I. Well, I'd like to thank them for the introduction and for inviting me to this wonderful lecture theatre. It's brilliant. Yeah. I want to talk about mathematical models of visual illusions. Now, if you've been watching the young lady spinning, you may have noticed that there is a visual illusion there. It's called the spinning dance or illusion. If you keep watching her after a while, she seems to change direction. She's going around one way and it might.
Might take ten, 15, 20 seconds and suddenly, hang on. She's going the other way. It's not a different movie. It's exactly the same loop being played over and over and over again. But our visual system interprets it in two different ways. And because it's a silhouette, we don't have all the clues as to which lake is in front of which and things like that. And so just as you keep watching, for most people, the direction will rather randomly flip, but probably five, ten, 15 seconds or so.
So. What I want to talk about is some work that's been going on over the last ten years or so. Some of it is by my research collaborator Marty Levitsky in Columbus, Ohio. He and I are working this a little bit in here to do this stuff we're working on at the moment. So this is kind of up to the date. Up to date mathematics. But I think it's a very interesting subject. So. If you want to know how to spell levitsky.
There it is. When he books restaurants, they elicit party because they know how to spell that. Okay, so there are some very famous illusions. The spinning dance or I've frozen her for the moment. There's the duck rabbit illusion. Is this a duck? Is it a rabbit? If you think that the stuff sticking out to one side, there is a beak and the head is facing in that direction.
It's a not very good duck. And if on the one hand you think that the thing sticking out at the side is, is it's an equally not very good rabbit. Okay. But it's. An image that can be interpreted in two different ways. Probably the most famous and the almost the simplest of these illusions is this. It's called the Nika Cube. After Nika who painted it, it was the first person to publish it. And. Certainly in our culture, you look at this, it's clearly a wireframe cube.
But if you look at it and ask which of the squares that you're seeing is a front, it flips. Sometimes it seems to be pointing out of the screen toward you. Sometimes it seems to be pointing the other way. And again, this is because there are two different objects which give exactly the same projection. So for an illusion, I want to draw a distinction between three kinds of visual phenomena one called illusions. One is impossible figures, and the third one is called rivalry. So illusions.
Both. I see the same information basically, although from slightly different angles. But that information is ambiguous. There's more than one possible interpretation, and the brain can't decide which interpretation is which. The brain is very busy. It doesn't just settle on something as like, Oh, that's what it is, it's fine done that end of it. Lose it for was it. It might not be that it might be something else. So you get fairly random changes but with some sort of average periodicity to them.
So Nikko published a paper and you put a reference out there if you want it. 1832 about the cube and if I shade one of the squares in it. This gives you a very clear the visual system. Certainly each of those is unambiguous as far as the visual system is concerned. It is also possible to say the whole thing is just a flat diagram on a flat sheet, but we tend to see this three dimensionality. So that's one example. Another one slightly sexist, but nonetheless, 1915.
You can see why. It's his wife, but it's also his mother in law. Now, sometimes it's hard to the wife. Am I going to get her? Yes. You just about see the red there. Okay. That's her nose sticking out. She's looking away like this. Yes. And. Yeah. Whereas for mother in law. This is not the chin for mother in law. This is the nose. The mouth is here. That's her eye. And she's looking this way. She has a rather large, ugly nose. Yep. So stare at it.
Could be his wife. It could be his mother. He doesn't say which ones. Which, of course. And the duck. Rabbit. That goes back to 1899. People have been interested in these things for a long time and in fact, quite a lot longer than anything we've seen. What about this? Say, is it a cat? He's a mouse. Now it's both. Is that a man playing a saxophone? Or is it a woman's face? Very heavily contrasted. It depends on which bits of the image you think, how you interpret them.
Yes. This one sometimes works on a screen. It doesn't always if you stare at it, the circles seem to rotate slightly. If it's not working for you and it doesn't always when it's up like that, look it up on the internet. It's not hard to find. Google images will do it for you. And in smaller scale, something about how their brain is interpreting their image is giving the illusion of movement. So that's a rather different kind. And then there's this lovely one phrase, a spiral.
