be parallels which which actually never meet I don't know if you call these parallels because they're they're parallels and finishes and then triangles you could see the angles don't add up 280 degrees they add up to something less and all that is well understood piece of mathematics but I sure had this very nice representation of it and you can see the way you you line up figures I get it right and something that definitely now I've also used this or let's say this one there's a nice illustration of a kind of geometry where infinity is represented you see the angels and devils actually for that in the West we leave this one as you can see the fish here the black ones are supposed to be our congruent in the geometry so you have to imagine that this fish and that fish they think they're the same the geometry is such that we represent it in this way so they gets squashed up towards the edge but as far as the fish are concerned the what they work once of the black ones regard it is completely homogeneous this geometry but you see also in this example with the eyes of the fish are exact circles and these circles remain exact circles no matter how close you are to the boundary he's done is extraordinarily accurately right down at the edge you can look at look at it with a microscope but you could still feel very very precise right down to the edge now you see this is useful I find it useful in my discussions of cosmology because you can use the same trick for space times this is this is a two dimensional space but in cosmology you talk about space and time together and three-dimensional space and run of time and you can talk about conformal representations just the same way and so I find it very handy here we see a picture this time going up the page and this is meant to be well you can only represent two dimensions of space but it's I've got this wiggly part in the back because I don't want to prejudice the whether or not the space is actually closed or not it might the open or closed I've allowed for it till we go around at the back but to have it look as that starting to be closed as useful because otherwise you can't convey the the the size of the space very easily but anyway there is the Big Bang here the universe expands slows down a bit and then starts to expand again with this exponential expansion which was recently observed and got the Nobel Prize and all that and you see here's the Escher trick you squash that down so it becomes a finite boundary this is a trick which is being useful to use talking about gravitational radiation and certainly I was in the in the 60s and 70s playing with that you can also do the opposite trick which is to expand out the Big Bang here yeah I was trying to find issue representations of this you did things rather like that here we have a picture as you go to the middle and again it's conformal and they go right down to the mid I think you have to cut a hole and stop it for a bit because the Big Bang is a it's really a surface rather than the point so you have to stop it off somewhere but it is very another a sure representation of this idea and here we have another illustration so you could think of the Big Bang is somewhere in the middle there and it's conformally represented but it's sort of opposite trick where you have infinity squash down and there you have the Big Bang stretched out this is a perfectly respectable thing to do in cosmology what's not quite so respectable is this model which is a model which which I've been playing with for a long time I think there's some evidence for it which I won't go into here because that's not the right talk but that's it I just want to show you this picture that you can do something very sure like which is join the the Big Bang stretching out of the Big Bang for the squashing down this is the this is our yawn I should say bit big bang to remote future this is the next day on after hours this is the one before hours and they form this nice continuous stretch with all these eons going right back to infinity in both directions okay that's enough of that I haven't played with the topology of the universe you can certainly have these homogeneous isotropic models and glue them up to the various ways to make topological things they should give them a straight topology in various ways you were just some examples see this is a bit like a mobius strip but that you have to go around four times I think before you get back to you started he did play with mobius strips as well and you can see these ants or whatever they are crawling over the mobius strip and there can be one way or the other way on top of each other nice things okay now let me talk about something else I did talk about tessellations of the plane and unfortunately what I'm going to show you came just too late for Asia he died before I was able to I think I wasn't quick enough off the mark and they say he died too early I wasn't quick enough off the mark these I want to show you this because it has some relation to the tiling that you see as you come in the building here and I think I want to explain how that what the idea is behind that tiling well this is the Italian I came across not long after she had actually died I think he would have appreciated this picture I should explain it's a well-known theorem of crystallography or if you like of the types of motions of the plane which have been talking about that the only symmetries you can have but you get translational symmetries but if you have a translational