Science & Technology Q&A for Kids (and others) [Part 136]

Sa. Hello everyone. Welcome to another episode of Science and Technology. Q and A for kids and others. And I see whole variety of questions here. I see one from Rudy. How is it that animal species all look relatively similarly similar across a breed? Example, dogs, golden retrievers, yet all humans have unique features? I think that that is purely in the eye of the beholder. I think that we humans have incredibly well tuned capabilities to distinguish faces. This presumably is of evolutionary value and you know, to tell who's who, so to speak. And we just don't have that for other species. In fact, even I will, I will say within humans, you know, if you are kind of, I don't know when I've, when I've traveled to other countries with, with I'm, I, you know, I'm kind of like, I can tell the, the, you know, the westerners who look a bit like me apart much more easily than I can tell other folks apart. I'm particularly bad at face recognition, but I think that's a pretty common phenomenon that when one has seen many examples of similar kinds of kinds of things, it's easier to tell them apart than when one hasn't seen so many examples. And I think in the case of, as I say, we humans really just have very well tuned capabilities to tell human faces apart. And I suspect if you're a, you know, if, if you are, if, if you're a person who keeps a lot of cats, I'm sure you can tell cat faces apart better than the average person. So I, I think it's, it's, I, I would speculate that if you were to make measurements on sort of cat faces, human faces, whatever else, that the same level of that we will be surprised at how small the variations have to be in the structure of human faces for us to tell them apart and that we just aren't as well able to tell apart other kinds of faces. Sabrina is commenting here that they have a fraternal twin who some people say looks identical, but others say look nothing alike. Yeah, even for identical twins. I know one collection of identical triplets where they, you know, to me they look the same, but to people who know them well, they absolutely do not. So I think that's one of these. It's in the eye of the beholder. Now there's this question of, I mean, this whole issue about even the case of identical twins with the same genetics. There are, and I think we talked about this another time, there are these epigenetic features that cause the actual final organism to be different and that's even true at the level of, you know, the fingerprints and the iris patterns in the eye. Those are things which are essentially sort of physical processes that form those particular shapes and so on that go beyond what is just the genome just says things like, you mix these two fluids together and that's how you'll make this thing. But the actual physics of how those mix is something that is determined by all kinds of detailed thermal effects and so on, and is not chemical reaction effects and so on, and has sort of additional randomness from the environment that is not carried in the genome. So there are these things that are features of the kind of just in the sort of construction of the organism as it grows. And then once, you know, in the case of human identical twins and things like that, once they're sort of out and about, then all sorts of differences in the environments in which they happen to exist in the way that they. You know, I suspect when there's a small change, that that change is often amplified because of the effect of that change on the particular personalities of the twins or whatever else. And I'm sure the same is true with litters of animals and things like this, anyway. So my feeling is it's an illusion that these things look the same and others don't. I mean, I would say that this whole question about what's kind of a feature of what we observe versus what's there in reality, this is something that is interesting in lots of different kinds of cases. Something that I think is deeply of interest in understanding how physics emerges in our physics project. All kinds of things like this. But that will get us into another whole rabbit hole here. Let me see the question here from Aaron about E, the exponential constant, about 2.71828, etc. And what it is and how it works. E is not what's usually called Euler's constant. That's a confusing thing. Leonard Euler did start talking about E, the exponential constant, but he has another constant named after him, the Euler Mascheroni constant, which is usually denoted gamma and is about 0.577. So the thing that's called E is the exponential constant. That's not the same thing as what's usually called Euler's constant. I can explain both of them, although actually this is one of these things where it'd be a lot better if I was actually showing you computationally what's going on. And maybe I can just have a try here, but maybe another time I had started talking about sort of explaining math computationally a Few weeks ago, and we got as far as algebra, and I promised I would do this again and get as far as calculus. And as we get into calculus, we will encounter E in an interesting way. But let me kind of roughly explain it. The question is, if you draw a curve and you say, let's say it's a curve, this value as a function of time, and it's some thing as a function of time, it might wiggle up, it might go down, it might do all kinds of things. But let's say it's going up like a power of something to the power of the time. So it's two to the power of the time, three to the power of the time, something like this. So every unit of time, we go from time equals 1 to time equals 2. We're going, let's say that our function is 2 to the T from time. Actually, let's say it's 10 to the T from time 1 to time, 2 to time 3. Every unit of time, the value of the function is getting multiplied by 10. So we'll get 10 to the 1, 10 to the 2, 10 to the 3. That's, you know, 10, 100, a thousand, et cetera, et cetera, et cetera. Okay, so we got this curve. It's going up like this. And we can ask the question, what's the area under that curve? And the answer is that the area under the curve will be some other function. Just like if we had a. If we had a curve that was just flat constant, then the area under that curve is a linear function. Because as we. As we just. We're sweeping out more and more of what's under the curve, and that's giving us sort of higher and higher area. So the area will just go up linearly. Well, in the case of powers, the area goes up like a power, but turns out that it doesn't. The. There's only one power for which the area goes up, and it's the same function as the original curve. And that's the case where. The base is this exponential constant. That's not a very good explanation. We got to see it on a computer. Sorry, this is much easier to understand on a computer. Let's see. Gosh, Brady is asking, in the spirit of Halloween, which just passed, is there any science behind ghostly appearances? Well, gosh, You know, back In Victorian times, 130, 40 years ago, people would constantly be doing things where they would talk about the aura associated with a person. And I think even they would. They would do things like say, let's have this piece of photographic film. We take a photograph and we think the person looks like this. But actually when we develop the photograph, there's all this extra stuff around the person. Now, that's really just, I think, a detail in most cases there of the way that the photograph was made and the way it was developed and so on. But you can ask the question, is there more associated with a person, so to speak, than we can see? And to some extent. Well, yes, because there are, for example, electric fields that are, you know, it's not very big, but, you know, whenever we