Frosty Fractals

This BBC podcast is supported by ads outside the UK. It's time to see what you can accomplish with Shopify by your side. So, we can now listen to your podcast. The power of science. I mean, do we always solve them? I mean, the hit rate's pretty low. But it is with science. It is with science. Happy New Year. Yeah. In advance. Thank you. Any plans? Yeah. No, God, no. Any resolutions? Oh, no, no. Only one that ever stuck. Go on. Give Blood. Really? Yeah, yeah, yeah. It was about five years ago I resolved to give blood. And so I've done that every three months ever since. Have you really? I have, I genuinely have. I think it's a really good New Year's resolution because A, it's a good thing to do. B, it happens once every three months. You've only got to do it four times. You've kept your resolution. Yeah, exactly. And so you've just kept that ticking over since then. Somewhere in the United Kingdom, there is somebody who has had one of the most traumatic days of their life, but now has just had some funny blood injected into their veins. Yeah, they do. A little hug from me. A little cellular hug. Yeah, and some red blood cells. And now they send you a thing going, they don't go it went into this person but they do go it was used in this hospital you know Do they? Yeah they do I feel like you could geolocate your My own blood Maybe what I do I'll swallow one of those kind of tags Swallow one of those tags in the hope that it gets removed with the blood and then just go around say people how is it? I do that like that It's your guy Yeah Your guy right here Yeah Unfortunately we haven't got an episode about either donating blood blood transfusions or inserting micro-trackers into your bloodstream. All of these would be great topics for a future series of Curious Cases. Instead, we're talking about frost. Yeah, fine. Frost is cool. You with me? I am. I think I like a bit of frost, a little bit of crunch underfoot. OK, I'll tell you what, listen to this question. Hello, my name is Jane and I live in the south-west of England in a village on the Somerset Levels. Early one January morning, I opened my bedroom curtains to see the most beautiful frost patterns on the roof of my car below the window. The fern-like patterns were so intricate. I had never seen anything quite like them before. They reminded me of a William Morris design. Why do intricate patterns like this sometimes form in frost and elsewhere in nature? Is it important to state that we put the music underneath that? That's not the soundtrack to Jane's life. If you do want to feature in Curious Cases you're going to have to have the high production values in the questions that you send in. Pick suitable music. I mean that was perfect. I mean she had a harpist with her when she made the call. You've also got to think about what time of year it's going out. She nailed it. She absolutely nailed it. Would you like to see a picture of her car? Oh I'd love to. Here you go. Oh wow. I mean that genuinely is intricate. I mean I've seen this sort of stuff before but well done her for taking a shot of it. It's quite reminiscent of that scene in Frozen when Elsa stamps her foot on the ground and then spiralling fractals all around. Oh, do you know what? I feel we're going to get back to that in some ways, but like whatever. But that is a nicely intricate, and I mean, I had to Google who William Morris is, by the way, because I'm not up on my arts and crafts movements, designers of the 19th century. What are you talking about? You're a middle class man in London. I know. I really got to tip up my game here. But my wallpaper choices are all very, I think it's paint we have. I don't think I have one. Oh, so common. I die, sorry. I mean, it's good paint. There's an and in the title. You could put this as a wallpaper though. Oh, you absolutely could. It's very beautiful, very beautiful. And she's right, it does look like two ferns spiralling outwards. An unusual pattern. I can see why she took a photo of it. Yeah, absolutely. But then asked as well, which is a beautiful thing to do. Can we answer though? Can we answer? Maybe. We on hand have a couple of people who are going to help us try at the very least. We have got Dr Thomas Whale, who is a physical chemist from the University of Leeds. Sarah Hart, who's a professor of mathematics at Birkbeck University. And Professor Christoph Saltzman, professor of physical and materials chemistry at UCL. Let's start by going to you, Christoph Saltzman, because you sound most like a character from Frozen. And that seems like a good place to begin this. Before we even get to the question of what's happening on our listeners' car window, let's just talk about ice generally and how ice forms. Basically, we need water and a cold temperature. That's right, yes. So either liquid water and you cool it down, you get ice. The other thing is you have water vapour and you have a cold surface and you get frost. But for something that has a very simple recipe, it comes in a lot of different forms. Ice is super complex. I mean, first of all, we currently know of 21 different phases of ice. So that's one of the starts. But even if we just focus on what we call the ordinary ice, that's what we call the ice one, then the ice one can appear in lots of different shapes and forms and snowflakes and needles, etc. So the ice one, that's the one that we're all familiar with. If you put some water into your freezer, you would get some ice one. So on Earth, pretty much everywhere, you just have ice one. Maybe high up in the atmosphere, you get more exotic forms of ice. But down here at ground level, we are only dealing with ice one. However, having said that, even just dealing with ice one, there's so much complexity in this material. OK, so right. So even within our limited amount of ice that we're dealing with here, why is it? What is it about the structure of ice that means that it can appear in so many different forms? I mean, as I said, ice contains water molecules, obviously, and it just turns out that H2O is a very versatile building block. So if I gave you a water molecule, there's just many, many different ways in which you can connect the water molecules and sort of build up three-dimensional structures. Now, it's