Isaac Newton Lecture 2025: Professor Sir Michael Berry HonFInstP

Well, good evening and uh I must start by saying what an honor it is to be uh invited to give this lecture and also what a pleasure it is to be in Newcastle again. It's a long time since I've been here and uh having a walk around yesterday, I realized I'd forgotten what a grand city it is. So, thank you very much. Now, over the years, I've occasionally uh spoken about rainbows in talks connected with optical phenomena in the atmosphere. But, I've come to realize recently that rainbows illustrate many concepts in physics, unexpectedly many. They either originated in people's attempts to understand rainbows or else can be illustrated by rainbows. And that's my theme today. It's the theme is connections. There'll be eight of them. That's an arbitrary number. There probably could have been more. Connections between rainbows and other physics and science more generally. Now, everybody loves rainbows. Wordsworth, "My heart leaps up when I behold a rainbow in the sky." Now, it's only our hearts. It's rainbows engage our brains as well, our intellects, our emotions as well as our hearts. Not that we can separate those, I have to say. Now, not everybody thinks the same. John Keats disagreed. "Do not all charms fly at the mere touch of cold philosophy?" He thought this would unweave the rainbow, would spoil it. I'm sure there are people who agree with with him, but I've never met any such person. So, uh let's begin. What is a rainbow? Well, I see a rainbow as the love child of light and water. And, uh there's always light and water. Here, there's light and water. You can't always see the water, but it's there behind um the the speaker. Now, it was very natural when people started to think about what light is and how to understand it that they would apply this their thinking to understand the rainbows such a dramatic natural phenomenon. You can't really say you've understood light unless you understand how rainbows form. And, it took a long time. So, Roger Bacon realized sunshine on a dripping cloud that, uh a rainbow, uh does involve light and water, in particular, light and raindrops. And, it became clear that to understand this, you must understand how light rays, geometric optics in those days, interact with water droplets, how rays bend when they encounter water. That's refraction. Now, the law of refraction was understood by Ibn Sahl in 984. It was, uh, rediscovered by Thomas Harriot in 1600 who didn't publish much, so he doesn't get the credit for it that he should. It was re-rediscovered by Snell in 1621, and we we speak about Snell's law. It was re-re-rediscovered by Descartes in 1637, but he was the person who really nailed the rainbow because knowing the law of refraction isn't enough to explain why there's a bright bow, a bright ring, an arc in the sky. And he did because he understood that if you have rays from the sun that come in uniformly, they get refracted in, reflected, and refracted out, and they come out non-uniformly. They come out concentrated in a particular direction, and it was his achievement to understand that. Now, it's an exercise a good undergraduate reasonable person has got an exercise to do the trigonometry and find what this angle is in terms of the refractive index of water. It's a little formula. It's about 40° to backwards. So, you stand with your back to the sun and 40° above the anti-solar point. Um now, Descartes knew trigonometry, and there were trigonometric tables, so he could calculate all these rays. But, the rainbow is a minimum deviation. And that requires calculus, which hadn't been invented. So, he couldn't do that bit of the calculation. So, what he did is this is my first connection, exactly what we scientists do now when we have a theory of something and we can't do the mathematics. He did a numerical experiment. We all know what that is. Remarkable. I don't say it was the first, but certainly when you read what what Descartes wrote, you recognize it as exactly what we do today. And what he did was he computed, he calculated the paths, the deflections of lots of rays that come in at different different heights relative we call different impact parameter, different heights relative to the raindrop. And I found his exact numbers and I reproduced his calculation. And here here they are. Here's the deflection in degrees and here are the points he calculated, many of them near the minimum to find out exactly where it is. And he realized that I mean he very charmingly said, "I took my pen and did the calculation." But those are his exact numbers. And why is this a rainbow? Because this minimum corresponds to focusing a lot of rays go into a little deflection. That's what focusing is, a lot goes into a little. And the name we have for this kind of focusing is a caustic. I'll use this term a lot in this conversation in this talk as we proceed. Okay, so he understood that. Excellent. Now, each raindrop emits a bright caustic cone. It's a cone because you have to rotate about this symmetry line. And when we look up, we see uh brightly lit all those raindrops on whose cones our eyes lie. Now, here's Descartes's picture and I've highlighted the rainbow ray. That's the primary rainbow. But of course you see often see two rainbows, that's one more reflection. Now, there's something not quite right about this picture because these um primary and secondary ray don't come from the same drop, they come from different drops. And Isaac Newton recognized this. We're now speaking 70 years later, 65 years. Here's his picture. The primary and