move our muscles, for example, that's an electrical phenomenon. And so there is sort of an electrical effect that you can in principle detect. In the case of the heart, we do ekg, which is detecting electrically the contraction of muscles in the heart. And we look at brain activity. With eeg, we're looking at the electrical activity of neurons in the brain. And in general, we can measure. If you put electrical sensors on a muscle, you can detect the contraction of the muscle, which is an electrically induced effect with those sensors. So one could certainly imagine one is not paying attention to any of the light coming from a person, so to speak, any of the reflected light. But one could measure just the electrical effects, for instance. Now, the electrical effects are small, and they're usually not measured except by actual contact with the skin and so on. But there is some small electromagnetic effect of kind of the things that cells do and the things we do in our sort of standard operations. That's an example of something where you could in principle detect that. And that would give you a kind of a different view of humans even looking at us in the infrared, where we're looking at the heat generated by us humans. Different parts of our bodies are at different temperatures because they have different blood flow to them, and they have different. Our blood is more or less a constant temperature because it's being circulated around in our bodies. But by the time when it can be cold on the surface because the. The heat is not diffusing adequately to the surface from the internal sort of core temperature associated with the blood. Now, I suppose in. Let's see, we're kind of wondering, you've got your haunted house, for example, and you've got kind of the. What's kind of left over from the folks who lived there before and what's kind of what could possibly still be there, so to speak. And I think much of that is a thing to realize. Okay, so here's a feature of that. You could say, well, you know, in when it comes down to the level of individual molecules, for example, there is surprising amounts of molecules are very small, and there's a surprising amount of mixing of these molecules, or even anything that's very small can get mixed a lot. Like, for example, in the pandemic that we all lived through, you could ask the question, all those little virus. Those virus particles were a hundred billionths of a hundred nanometers in size. They were 1/10 millionth of a meter across, and so very small. And you could ask the question, all those viruses that infected all those people around the world, if you just put them in a box, how big would they end up being? And the answer? Well, I remember early in the pandemic I worked this out, and it was about the size of one's fist. I think probably by later in the pandemic, probably got to be the size of a small box like this. But because they're so small, by the time that you can get, and because just a few virus particles are enough potentially to cause an infection, presumably you can take that kind of box of stuff and you can enough virus particles in there that when you spread them out across the world, so to speak, you can sort of, you can get everywhere with those virus particles. So it ends up, you know, these statements like if you take, oh, I don't know, the. Any old, you know, glass of water that somebody drank at some point, then the molecules. There are enough water molecules in that glass of water, and water gets mixed enough in the kind of what happens in the climate and the atmosphere and the oceans and so on, that in not too long, the individual water molecules from that particular cup of water, the next cup of water you drink is probably going to have one of those very same molecules in it. Just because of the way that things mix around. I suppose the. Yeah, so I'm not really doing very well on making my Halloween theory here, but that's perhaps a few comments. Let's see. Giovanni asks, if an advanced civilization lived on Earth 1 billion years ago, would there be any signs of their existence in today's time? Interesting question. I think that. Well, I would say that let's assume it's our civilization. So let's say we took our civilization and we transported it back to a billion years ago and we built our roads and we built our skyscrapers and we built all these kinds of things. Would there be any sign of this a billion years later? I think the answer is yes. Just as rocks basically last a long time, and when things get turned into fossils, they get Turned into fossils, usually because there's some soft thing. And that soft thing kind of makes an impression on where the rock is going to be formed, and then the rock really stays there for a very, very long time. And rocks. We certainly have rocks. Well, on the moon, the rocks from. From when the moon was formed 4.6 billion years ago are still exposed on the surface and the Earth for a couple of reasons. One, because of the atmosphere and erosion and so on, and two, because of the liquid core of the Earth and kind of volcanic activity and so on, the rocks tend to get sort of turned over. And there are fewer places on the surface of the Earth where there are rocks from billions of years ago exposed on the surface. But the rocks still get. They maintain a lot of their integrity. And that's why we can still find fossils going back, okay, it's microfossils back from 2 billion years ago, but it's still. We can still find sort of evidence of what was happening, sort of as defined by its impression left in rock, because rock just stays a long time. And the things we've built, the concrete in our buildings and things like this, the stone walls we built and so on, it's just rocks, and rocks last a billion years. And so I think we would see that kind of evidence of what our civilization looked like. I mean, if we imagine that sort of all the people disappeared and everything was left derelict, so to speak, oh, all sorts of things would happen. I mean, if the plants are still around, there'd be a lot of encroachment of plants on the infrastructure that we leave behind. A lot of things would rust and decay. But the things that are rock, like concrete and so on, or for that matter, stone buildings, would kind of stay up to the point where things, you know, there might be earthquakes that would make the stone buildings fall down. There might be gradually. You see, in places I've seen in very old stone constructions where trees grow through the stone construction and eventually push things out of the way. But that's a fairly gradual process. And my guess is that a lot of the things we've made that are sort of stone, like, would stay for a really long time, and that the things we've made would be preserved. Now, people worry about that, examples of things which one wants to be preserved. Like, for instance, in places where there's radioactive waste that's been tried to be stored, it's like, can we make a storage system that will sort of stay, keep its integrity for. Well, then it's more modestly, 10,000 years or something, and how will that work out? But I think the more challenging issue is, okay, given our civilization projected back a billion years, and would we, you know, would we be able to find sort of our civilization? I think the answer is probably yes. But if you say, could we find that there was a civilization or something that we would call a civilization from millions of years ago and, you know, could we tell that that happened? I think that's much less convincing. I think that when we look at the dinosaurs or something, and we see that there's some place where there's lots of dinosaur footprints and we find them, and then you could say, well, was that a ritual dance of the dinosaurs where they were worshiping some dinosaur