[MUSIC] Hello, I'm Fred Wan from the Department of Mathematics at UCI, UC Irvine. Our discussion today is about complex patterns and how everyday in our everyday life there are plenty of complex patterns around us. As you can see some of them in back of the screen behind me. These complex patterns, some are interesting, others are beautiful and they are as beautiful and as interesting as my camera skill allowed them to be. So we do know quite a bit about these patterns, especially the physical ones that we see like water waves. Water waves in the ocean, the rainbow in the sky and so on. But we know a lot less about patterns associated with complex biological systems. So today, we would like to share with you whatever we do know about them. And although it was not complete it does give us some considerable insight to the pattern why some like the elephant here have no pattern, very boring, and others are much more interesting. We know a lot more about the physical patterns, like water waves in the ocean and rainbow in the skies. We know where they come from, how they are generated. We also know maybe how they will evolve as time goes on. But we know a lot less about biological systems there's the pattern that's associated with them and for them we see some of them have very little patterns and others very similar like the lions would have very little patterns and the leopards would have lots of spots and so why are they different? And this is the elephant and this is the leopard. And how about the pattern changing with time? That's another question, like the Penguins. When they were small they are kind of uninteresting. And when they grow, they become much more complex. So while we do not have answers to all these questions, we do know something about them. And we wish today, I'd like to share with you what little do we know about them, especially those come from theory formulated my mathematicians. But before we do that, let's give you a little bit background. The people, we have a lot of people working in this area but the person who's contributed the most and at the beginning of it is somebody named Alan Turing, the Mathematician from Manchester, England. The name by familiar to you as you probably have seen The Imitation Game, maybe don't, you probably should see it. So Alan Turing is the one who cracked the nasty code, run the marathons, build computers and do many other things. Among other things he did is in a later day in his life about 1950, he decided to get interested in patterns in biological systems. To appreciate Turing's amazing accomplishments, we need to begin a little bit with some background information that led up to Turing's remarkable theory on pattern formation, and that is this. It is known mathematically, mathematicians have proved this over the years, that any complex pattern like the one that we see in the back of me, how simple or complex a combination of set of basic units, components you might say. And these component may be cosines like this one. 1, cosine(x), cosine(2x), cosine(3x), we're familiar with them. From trigonometry in high school, and yet the triangle in the back of the very top had nothing to do with this curvy things. A lot slated than curvy things and yet mathematics tells us that the triangle is a combination of this other entities, these other components, the cosines. Now, cosine is not the only set of basic components we could use for representing a given pattern. Here's another one, a bunch of sines. We know sines. sin(x), sin(2x), and so on. And they two are different from the cosine that we saw before and yet under certain circumstances they would be most appropriate as a combination of complex pattern. You can generally a complex pattern as a combination of them. So just which one do you want to use depends on the problem. For example, we can have this triangle and we can use a whole bunch of cosines, combination cosines. If you just use a few like the one, if you use one for example, you got the blue curve. Which is just a simple cosine and they are kinda different from the triangle but if you add more to it then you see that they're getting the red curve which is a little bit closer to the triangle and as you add more and more components you get closer and closer to the triangle. In fact after while you couldn't tell the difference between with your naked eyes between what the combination is and the triangle, you write on the same piece of paper. So another one is a more drastic one this is the so called step function a stair case. And that's quite different from the sins, the sine function or cosine function and yet if you put a bunch of cosines together, sine function together, you get closer and closer to the step function. As you see, you just use one sine, it's like the blue one. You add more to them, you get the red one and then so on. And as you get more and more, you get closer and closer to the step case. Now, on the other hand this one, it doesn't go there as quickly as the cosine for the triangle and there are reasons for that. But nevertheless, the mathematicians assure you that you will have the real thing as you take more and more terms in the combination. There are other, but we don't need to go into it. The basic idea is that any complex pattern is a combination set of basic components. There's a remarkable things in this statement. It's not as obvious as it may seem, but we don't have time to go into it. T here in high dimension, you have x along the line but at points along the line it can be capitalized by x. But if you have two dimension, like a plate or a table top or something. You might need two dimensional characterization and in this case we can have cosine x times cosine xy that will be one component. Cosine 2x times cosine y is another and so on. Now this can get more complicated if you want to visualize here's one possibility. Here's a tabletop that you might want to characterize by a bunch of basic units. And the one unit of cosine x cosine y is right beneath it. So you can think a little bit different perspective from below. So you can see what they look like. There's only one component now. You wanna see more components? We're gonna have the next one, two of the components, the orange one and the green one, the two different components. And you have a bunch of these and add them together, you get the table top so that's the idea. And there's a name for that, it's called the Fourier Series representation of any given pattern. And mathematician assure you that you can do that. In fact, Fourier is one of the first person who tried this and what he did becomes what benefit us a great deal today. And all the engineers and scientists use it forever and ever and get to enable us to do many scientific things making many products and so on. So one of the thing that for you use it for is to use it to analyze a temperature change in an object or along a wire that you pass current through it. You can heat it up or in a room where you try to heat up warm up people from when it was cold. So the idea again is that the temperature in a room or along a wire, look at that. Is that the bottom black line is the wire and the red above it is the temperature at different points along the line, and very high in the middle and not, and zero temperature at the end say zero Celsius. So what we like to do is, as time goes on, we want to see how this temperature distribution's in the triangular red pattern evolved. We do this with the understanding that the two ends, the y, we maintain at zero temperature. So allow heat to get in and out, we keep it that way and then see what happened to the temperature distribution along the y as time goes on. And what Fourier found in using his four year decomposition, you might say. Breaking the triangle down to a bunch of basic units. He find that as heat defuse's, which is that's what it does because that he likes to even things out. The triangular temperature distribution evolved in an interesting way. They of course, decrease because the heat leaks out of the two ends. And so gradually, you get nothing but a cosine curve, the blue curve. And more and more like a blue curve and the magnitude gets smaller and smaller. All the other components seem to have disappeared. In fact, what Gloria found in bioanalysis is that the component that has the highest oscillation would disappear first. They dissipate away and so while the triangular temperature distribution. It can be broken up into a different component the high with the highest oscillatory behavior dissipates first they go away first. So what your left with is at the end a sort of a consign one hump getting smaller and smaller as time goes on. Eventually all heat disappear from the wire when it gets cooled down and the whole thing flattens out. So in this sense, the situation's not very interesting. Because what had start off as being a interesting triangular profile you now gradually get down to a uniform distribution of zero temperature. Uniformity is never interesting as far as being a complex pattern is concerned. So that's the interesting thing about one single entity like heat. If you allow it to evolve through diffusion or otherwise, it had this behavior of the fastest oscillation. A component goes out first and this is something that Fourier discovered and we still use it together and we'll have an interesting application maybe hopefully we'll get to it in the end. So similar if you have a temperature profile that's triangular like the step function you see that by Fourier decomposed the step function into its different components. Again, the one that oscillate the most disappear, dissipates first. And again, the sine function different component has different level of oscillation. And the one that survives is a blue curve with hardly any oscillation at all as time goes on. So this is the conclusion. Pattern for scalar force phenomena like temperature, heat, in a wire or in a flat plate would as time goes on, the most oscillatory component dissipates sooner, goes away first. So that's the kind of mathematical result. That would serve as a background for our introduction of Turing in contribution.