secondary rainbow different, also the colors different. We'll come to the colors in a minute. Now, we learned something from this. It's my second connection. That rainbows are illusions. There is There's arc in the sky. Each person looking at the rainbow sees different drops of water because their cones reach the eye. Different people see different cones. I remember 50 years ago standing at the border between Zimbabwe and what later became what soon became Zimbabwe and Zambia at Victoria Falls. So close was the was the spray of water to me that I saw different rainbows with my two eyes. I close one, close the other, the rainbow shifts. So, rainbows are illusions. And that's a a large subject now. I give a different talk about geometrical optics illusions intended to explain that geometrical optics is very old physics, but it's still alive and can explain qualitatively unfamiliar phenomenon has done in the last few years. Anyway, illusions. You know, when you look in a mirror, we don't think about this, but there's no copy of yourself behind the mirror. If you go around there, you don't see a copy of yourself. It's virtual images are illusions. Good. Um colors. They're often wrong. Most egregiously in one of the most famous books on optics of the atmosphere by Marcel Minnaert, uh written in the 1930s, translated and reproduced by Dover Publications, who will never be forgiven by getting the colors in the wrong order. It's red on the outside. There are many such. Here's the BBC Learning English. The colors are wrong. Um here's drama therapy for children. The colors are wrong. Here the colors are wrong. The pink the the the the pink is in the wrong place. The yellow is in the wrong place, and so on. Many postage stamps throughout the world have the colors in the wrong order. It's really unforgivable. Even my 4-year-old grandson can get it right. Um we come much more about the colors later. Oh no, sorry, another one. This famous picture by Millet, The Blind Girl with the two rainbows. This was shown in the 19th century. It was displayed at a dinner. Millet was there and Stokes was there. And Stokes pointed out to him, "You've got the colors in the wrong order." And he said, "Thank you for telling me. I will go and correct it." This is the correct picture. Unfortunately, color photography hadn't been developed, so we don't know what the original looked like, but still there's a history in these wrong colors. Now, Isaac Newton made a big step towards understanding the colors. We all know it's dispersion. The refractive index is different for rays of different colors and the red is refracted least. It's a small effect. It's about uh 1% difference in refractive index between less than 1% between the red and the blue. So, here's the picture. The red is refracted least and the blue is refracted more. Now, this is only half the story. And to understand why it's only half the story, we need to go a little deeper. Here's Isaac Newton's house with a rainbow, an iconic picture, a rainbow captured by uh Roy Bishop over Newton's house. But look more carefully at the just the primary rainbow, not the secondary. You see there's a little additional fringe in there. Can I see it on the screen? Yes, you can see it. It's clear. Zooming in, you can see it more clearly. That was understood 100 years later by Thomas Young. And it was one of the examples he gave why light is a wave phenomenon, cuz it's an interference fringe. They were called supernumerary bows. Supernumeraries unwanted, surplus to requirements, didn't fit Newton's theory. There's no evidence that Newton ever saw a supernumerary rainbow. And I'm you know, because he was a very careful observer, especially of things that didn't fit his theories. So, that's anyway a supernumerary rainbow. Now, um what it's showing you is that ray theory is an approximation. At a deeper level, of course, light is waves. We we know this. Um at a deeper level still, it's polaroids, electromagnetic fields, deeper still, they're quantum states, and so on. But, it's a hierarchy. Um Now, so this is ironic as well as iconic, because it's showing something that Isaac Newton were unable to understand. Um What's interfering? Well, in every direction inside the bow, two rays emerge. And they have different path lengths, so they interfere with each other, and that interference changes as you look as you look at different directions, different parts of the rainbow, and you're seeing different pairs of rays that emerge in the given direction. So, that's the That's the explanation that Thomas Young understood. Well, you can sometimes see more of these, and here you can see a great many more. And this is my next connection. You're seeing something very deep when you see these supernumerary rainbows. You're seeing ray physics failing and being replaced by the deeper wave physics. Now, this is something that occurs throughout physical science. We often find we have theories, classical mechanics, which when you look more closely at particular phenomena, you realize it's an approximation, there's something deeper, quantum mechanics on the one side, relativity on the other. It's thermodynamics. If you look more carefully and you look at fluctuations, you realize that statistical mechanics underlies it and reveals thermodynamics as a very powerful approximation, but an approximation nevertheless. So, you're seeing something very deep scientifically with your naked eyes when you um when you when you see these supernumerary rainbow fringes. Now, the connect the fourth connection is this. You don't see these supernumeraries very