God, or was that just a herd of dinosaurs doing something like what we think modern animals are doing? I think that's really hard to answer. You know, if we say, did the dinosaurs make artifacts that are recognizable to us? That's something where we can say, well, doesn't look like it. But we say, artifacts recognizable to the dinosaurs or sort of a. This is a very challenging thing. I mean, we see that now, oh, for example, these pufferfish that spend a week kind of making these very elaborate mandala, like patterns in the sand, and then they destroy them again, but they make this very elaborate pattern. And maybe those get fossilized. And you look at one of these things, you say, is that a mandala that was made by the, I don't know, Buddhist monks or something? Or was that a pattern made by this pufferfish? And you say the Buddhist monks are doing the thing which is very much derived from a whole elaborate course of civilization and so on. And the pufferfish. Well, we assume that there isn't an elaborate sort of oral tradition of pufferfish passing down the how to create this mandala type pattern, but it's some kind of genetically determined feature of pufferfish that they can do that. But I think it's a really subtle thing to know. It's a very interesting question, you know, know, how does one know was something created for a purpose? Really? That ends up only being defined when we start sort of having a way of connecting the purposes that we internally understand to what we can see happen in that system. And, you know, this relates again to things like AIs, where the AI is just doing something and, you know, it's computing something. And we look at it and we say, is it trying to do something? Or is it just randomly flapping around doing sort of arbitrary computations? And that question of sort of Is it aligned with our human kind of conceptions of what one should be doing? Is. That's a challenging thing to define, but it's an important thing to define if we're trying to say we want AIs to only do things which are aligned with certain things humans want want to do, versus we'll let them do arbitrary computations and so on. So just a few thoughts about that. Let's see. Well, all kinds of questions here. Peter is asking, how does photography work? How are we able to capture an image so easily, whether on film or on a phone screen? So what's happening in photography? Basically, we're trying to take light and we're trying to somehow have the fact that light fell on something, be a thing that's permanently recorded in the thing. So what is light? There are different ways to think about light, but we can think about light as a stream of photons, a stream of particles of light. And we can also, in a sort of complementary way, we can think about it as this kind of electromagnetic wave. But for these purposes here, let's think of it as just a stream of photons. And so the question is, a photon falls on something, how does that change the nature of the thing? Okay, so what was discovered back in the 1800s was that if you had a particular chemical silver iodide, that when a photon of light hit that piece of silver iodide, it would take that molecule, which was a silver, and an iodine atom, and it would somehow kick the silver atom out. And so it would make a chemical change basically to that molecule. So you would cover a piece of paper or something with lots of silver iodide, and then you would have. Then you have the photon of light would. In some places it would hit that. Whenever, whenever there was light, it would. Whatever light fell on that particular. On that piece of paper, and it. It would hit this sodium silver iodide, and it would kick the silver atom just a little bit out. Okay, so then what you had to do was to develop the film. And that's a chemical process that would take the. Take that small chemical effect and make it actually show up as a change of color. And I'm now forgetting how the change of color worked. But basically the idea is you've made a chemical change, and that chemical change can have the effect of changing the color of what's on the page. Well, what typically was done was in places where the light fell on the film, the iodine was kicked out and the film went black. In places where the light didn't fall, it would stay white. So if it was a film, so it was transparent. And so in places where the light didn't fall, it would stay transparent. In places where the light did fall, it would go black. Now, that's obviously the negative of what we want. We want it to be lighter where the light fell and darker where the light didn't fall. So that meant that what you would get initially is a negative. And then what you had to do was to make a positive from the negative. And you would do that by essentially using this same photographic process. Again, you would take the negative, put it on another sort of piece of paper, and then you would shine light through the negative. And then the places where the light didn't fall would stay. Is that the right way? Is that right? You would. Gosh, I even did this when I was a kid a bunch of times. You would take the negative film, which was what was sort of in the camera was this film that when exposed to light, would turn black, when not exposed to light, would turn white. And then what is the process? I mean, it's another chemical process that allows you to sort of reverse that again. I think it's a very similar process to the first one that allows you to reverse that again and get the sort of positive picture. And that was for black and white photography. When you're doing color photography, the idea is that, well, for the human eye, we really just detect the redness, greenness and blueness of light. Light can have any kind of sort of configuration of energies of photons and so on, but we just detect the ones which are kind of in the red bucket, green bucket, blue bucket. And so one way to deal with that is to just say, well, let's make three. Let's just have kind of three separate. We'll just use a filter. We'll have a blue filter, a green filter, a red filter. And those filters will let through only the red light, only the green light, whatever. Then we have. On our final film, we have. Well, one way to do it was to actually have three passes where we're having something which will put the red color, the green color, the blue color down. I mean, in a sense, that's what happens in, like a laser printer, color laser printer. You've literally got, you know, the granules of. Well, it's actually for what you. When you print, it's not red, green, blue, it's cyan, magenta, yellow, which are the complementary colors. But you've literally got, you know, a thing of toner granules of something in Each of those colors. So, okay, that was a little bit on how photography with film works. In the case of something like a digital camera, the way that works is what you're doing is instead of getting an actual sort of image fixed on the, on the piece of paper or the film, what you're trying to do is you're trying to say we have all these pixels in our image. We might have, I don't know, in a modern camera, there might be 10 million pixels. So we're dividing the, you know, the image. The lens of the camera causes what you see in the outside world to be an image of that to be formed on essentially the retina of the camera. On the, on the, on this detector that's an array of little tiny cells, let's say 10 million of them. Okay, so how do the individual cells work? What they're trying to do is produce an electrical signal when light, when a photon of light falls on that particular square on that particular pixel in the array. And these are usually, they're called charge coupled devices. And that's the most common type of detector. But essentially what's happening is it's a