How Mathematicians Think About Patterns – Professor Ian Stewart

yeah I want to talk about how mathematicians think about patterns well I want to talk about how I think about patterns and how quite a lot of mathematicians think about patterns this isn’t the only way to do it but I want to bring out two features of patterns which are particularly important which is the role of symmetry and the role of what’s called symmetry breaking which is clearly related and I want to show you how patterns can be recognized classified but also understood to some extent in terms of those two basic mathematical ideas and I’m going to go through four basic types of pattern snowflakes animal markings now I gather there was an earlier talk in this little series about zebra stripes I don’t know exactly what was done in that but there will be a little bit of overlap and I hope I won’t contradict what support it said I hope it will be a complimentary view there are many ways to think about these things and in the Christmas lectures we actually brought one of these animals into the lecture room you can find it on the Royal Institution website you can actually download and watch the lectures and I think was lecture number five so we brought a six-month-old tigress into the lecture room so people on the front rows we should imagine a tigress being brought in on the end of a chain with two burly young men very well behaved tigress but it was basically I felt that was the pinnacle of my lecturing career I’ve never been able to match that I would have talked about sand dunes and I want to talk about animal movement okay so let’s start with snowflakes so here’s a couple of modern pictures and one rather older picture of the remarkable structure of some not all but a lot of snowflakes and the detailed structure is very very different but you can see that there is a remarkable six-fold symmetry in the snowflake I was watching television program the other day and there was somebody with an experiment I said they don’t do that what they didn’t do that in his experiment but I think that’s because it was so carefully controlled that actually I think you need a little bit of random noise to smooth things out a bit to get this kind of symmetry I’m not quite sure why his experiments didn’t work but a lot of other people’s experiments do work so there’s still some mysteries about snowflakes in fact and I just want to emphasize that the six-fold symmetry in this perfect form is not the only thing that can happen here is another form of ice crystal and you can sort of vaguely believe this is some fancy relative of the six-fold symmetric snowflake but it’s clearly not as regular those are very interesting questions you could ask about this picture but let’s just go to this this is a real snowflake chosen I’m sure to be pretty close to perfect if you look you can just see little bits where it’s slightly melted you can you can see if you look closely that this is not faked up this is genuine and what puzzles everyone is this mixture of the symmetries of the thing and I’m going to explain those now but also the remarkable variety within this range of symmetric structures so the way mathematicians think about symmetry since about 1830 is in terms of transformations in terms of ways of moving an object so that when you’ve moved it if somebody looked away while you were doing it to look back again they wouldn’t realize you’ve done anything so if I take this snowflake and I rotate it through sixty degrees you really wouldn’t see any difference it would look exactly the same shape in exactly the same place and if I rotate it through 120 degrees 180 degrees multiples of 60 degrees the same thing works in addition to these rotational

symmetries the snowflake has six reflectional symmetry if I put a mirror in the right place and flip it over it looks exactly the same and so there’s three red lines showing three mirrors which in this picture go between the big lumps of the snowflake and then there are three more green lines which run down the center of the protrusions of the snowflake and that gives us six more so this snowflake has six symmetries that are rotations another six that are reflections twelve in all and what really matters mathematically is not just how many but the way these different symmetries relate to each other what happens when you do several or thought of them in turn what do you get and of course one of the features that is automatic from this idea of symmetry is a transformation that leaves things unchanged is if you do two of them in a row you leave it unchanged and then you leave it unchanged again so what does it look like it looks exactly the same as it did to begin with you’ve moved it twice but that must be equivalent to moving it once because the twelve I’ve shown you are the only ways to move a snowflake around by rigid motions and make it fit in the same position so mathematically you can take two of these transformations and combine them together to get another one so it’s like a little multiplication table if you wish this transformation followed by this other makes this third transformation this is called a group and group theory is now a major major part of mathematics and how we analyze symmetries but I don’t want to teach you in a group theory I just want to alert you to its existence and just to to make this point that this snowflake really is very symmetric the top left-hand picture is that picture of the snowflake – Runk down the next five to the right I’ve rotated it through 60 120 180 240 and 300 degrees actually did this on the computer I just took the picture and twisted it I did it clockwise which is the opposite of the mathematicians usual convention but no matter and then the pictures on the bottom are the same thing but I reflected it first so we have 12 these are the 12 images of that snowflake under its 12 symmetries and you have to look very