often. Why don't you? That's another concept which has come to prominence in recent years. It's uh decoherence. Now, this um was really discovered by Thomas Young because he had to explain, to respond to very polemical, very bitter anonymous critic of his wave theory, who essentially wrote, "You talk about these rainbows and you have this experiment with two slits and show us these little fringes and the this is supposed to tell us that light is a wave. Why don't we see these fringes all the time?" And Thomas Young realized it's because they're delicate. They you can they can easily be destroyed by sources outside the control of the experimenter, sources which dis- which suppress interference. And I want to talk a little bit about that. 1 + 1 = 2. Two candles, two flashlights are twice as bright as one. Now, um that's because flashlight is messy light. With pure light, intensities don't add. Instead, 1 + 1 is not equal to two for intensities. Waves add and waves, of course, have the additional property of phase. Specific phase, I mean very generally specifying the stage of anything that oscillates. You know, the phases of the moon is the shape of the moon over a month, but these are phases of of waves. And differences of waves cause interference. And the intensity one intensity one intensity two could be anything between zero and four. In particular, it can be zero. And we we're familiar with perfectly destructive interference and we tend to forget how amazing it is. And Arago, in his obituary of Thomas Young, wrote, "Who could have imagined that darkness could could be engendered by adding light to light?" Really very a very amazing thing. So, we need to know why two flashlights or two torches are twice as bright as one. And of course, the reason is that in messy light, the phase varies very rapidly. It also varies varies rapidly even in pure light, but very randomly, and we can't follow these variations. So, what we see is the average intensity, and the average average over messiness or over time is is two. So, 1 + 1 = 2 is quite a subtle thing in wave optics, and you res- only achieve this if you involve decoherence, inability to detect or external or causes scrambling the phases. Um So, something deep about 1 + 1 not equals 2. As a piece of mathematics, it's elementary. Not quite, because if you read Russell and Whitehead's epic Principia Mathematica building up mathematics from logic, 1 + 1 = 2 is their first worked example halfway through volume two, but still we all know it's a trivial theorem. But in applied mathematics it's more complicated because it has to actually apply. So, two people produce a child, 1 + 1 = 3. Two raindrops slide down the windscreen of your car and coalesce, 1 + 1 = 1. You know, in applied mathematics it's different from pure mathematics. It actually has to apply, otherwise it doesn't work. Just a perspective on this. Anyway, Thomas Young realized that if you have a wave near a caustic near the rainbow, there has to be a wave function and he couldn't calculate what the shape of that wave is. That was done by Airy a few decades later and uh here's the intensity that he um of what he calculated. Um the intensity is the square of the wave function. I call it the squarey function. We call this the Airy function. And you see there's the oscillations on the bright side. This is for a single raindrop with a single size and with one color. That's the shape. And on the dark side where you have no geometrical rays, you have an evanescent wave penetrating. Good. Now you have to stretch this and shift it for all the wavelengths and all the drops to get the actual colors. We talk about that a little bit later. But uh anyway, here's his Airy function. Um and the next connection is um universality. The Airy function interference is one of the ubiquitous mathematical patterns. Airy himself realized this because he didn't use word rainbow in the title of his paper. He said interference in the neighborhood of a caustic. And uh here it is. I'll tell you what where this comes from in a minute. Um but going across here is Whoops, excuse me. Go across here. This is crossing the rainbow. And here's the Airy function with its characteristic pattern of oscillations. I want to give you some examples of how widespread this is. You can have water droplets that aren't spheres. You can produce them by acoustic levitation. They're flattened. They're oblate. This was done by Philip Marston and a colleague many years ago. This is a a recent paper. And my colleague John Nye in Bristol explained it. These people used a ultrasonic levitator uh which is a very simple practical kind recently developed by my colleagues in in in Bristol. Well, um here's a an oblate raindrop. And uh a drop of water with its height and its uh its its diameter, its horizontal diameter. And here are the patterns of the caustics. You see, here's the ordinary circular rainbow. And it persists for a quite a long time. It's one of the universal patterns. But um after some time it doesn't it interacts with this cusp that emerges in a particular structure. The name doesn't matter, but it's it it's it's it's a it's a different structure. It changes. When does this happen? It happens when the the oblateness is about 30%, which is more than happens in actual raindrops as they fall, which also aren't quite spherical. But the Airy function persists as they showed. And here you can see uh as you increase the oblateness. And here's the critical case where where it's destroyed. But until then, you've got this nice Airy function. There it is. If I take a piece of bathroom