little bit like the silver iodide story where the photon comes in and it kicks the silver atom out. In the case of these kind of charge coupled devices, what's basically happening is the photon comes in and it kicks an electron out. It's essentially the. Well, it's kind of an analog of the photoelectric effect, which was the big giveaway that photons existed that Einstein talked about in 1905 and so on. But it's sort of the same idea. Photon comes in, it sort of kicks an electron out. We then get to detect that electron being sort of kicked out as an electrical signal. And so then we know that the pixel at this position kind of had a photon fall on it. So that's. So light has fallen there. And the way it works with color, color cameras is there's a bunch of pixels, and some pixels have a little red filter in front of them. Some have a green filter, some have a blue filter. And you're just looking at which pixels did you get? Did you get in a particular place? Did you get. Was it the red pixels that had the electrons kicked out of them, or was it the green pixels? And that tells you what the color is? Because it's. If it's like, you know, red and green together, you've got yellow light that will produce both red and green will be active and so on. And the. So in fact, what that Means is that, for example, if you change this kind of filters that are in front of these pixels, you could actually detect a larger range of colors. For example, you could more finely detect what the actual photons that are arriving are. And, and that gives you what's called hyperspectral imaging, where you're able to say more than just saying, oh, it's roughly red or it's roughly this color as humans perceive color. And that's relevant if you're trying to identify a material. Lots of things where to us it's just, oh, that's white. Well, actually it's not really white. It's just that it has roughly the same amounts of red, green and blue with our definitions of red, green and blue. But if you were to look more finely at kind of the frequencies of light that are produced, you would see some whole complicated pattern of that and you would be able to much more tell what that thing that just looks white to us is made of. Just like we can tell some substances apart by the colors that we perceive, there's a sort of finer grained version of that. And it's been a thing where people haven't yet produced these in a really industrial way. Hyperspectral cameras. You can get hyperspectral cameras, they're used particularly for things like telling whether food has gone bad or not. Because it's like, well, I've got something and it looks like this piece of meat. And it kind of looks, to us, it looks kind of the same, it doesn't look any different. But if you look in a more fine grained way, in a hyperspectral way, then you can tell that it really changed color. And people had imagined that you could do that for things with dermatology, telling whether, know that spot on your skin which just looks brown to us, what its precise spectrum is, and whether you could tell was that a bad spot or a spot that didn't matter or whatever else that didn't, I think pan out quite as well. But certainly for things like food, telling whether you know, food's good or not, hyperspectral imaging is used. Hyperspectral imaging is used for some other kinds of material identification, telling where the crops, well, the things like where the crops are growing, you tend to more need to use things in the infrared and so on. But again, this idea of you put this kind of mask in front of the actual pixel detectors and that tells you the color. So that's how that roughly works. And by the way, it's incredibly similar to the way that our eyes work because at the back of our eye, in our retina there is an array of photo detectors of photoreceptor cells. It's roughly, they're all, they're kind of roughly circular cells and they pack them in and there's kind of a hexagonal array because just like coins, you squash a bunch of coins together, they'll tend to squash into a pattern that's roughly hexagonal. And same is true of the cells in the back of our eye. And they're sort of this array of photoreceptors. And depending on what position the photoreceptor is in on one's eye, that's saying, oh, there's a piece of light that came from that part of our visual field or a piece of that came from that part. And in the case of our eye there are, we have three kinds of so called cone cells, three kinds of photoreceptors that detect different frequencies of photons, different colors of light, and those correspond to the red, green, blue, sort of buckets of photons. So that's basically the idea. And when it comes to other kinds of other kinds of ways that you would sort of take visual input and do things with it. Another case is things like photocopying or laser printers. What happens in photocopiers is you've got your page and what's done is that you use, I think it's usually selenium, you have a kind of a slab of that material and you give it. How does this work? I think you give it an electric charge. And then whenever light falls on a particular place, it conducts the electricity and so it gets rid of the electric charge in that particular place. So you've got this plate and wherever light falls on the plate and you charge up the plate. And then whenever light falls on a particular place on the plate, that charge is conducted away. And when it doesn't fall, the, the plate stays charged up, stays electrically charged. And then you have this thing called toner, which is essentially plastic, tiny plastic granules. And you're kind of dusting those across this plate. And in places where there's an electric charge, they stick. In places where there isn't, they don't stick. And so you get this plate that's then kind of the image of your original piece of paper or whatever. And you've got these pieces of toner that stick on the plate in places where. Stick on where still electric charge. So you've got this thing, it's still very fragile because just A bunch of plastic granules sticking on top of this plate. And then what you do is you bake those plastic granules onto a piece of paper. So you put a piece of paper in there, and they're plastic granules, and they're sort of sitting there. And then you heat it up, and those plastic granules melt onto the piece of paper. And that's how you get your results. And so that's why when you have a laser printer, the paper comes out hot because it just had those plastic granules baked onto the paper. And again, this idea, if you look inside a color laser printer, you'll find there are. Well, they're usually four little receptacles of toner. There'll be cyan, magenta, yellow, and black. Those are the different colors that are used that you can make sort of. The black is sort of used for convenience that you add that in. In principle, you could make that by combining the cyan, magenta, and yellow, but it would be a muddy black, typically. And it's better to have just a plain black. By the way, it's reflecting very much the way that our human visual system works, because in addition to the cone cells that detect red, green, and blue, we also have rod cells that essentially are detecting black and white. There. So that's a rough introduction to how photography works. Let's see. Question from Jamie. How do mirrors work? Why do they reflect things? Mirrors. What's happening in a mirror is photon comes in going in one direction, photon goes out in the opposite direction. So normally, anything, any material is made of a collection of atoms. And when photons interact with atoms, they do it well. Okay, photons, I should say. Okay, in an atom. It's a little bit more complicated here. So photon is coming in. It's interacting with material. I should have said. And sometimes the