symmetry that's sliding it along without any rotation together with a rotational symmetry then the only rotational symmetries you can have are twofold threefold fourfold and six-fold in quite easy to prove that but then here you have a pattern which seems to be very uniform it's just indeed very uniform as I've explained but it seems to have a five-fold symmetry you see it's made out of Pentagon's or pit actually four different shapes Pentagon's these little rhombuses you have these Pentacles of five pointed star and just as caps as I call them sort of half stars it's quite interesting to look at even without knowing how it's built if you um set your eye along sort of where I'm standing better than where you are and you can see the things line up every little line continues so with an infinite number of line segments whichever line you choose the density of those line segments all the way along is the same you also have these interesting decagons all the place you see her regular decagons regular Dinkins whenever you see a regular second ring like this one here Isis not what it used to be so I have to trouble finding these things the decagons are always the same three Pentagon's one jester's cap and two rhombuses wherever they are and whenever you find one of these diamonds it's surrounded by a ring of ten Pentagon's so I always rather like those rings I'll come back to them in a minute the pattern is very uniform in fact although it never repeats itself a cut that would violate the theorem it all almost repeats itself that is to say if you give me say any percentage less than 100 percent say in 99.9 percent then I could give you a motion translational motion of this such that the pattern agrees to 99.9 percent and also certainly rotationally by full rotation to 99.9 she said and then you say well maybe what about 99.99% he says yeah you can do that to any percentage less than 100% you can find a motion which will send this person into itself but it's never quite it symmetric and that's how it legal super theorem let me explain how this is constructed here we have I should say it's a sort of hierarchical arrangement and here we have a bigger version of the same thing here we have a this is how I arrange so another bigger version of the same thing so that is the sort of organization it's quite useful to show that and then ask somebody can you find these Pentagon's it's not so easy which is a way of illustrating we are uniform it really is okay now this pattern I think I'm renting here's the second set of hierarchy this one here is a little easier to explain what I'm going to say yeah I can see the patterns more easily to here here you have the the decagon and the rings I always like these rings you see I'll come back to that in a minute but this is made out of four different shapes where the Pentagon's actually well that's not going to then there's another way of doing the same sort of thing these are kites and darts and the relationship between the kites and us and the original pattern is this and you if you look carefully in the first done it right you see each kite it's the same set of lines in it and each as each other card and each dart has the same set of lines in this picture and if you match the lines then you get this arrangement here in fact the only way of fitting the lines together so that you match them is to make that pattern so these kites and darts marked with the lines that you see here can only be assembled in a non-periodic way and so this example was the first known example or you have two shapes which will only tile the plane in a non periodic way you can also assure eyes this and this is the sort of thing spectrum as she wasn't able to see this I'm sure he we have done wonderful things I shall try and find the right spot here which is difficult that's not bad yeah I don't know how well I done it but you can see there this these are kites and darts but modified so she looked like birds so this is what I mean by sure ization I this is my poor attempt trying to do something that s you might have done and she lived long enough but other people have done other things like this too so it's interesting that you can do something which I'm sure actually the rebel in if you never had the chance right let me go back do the kites and darts and so on what are the initial this one here there's another way of doing it which is these rhombuses and again they are related to the kites and darts with lines drawn on the rhombuses and if you match the lines then you get the original pattern back again ah and you can do it directly between the kites and darts rhombuses choice you can mark the mark one of them they get to the other one and so on I want to try and describe the tiling outside the front of this building because those are kites and darts but what are the markings on those tiles well that came from the following and I've been don't get these things upside-down when I pull them to the side here we have the original Pentagon pattern and as I say I rather like these rings of Pentagon's so it's nice to mark them there we have these green things go around the through those rings of Pentagon's now that doesn't quite fill up the regions of being Pentagon's so let's do a little more to that I'm going to add some more green lines and the virtue of what I'm doing this is that now I have a pattern which is really the same as that pattern but it's a distortion of it so you see here we have the Pentagon's you've got three different