window glass where the undulations are smooth and on the scale of about a millimeter and shine a laser beam through, then this is onto a screen, this is what you see. Typical pattern. You see a variety of patterns like this which change as you move the laser beam through the glass. Um Well, you're seeing caustics here and they're the regions bounding different numbers of The regions on the screen reached by different numbers of rays. The boundaries of those regions are the caustics and you see the Airy function everywhere. Um and that's it's one of these places I can't remember which one it was. Maybe it's this one here which is the picture I showed you earlier. Good. Um Well, it needs a little bit of explaining. Um you can make a circular caustic. Mathematically, it's a Bessel function. And uh This is a caustic. You can see it looks a bit like an Airy function, but it's but this is a different piece of mathematics. It's a Bessel function. And the underlying rays enveloping a curved caustic, there are things called Airy beams illustrating this. But here they are and this was known from 1960 and here's the underneath here's the Airy function showing you the interference and the dark region inside. But uh They look like Airy functions, but where are they? This is a Bessel function. Here is a beautiful piece of mathematics. You indulge me for a moment. Um There is something called a technique of uniform approximation. How to get Airy functions hidden in lots of different mathematical structures. Um starting in the 1930s and was applied to physics in the 1960s, quantum and optical. Um It's fantastically good approximation. I mean, I've I've Here Here are the dots, which are the approximation. Well, what is it? Well, you have a Bessel function, and then you've got the approximation. It looks complicated, but here's the Airy function of some slightly different variable. Actually looking quite complicated, but it's just to do with the difference of the paths, the same as in the rainbow. Now, you might think, "What's the purpose of something that looks very complicated instead of something that looks quite simple?" The reason is this. This depends on two variables, X and N. This depends only on one. And the value of an approximation reduces the number of variables and illustrates the universality. Um I spent an enjoyable year collaborating with Alison Stott, who's a glass artist who lives in Bristol, and she came to me because she was fascinated by glass and light. And she made this structure, it's about this big, which is on the wall of our coffee room in Bristol. Behind this object, this lump of glass, is an LED. And on the backside, there are undulations which she, with her cleverness of glass blowing, has produced. This produces caustics. Here they are, on the front, which is ground glass, which acts as a screen. So, you get this lovely pattern of caustics. Now, where's the Airy function? Well, this is There's a lot of decoherence. This is white light, so of course you don't see fringes. Moreover, the undulations are much bigger than raindrops, so you you the the the the um the fringes would be too small to see even if it was one color. But she's clever, and she had this idea. This has to be connected to the mains. So, she arranged the the cable in the shape of the Airy function. It was her idea. Good. Um Neutrons, quantum particles. If you have a source of neutrons with fixed energy spraying upwards, they spray out and they fall down under gravity, these slow neutrons. And they envelop this caustic. A paraboloidal caustic when you rotate it. Well, it has a name. It's used for something. I'm not going to tell you about it. Gravity focusing spectrometer. But it's a These are quantum particles. So, you expect to see Airy functions. And question is, how big are the fringes? And this is a surprise. Um The distance between these first two fringes, well, there's a formula. It involves Planck's constant cuz it's quantum, mass of the neutron, and G because it's falling under gravity. Now, what it doesn't involve is the energy which determines the de Broglie wavelength of these quantum waves. So, these are interference fringes which are independent of the de Broglie wavelength of the particles. Now, that sounds paradoxical. It isn't paradoxical, but it's a It was an unexpected thing. And I'm waiting 14 years now for somebody to do the experiment. The difficulty is getting the neutron sufficiently monoenergetic not to blur out the not to blur out the fringes. Um Well, how big is this? It's a number, you know. And it's it's almost macroscopic. It's about three three microns. 10 times more, you'd just be able to see it. Uh if this experiment could ever be done, it would be the world's worst way of measuring G. Good. In water, you've all seen this V of the wave behind a moving ship or swimming swimming ducks, these V shapes. Well, this V is a caustic of the rays of the water waves. It follows from the dispersion relation for water. Well, waves on water. So, across here there are Airy functions. Indeed, on my home page well, no. On the digital library of mathematical functions, this picture is illustrating the Airy function and uniform approximations and things. And uh it uh somebody wrote to me from Google Ocean saying that he intends to use it to model the wakes behind moving ships on the ocean. It hasn't happened yet, but that'll be nice. Anyway, it's calculated using Airy functions. Now, imagine you're in a small boat. Here you are. And a big ship passes by. Eventually, its V will reach you. Twice as long as you think because the group