photon is kind of interactive. The way photons interact with matter is through the energy of photons. That corresponds to visible light is through kind of kicking the electrons that exist in the matter. The nuclei of atoms are not really. They don't really pay attention. The energy of photons of visible light is not enough to do much to the nuclei of atoms. If you have gamma rays, X rays, things like that, you can do things to the nuclei of atoms. But with visible light, you only get to sort of mess around with the electrons in atoms. And so what happens is, in a typical material, one always thinks of an atom, it's got its nucleus with protons and neutrons. It's got electrons kind of outside the nucleus, kind of Going around the nucleus, in some sense, the nucleus is 100,000 times smaller than the typical sort of distance from the nucleus to the electrons. And so. In different kinds of matter, one has this picture of nuclei of atoms with electrons just hanging out around that nucleus. Or a different thing that can happen, which is that the atoms can share their electrons. And so in general, when you have, I don't know, a molecule like water, what's kind of the electrons are shared between the oxygen and the hydrogens in the water. And that's kind of how the water is, how those atoms are kind of glued together in the water molecule. Now, metals have this feature that in a metal, the electrons are kind of shared across all the atoms, or at least some of the electrons are shared across all the atoms, the so called conduction band, where instead of the electrons being localized to particular atoms, the kind of atoms are close enough together that sort of all the places where their electrons hang out, they all merge. And the electrons kind of are free to float from one side of the material to another because they're not attached to an individual atom. And that's why metals conduct electricity, is because their electrons can kind of roam freely. And that means when you put a voltage, which is kind of a push to the electrons, electric field that pushes the electrons, they can kind of roam freely through the material without getting stuck atom to atom. Whereas an insulator, as opposed to a conductor, something like a, I don't know, a piece of plastic or some piece of wood, something like this, the electrons are stuck within their atoms, and try as you might to push them out of their atoms, you can't do that. Eventually, if you put a strong enough electric field, the material will break down and the electrons will be ripped out of their atoms. But in the normal course of things, with just a little battery or something, the electrons will be stuck in their atoms and you won't conduct electricity. Okay, what does this have to do with mirrors? Well, in a mirror, you've got a metal, and what's happening is that the electrons, the photon comes in and it's kind of. Let's see if I can actually explain this. It's. This is a little bit subtle. The. Well, all right, let's try this. All right, so first of all, we have to admit that light isn't quite just photons. We can think of light as being a stream of these particles of light, but kind of another way to think of it is as an electromagnetic wave, where we have electric fields and magnetic fields, and they're kind of continually Kind of oscillating in free space. And that's. That's an electromagnetic field is light can also be thought of as this wave of, you know, magnetic field gets big, electric field gets big, magnetic field gets big, electric field gets big. And that kind of wave of electromagnetism is another version of light, as opposed to thinking of it as it's a sort of a blob of electromagnetic energy that's in this photon. These are two complementary ways to look at light. They both, you know, you can kind of see that they might have some correspondence because the photon is this kind of blob of electromagnetic energy. The wave is this kind of description of the light as this kind of whole collection of pieces of electromagnetic energy. And you can imagine that the sort of stream of photons that a blob. Blob, Blob, blob of photons is sort of something that you could view as being a bit like this thing that is the electromagnetic wave. This correspondence between these two things is something that gets more clearly understood in quantum mechanics. But we don't need to go there for these purposes. So we've got this electromagnetic field that is the light that hits this. This metal plate, let's say, this mirror. And what happens is that the electrons, as a result of when this electromagnetic wave impinges on the mirror, it's causing the electrons to be moved around in the mirror. The electric field is pushing the electrons in a certain direction. Magnetic field is moving them around. What ends up happening is that, gosh, this. This relies on several different things. Okay? So another fact is that when you wiggle an electron around, it produces an electromagnetic wave. So that's how radio waves work. You're taking in the antenna of your cell phone or whatever else. What's happening is, in the case of a cell phone, like a billion times a second, you're, you know, you're taking a voltage, you're reversing it. You're taking voltage, you're reversing it. It's making the electrons go back and forth, back and forth inside the little wire. Essentially, that is the antenna in the phone. And every time the electrons are made to wiggle back and forth, that wiggling process causes an electromagnetic wave to be produced, causes a radio wave to get generated. The same is true with visible light. The same is true with electrons in the metal. They're kind of wiggling around, and they're generating an electromagnetic wave from their wiggles. But what's happening is the. The electromagnetic wave that comes in is generating wiggles and those wiggles are now generating another electromagnetic wave. And it so happens that the way the math works out, the wave that comes in produces wiggles that are precisely the ones that you need to get a wave that's going back out in the opposite direction to be what those, the wiggles that are induced by the wave coming in cause a wave going out in the opposite direction to be produced. Another way to think about this that maybe is not quite as accurate is to just say they're photons and they're hitting this mirror and they're bouncing off. Now, you know what happens is in a mirror, let's say it's the photons, they exactly bounce back in the same direction that they came from or they rather actually more accurately if they're, if they hit it just straight on, they bounce back in the same direction. Otherwise they follow the laws of momentum basically. And if they hit at an angle, they bounce off at sort of the opposite angle. It's kind of like a, you know, it's an elastic, you know, the billiard ball that hits the side of the table and bounces off, reversing, kind of coming back towards you, but continuing in the direction that it was going in. So that's called specular reflection. And that's what happens in a metal. And other materials are more complicated in the way that they reflect photons. And for example, a typical thing is that the photons come in and for instance, so called Lambertian reflectors, materials that aren't shiny, that aren't mirror like they tend to. Photon comes in and it's like photons on average come out at all possible angles. That's kind of like a diffuse, it's kind of like you can see it because there's light being reflected, but it's not being reflected in this mirror like way. So that's how it's roughly. And so there's a whole story, well cataloged in computer graphics of the different ways in which, for example, light can come in. And most of the photons come out in the specular direction, but some of them don't. And they come out in