also been studied for hundreds of years. It has been, yes. Kepler was one of the first people to ask, why when something crystallizes, do you get symmetry emerging? And he just took a walk on a winter's evening, and then suddenly the snowflake fell on his coat, and he asked the question, why does it have six branches and not maybe five or seven? And he started thinking about it, wrote a paper. I mean I got a demonstration on that here as well So he did it with cannonballs We thought cannonballs are a bit tricky to bring here So we got these billiard balls instead representing molecules And obviously in the liquid it looks like a disordered structure The molecules are all moving around, more or less randomly. However, then it's a bit like musical chairs. So when the music stops, they will have to start finding their seats and arrange. Got one in the center and six around. and we've just established the billiard balls would crystallise to give us a structure that has a six-fold symmetry. It's not five, it's not seven, it's six. If you rattle around some billiard balls inside the triangle and then once you stop rattling, kind of get them to hug in close to each other, you're going to find this structure where you have one in the centre and six around the outside because you cannot fit in a seventh around the outside because with spheres, I mean, they sort of tuck into the gaps left. Exactly, yes. And so however many you have, you've got a sort of central key character and then sixfold going off it. And it doesn't matter which one you pick is the key character. That's the way that they pack tightly together. Absolutely. So there's an analogy between that and what happens with water molecules. Absolutely. Even though it's not precisely the exact same. It's not precisely the same, but the principles are exactly the same. So, but snowflakes, which are again on a larger scale, they also have a sixfold symmetry. Absolutely. And that comes down to what happens at the molecular level. Right. We've got this building principle and that will affect the shape of the crystal. And this oft-repeated kind of art about there are no identical snowflakes. Yeah. I mean, it's clearly obviously unprovable in one sense. But it's a probability game, isn't it, that there's just so many possible shapes. Is that it? Yeah. And when you start with this tiny initial crystal, which has this hexagonal structure, then you've got this tiny little thing is floating through the air and it is encountering different atmospheric conditions constantly. And those different atmospheric conditions have an effect on then what the crystal structure does. So you start with a six-pointed thing, a hexagon, but then one snowflake goes into this little bit of atmosphere that's slightly colder or warmer or more humid or less pressure and at different pressures and temperatures you get variation between crystals that are forming that are kind of plate-like and flat or ones that are more needle-like, sort of like tree branches. And the individual snowflakes pass through the atmosphere as it comes down to Earth. Every individual snowflake will have a slightly different path. So every individual snowflake will have different conditions. If you get a properly identical, does the universe fracture into two different timelines? is an evil one and a good time light. OK, well, so far we've been talking about snowflakes floating through the air. But when we go back to our listeners' question, and specifically this car roof here, what's different here? I mean, this isn't a snowflake, Thomas. No, no. If I describe it, it looks kind of like ferns of ice grown onto the roof of a car. What I would call dendritic growth. So lots of kind of, I suppose, almost like needles or fans fanning out from a point. And it looks like there's a sort of initiation point in the middle of all this. So what I think has happened is a film of supercooled water has one way or another formed on the roof of this car. When you're below zero atmospheric pressure, the water would rather be ice. However, it can persist as supercooled water. Because it is one of the things that you learn in school. You have to unlearn to a certain extent that the change of state aren't like a switch that you hit and bam, suddenly you're ice. or suddenly your steam, that there are states where the temperature can go beyond those boundaries that we're taught for some time. And that's what supercooling is, isn't it? Absolutely, yeah, yeah. So I think there was a note from the listener that it was just below freezing when they observed this pattern. So I think what's happened is that all the water, so that sort of whole film of water on the roof of the car has gone below zero degrees C. And then at some point, a nucleation event occurred, probably at this point where you can see it looks like all the ice crystals have grown out. and then you've got this very rapid dendritic growth that has formed out away from that point. Okay, you've got super-cooled water. Something has happened, maybe like, I don't know, like a grain of something? Yeah. A bit of mud got kicked up? Yeah, just a bit of dust or, you know, maybe often we talk about bacterial ice nucleators that work at really warm temperatures. There could be a bacteria that happens to nucleate ice really well there. Just something that is capable of nucleating ice at a very warm temperature, so very near the melting point. So hold on a second. and then this very rapidly froze. Very rapidly froze. So what you're saying is this is exactly like that scene in Frozen where Elsa stomps her foot down and then the ice, whoosh, creates this pattern beneath her. I suspect it would have looked a lot, yeah, yeah, yeah. There are frozen fractals everywhere. Yes. Can we be certain it wasn't a miniature Elsa? Christoph, maybe you should take that. We can't fully rule it out, right? Good, good, good, good, good. But you've run some of your own experiments, haven't you? Yeah, yeah. So when I was contacted about this, I looked at it and thought, I'm pretty sure this is what's happened. But I wasn't convinced enough to come on the radio and say it. So I thought I would try and replicate the effect. And what I did was I waited for a day that was cold enough, so below zero in