hard indeed at these pictures to see any difference was if I did it 30 degrees you say that’s pointing in the wrong direction okay so so the puzzles are where where does this symmetry come from what’s the physical basis for this and then after that the other puzzle okay that’s this particular snowflake but there’s all these other fancy looking snowflakes lots of different structures but they have this same group of symmetries way back good many hundred years ago Johannes Kepler was thinking about this stuff and he wrote a little book called on the six cornered snowflake de nieve a six angular I think it is in Latin it was a Christmas present to his sponsor who was paying his salary basically and in it he just did thought experiments about this puzzle of the snowflake and it’s six-fold symmetry and he worked his way through and he thought about packing balls together and you know if you put a lot of coins on the table and push them together you get a honeycomb pattern and he kind of worked his way towards what we would now recognize as the idea that ice is a snow crisps snow is an ice crystal get it right way around and it is something in the atomic structure of the ice crystal that is governing this six-fold symmetry and this was back in the time when atoms of something the ancient Greeks are thought about and and mostly dismissed again and it was all kind of forgotten and and it wasn’t until about 1900 that physicists actually started to believe in atoms it’s remarkable how recently that happened so Kepler worked his way towards the beginnings of the way we would now tell the story which is snowflakes are formed from ice ices of crystal form from water this is a

picture of the particular form of the ice crystal that is occurring in normal ice at normal temperatures and pressures and you can see the two different kinds of atoms the hydrogen and the oxygen hydrogens are black oxygens are white and there is a sort of honeycomb structure to this it’s in three dimensions it’s not completely flat but it’s within this structure of the ice crystal as it grows that there is among other things there is this group of 12 symmetries the hexagonal symmetry now when you actually ask under what conditions the different forms of snowflake occur this is experimental work and the two important variables are the temperature which is in degree centigrade along the bottom and the super saturation the amount of moisture up in the cloud where the snowflake forms which goes at the left hand side and depending on the combination of those two variables we get lots of different shapes and at the top in the middle and the plates you see these very flowery plant like things called dendrites you see sectored plates which are like hexagons with more hexagons stuck on which is what we were looking at just now below that you see just thin hexagonal plates and further down it grows a lot more in the third dimension and you can look at this picture and see all of these different kinds of shapes and some fancy ones that are not particularly hexagonal at all and if you start playing with other variables you can get more structure but the source of the variety of a crystal grown as a uniform temperature and a uniform pressure you could sort of predict what roughly what kind of shape it will be and it’s classified by this why then do we see such variety well let’s just take a closer look at dendrites growing so here is a little piece of a snowflake growing this tree like structure and in fact what you see is that as the side branches get bigger they put out their own side branches very like a tree as the tree puts out branches the branches put out smaller branches you get down to twigs the snowflake does this as well and it does it because of what physicists were called an instability if you have a flat edge to the crystal but there’s a little bump on it which is going to happen just because a little bits of a few bits of ice are going to stick to it under those conditions of temperature and super saturation that caused this that bump grows faster than the flat bits so once you’ve got this little bump it gets bigger so this instability of the flat edge amplifies so the conditions of temperature and pressure where the snowflake is affect the immediate pattern with which extra bits of ice get added on yeah well now imagine a little snowflake going around in a big storm cloud it’s tiny little snowflake ice is forming on it at any given moment at all of the points in this honeycomb six-fold symmetric pattern at all of those points the conditions are pretty much the same at a given instant so you go to a little diagram of what shape crystal you should get and it does whatever it should do at that instant so it does the same thing at each of the six corners but a split second later it’s whirled around in the cloud and the conditions are different so now you kind of reset and say what do I do now oh I grow this structure at all six corners so because the snowflake is small so on a very small scale the conditions at the corners are pretty much the same but on a larger time scale the conditions at those six corners keep changing you could actually kind of draw a picture of how the snowflakes temperature and super saturation move around that diagram and the rules for growing keep changing as it moves now there’s an awful lot of ways to draw Wiggly curves through that diagram so there’s a lot of different detailed shapes for your snowflake but the six