this kind of cone that's close to the specular direction. They're usually some numbers that characterize how much spreading there is and those very much affect our perception of different materials of is it shiny? Is it not, Is it these different kinds of things? But the question of whether what we see in a mirror is what's really there. Yeah, we don't have a lot of choice in the fact that the photons are essentially being sort of reflected, or the electromagnetic waves are being sort of regenerated, going back in the opposite direction. And there's nothing really that can be done with that. Now, having said that, there certainly are things that you can imagine happening. So, for example, let's say that that mirror was not really a mirror. It was really something where every little tiny piece of that mirror is like a pixel. And it's actually, for example, a mems, for example. Oh, my gosh. What does that stand for? It's a. Okay. So some materials, if you put an electric field on the material, it will change the mechanics of the material. So, for example, you put an electric field, the material, the distance between atoms will get slightly larger, or you put an electric field. That's a typical kind of thing to have happen. And so there are mems, which I've now forgotten. Micro. Oh my gosh. Micromechanical, but I'm now forgetting what the acronym stands for. But anyway, micromechanical devices where electric fields, voltages can change the mechanics of the device. So for example, the. The thing that. So some projectors, computer projectors, work this way. They have a mirror, and it isn't really an ordinary mirror. It's a mirror that's broken in tiny little pixels. And each pixel has this kind of electrical effect. So the mirror can effectively be turned by voltage. Just like you could change the switch on or off a pixel on your screen by changing. In that case, you're changing the direction of molecules in a liquid crystal, for example, or you're doing something else where little voltage for that particular pixel is making a change. So similarly, on this mirror, you can have a mirror that isn't really a mirror because every pixel is sort of active and being addressed. And you can put a voltage there or not put a voltage there. And the mirror doesn't show what the mirror could have shown. Now, another thing that can happen with mirror like things is holograms. So in a hologram, what you're doing is normally when you reflect light, it's like whatever light comes in, it just gets reflected. But let's say that there were details of how you reflected the light that were small enough that they're like the wavelength of light. They're about the size of the wavelength of light. So normally light follows. We're used to light following what's usually called geometrical optics. You see it very clearly with lasers. You have a beam of laser light. You reflect the beam of laser light from a mirror, there'll be another beam that comes back from the mirror. Most of the time, light just goes sort of in straight lines and it gets reflected from mirrors and it's does all these kinds of things. Okay, so because light can be thought of as made up of this wave, there is also a diffraction effect, interference effects. And you can see that. Convenient place to see that is if you look at ripples in water and you see, let's say that you've got ripples and you've kind of got this baffle where you're trying to prevent the water from getting through. So you've got these ripples coming up to this baffle, and then the ripples go out. And in the middle between those two sides, the ripples will just keep going straight forward. But if you look at an angle sort of coming out from that opening, there will be little ripples that go off to the sides. So in other words, you wouldn't be able to. They're not going in a straight line. The ripples are not just going straight out of the port, whatever. If that was in the ocean, the ripples are kind of going to the side. And that's because. Well, because effectively, the way you think about it is there's a wave that's kind of wiggling the water. And at every point, that wave, when the wave arrives somewhere, it makes a wave that's actually a circular wave. But when you combine those circular waves together in the place where all the waves were kind of lined up, they just keep going in that direction. But in places where there weren't other waves, a little piece of sort of circle gets left over and you get kind of their ripples going out in the direction that wasn't the straight line direction. Same thing happens with light, but it happens. That effect happens only on length scales that are of order the wavelength of light. So typical light, let's see, red light is 500nm, which is. Half a millionth of a meter is the wavelength. So when we talk about these electromagnetic waves, they are the sort of. The distance between the wiggles and the wave is to make red light is about half a millionth of a meter. And so that. And so this diffraction effect only happens when you have like, openings that are comparable to the wavelength of light. And so diffraction is. And so what you can do is effectively have a. If you have something which is like a. Like a mirror or like a piece of film that can transmit light, if you have features there that are comparable to the wavelength of light, you won't. It won't be the case that light either goes straight through or gets, or gets blocked. It can be the case that it diffracts. So it has this, it can sort of go not in that straight line direction. And in any case, the way that you make holograms is by essentially picking up, well, it's, it's kind of picking up the wave structure of light and you can make kind of a mirror that essentially causes you to reconstruct the exact pattern of electromagnetic waves that would correspond to some three dimensional object. I'm going to have to explain a bit more to really get this clearly explained. But just to say, just for sort of, for fun, you could imagine. We don't know how to do it yet, but one can imagine making a hologram in real time where you have these little micro mirrors and they're so small that they're comparable to the wavelength of light and they're all wiggling around. And they're wiggling around fast enough that you can actually make this sort of 3D movie projected out into empty space by making sort of the pattern of electromagnetic waves correspond to the pattern that would be there if the actual physical objects were sitting there in the space. So, ah, mems. Microelectromechanical system. Thank you. I knew there was micromechanics in there, I just couldn't remember. Microelectromechanical. Thank you. I'll try to remember that. Let's see, maybe one more question here. Timescolor is asking, is there a number like E or PI that instead of being small under 10, is big? How do these numbers get specific notation and names? Well, you know, you do lots of things in math and some things show up repeatedly. Like PI shows up all over the place. It's the ratio of the circumference to diameter of a circle. It's the thing that shows up. You know, if you say, are two numbers relatively prime, do they have divisors in common? Things that divide them in common? Numbers that divide them in common? Well, the fraction of numbers that aren't relatively prime is 6 over, that are relatively prime is 6 over PI squared. PI showed up again. PI shows up all over the place. It's a really common thing to show up. E shows up pretty often. There are other constants I think I mentioned before. The Euler Mascheroni Constant Gamma is 0.577. That shows up much more rarely. I mean, I'll tell you where that shows up. If you take the sum of 1 over n with n going up to infinity. Well, that Sum diverges, it's infinite. But if you subtract log N. Oh, I'm