velocity with which the pattern moves is only half the velocity with which the waves move. So, you think it's going to reach you sooner, but it does reach you and you rock up and down. As you rock up and down, you're feeling in your body the Airy function, the same one that you see with your eyes when you see supernumerary rainbows. It really is a universal pattern. Um tidal bores, near Bristol we have one. Um that's a caustic in space-time. Uh it's actually it's almost an Airy function. Talking mathematically, it's the integral of one. But, here it is. You see the the front and you see the oscillations. Um the biggest tidal bore in the world, it comes when tides they come when the tide takes a long time from an open ocean moving up a river which gradually gets narrower and shallower and then the incoming tide instead of taking 6 hours to rise the 6 hours to fall concentrates into this universal wave form. Uh the biggest in the world is in China on Qiantang River between Shanghai and Hangzhou and here's one that I saw a while ago. It's it's coming and you see the the waves behind. It it passes you and then it's passed. Very dramatic, very beautiful natural phenomenon. Tsunamis are a similar phenomenon moving across the ocean again described by an Airy function. So universal. Now let's come back to the colors. The connection with color science and of course it was with rainbows that people were interested in when they began to try to understand color. And then the important thing to realize is that color is not wavelength. Color is perception in the eye and the brain. Wavelength is physics. It's the physics of the light incident at each point on the retina. To specify you need to specify infinitely many numbers. How much of this wavelength, that wavelength, that wavelength and so on. Um Isaac Newton understood this. To speak properly he wrote the rays are not colored. And somewhere else he wrote I've no intention of trying to understand the causes of color. Did he realize it? But again it was Thomas Young, the same Thomas Young who understood from some experiments that he did with color that actually we only see three numbers. There are three diff we now know they're cones in the eye. They different sensitivity to wavelengths, and so any pattern of wavelength that comes from physics that strikes the retina at each point on the retina from each point in the scene, um excites these three cones differently and sends three numbers to the brain. So, color is perception from the infinite dimensional space of wavelengths infinitely many wavelengths have to be specified to the three-dimensional space of color cone excitations. So, that's what color um perception is. And to test this, now we've got these computers with these nice screens, to test this, can you reproduce rainbows? And uh well, when I wasn't the first person to do this, but I wanted to learn how to do it about 30 years ago. Then it was difficult. Um it was I used to say this way, it's easy to use these computer screens to uh show colored pictures or to draw um graphs where you have different colors for different features and so on, but it wasn't easy to use color on a screen to represent color. There were lots of tricks you had to learn. Now you can see it all online. Um you have to convert these three numbers to tell the computer which RGB pixels to excite. And there are all kinds of tricks that you have to use to get that right. It's non-linear and so all kinds of things. But anyway, you can do it. And so here's Newton's bow. This is pure ray dispersion. You don't see any interference for big raindrops. There we are. Now, this is idealized because I'm assuming the drops are all one size. There's no decoherence. And also, the other cause of decoherence, I didn't mention it, the sun has a width of half a degree. So, you have to That's not in this these simulations. Okay, other people who are Philip Laven include all these features. Smaller drops, 500 micron radius. You begin to see some fringes, very small ones. 200 microns, 150 microns, 100 microns. So, you can reproduce these colors more or less realistically, although they don't have these decoherence features that so often spoil attempts to see supernumerary rainbows. Now, um actually, real rainbows are very weakly colored. That's a surprise. I need to talk about that. Raymond Lee and Alistair Fraser took photographs of an actual rainbow and carefully measured the color distribution across the rainbow. And And here's their result. You see, this is the space of colors. I told you you need three variables. One of them is the intensity, so you make that one. So, you need two. This is a standard representation of color. Spectral colors around the boundary, and then you go between blue and red along here. But, your TV screens, they occupy a triangular region with the RGB somewhere there or more. So, you can see almost all possible colors you can see on the television. But, rainbows occupy this tiny region here, the rainbow they measured. Why is that? It's because the rainbow colors were diminished by the fact that there's background clouds, which are much brighter than you think. So, although our brains process these rainbows and they look very intensely colored, actually they're not. This is just a few percent of the total area of the color diagram. Now, physicists are playful people. And I want to tell you and show you an anti-Newton bow. Imagine water didn't have a refractive index that depends on color. So, Newton would predict the rainbow would be white. No, because there would still be interference colors. So, let's