fold symmetry is actually preserved at all stages and so the thing grows into this perfectly symmetric shape barring accidents barring this is all you know this is not absolutely literally perfect all the time but when you get one of these symmetric snowflakes that’s pretty much what’s happened so the symmetries of the crystal lattice and the way snowflakes grow as ice crystals combine together in the cloud to give the variety but keep the symmetry okay let’s move on to my favorite symmetric beast I’m a cat fan the tiger Tigers are stripy and the stripes are actually quite elaborate and they are not perfect mathematical stripes if you look at the ceiling of this building you see perfect mathematical stripes in the in the beams on the roof they’re just absolutely like this they’re identical Tiger isn’t quite like that but then the tiger is not just a nice flat mathematical plane but if you look at the the rear end of the animal the this bit technical called a tail and you actually see are very nice rings around the tail fairly evenly spaced they get they shrink a little bit towards the end and then it’s got the black tip but that’s that’s the sort of most mathematical part of a tiger I think the the tail so can we understand where those patterns are coming from I want to convince you that it has a relation to two symmetry now Tigers are not the only cats with patterns there’s a leopard very nice leopard doing typical leopard things sits in a tree dangles it legs down a lot of the day it spends like that cheetah runs around a lot that is a Jaguar South American and this is a clouded leopard such a beautiful pattern that it is an endangered species much more in danger than most of them just because some people think this is such a pretty pattern in the fur that they want basically to make something out of it and wear it and of course Tigers are not the only stripy animals around very much not fish also some fish have stripes some fish have circular stripes some fish have sports when we did the Christmas lectures in Japan in the summer we couldn’t get a tiger so we got a cat and this was a cat with a circular stripe on its side a bit like this fish but it wasn’t blue a beautiful animal well now the our understanding of this mathematically oh yeah and you can have all sorts of interesting combinations of different spots and all sorts of things okay this gentleman who’s centenary we celebrated last year Alan Turing now here’s the person who founded the mathematical theory of animal markings and it’s kind of gone in and out of favor with biologists ever since and everytime they convince themselves and the mathematicians that it doesn’t really work or the latest incarnation doesn’t really work a few years later some other bunch of mathematicians or biologists or fairly recently people in the dental school come along and say yes it does there are you know so there are places where it works there are places where it does don’t take it too literally and it doesn’t completely match the biology think of it slightly metaphorically as a type of model and it works rather well so cheering was actually interested in I’m sorry in this animal initially the cow which has these big blotches on the side they are so iconic that you see them driving around on the side of lorries all over the world we were walking past the local grocery store and there was a there’s always a one of these lorries parked outside but last week it was a different company and it didn’t have the black blotches on the white background and and we’re actually quite disturbed by this suddenly the

world had changed no our our ten-wheeler cow had disappeared so Alan Turing in his papers were discovered these plots they’re little bit of a mystery but they are his they are clearly his mathematical model which I’m going to talk about of how these blotches on a cow or that kind of pattern forms and we’re getting some nice blotches it’s based on computer output because if you look very closely at the little dots they’re actually numbers in I believe hexadecimal notation which is what computers were using he’s clearly copied out all the numbers by hand and drawn contour lines through places where the numbers are all the same the sort of way you would get on a weather map you have these lines where the pressure is all all the same imagine doing that by hand on a grid which just has the numbers for the pressure well this is what he’s done instead of pressure it’s the concentration of some chemical but he did also write and publish a research paper on this and he was interested in stripes spots blotches on cows it was Russian also interested in this shape of creatures like the Hydra which is a little microscopic creature with a lot of tentacles and he worked his way to a theory that what must happen and this was kind of before modern genetics and that’s one of the reasons the biologists stopped being happy with it because it was on a level where that the influence of the genes was not explicit but the idea was think of the animal when it’s still an embryo fairly simple the basis of the eventual pattern of markings is laid down chemically in the cells while it’s still in an embryonic stage you get what’s called a pre pattern you can’t see it but if you could detect the relevant chemical you will be able to and he said I’ll call this chemical the morphogen this is the typical kind of word scientists use for I know what it does but they don’t know what it is morphogen form generator yep or shape generator or in this case marking generator it’s a thing that generates the pattern and the idea is the pattern itself comes about as a result of two different processes chemicals reacting with each other which changes the concentration of those chemicals and diffusion which spreads the chemicals slowly across the embryo so here a lot of chemical at one place um wait it will slowly spread and so these are called reaction diffusion equations so you have complicated local chemical reactions which are nonlinear in the jargon they obey quite complex mathematical equations and then global diffusion across the whole thing and the diffusion kind of organizes what the local nonlinear stuff is doing and the result of this is you get patterns and this was cheering’s big inside you get patterns and these are experiments at the top and computer-generated