sorry. If you take that sum up to an upper limit of, let's say K, and you subtract log k, the limit of that, as K goes to infinity, is that 0.577 number. And that number appears somewhat often, not as often as e and PI. And there's this sort of ranking of which kinds of numbers, which sort of packets of mathematics occur repeatedly. And that's really what determines the things we give names to in mathematics. It's very similar to the way we give names to things in natural language. By the time we've seen a chair a bunch of times, it's worth giving it a name. By the sum we've seen a live stream a bunch of times, it's worth giving it a name. And so it's the same thing in mathematics. When things in the natural course of mathematics show up repeatedly, they get given names. So the addition, very common operation get given a name. Exponentiation number raised to a power, it's given a name. The operation of repeated exponentiation is much rarer. It's usually given a name. It has various different names like power towers and tetration and so on. It's kind of on the edge of having a name because it's on the edge of being widely enough used that it's worth it having a name. And in a sense, any quantity you can imagine, you can work out any number 0.57, 2, 4, 8, 6 4th blah, blah, blah, blah, blah, going off to infinity. You could give that number a name if you wanted to. There are obviously an infinite number of numbers like that. So you'd end up with an infinite number of names, and that wouldn't be terribly useful. But the place where you give names to things is where they appear repeatedly. Now it's a question of the structure of mathematics, sort of how common a particular name is. Here's an interesting example. If you go to WolframAlpha and you type in some number 0.52, 3, 4, 6, whatever it is, one of the pods it will produce as output is it will say possible closed forms for this number. Okay, so let's say the number we typed in was 0.75, okay? So most people would know that that's three over four, three quarters. Let's say we typed in the number 0.77, okay, that's 77 out of over 100. But maybe 0.77, I don't know. Let me give you another example 3.14159, okay, so you could write that as 314,000, blah blah, blah, blah, blah, divided by some denominator. It's also roughly PI. And so our little thing for estimating, sort of guessing what the closed form formula for a number is, it tries to trade off, oh, it's this thing 3.314159 divided by a million or something versus it's just called PI. And it's sort of an interesting thing because any given number, there'll be some formula for that number, at least to the number of digits you've specified. It might be the square root of the cube root of seven minus the fifth root of six, et cetera, et cetera, et cetera, some very ornate thing. Or to that precision, it might be a named constant. So there aren't that many named constants. I mean, the ones that are really quite common, PI is probably the all time winner followed by E. There's also the golden ratio, square root of five plus one over two. That's somewhat common. The Euler Mascaroni constant, a thing called Catalan's constant that shows up in working out counting the number of trees that you get. There's another thing called Kinshin's constant that comes up in continued fractions. But by the time those are the ones that are in Wolfram language and Mathematica is built in things because those are the ones that we're really tailing off in how common they are. By the time you're at Kinshin's constant, it's really pretty rare. It's not a very commonly occurring thing. So this question of what gets a name, it's things that occur commonly. There's a big debate on whether PI over two should have its own name. Sometimes people say, well, PI is this thing with two legs and maybe we should call tau the thing with one leg, namely PI over two, so to speak, the half of the PI. And actually we did an analysis a number of years ago of whether that actually makes sense. Because you can just look at all the formulas that are out there that people have put in papers and things like that and say, when PI occurs, does it occur divided by two or not? And the answer was it's not really that convincing. PI occurs more often as three times PI, this thing, not always PI over two. So that's kind of the story. It's sort of an interesting thing to ask. The question sort of, what can we learn about mathematics by looking at the distribution of these packages of mathematics that occur repeatedly similar to what do we learn about human language? We know in human language there's this thing called Zipf's law that says that if we rank words in English in order of how commonly they're used, like the. And a and so on be very common words. And you know, rhinoceros would be a less common word. And what is that word? Flossy, pocky, nihilification, which is some semi made up word, but that will be a very rare word. But it's in many dictionaries, I think antidenarianism, that's a common long word. That's pretty rare, but lots of words that are rare. And if we rank these words in terms of the most frequent word to the least frequent word out of the 50,000 words that are common in English, for example, it turns out that the nth most common word in English has frequency roughly 1N. It's called Zipp's Law. Nobody really knows why. It's true. There are some possible theories for it, but that type of distribution is very common in lots of places. And I remember we worked this out, oh gosh, 20 something years ago, we worked out that same distribution for mathematical constants and. No, actually we worked it out. I'm sorry, we worked it out for letters that people use like alpha, beta, gamma and so on people use in mathematical formulas. So I'm not sure. I don't think there are quite enough mathematical constants to be able to be able to work that out. But my guess is that if we go from PI to E to these other things, that we'll see a similar kind of less and less frequent use of that particular thing. So it's an interesting question whether actually it's kind of interesting, I should think about this. The. If you look at mathematical formulas that show up in practice and you look at sort of clumps of stuff in those formulas, what, and this is like the PI versus Tao question, what are the lumps of stuff that appear repeatedly in these formulas and what's their frequency? And so on. And I think that's. And does that give one some indication? Is it the case that if you have a meaningful mathematical paper, it's probably going to have this particular distribution? Let me give you an example of something like that. This is a very weird fact. There's this thing called Benford's Law. And what it says is this. If you have a bunch of numbers that show up in, I don't know, accounting statements, you've got numbers of dollars or you've got some other, or you're looking at a table of physical constants, and there's different constants that appear and so on and so on. Okay, Weird fact, the leading digits in those numbers are not equally frequent. So, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Well, up to 9, they're not equally frequent. One is more common. And there's a distribution where one is the most common and it goes down to 9, which is the least common. And so this is something where when you just take numbers and you're adding them up and you're doing computations on them, the reason it happens this way is because when the number gets bigger, okay, so you're doing things on a number like seven, and eventually you'll add four to it and make 11. And now it's got leading digit one. And this whole process of sort of adding things to numbers and so on, it means that the numbers tend to be, well, they're kind of. It's not the digits that are equally distributed. It's kind