And we have this beautiful series of spirals all converging to the centre, except they're not spirals, the circles, because each one of them is a circle. But because of the way that the circle is shaded, the eye is sort of drawn to the centre and gets this into this sort of impression of a spiral. So that's so nice. Would do it again. I mean, I know that's not a spiral. I've sat there and traced my finger around it and it comes back to our star to draw and I still don't believe it.
And even when I put up the pink circle, I don't believe it. And you can put up as many circles as you like, but if you get enough circles, it kind of hides all the spiral bits. So then you start to believe it. And what about this one Stone Applied Illusion? What are we seeing here? Well, what I'm seeing at the moment is a series of white diamonds moving vertically.
What I'm seeing now and some of you probably were seeing and have now changed your mind is a series of stripes going through each other like this? Yeah. You've got one not going at y one not going that way. And as you watch, suddenly it turns into diamonds going up again, and then it goes back to these moving stripes. So all of this is is telling us something about the visual system, but it's not at all clear what.
But there's a quite, quite a good scientific principle is a useful way to get extra understanding of a complicated system is is to see what happens when it kind of goes wrong. You either try and break it or you try and find places where it's perhaps giving. Clues as to the internal processing that must be going on behind the perception. I mean, we just look at things, oh, that's that's a cat, that's a dog, whatever. The brain is doing an awful lot before we recognise a cat or dog.
Second category impossible figures. This is the Penrose triangle. Each corner of this looks like a perfectly reasonable right angle. Bend in three dimensions, but if you follow it round, you realise you couldn't make one. You could make an object that looks like this. But if you move it very slight, you realise that it's completely different. But this is not really a possible object in ordinary three dimensional space. Locally, it makes sense. Globally, the pieces don't fit together.
So this isn't an illusion. It's worse. It's something that can't exist at all. But hang on. I'm just looking at a picture of it. So for these both eyes, get the information again. But it's impossible. There is no consistent interpretation. But you see, what happens is your attention focuses on one corner or another corner or another corner, and you check and it makes sense at each corner. And every so often you kind of stand back from it and think, Yeah, but the whole thing doesn't make sense.
There's lots of these. This is a sample. Yeah, the rectangle version of the triangle. There's a thing that at one end looks like three cylinders and the other end looks like some sort of square shaped U-shaped thing. And it doesn't fit together properly. The staircase in the middle at the bottom goes up and up and up and up and around. Around forever. Well, if you go the other way, it goes down, down, down, down, down. Forever. And how many feet has the elephant got? I like that one now.
More Sasha. Artists, well known to mathematicians, used some of these impossible objects in his artistic work. And I just show you a couple. So the. The monks on the roof of the monastery there are going round and round one of these neverending staircases and one lot of them going up and the other lots are going down and they're going past each other. So he he found artistic inspiration in these things.
And the third kind of phenomenon I want to talk about, and I'll start by saying a lot more about that, is called rivalry. This goes back to at least 1593. I suspect Aristotle knew about it, but early in 1593, Porter put two different books, one in front of each eye, and discovered he could either persuade himself that he was seeing one of them or he was seeing the other one, rather than seeing some sort of superposition of the two. And he mentioned this.
So I want to talk about some rivalry experiments which are slightly puzzling. So in rivalry, it's different from illusions and impossible objects in the sense that each eye gets different information and that information is conflicting. It's not like when we normally look at an object with both eyes, see slightly different versions of the same thing, and put together a three dimensional interpretation is that the two things we're looking at to just can't possibly coexist at the same place.
So there's an experiment done where the two images are vertical pink and grey stripes on horizontal green and grey stripes, one presented to the left, either x l e, one to the right eye. And there's another one which generally known as the monkey text experiment. And I'll explain a little bit more. But the images there is it's like a jigsaw puzzle of it's some sort of, I think, not really a monkey and some sort of blue text on a green background.
And they've been cut out like a little jigsaw puzzle and then pieces have been swapped between them. So you've got these two mixed images and you show these two one to each eye and get people to in various ways tell you what they're seeing. Okay. So with this one, the surprise was. You only show them two images. And what a lot of people do is they say, oh, it's the vertical stripes. No, it's the horizontal green ones. No, it's the vertical pink ones and so on.