versions of the Pentagon's they're very much distorted but three different Pentagon's curvilinear Pentagon's now and you see the justice caps here there's one nearly pointing this way yes justice Cap'n this is just that this justice cap and so on and so that is the same pattern but somewhat distorted now can you get these onto a rhombus tiling well you can about the trouble is the rhombuses aren't all quite the same so you need to add a few more lines having added a few more lines you know I'll see that every fat rhombus has the same markings on it every thin rhombus has the same markings on it and that is the tiling that we have outside the front so if you want to know what it is that's what it is it's it's the rhombus tiling all these mean the markings there I wondered for quite a long time whether they you have places where these rings close up and there are four different ones where they close up and you can see all four of them did with different patterns on outside but there was an but are there any of us well the others would have to be ones which actually cross each other cross them and are there any which close up and cross each other I don't think there are in the pattern there and I sort of conjectured that none of them did I realized the edge of that complete conjecture is completely wrong I did find one which Wiggles all over the place much too big for the pattern outside the building here Wiggles all over the place and finally gets back to itself so that conjecture was wrong to have a new conjecture now is that that almost all of them do close so you can certainly have passes where where one or two of them don't close but if it's more than two I think probably real close that's a conjecture I don't know if it's true or not okay that's that and this in fact is the pattern that I gave to people so they would get it right now I want to give you something else I've been playing around these these things quite a bit and there are other things you can do too here is a set of three shapes not just two and they're quite simple shapes square irregular hexagons and irregular dodecagon and these markings on them are the things you have to match and those interested to know if when you match them what you do you get some feeling for the pattern as a whole I tried this with something before and I got no feeling whatsoever but Joshua is sock Allah I gave a talk where I mentioned this at the end Alcala and a computer program which was related to this and you can see there is a person over there I'll show you the next picture which is the next stage of the hierarchy and I was really quite surprised how it is really a very attractive pattern it's one of these things which never actually quite repeats it's not quite right in that those little Wiggly a tiny rings or to be exact circles but that's the only thing that's not quite right about it they should be a little tiny one should be exact circles they actually look poly polygon more anyway somebody perhaps could make a building then they would have these three tiles and there they leave a lot of them to see that pattern mask okay anyway that's enough of this now when I told you that I did go and see the an Escher exhibition when I was in Amsterdam at the International Congress of mathematicians and it was in Bangkok Museum one of my lecturers Sean Wiley who taught me algebraic topology and he had a picture of the on the catalogue which had the Asha picture night and day I'm haven't shown you what that one with birds flying off in different directions and spaces between the birds became the birds going the other way one of them it's a night scene on that day scene on the other side and so I went into this exhibition and I may remember remember being particularly struck by this picture called relativity all these different staircases and creatures going up staircases with gravity being in different directions and I was really very struck by that and of course many of the other things he had there so I went away trying to think I would try and do something impossible myself not with the skill that this you had of course but at least with an idea that I hadn't actually seen and I produced things with bridges and rivers and that going off in different directions and I whittled it down to something which I thought illustrated the impossibility in its simplest sort of reduced to its simplest form and I showed this picture to my father and then he started producing staircases buildings and all that sort of thing and we wrote a paper where we showed some of these pictures we didn't know what the subject was but my father happened to know the editor of the British Journal psychology so we decided it was psychology probably they could get it published because he was good friends and indeed that's what happened so it was published and we then set the copies to Asha giving reference to his catalog and saying appreciation of his work and in the meantime Asha had actually produced something of a similar nature which is this picture Belvedere II where you see the top part of the building is joined up in the possible way to the bottom half he after seeing the paper we has in the British Journal of Psychology Asha then produced his own version of the staircase which is a very famous picture now the monks going around in both directions very beautifully done and actually was very generous and giving credit to me through our paper I this was the occasion night when I said I'm s at Mehta my father and Asha had