of the size of numbers is equally distributed. But when you turn that into digits, it is more common to have the number start with a one, because that's the sort of. More how to explain this. They're sort of. By the time you've got to nine, you've almost got a bigger number. And then it would start with a one. Again, not a great explanation. But in any case, the result is this law that is true for many numbers that just sort of appear from doing arithmetic and things. It's not true for all numbers. It's not true. For example, the primes, I don't think follow Benford's law. But it's true for a lot of numbers that are just made by doing standard arithmetic operations. So that means that it's much more common to. For example, if you're trying to work out, did somebody cook these numbers up? Did they fake them? A good test is, do the numbers follow Benford's law? Actually, rather embarrassingly, during the pandemic, there were places where you could look at the reported numbers of cases and things, and they definitely didn't follow Benford's law. And this is a bad sign of these numbers weren't really what they should have been, so to speak. They were just kind of made up. But this is a thing that people use as a kind of test of whether, you know, when you've got a bunch of accounting data, you know, were these numbers just made up or were they actually the result of the actual operations of a company or something like this? But that's another place that's a place where just because of this sort of structure of numbers with digits, you get this different number. So, you know, ones are more common than nines and so on that. Anyway. So this whole question about sort of how packages of mathematical functionality occur, when they occur repeatedly and so on is something one could study, and it's been studied a bit. There's probably more that could be studied about it. In a sense, this question of when do repeated lumps of mathematical work, computational work, occur? That's in a sense the role. Well, in human language, we're making up words whenever a repeated concept occurs in mathematics, we might make up a notation when a repeated concept occurs in computation. That's the role of language designers like, well, me to figure out in a computational language, when are there repeated kind of lumps of computational work which we should give a name to and implement a function to do that. And that's kind of the. And again, with Wolfram language, there's also a distribution of how common different functions are in typical programs. And I think it's a Zipf's law type distribution again, where things like table are really common and things like fold, pair, list are much less common. Why exactly this happens, it's really unclear. The fact that there are these kinds of quantitative laws of meaningful stuff probably is very related to why there are all sorts of scaling laws and so on that happen in LLMs in large language models and the innards of chat, GPT and so on. There are all kinds of quantitative results that are known that are important in figuring out, for example, how much effort it's worth going to, to train the model and how long it's going to take and all those kinds of things. Things. And I think. That's kind of a. So this kind of the frequency of things when things are meaningful is something we don't understand very well. But there seem to be some sort of universal laws of that. Okay, Robot Watcher is asking, how many digits of PI can you recite? At this moment, not very many. 3.141592653, blah. So I only made it to 809. One question is, why are those digits so random? And we know that the digits of PI will never repeat. That was proved about 150 years ago. We know that the digits of PI will never correspond to things like square roots, cube roots of numbers, and so on, but we don't know much else about the digits of PI. We don't know for sure that, for example, every digit 1 through 90 through 9 occurs with equal frequency. It probably does. People have looked at that up to now. There are trillion digits of PI that have been computed, and it looks like all the digits occur with equal frequency. But nobody's ever been able to prove that. And for example, there are places in PI where you have the 00000 for a bunch of zeros, and there's a certain probability for that to happen where there are n zeros. The probability is roughly 10 to the minus n. So it seems, because the digits seem to be being generated randomly, actually the fact that you can kind of get every string of digits in PI leads to some fun stuff. Like a few years ago on PI day of the century, that's three 3-14-15. March 14, 2015, we made this website still running, called mypi day.com, where you type in your birthday and it will find where in PI is your birthday. So, for example, if Your birthday was 3, 14, 15, be right at the beginning of PI. But if your birthday was whatever, 8, 29, whatever, it would be much later in PI. And for any birthday, you can find it in PI. And it's believed that for any sequence of digits, you can find it in PI. But nobody's ever proved that. Nobody knows that for sure. It's one of the. It's an old unsolved problem in mathematics to prove that. And it's not known for any of those kinds of numbers, like square root of two or the exponential constant e, any of those numbers. It's not known that the structure of the digit sequence is not known that well. And it's kind of an interesting thing because it's sort of a case where even though the description of PI is quite simple, the digits once generated seem quite random. This is a phenomenon that I've studied a lot in computational systems, actually a very common phenomenon that even though the rules are quite simple, the things once generated seem random. But any case, the thing that happens with PI, the reason it's hard to recite digits of PI or whatever is because they're kind of random. There isn't, you know, we can't really make a pattern to them. It's kind of like in order to make a pattern to them, to help you remember them, you are sort of imposing that pattern on them. It's kind of like constellations of stars in the sky. There really isn't a bull in the Taurus constellation or something that arranges the stars. It's just that we humans say, oh, well, it's kind of reminiscent of this thing, this thing that exists in our world that sort of happens to correspond to this piece of randomness. And I'm sure people who memorize digits of PI, they're probably remembering. They're sort of chunking it up by remembering. The rhythm of this collection of digits is a bit like this rhythm that I remember from this other place. And that's kind of. That tends to be how our memory works. But it's a good example of one of the many kind of, sort of easy to state, hard to establish results in mathematics is the apparent randomness of digits of PI. And there's much that's been made, people have noticed that sort of some apparent randomness in which numbers are prime. But the regularities there are regularities that have been found in the distribution of primes, and those regularities have been much studied. That leads to a thing called prime number theorem, leads to the Riemann Hypothesis as an unsolved problem about the randomness of the primes and so on. But this is sort of a characteristic frontier of mathematics that really hasn't moved much in the case of the digits of PI, that hasn't really moved much in 100 years to figure out more about how that works anyway. So that's my excuse for not being able to recite more digits of PI, as they really are quite random with respect, or they seem to be, even though they have a rule to generate them, they seem random once generated. All right, well, I should stop there and thanks for some interesting questions here and see you another time. Bye for now.