But some of them say it's horizontal green and pink stripes. Or its vertical green and pink stripes. The two on the right. So. Some people see the images that were actually shown to the eyes, but they alternate in some way. Some people see images that were not shown to either eye, but were kind of assembled from bits of the information. But they put it together to create an image that they never actually saw. This is called colour Ms. Binding.
The colours are bound to the stripe somewhere in the brain, but some of the colours have been assigned to the wrong bit of the image. So this general impression of Stripe in this vertical, horizontal, green, pink. But now you can put those together in several different ways. So the observed images may actually be things that were not presented at all. And this happens with the monkey text in a sense. I think it's set up to do this.
A lot of people will just see the two actual images, these mixed jumbled up images, alternating. They'll see one and see the other. But sometimes they will see the complete monkey and the complete text, and they tend to alternate as well, which is what's interesting. So the brain is putting the pieces together, but there's several different ways to do that. So what we want to do is try and get some insight into what the visual system does when this happens.
Compare this insight with the observations. But in science, what you'd like to do is predict new phenomena. You want to predict things that have not been seen. Compare them with observations. And you hope. That they will, in fact, be seen. And if not, you go back to the drawing board, start again. So this sort of seven step process model, what's going on?
And I'm going to do this using a very simple little network, which is not really meant to be an exact representation of something that's happened. Things that exist in the brain is more a kind of schematic representation of what? Collections of nerve cells in the brain might be doing. Use some mathematical techniques to analyse those networks and see what in general terms, what would you expect them to do? Plug in some rather more specific equations so that the models match the neuroscience.
Better ask about various things that are important to people in dynamical systems, such and indeed in the real world. Is it stable? You can have mathematical solutions to things which won't actually exist because they are too easily destroyed. A pencil, in theory, will balance on its point. You try it to balance on its side very nicely. It's stable on its side. It's unstable on each point. But there is a mathematical balance point there.
Compare with observations. Invent new experiments. Make predictions. Get someone to do the experiments. I'm a mathematician, so my friend Marty, we can't we can't do the experiments. We don't have access to the equipment. We need to sign all the forms so that we can use graduate students or something as experimental subjects, you know. So but fortunately there are other people who do that kind of thing and then go round around this process.
Okay. And the starting point for the mathematics is what we call the Wilson Network. Hugh Wilson, who works in mathematical neuroscience, was interested in modelling how the brain makes decisions, and he wanted some sort of schematic, high level model, not down on the level of individual nerve cells, but just how does the brain. Given some alternatives, how does it decide? What? What to do?
What it's saying if there's an election and you want to vote, you've got a choice of candidates and you have to decide who to vote for. And you feed in all sorts of information. These are their policies. Yeah, well, they say those policies, but I don't believe this person any way or whatever. You put all this together and eventually you go and you, Marco, across on your balance sheet and somewhere in the brain, all of this stuff is being put together and a decision is popping out.
And he, Wilson, said, Well, let's take a very simple example. I look at an adult. And it might be read, it might be green, it might be blue, and somewhere or other. There is a circuit that recognises red, green and blue. So as well as these dots representing what you're looking at, they kind of stand for bunches of nerve cells, some of which have been trained to recognise red, some to recognise green, some to recognise blue.
A very important feature of such a network should be that it comes to one decision, and these black lines with dots on them represent what are called inhibitory connections. In the neuroscience jargon, that is an inhibitory connection is where you've got a bunch of nerve cells that think one cell nerve cells are active. They will fire, they will send electrical signals, and they may be not very active or they may be much more active. Perhaps they send more more signals per second, whatever.
Okay. And inhibitory connection is it this one's active and it sends an inhibitory connection to this one. It shuts this one down. So basically it's telling the next sell a long stop. These connections go both ways. Each of these things are telling the other one to stop, but the dynamics is set up so that with I if I input something into this to if I input something that kind of looks red, that increases the activity, the red one so much that it shuts the other two down.