Well, it's not quite easy to simulate once you have this software, once you write the programs, you can imagine as I said, water is not dispersive, you imagine that. Um raindrops are all the same size, the sun is a point, and there's no background light. So, I'll show you in a minute what the rainbow would look like. But, I want to talk about this kind of playfulness. It's a bit reminiscent of Wallace Stevens. He wrote a poem inspired by Picasso's old guitarist, and he said, "They said you have a blue guitar. You do not play things as they are." The man replied, "Things as they are are changed upon the blue guitar." Well, in this is in the spirit of the anti-Newton rainbow, and there it is. Um only interference colors, this is what a rainbow would look like. And now comes my seventh connection. You see, this was just playful. But, there's Gell-Mann's totalitarian principle, which says, "Anything that physics permits will happen somewhere." Well, where might this happen? With gravitational lensing, because you know, it's a consequence of relativity that mass bends space and effectively makes it have a refractive index. It bends light. And that's his Einstein's famous light bending as it passed by a star. Now, Einstein realized that this would uh mean that if you have a distant star and a star in between, the image could be distorted in interesting ways. But he said, "This will never be observed. It's much too small an effect." Well, we now know gravitational lensing is a very central part of modern astronomy. It's used repeatedly for many, many purposes. It's a small effect, but it can be detected very accurately. Light from a distant object, the source, distorted by mass in between, which may be maybe dark matter, and you can infer its structure by measuring the distortion. Now, the point is that gravitational lensing is achromatic. The refractive index of space doesn't depend on wavelength. Good. Well, that means that if you would see interference near a caustic, and caustics are very common in gravitational lensing because this is not the type of imaging that you get with a point source that you learn at school. You get the these complicated caustics, like the one I showed you if you shine a laser beam through bathroom window glass, different. So, um you would expect to see near caustics By the way, I didn't say this, but uh at near caustics, fringes are bigger, as well, so more likely to be observed. I've seen no evidence yet that uh uh this has been observed. And indeed, it's possible it's been observed and not realized because the achromaticity is one of the reasons why people identify different spots in the sky as different images of the same source because they have the same spectrum possibly red or blue shifted. So, the achromaticity of refraction geometrical optics of gravitational lensing is is used in practice. But, if I've never yet seen any recorded definitive examples of interference from gravitational lensing near acoustic. But, if that's observed, first of all, it give interferometric precision to the inferences that are made in gravitational lensing. And uh secondly, they would illustrate this gravity's rainbow. Change according to the distribution of light. It needn't be in the visible region. It could be microwaves. It could be radio, whatever. But, still that's an idea of the of the um distortion of colors that you would expect near acoustic. I've written about this. Okay, so that's number seven. Now, the last connection I want to make is mathematical. And you please bear with me. Um you My attitude to mathematics is same as Peter Atkins, a writer on He wrote a textbook on theoretical chemistry. Determining where mathematics ends and science begins is as difficult and as pointless as mapping the edge of the morning mist. And we theoretical physicists spend our lives moving in and out of this mist. So, you know, we we're not so insistent on difference between mathematics and physics. I think the difference is a largely cultural, sociological. They have different styles. They like to prove, we like to use. It did different, but it's not really fundamental. Um and I will illustrate that now. See, Airy had his mathematics. It was an particular integral for his Airy function. But he couldn't calculate it. He was in the same position as Descartes of several centuries before who had this theory of deflection minimum deviation, but didn't have the calculus. Well, Airy couldn't calculate except very close to the rainbow. Here's his function. He could calculate this bit that this much by some clever numerics. And he was frustrated. You see, at first he he thought maybe his function is some already known function in disguise. And he tried very hard to identify is it a is it a Fresnel integral? Is it a logarithm? Is it something? He realized it it wasn't. And he it's frustrating him because he couldn't calculate these oscillations. People had measured them in the laboratory with globes of water. Um so he was frustrated that he could just measure this bit near the near the near the maximum. And uh it was Stokes a decade later who wrote a famous letter to his fiance. He said, "I've been doing something that you probably won't let me do when we're married, which is staying up all night working hard on a mathematical problem. A few days ago I returned to a problem of Airy, which she would understand. She was the daughter of the astronomer royal of Northern Ireland. And at last, after two or three days fight, I found the answer." And he was able to um uh approximate the Airy function away from the region that Airy himself