a quite a correspondence backwards and forwards and I had Ash's telephone number I was travelling in in the Netherlands I think in I put apeldoorn near to Bonn where she lived and I phoned him up and he said yes come along and he said come along for tea which is what happened so that's how I got to know him but the triangle which I want to show you I'm going to say something about the mathematics that this is involved with but before getting to that let me show you I was a psychology conference and I thought I've try and go one step better than some of these pictures and produce an invisible impossible object so this invisible impossible triangle this is also an invisible possible triangle which one is better you can put the two together and maybe that works best I don't know anyway you can probably see the invisible triangle it's not really there but it's still impossible even if it's not there I was at a there was a film being made for some reason about twisters and things like that and the people give us a shirt I think it was on the BBC 110 a long long time ago and they were stuck to me making this film and talking about twisters and at one point and they've finally asked me what's it good for you see so I said well one thing you can use twisters for solving Maxwell's equations and then I said well well you had this you have to use a certain idea that I can't really explain it it's Co homology I can't really explain that to you but it's they've got very disappointed about that so then I went home and I thought gosh I can Kamali's yeah of course on they can explain it here we are you see this is a nice way of producing that you see Omak imagine that you have some pieces which you have to assemble to make that object and these pieces can consist of a lot of corners like that maybe they made out of wood you see and you make these out of wood and you have instructions about how to blew them together that's supposed to glue to that one and this is to glue to this one and that glues to that one and you have to look at these instructions and see whether the instructions allow you to build the object or you're going to have a little problem with it and then I rise that's actually is an element of comala G if the caramel is a element vanishes you just see putting you see that these instructions give you a check representation of an object and this object you're using the freedom you have of the distance from the eye and so that's what enables you to struck the kamalji non-trivial color multi element and if the instructions have a non-vanishing code kamalji element then that's something you can't build comala element does vanish than you you can build it so I thought this was a nice way into that notion of comala gee I see it's it's the measure of it's a check representation of kamalji I was talking about this to in at a conference in I think it was in Rome it was on that mathematics and arts and I can't remember something like that and I was talking to an American mathematician unfortunate forgotten who it was and I was explaining that my talk the next day was going to be about what comb ology and how this represents that and he said well is there any other group you could represent so well I hadn't really thought about that this is the sort of you could take the distance from the eye and you take the logarithm of it and that gives you a an additive group so you can then talk about the curl mangy in the ordinary check way that has to be explained in more detail in the book of course but then he said in any other grips I don't well yeah there is you know I could think of z2 and so in fact I drew this picture out and I gave it to my talk the next day I think it's a little bit easier if I don't show you that one it's something which I developed a bit later it's the same idea and what is this you see well this is an idea that Asscher use and let me show you this picture here it's the only one I know of his which I'm not sure it's quite Kamala G I haven't quite figured it out yet but you see on one side of the picture I mean actually use this idea for other things where you have this sort of Necker cube paradox the thing could be one way around or the other you can see a cube edge on and you may see the back of it as well you don't know quite with it okay you see the Kuban don't know they're looking inside or outside and you you see this you get it one way around and then that tells you the rest of the picture which way around it is so Asha has us here and this is clearly one way around and this is clearly the other way around but in the middle it's ambiguous so the whole thing is impossible because you get the ambiguity in the middle which if you make one choice which is this choice that contradicts that and the other choice contradicts this but the thing I had here I think a she would have done wonderful things with this too but you see the staircase you could imagine walking around the staircase and well it's impossible I did show it to her I did show it to an expert on impossible objects who was at the conference I showed in this picture and he said can you see what's wrong with this it's nothing on that picture could you make that out of wood you think yeah sure as I could make this add wood are you really sure you could you have to work because if trouble is your eyes you go around it it flips to the other one without your noticing that's what's happening so you certainly couldn't make that out of a piece of wood anyway so that's that can