And when they're shut down, they can't change the red one. So we decide on red. If something comes in that makes the green one excite, it shuts the other two down and so on. So it's a kind of winner take all structure, and it comes to one decision. Okay. Now, that's the easy bit. The by convention, I'm going to draw these things as vertical columns, so called.
And in each column, each column represents an attribute of the image, some feature of the image that we're going to focus on and the different possibilities we'll call levels just because that's what they look like. Got to have some word for this. So winner takes all links between levels in one attribute column. And then you do the same kind of thing, but for different attributes. Okay. I've drawn them all looking the same just to emphasise the structure.
But for example in the column is binding experiment. There are two kinds of attributes. Are the stripes vertical or horizontal? Is the colour pink or green? Should have one column with pink green. One column with vertical horizontal. So each column is trained to recognise which of those levels is occurring. And then you show each an image and in each column you pick out the attribute corresponding to the image, and you link those cells by excitatory connections.
Those do the opposite. If this cell starts up and it excites this cell, then both of them are going. So as soon as one of these purple cells lights up, it makes all the others active as well. And you buy them all together with a different kind of coupling. Neuroscience is very strongly focussed on this use of inhibition and excitation between cells, and it sets up kind of computer like switching circuits and all sorts of things.
Okay. But you when you train your Wilson network to recognise a particular type of image, what you're telling it is the component attributes, what the level should be. And then you put that in as right excitations between them. Okay. Now then the mathematical question is, having set up a network, you know, intuitively, you've kind of got the idea. It can choose levels and it's been trained on certain combinations.
But what does it really do? And there are two philosophies on how to do this, which are call model independent and model dependent. Okay. Model independent is the structure of the thing means certain things are going to happen pretty much independently at the precise mathematical equations that you write down to the dynamics. There are lots of things in dynamics that don't actually depend a lot on the equations that my arm swings to and fro. Like a pendulum that works with a real pendulum.
It works with my arm, it works with the other arm. The equations are actually slightly different in each case, even for the two arms they weren't quite the same. But nonetheless, swinging to and fro in a periodic fashion is a very common sort of model, independent behaviour, model dependencies. I'm going to write down detailed models, analyse the heck out of them and tell you what they do. So. Let's go for the simplest possible set up.
So this is going to be my model of the brain looking at the rabbit duck or the neck, a cube, particularly in the neck, a cube. And also there are some simple examples of rivalry. So even this little two cell network, one column, two levels, can't get much simpler than that one called. On one level, there's nothing to choose. Okay. Well, with the next cube, we could say one of these represents perceiving the cube with the blue face poking out in front and the green.
So there represents the alternative version of the cube where we perceive it with the other square face poking out in front of the eye isn't shown, the blue faces is just shown the next cube. But there are two interpretations. So what would this sort of network do? Well, it's real mathematics. There are some formulas. Here is a model dependent analysis of this thing.
You write down equations, the ones that people like in this context, these equations for the the rate at which a neurone is firing, it sends out a series of pulses and if it sends out very fast pulses, then it's firing at a higher rate. And that actually means it's got very excited, thinks it's seen something. And what these equations do. There's a thing called the gain function. It looks like this. So it goes from zero up to one zero means of one means on.
And it kind of transitions between. Okay. So what the equations say is basically that one of these cells receives input signals from the other cell. And the level at which that input signal occurs affects the behaviour of the red. So the red cell gets an input from the green cell. And response to that and the green so gets mathematically the same formula with some ones and twos into changed. Okay don't worry about it but if you want to know this is a published paper on this.
This is the kind of stuff you write down. Okay. So there's two variables for each cell and there's a very standard form for the equations. When you get used to it, you can write these things down. I'm going to show you some much more complicated networks a little bit later. And I just remind you again, the game function is this sigmoid thing, as they call it. And if you shove this on the computer and solve the equations, this is the kind of picture you get.
And this is done by John Renshaw. And the blue curve actually would be the green cell because he used different colours from me. But what you see is the red cell fires and then the blue light is green. So if the red is at the top and the blue is much lower level of activity, and then suddenly the blue has a high level activity and the red is low, then the red is high in the blues high, then the red sign and the blues high.