pictures at the bottom of these reaction diffusion equations and we see spots and stripes and more complex looking spots and actually very complicated rather irregular structures as well and just to drive that point home here’s a fish called a box fish with one of these rather it’s irregular but the spacing between those yellow stripes is pretty much the same everywhere there is a regular spacing to it but the details are irregular and the computed tearing pattern which is qualitatively similar I’m not suggesting these match in detail but they’re the same kind of thing okay now cheering explicitly in his paper explains this in terms of what he called the breakdown of symmetry this is my symmetry breaking that I was talking about and he says basically those are his words but let me summarize it there’s a problem there’s what looks like quite a difficult problem the embryo at the time the patterns are formed down before the creek the animal grows is essentially spherical it’s

spherically symmetric simplify at least if it is like a ball if I rotate it you can’t really tell in which direction it’s pointing ok now biologically there are actually some markers for front back left right and top bottom already at that stage but nonetheless look at it it just looked around and it’s obeying laws of physics and chemistry everywhere to produce the eventual animal and if it was like a snowflake which is six-fold symmetric and therefore does the same thing at all six corners a sphere should do the same thing everywhere hasn’t got corners every point on the sphere can be moved to any other point by a symmetry shouldn’t it just grow the same pattern everywhere in which case your horse that eventually emerges ought to be a sphere but it can’t result in horse horse is not spherically symmetric he has an answer to this objection and okay people wouldn’t necessarily phrase this in terms of embryo turning into horse but there is a quite a period in the history of science where people didn’t quite understand this point and the idea was if you have a symmetric system its behavior will be equally symmetric ensuring says no this is not true this doesn’t happen and it doesn’t happen because the symmetric state can become unstable very tiny perturbations little bumps on the sphere in the case of the horse changes in the chemistry at one point or some region can destroy the entire symmetry of the object because those small changes can grow instead of damping down and everything staying symmetric they can grow they can take over the structure and the whole thing changes and it loses some of the symmetry that you thought was there and he explains this terrine obviously had a good sense of humor I mean this is in a journal paper example if a rod is hanging from point a little above its center of gravity so you’ve got something like a ruler which is pivoted just above the middle and it swings it will hang vertically downwards in stable equilibrium now when it does this it’s essentially symmetric from left to right for example it’s hanging in the middle now says imagine a mouse climbing up the rod if the mouse keeps climbing out the rod and goes past the middle and up towards the top at some point the whole thing’s top-heavy what does it do it flips right over and so this equilibrium becomes unstable instead of just hanging steadily it flips over and it starts swinging from side to side when it swings from side to side in fact it’s the time symmetry it’s in the same place at all times that gets lost it’s not in the same place at all times it’s moving but it’s moving periodically it’s repeating the same motion over and over again assuming there’s no air resistance or friction so after a certain period of time it does the same thing does it again does it again does it again now if you actually draw a picture of the position of the pendulum with this mouse hang on for dear life swinging from side to side you draw it against time you get some sort of curve which repeats the same structure over and over and over and over again it’s like stripes in time yeah so we’re seeing the pattern but in this case we’re seeing it in time so if I only realized Turing had written this after I’d prepared my version of much the same thing so instead of his rod I have a rod as well but it’s pivoted at the bottom and held by a spring and the spring is such that it likes to sit vertically and if I move it to one side or the other I stretch or compress the spring and the forces that the spring exerts to push it back again are the same on both sides so although I’ve only drawn the spring on one side this whole thing as a system is left-right symmetric and when it’s vertically upright it’s in a left-right symmetric state if I flip the whole thing over the rod still appears to be

in the same place okay now I don’t use a mouse if I’d known Turing had written about a mouse I might have put a mouse in but it’s more dramatic if you take an elephant and put him on top of the rod and you don’t have to be an engineer to look at this and say that’s not going to work if that’s a really heavy elephant strong rod but the spring is not going to be able to support the elephant so it’s going to do something like that well now the whole thing has moved off to one side and the state of that rod is no longer left-right symmetric if I flip that picture the rod goes over to the other side yeah so although the mathematical equations the whole system has this left-right symmetry the state that you observe does not why not because that vertical upright state although mathematically possible is not stable slightest breath of wind and it will start to move and then the weight of the elephant takes over and down it goes until eventually you compress this amazingly powerful spring that will resist the downward force created by the elephant so where’s the symmetry gone there’s another solution that looks like that yep so you can