had calculated. Look how accurate his approximation is here and also on the dark side. Different approximations on the bright and dark, which I want to explain. So, on the bright side, cosine trigonometry oscillations. On the dark side, an exponential which decays. Um any lesser person than Stokes would be very happy with this. But, he saw a problem, which I will explain. It's the physics of this mathematics. Well, here it is again, bright side, dark side. Here is an exponential. Here is a cosine. A cosine is two exponentials. One complex and another. Now, in physics, we represent waves by complex exponentials. There are two waves. We know they're the waves that interfere. That's the physics of this little formula. And on the other side, there's one, which is a geometrical wave. It's a decaying wave. But, Stokes was very puzzled. Two waves on the bright side, one on the other side. How can one wave become two? He was very puzzled. This is the same mathematical function. I haven't written it to you. It's an integral that Airy wrote down. How could it be so different on the two sides? Now, this is a deep problem. It's now called the Stokes phenomenon. He realized over the course of his life that this applies throughout mathematics. There's a great many cases where you have different approximations in different regions, different numbers of what we could call waves. Um and he realized that these this phenomenon arises because the pre-calculated approximation were just the first terms of infinite series. And I will show you this is the most technical thing in the talk. On the dark side, a single exponential with a infinite series, and I've just showed you before the first term. And he calculated these different terms, and he realized that this series has a property which he didn't like. It's divergent. It starts for large Z far away from the rainbow that terms get smaller and smaller, but after some time they get bigger and they diverge. And he puzzled over this for a very long time. That's what his night time confession he made to his fiance in a 55-page letter, by the way, of which only a few pages have survived. Um and he realized that he traced the divergence. I don't have time to explain this. And the birth of the second exponential to the connection between the dark and bright sides of the rainbow in ways which he sort of understood very deep and he understood exactly where you have to go round in the complex plane, never mind what how. And it was only um a few years ago, a few decades ago, that it was possible to understand exactly what happens and how the divergent part of the series gives birth to the second exponential. And this is now a large area of research in not just in in optics, but in quantum physics, and especially quantum field theory and string theory. So, it's a large area of research. There are conferences based on Stokes phenomenon and the and and the origin of it. We shouldn't forget that it started with trying to understand the rainbow. Right. So, now I will tell you these connections that I've described to you. Numerical experiments as Descartes couldn't do find them couldn't find the minimum. He didn't have calculus. Illusions, there's no arc in the sky. Ray physics is inaccurate and is replaced by the deeper physics of waves. It's a model for how one theory approximates another. You can see it with your naked eyes. You don't see interference all the time. Decoherence, it's a theme now very common in one ingredient in how the classical world emerges from the quantum world. Airy function interference is universal. It decorates all smooth courses, one of the great underlying mathematical patterns. Color theory, and so I've explained it in pure space of wavelength which projects by our retina to the three-dimensional space of color. This Gell-Mann totalitarian principle leading to um prediction of what will happen in gravitational lensing. And uh mathematics of divergent series, lots of things. All goes back to uh understanding this beautiful natural phenomenon. Now, I've described a lot of history during this talk, but it's a kind of history that professional historians don't like. They have a name for it. They did It's called Whig history. That comes from politics. It doesn't matter what it where the origin was. But I'm unashamed of it. It's explaining It's understanding the past in terms of the present. Now, they say, professional historians, you mustn't do this. You must only study the past using sources available at the time. Now, this is clearly very important to cuz it's very easy to get distorted versions of what happened in past times, and they can get this right. But it's not intelligent to restrict your attention only to the past when thinking about the past because the reason why we're interested in those scientists well, they were cleverer than their contemporaries and mostly cleverer than us and it's what they led to that's the reason why we study them. And of course, as we understand more and more about what they did in terms of what we do today, we learn more about the it's a unifying element of our of our culture. So, unashamed Whig history. I'm in good company. Steven Weinberg wrote a book on the history of physics and he said uh uh this is unashamed Whig history. Good. Um Charles Frank, my late colleague emphasized the importance of connections. They're not optional extras. Physics is not just concerning the nature of things but concerning the interconnectedness of all the natures of things. So, summing up, rainbows are a kind of metaphor for how we do physics and what we find. Thank you.