So the way you interpret these signals is the cell that's firing at the highest right winds. So when the red is at the highest rate, you think that the next cube looks like the top picture. But when the other one is at the highest rate, you think it looks like the bottom picture. So what the mathematics predicts is it will move from the Q pointing one way to the other. Way back again, back again, back again, back and flipping between them.
In this mobile, it's a perfectly irregular flip, which is not quite what's observed. And what actually usually do is then add some random terms to the equations to make the flips. Not completely irregular. But I'm going to be happy with regular flips, bearing in mind that we're not getting too close to too worried about the details. Let's get the big picture, broad picture. Okay.
Now, these things switch on by what mathematicians call a hop bifurcation, which is a stable state, which, as you turn up some input to it, suddenly starts to wobble to. Okay. It's called a hot five fixation. His his little animation of one that occurs and then disappears again. You know, there's a the blue thing is the steady state which expands to something periodic and then shuts down again. And you can ask what sort of hot bifurcation is occur in this two cell model?
And if we do a model independent analysis, the important thing is not really the dynamics, it's the symmetry. If I flip the two cells, all of the equations flip as well, and the equations for the two bits look the same. It's just that the variables are swapped. And there are two kinds of hot bifurcation that occur in this kind of system. And I can demonstrate. Okay. So here are the two cells. Either they both do the same thing. At the same time or they alternate like this.
Yes. This is a child hopping. This is somebody walking. So these two patterns are what you don't see except with very special circumstances, which you have to fiddle the equations to make them work. Would be this one doing something and then this one doing the same thing, but perhaps just a little bit behind it. I can just about manage to do that, but my arms want to go back to this much more stable alternation. Yeah. So the math tells you that this is what you should expect.
And of course, the out of face case is the one that John Rizal's computer program was picking up. And in fact, in these inhibitory networks, basically inhibition says they shouldn't both be firing at the same time because they're trying to shut each other down. If one of them wins, the other one is happy, but then the other one could die down. And then the second one start up.
So the model independent analysis tells you you should expect to see this half period out of phase alternation in that particular network. And what we want to do is mimic that in more complicated cases. What really kicked this whole thing off from our point of view. People like John Ensign have been working on this for a lot longer. My friend Marty was he was and recently stopped being director of a thing called the Mathematical Sciences Institute in Ohio.
And as director, he got the job that Alan had there, which was introducing the speaker and sometimes sitting and listening to the lecture. And somebody was giving a lecture on the rivalry and mentioned the monkey text. I'm Marty Thornton. I wonder what the Wilson Network looks like. And the answer is the simplest way to set it up is there's a picture of the jigsaw puzzle.
Each of those pictures is essentially divided into six pieces, and you take the white pieces and the blue pieces, but then you swap the corresponding sets of three between the two images. So you you basically you can if you take the left hand picture, it's got. It's got text in the blue region. A monkey in the white region was the other picture. It got text in the white bridge and the monkey in the blue region.
So your attributes, your columns in the network are what's going on in the right white region, what's going on in the blue region? And in each region it can either be monkey or ticks. That's the two levels. So if you ignore the cross in the middle, you've got just the two columns with these inhibitory connections. But you're trying your brain by showing the two eyes pictures in which monkey at one level goes with text at the other level.
So you get this excitatory connection swapping levels, and then text in one level goes to monkey in the other one. So you get a crossing that goes the other way. So we get this excitation across inhibition up and down. Well, you can look at that and say, what? Are the patterns of oscillation. What are the analogues of in phase? Out of phase? The answer is there's four of them. Everything can be in phase. The stuff on the left side can be out of phase with the stuff on the right.
So the two things on the left oscillate, the two things on the right oscillate, but they're out of phase with each other. Or you can do that diagonally or you can do it top bottom. Okay. These are the four possible half by fixations in that network, but the inhibitory connections rule two of them out. Inhibition means you shouldn't have the same colour top and bottom. So that was not good. And that was no good. But the other two so are having the same colours left and right.