you make these impossible objects well that's a good question you could certainly make the pieces one thing you can do is this in fact this is a model of something which I believe somewhere Wolfson College it's still there I'm not quite sure where there was a bigger version of this I can't show it to you all at once everywhere because you see this is the impossibility it's not a Czech version this is more like a like a continuous version suppose you can have and that's one way of doing it but let me show you something else ah no that's the picture we just had isn't it this we've done that yes Wow look it still works that was lined up just right so you have it's a possible impossible triangle and of course if you want to see what it really is you give us a little twist well first of all I think the best things but this we had there we are oh we give us a little twist and then you can see what it is okay well thank you very much [Applause] thank you very much for this wonderful lecture and now for the second part of the events I like to please had me welcome a Nick with us who is going to take and present the awards okay so in a moment I'm going to call on Lavinia clay to present the clay award for the dissemination of mathematical knowledge to Roger Penrose so dissemination of mathematical knowledge is a very important part of the clay Institute's mission alongside supporting mathematical research at the very highest level and of course giving very large prizes the this award recognizes the achievements of people who have made exceptional contributions under both headings Rogers case the award is in recognition of his outstanding contributions to geometry relativity and other branches of mathematics and of his tireless work in explaining mathematical ideas to the public through popular books public lectures and broadcasts but of course Allen the beginning gave a long list of his contributions within relativity within geometry and that his tics vary the first box very very clearly but he is also as you've seen a brilliant expositor of mathematical ideas and it's the combination of these two which leads to the award so we've seen the sum of these contributions to geometry and the Penrose tiling but there's another aspect of his work which is really very important in which bridges the truth the - which is the passing from one area to another so he begins by thinking about something impossible so you start thinking about the idea of tiling the plane with Pentagon's well most people might have occurred to the learner to think whether that's possible and they decide very rapidly that it's not possible well Roger thinks about that and what does it lead to it leads to the Penrose tiling and that has an impact a huge impact outside the area in which he thought about it and in that case an impact in the visual arts and and beyond he starts thinking about impossible objects where does that lead to leads to a paper in the British Journal of Psychology which is there are other examples of this I'll mention briefly the more Penrose inverse well again you you learn at school about taking the inverse of a square matrix well what about a matrix with isn't square well you give that a few seconds thought and that's impossible so you go on what if you're Roger you think about it some more and that leads to this generalized inverse and again that has a huge impact in fact his paper on the generalized inverse is his most cited paper it's a hugely useful tool in numerical analysis that's had a big impact on computation and other things where he's crossed the boundary perhaps is it was Roger who this one's for the computer scientist who first made the suggestion that the church is lambda calculus might be a useful tool in thinking about programming languages and that again had a huge impact but he's best known outside the scientific community for his popular books which are wide-ranging highly original highly controversial in the very best sense of being controversial in that they provoke debate and get people talking so they have a very unusual feature for popular books in science in that they're addressed not only to the general public in explaining mathematical ideas but also in engaging the public in the development ideas of ideas he's putting forward his own scientific ideas original ideas in these books in a way that many other people do not his books have been translated into many different languages they've sold in the hundreds of thousands and they are a remarkable achievement well you may have noticed that Roger doesn't use the most up-to-date of AD technology it's public lectures but notwithstanding that he does have a very significant online presence if you look on YouTube you'll find some 30 expository visit videos recording lectures and and interviews many of them have been viewed tens of thousands of times or in one case over 300,000 times so that's a huge impact there so I think it's clear that he very amply fulfills the criteria for our award both in the contributions that he's made in the development of mathematical ideas and the incredible work he's done in the dissemination of mathematics so I'll now call on Lavinia clay to present the award to him [Applause] you

Channel: Oxford Mathematics

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Keywords: Oxford Mathematcs Public Lectures, Roger Penrose, Escher and Penrose, Maths Lecture, Math Lecture, Maths and Art

Id: VlkH9v4dP7w

Length: 70min 9sec (4209 seconds)

Published: Mon Oct 01 2018