have a symmetric system with a symmetric state but if that loses stability the system goes to an asymmetric state but there are traces of the symmetry still remaining there is a symmetrically related state that’s still there now let’s look at the same idea but with a richer range of symmetries symmetries of the whole plane again you can look around this room and see some examples I talked about stripes in the roof you can’t people online won’t bear to see these stripes but imagine a whole pile of stripes okay so you have the symmetries of that sort of striped pattern are if you move it one stripe along it still fits movie 2 stripes along it still fits but move the stripes into the gap between other stripes and you’ll notice it’s changed so it has symmetries but they’re not slide it to any position they are move it very specific distances at right angles to it stripes or slides as long the stripes you won’t see any change so symmetries of the plane are a much richer source of patterns and there are basically four kinds three major ones and a another one I might as well mention so take some shape in the plane here it is a it happens to be a schoolteacher dragging a donkey for reasons that have nothing to do with this talk it’s a shape and you can rotate that shape you can reflect it in some mirror not necessarily left-to-right but it can be or you can also do a thing called a glide reflection which pushes it along and flips it over okay now these are the basis of symmetries of patterns in the plane so let’s go back to Turing and the tiger and animal markings Jim Murray a good many years ago put Turing’s equations onto a domain that he shaped a bit like the skin of an animal now it’s not entirely the case that when the embryo grows that’s not exactly the right shape of domain but there are a number of reasons in the way the patterns form that actually kind of unwrapping the animal into a a blog with four bits sticking out and a tail end is is not such a bad thing to do and what he discovered was you can get interesting patterns forming and you can also get these there’s one at the bottom that has one stripe it’s completely black there’s one just above it that has well it half the stripe if you wish one of it has a black at one end and white at the other end I complete single unit of stripes there’s one that has black at two ends and white in the middle and you can actually find sheep for example which and goats which have that sort of pattern and then as you change the mathematical numbers you get all sorts of different patterns and he also looked at thee as I remarked the the rather symmetric bit that’s on the back end of big cats and he found you can get stripy patterns is a tapering

cone so you solved earrings equations on the tapering cone and you find stripes and you find spots and you find a wonderful set of patterns which are stripy at the thin end and spotty at the thick end and he says the mathematical reason is quite straightforward if you’ve got stripes going round they will be stable until there’s enough room for them to do something different and the main thing they can do that’s different is stripe breaks up into a series of equally spaced spots so as the amount of room increases you can get stripes breaking up by symmetry breaking into spots so he proposed a theorem here here is a computer calculation showing this kind of thing but with a perfect cylinder you get stripes but it is possible for them to break up into a pattern of spots it’s actually a hexagonal pattern that’s very interesting about it if you the successive stripes break in alternate places one breaks like this the next one it’s down here then it’s back like that that’s back like and if it’s tapered you can see similar things occurring so Jim Murray has a theorem here’s the theorem spotted animal can have a striped tail but a striped animal cannot have a spotted tail okay proof in order to get stripes breaking up into spots you have to have a larger region of the domain on the whole the body of the cat is a lot bigger than the tail and indeed if you have stripes and spots mixed up at the rear end of the tail where it’s nice and clean you will see the stripes and then they’ll turn into spots and to demonstrate this theorem here we have a spotted cat with a spotted tail there we have a striped cat with a striped tail and here we have a spotted cat with a striped tail cheetah and I don’t have a picture to show you of a striped cap with a spotted tail okay so yep so Jim is right okay let me move on my wall let’s skip over a little bit of stuff here yeah so I want to tell you about sand dunes where all of this and the relation to symmetry is it’s in some ways easier to understand because we understand what the sandy looks like a pattern of chemical concentrations is something that we’re not quite so familiar with so what I want to show you is sand dunes actually have very similar patterns to animal markings and the reason is they got basically the same symmetries their patterns in the plane one of the commonest patterns for sand dunes are called linear dunes they’re basically stripes the stripes across the desert but instead of colors it’s whether the sand is high or low they’re formed by strong winds that blow in a constant direction and these are an algeria photograph from the satellite here’s another example of essentially the same kind of thing that’s whereabouts in Algeria you find it this is the biggest set of linear sand dunes or linear dunes they’re not sand in the solar system and it’s on Titan its photograph from the Cassini satellite and there was just this vast set of linear dunes on Titan so these things don’t just occur on the earth now there’s also these wonderful crescent-shaped or buckin dunes these are in Egypt those are on Mars here’s a more complex field of these dunes and in the Namib Desert and the way they form is you get essentially get a pile of sand builds up as the wind blows and the wind blows over the top and it forms a big vortex that sweeps out the bit in the middle of the Crescent the region in if you’ve got a crescent the region in the middle is where this vortex is sweeping stuff out