But they shouldn't be in the same colour. What are those? Two patterns of oscillation. One of them is the original pair of images, learned patterns. We're getting a rivalry between those two images. The other one, you see the whole monkey and the whole text. And if you go back and look at the diagram, you can actually check. That's really what you would be seeing. So this is rivalry between two patterns that the I never saw but the brain put together.
It's not greatly surprising the brain puts it together. We're all quite good at jigsaw puzzles and things. But. The same thing is going on in the Chevelle column is binding experiment instead of monkey text. What you've got is essentially a horizontal vertical going with the colours. You can use a similar network, although later I'll show you a slightly better network for. Okay. And here's another pair of rivalry experiments that do the same thing.
These are either beautiful. So if you show the left eye concentric circles and the right eye horizontal stripes, people sometimes see something which is circles in the left half of the visual field and stripes in the other half. Excuse me. Or the other way around, and they will say alternating between the two. Or if you show them this hourglass shaped thing and the hexagon again, the brain may put together half a one with half of the other and then flip between the two.
Exactly. The same network will deal with these. So now we can make some predictions, unfortunately, and I'll give this away before you get too excited. No one's yet done the experiment because it's not entirely straightforward. But suppose we showed three red dots to one eye and three green dots to the other line. Well, you could get alternation between those patterns, but you can draw up you will some network. And if I rearrange this a bit, it's the same network.
I'll just move things around. It's got symmetries of the hexagon. So, of course, the dynamic assistance people know all about bifurcation with the symmetries of the hexagon. And there's actually quite a lot of possibilities, but a rather nice one is what's called a discrete rotating wave. So a, b, c, D, e, f are a sequence of images that would be consistent with the dynamics of that network.
And what's happening is essentially that the two red dots are rotating round the triangle in pairs, except sometimes it's only when this transition there's one red dot in two green dots. So the green and red dots chase each other round around the triangle. Now, it would be lovely to do an experiment that shows that happening. But as I say, unfortunately, no has actually done that. Okay. Now there is actually an algorithm for constructing these things.
And the network that actually makes most sense for the columnist's binding experiment is a slightly more complicated one. I won't take you through the details. I'll just prove to you that this is there now a problem with the two cell model of the next cube, as you've kind of put into it, what you expect to get out. You've put in cube facing one white cube facing the other way. Okay. Well. That's kind. It would be nice to have something that that doesn't make that assumption.
It doesn't even know for certain it's a cube. So what we thought was. It's reasonable that the brain should dissect this image into a series of lines, a number of the lines. There are eight of them. I haven't worried about the vertical edges because I'm not going to do anything with those. It's where those lines cross that the ambiguity arises. Now, suppose I set up this network. Okay, this is getting more complicated. Okay, let me explain where this network comes from.
So the idea is this is a visual illusion. It's not rivalry. So there isn't the idea of a learned set of patterns. Instead of that, we have the idea that the brain has already learned to recognise certain shapes and images. This seems pretty clear, actually, that somewhere if I look at something and see that it's a cat, I know it's a cat because I've been told over and over again those things are cats and the brain has figured out that anything cat like is a cat.
So when we see one of these pictures, we try to match it to an object that we already know about. Now, objects we know about must actually be possible in three dimensions. We're seeing a two dimensional projection of a three dimensional object. The question is which object is it?
So there should be some sort of geometric consistency so that if I have if I see a line pointing out that way and I line pointing out that way, the stuff on the end of them should not be at the same place and things like that. So that's what allows us to draw up this particular very complicated network. And we only make three assumptions. Vertical lines in the image represent vertical lines in reality.
Vertical is special in visual perception. It's very special lines that seem to be parallel in the image should be parallel or nearly so in the object. It's a reasonable assumption. It's not necessarily the case, but let's assume that. And similarly, lines that are not parallel in the. Image should not be parallel in the real thing. The only is not being given something designed to fool it. Well, if you then you can then draw up a model because of the.
What I was calling columns are now arranged horizontally for each line. There are two directions it can point basically forwards or backwards relative to the plane of the image. And if you draw up the network, that corresponds to the way the cube goes. We get this quite complicated thing. The inhibitory pairs go across in each of the eight connections go across in each of the pairs.