and funneling it out to the edges so the wind actually blows from the if the present is shaped this way the winds blowing that way so the arms of the Crescent are kind of being pushed along and in fact the entire dune moves so as the wind blows over the top this is the standard way or sand dune patterns form sand gets moved around the shape of the dune affects what happens and it also affects how the wind blows it affects the aerodynamics so there’s this interaction between the wind and the sand and even if it’s a steady wind once the sands change shape from flat the effects of the wind are different in different places so sandy roads from the wind would face gets pushed over the top dumped on the other side the Leeside and the sand moves the dew moves and people who study this kind of thing did you they classify dunes into six main patterns there are others but these are the main ones so there’s the Vulcan or Crescent dunes linear longitudinal basically they point in the same direction as the average wind speed transverse Tunes which are like stripes as well but they are right angles to the wind speed parabolic dunes which form usually at the edge of where the desert meets water and they kind of pinned in place by the vegetation black annoyed dunes which all wobbly and star dunes which are dotted all over rather randomly so we can model this mathematically by saying well we’ll take a model desert with flats and wind blows at constant speed picks direction and the desert is infinite the symmetries are all rigid motions but we’ve got a wind so it’s rigid motions that preserve that wind direction so I can slide the desert but not rotate it I can reflect it but only in a mirror in the the points the same way as the wind and then I can look at the group of symmetries and say if symmetries break then that means some of the symmetries no longer apply to the state of the desert but some smaller set of them may remember I was talking about the mouse on the pendulum that’s saying if you draw a picture if that pendulum was always in the same place the picture of where it was it’d just be a horizontal straight line and you could move it to any position through any distance any time and it would look exactly the same but if it’s oscillating periodically I can move it by one period two periods three periods – one period go the other way so a continuous set of symmetries in some direction when it breaks what it tends to do is it breaks up into discrete translations in that direction through some multiples of some fixed distance or time and this is what we see in the dunes think about transverse dunes the wind is blowing vertically the symmetries could slide those dunes up and down to any position you like they could slide the desert sideways to any position and they can flip it left right what symmetries does this pattern of dunes have any translation at right angles any distance whatsoever fixed distances along the wind direction so the symmetry along the direction of the wind has broken and you can predict this pattern of stripes just from the symmetries you need to know when that pattern is stable but you can predict that’s a possible pattern now with the bark annoyed dunes where it’s like that except they’ve become wavy we’ve broken another set of symmetries the left-right translations they’re not through any distance anymore they are through specific multiples of the wavelength as well so a second group of symmetries has broken to get the buck annoyed dunes and in fact if you imagine taking the sharp pointy bits and just pinching them off you get a pattern of spots which would again have specific translations in both directions yep linear dunes are very

similar to the transverse ones in terms of the symmetry except it’s now the translations of right angles to the wind that have broken to give stripes the ones in the wind direction remain barking dunes you can imagine some kind of lattice of copies this is a very very idealized picture or you might imagine that they’re perhaps staggered with respect to each other so there’s one then one then one that one then one like that so again they’re symmetries in both directions have broken parabolic dunes they’ve lost the symmetry of translation all together in one direction there’s nothing there you can’t move them but you can still move them specific amounts left to right and finally the star dunes which probably our best thought of is not really having much symmetry at all everything’s broken they do have sort of local vaguely rotational symmetries star dunes form when the wind on average doesn’t go in any particular direction one day it’s coming from the north another day it’s coming from the east another day from the south southwest whatever no prevailing direction so the pattern has no prevailing direction either so now we see that the same symmetry principles are applying in two different mathematical systems or two different physical systems the markings on animals where it’s a chemical pattern which we now know is related to the genetics but that’s another story and the pattern of sand dunes where it’s a pattern of sand and roughly speaking any physical system in the plane that is starts out being symmetric under lots and lots of rigid motions in the plane you’ll get the same kind of range of symmetry types of patterns so the same mathematics will apply to lots of different systems and that’s what makes this very powerful ok I’d like to end up with something I’m pretty sure I talked about aggression College about 18 years ago possibly using the same slides to some extent which is the animal movement okay here is a lovely sow trotting along yep what’s mathematical about that I was actually giving a lecture to some schoolchildren in Walter and this is the point at which they woke up once we got the trotting Pig then I had their attention so I probably should have put what I did put Pig quite early but then we forgot about it anyway so you probably know the poem yes centipede was happy quite until a toad in fun inquired which leg goes after which this raised its mind to such a pitch it they distracted in a ditch considering how to run I’m not going to talk about centipedes they are a little bit too complicated although the mathematics I’m going to tell you about does seem to apply to them with a few interesting predictions but the main point here is that animals move in lots of different ways yeah we’ve got a rabbit the rabbit was a hare actually the hare is bounding the horse there is trotting the elephant is walking and I can be pretty certain about that because essentially there’s only one pattern that elephants move with that’s not quite right but it’s close and the insect has six legs and that one I think is doing what they call a tripod gait which is essentially that the legs move in groups of three but to one at the front one at the back and the opposite one in the middle if he’s not doing that it’s doing a thing where essentially it sort of it’s like a walk it’s this leg that there this so that this said that very and then all well okay meta chronal go that’s called um and I’m not quite sure what the bully is doing the cheetah is doing what’s called a transverse no it’s called a rotary gallop for the cheetah it’s it’s getting so it’s going so fast that actually the back legs have got in front of the front legs you can see it’s catching up with itself basically and the millipede which is an even bigger challenge than the centipede okay these things are called gates and it’s actually useful there are lots of applications of this kind of it’s a mixture of mathematics and physiology basically so there are medical reasons for wanting to understand human gates because people have problems wrong with

their knees and hips and you know tell me about it and it’s good for sports science mophir I didn’t just win his medals by being very good at running that there’s a lot of analysis and training is aimed at specific things that the sports scientists have understood about how to run fast or you can use it to analyze dinosaur tracks and understand how the dinosaurs move or you can use it to look at human evolution nice picture from nature not too long ago suggesting that running was a major factor in the way the human body has evolved or you can look at robots with legs how can you make them work very useful to the US military who want to defuse bombs on firing ranges and things like that or you can use them to explore planets I mean we just lend it another wheeled robot on Mars but that’s because the engineers really understand wheels legs will be the coming thing but not quite yet because the wheel is a little more reliable but they’ve they’ve gone to some links with the latest Mars rover to make sure that it’s wheels don’t get stuck and so forth okay and future applications send out robots all over the solar system if not beyond and they may not have wheels and they may not have legs either they may have all sorts of structures okay so the subject of gait analysis was started by this chap born Eduard Muggeridge rapidly became Edward my bridge sandy Potter and he was looking he took photographs of animals moving and let me run you through one or two of these so when the elephant walks and indeed when any other four-legged animal walks the pattern is this look at when the left rear leg hits the ground it will be followed by the left front then the right rear then the right front if you look closely at that picture this is what you see that put that foot that foot back foot if you ever see them and I have sometimes seen animations on television they’ve got better now but in the early days of animations of dinosaurs and things sometimes it was that foot that foot that foot that foot back leg back leg front leg front leg no they don’t do that back front back front that’s how it works our Pig like the horse is trotting and you can recognize a trot because the legs are look together in pairs a diagonal pair hits the ground and then the other diagonal pair it’s the graph those two then those two okay and all of these have patterns if we go to the canter now that’s rather complicated lifts rear then a diagonal pair together and then the missing leg hits the ground after that and that’s why you get this rather odd looking structure in the legs transverse gallop and elk here is doing it left rear and then right rear but with a slight delay then left front and right front with a slight delay those two in those two clippity-clop a TT right a camel wrecking or pacing I still don’t believe this both legs on the same side move to go then the other two do what I really don’t believe about it I can just about blue with the camel giraffes do it it’s enormously tall animal it picks up both legs on the same side it’s kind of fall over we were in Botswana or a few months ago no they don’t then they don’t move they can move fast but they can also be quite slowly and they do it the same way it’s a it is quite a puzzle so mathematically we represent these as a pattern or what are called the phases of each leg moves periodically the phase is the timing of when it hits the ground so for the walk you have this pattern hit the ground at time zero a quarter of a period later the front leg on that side half a period later then three-quarters of a period the legs the other side yes you can see what these pictures show and you can look at all of the different