Everything else is excitatory connections or inhibitory ones, which essentially encoding this geometric consistency condition. So we got what mathematically is quite a pretty object. It's actually got quite a few symmetries. There's obvious ones left and right or flip top and bottom, but you can also take the two sets of pairs and flip them simultaneously. So you've got eight symmetries. So don't worry about the details. There are eight possible patterns of oscillation.
Okay. Now, I'm not going to spend time explaining what this picture encodes, but you can sort of say this is a list of the eight oscillation patterns. But again, we can get rid of quite a lot of them because of the extra structure of inhibit inhibition and excitation and other things. We can get rid of everything in the top row. We can get rid of that one. That one and that one. Those are all inconsistent with the extra structure. And we're left with just one. What does that look like?
Well, you can sit and stare at this. And if we had half an hour, I could take you through it and figure out. But let's look, for example, at. One and five here. The shaving is meant to say if if we think line one is pointing forwards, then we also think line five is pointing forwards. If line one is pointing out of the page, so is line five. And while we're at it, line four is also pointing out of the page and so is line eight.
Okay. And similarly for the pairs two, three, six and seven, they're all pointing in the same direction, but they will flip between forwards and backwards. So if I were to solve the equations and here here's a picture of what's going on, and we can see some of the colours. Some of the colours are missing because the curves are hidden behind the other curves. They synchronise in pairs. But let's look at green and brown. Green is three forwards. Brown is seven forwards.
Three forwards. Seven forwards. Yeah. Three. Yep. And. What we're seeing alternation between the colours or here the purple and the magenta. And if you actually look at that, what's happening is the whole image seem to flip from point to point in one way to the cube pointing the other way. But if you look very closely at where it changes. Things don't quite cross. You get on the left hand picture at the bottom.
The curves are not all crossing through at the same point. So the flips occur very slightly different times. And what the brain is perceiving during that very short flip is actually an impossible figure. Bits of the cube are fitting together like that impossible rectangle. So the mathematical model was the time where, okay, for the moment, mathematical model is has this little period of confusion where the brain thinks part of the image is flipped.
And the other part hasn't, but that doesn't make sense. And that somehow triggers the other one flipping and then the whole thing's consistent again. And the same thing happens in the rabbit duck. If I set up a network for rabbit and duck, two columns, one is it is or beak is the head facing right or left. Draw out my network, putting what I know about the images. And again I get this alternation between. Beak is right left, but the beacon is flip to slightly different time from the head.
So I'm looking at the image and I'm seeing a duck. And then I think, no, those aren't beaks, those are ears. And momentarily the heads facing the wrong way and then the head must be facing the other way. Then it looks like a rabbit. So that's kind of prediction there. And I think I will. Finish at this point to allow time for questions. There is more. There is a cube with three possible interpretations.
There's this just look at the spinning dancer again, and I'll finish with a little more analysis of the spinning dancer. Okay. The kind of. Attribute that we should focus on things like is the head facing forwards or backwards? Is that the left or right arm? Is the other one the left or right arm? Which leg is in front? Is that the left foot? Is that the right foot?
And if I were to draw up a network for that, I'd have a whole pile of columns, possible positions of the head, which arm is which which leg is in front, which foot is which. And I have a whole pile of excitatory connections. So the things that make sense, we know it's a dance, so we know this is a human body. We don't have the top half spinning one way and the bottom half spinning the other way.
You could do that with a model, but you can't actually do it with the doll or something if it had a pivot in the middle of it. You can't do that with a real person. It wouldn't it wouldn't last very long. And if you simulate that, what you see is essentially that there is the same effect as with the next cube, that the all of that all of the levels that go together with clockwise spinning flip at almost the same time to everything consistent with anticlockwise spinning.
And then it flips back again and again if you look very, very closely at the transition. The different columns flip at slightly different times, but it's so close that in fact, in the mathematical pictures you just see that the waves flipping one after the other, everything flips together. Each one of these pictures is actually five curves sitting on top of each other. They're so close together, it looks like one curve. So that's the spinning dancer.
There's a similar way of dealing with the moving plaid. And so that is a little introduction to the mathematics of visual illusions. And it's the.