gates in terms of these patterns of phases so for the trot it’s north on one pair a half on the other pair for the pace naught on one side half on the other side for the bound the hair which I haven’t shown you pictures of back in front yeah I think the dog moving really really fast then it gets more complicated for things like the transverse gallop the rotary gallop which is what a cheetah uses which is very similar but one end is switched left-right compared to the others the canter which is really quite bizarre but has this one diagonal pair is going out of phase nor to half nor to half now the para hitting the ground together which is the two basic two-legged gates human beings do this when we walk it’s left right left right left right when it’s children hopping along its both feet together you try and do that when you get older it doesn’t work right and the horse cantering does both at the same time it’s remarkable and then there’s the proc not not not more that’s a prompt for you whole animal straight up in the air okay so here’s the prong so we have at least eight they’re actually more patterns and five of these are very symmetric the four along the top and the conch is the most symmetrical wall in fact the most symmetric even more symmetric than Pronk is simply stand there do nothing all the legs are doing the same thing and they’re doing it at all possible times the time symmetry has not broken if then and wants move around it breaks that symmetry and the simplest thing is to move periodically and if it’s not what anything else to do it’s kind of wanders around in a fairly rhythmic movement so what is it that’s creating these patterns this is kind of reverse engineering the physiology from the behavior and to cut a long story short they are produced by something called a central pattern generator this is a network of nerve cells that creates the basic rhythms there’s a whole pile of other stuff goes on to control whether you just whether you stop and have a look around and where or whether you see the line coming and run away at high speed whatever but the default rhythms come from a network of nerve cells somewhere in the spine it sends signals to the muscles in the legs it’s not in the brain it’s not in the legs it’s somewhere in between and you can deduce what it probably looks like is actually very hard to dissect out by studying the mathematics of coupled oscillator systems so let’s take the simplest one if I have two mathematical units coupled together there are two common patterns that you will see they both do the same thing at the same time or they take turns and I’m showing here with two pendulums you’ve got pendulums in two swinging like this or pendulums doing that it is possible to do other things but it’s much rarer and this is to do with the symmetries interacting with the phase shifts on the different oscillators and you can do the same thing with four oscillators in a ring and then you find patterns like all four of them do the same thing or they alternate left right left right in terms of North half naught or half and so on you get different patterns so how draw pictures of the phases in fact these are the four patterns you expect in this ring all 0 naught and a half alternating a quarter shift as you go anti-clockwise a quarter shift as you go clockwise and these are a bit like that’s like the first ones like the prompt the second one is a bit like a pace or a trot particularly and these two look like a walk and one of them is walking backwards and one of those walking forwards now if you try and mix that up to the animal it doesn’t actually quite work and what about the others what we think is going on at least in a the simplest possible network that can do this is you don’t use four oscillators you use eight and you use two for each leg and physiologically we

think one of these essentially pulls the leg forwards and the other one pushes it back again they control muscle groups that have that effect because muscles pull they don’t push yeah but you can think of the patterns by just looking at the bottom for these that these are perhaps the muscles that push the leg and then above them are the ones that pull and so if you write down all the patterns that are possible in this kind of network for example you can get a pattern where the phases go not the quarter half three quarters at the left hand side and then feed back and keep doing it but on the other side it’s delayed by half the period this is one of the patterns there are eight patterns for this network basically well if you just look at the bottom four that’s the walk that is exactly the walk whereas if we have this pattern which can also occur and just look at the bottom four you recognize the trot that’s the band noir path North half that’s the pace that’s the prop you’d expect to do it with four but there are some subtle mathematical reasons why that doesn’t seem to be correct and then these things can occur as well but it’s a more complicated story now if when we did the maths we realize there is one pattern we hadn’t noticed anywhere which looks like this not a quarter half three quarters like a walk but the same on both sides now it’s not the bound the bound is back front back front at equal intervals of time not half naught to half this is not a quarter what’s happened what’s happened naught the quarter something what didn’t know then we went to the rodeo and we saw the bucking Broncos and then of course we realized that good old my bridge had seen all this before the back legs hit the ground then the front then actually the whole animal hangs in the air for quite a long time look at that picture from top to top to the second row it’s done one and it’s still not coming down completely and when we looked at the video of the thing we saw in the rodeo it is almost exactly naught a quarter for the legs and then the animals sort of getting itself back on the ground again so it’s not 100% confirmation of the theory but it is at this consistent and I’ll stop there thank you very much / and slightly sorry about you