Leeds. And then I took some tap water. And I probably wouldn't recommend doing this because you might damage your windscreen. But I took some water from my kitchen tap and poured it on. And I got a sort of a nice transparent film. Nowhere near as nice to your listeners who's got these really nice crystals. but we've got these pretty clear frost ferns, so I'd say it's pretty clearly the same phenomenon. It looks very, very similar to the one that you see here. So we have an answer, essentially, as to why this happened. Yeah, I think so. Certainly we could replicate the effect. Let's pick up on this idea of supercooling. Why doesn't the water just freeze when it goes below zero? If you imagine you have a volume of water and you cool it down, you've got all the molecules flying around. In order to get a crystal where they aren't moving, you need to first create a sort of an initial crystal body. So you need to have a bit of, you know, crystal form. And in order to do that, you have to create an interface between that initial tiny crystal. And this is something that is on a nanoscale, so billionth of a meter, very, very tiny. You have to create an interface between that and the supercooled water. And it turns out there's a big energetic penalty to that. So it's something that it's just very unlikely will happen. And it means that we can supercool, you know, really pure water in fairly small volumes down to about minus 38 degrees before that occurs. It's all to do with this difficulty of creating interface between the tiny initial ice crystal and the supercooled water. So, hold on, water is happy being in liquid form, unless subjected to by an external nucleation point, to minus 38 degrees. Yeah, I mean, and if you had all the water on Earth and you cool it down to minus 20 degrees C, you wouldn't expect to get one homogeneous nucleation event in the lifetime of the universe. I'm sorry, what? Yeah, because it seems to happen a lot. We seem to have a lot of ice around. Yeah, we do. And we haven't quite lived the age of the universe. No, we haven't got there yet. And what this tells us is that essentially all ice that we see in nature is formed. You have a surface that induces that ice formation caused by some sort of material. Basically, ice is contagious. I think that's... If supercooled water is in contact with another substance, that will often kick it off. But different substances have different capacities to kick off ice nucleation. But if you just have, well, hopefully I'll show you, but if you just have a bottle of water where all it's in contact with is plastic, we might get it to minus 10 before that plastic causes freezing. OK, can we go and have a quick go? So in the corner of the room, we have a polystyrene box, the type you might find a lobster in. It is filled with ice. And before we started recording, Thomas chucked a lot of salt on top and then shoved a bottle of water in the middle of it. He is now checking the temperature of this bottle of water Not cold enough he says We come back to it So this is a super solution of a salt in water And it's liquid. But this really shouldn't be liquid. This actually wants to be a solid. Because it's a super-cooled liquid at room temperature. What liquid is it? It's sodium acetate. So we need to induce nucleation here. And for doing that, there's a chip in here. A little metal button. A metal button. And if I flick this, whip, there you go. Oh, wow. You can see it crystallizing. It's suddenly turning into solid. Can I hold it? Yeah. Because this reminds me of, you know, those hand warmers. It's suddenly got hot. I mean, to be honest, this is a hand warmer. Right. But that, hold on, hold on, hold on. Okay, so hand warmers where you click the button and then suddenly it turns solid and then gives off heat. Yeah. What? What? They are just super cool solutions of the salt in water, right? And in principle, they want to crystallise, but it needs this clicking of this chip here to induce the crystallisation. And then so that's why you can bring them back to their original state by reheating it. Absolutely. And the act of that giving off heat is because the process of crystallising... Crystallisation always releases heat. Aren't people clever? Isn't that clever? It's reusable. The nuclear aeration event. How big an event does it have to be? I mean, because the phrase feels like it's a big event. But it is just some piece of dust. It is just some piece of dust, yeah. So could you detect, by looking at someone's windscreen eyes, how clean their windscreen is? Are people judging us? Yeah, maybe if you sort of video the whole street, you could tell who has the clean windscreen. This might matter. Although it might turn out the soap nucleotides ice. I think we don't have a sort of... Damn it, too clean! This is what a lot of my research is about. We don't have a great picture of what nucleotides ice well and what doesn't. We have sort of empirical examples. We don't have an underpinning theory. Speaking of which, shall we check out the soap? Go on, let's try. See if it's actually happening. OK, so you now have a bottle of mineral water. When you say cold, how cold is that bottle of mineral water? It's about minus four. OK. In my old primary school science head, that's going, that should now be ice. It's not ice. Yeah, yeah. So you'd hope, yeah, yeah, you'd expect that to be ice. So we've got... Don't knock it. Don't knock it because microscopic events. Everyone keep your bacteria to yourselves. So this is a mineral called feldspark. This nucleates ice really well. And that's just water in there. That's not water in salt. Yeah, just water. And hopefully, I pour it out, you see it immediately turns into ice. Yes. Oh, that's fun. As you pour the water on top, you're creating this mountain, an entire mountain of ice. Keep going. Make this mountain. Keep going. Yeah, so you can see as soon as the super cold water contacts existing ice, it just transitions straight away. Oh, wow. Absolutely brilliant. Stalagmite in real time. In real time. I mean, the other demo we do here, so if you take that and just give it a really sharp shake. Sharper. It just all brings it. Oh, wow. That's an excellent magic trick. Yeah, it's good, isn't it? I mean, there's a lot of build-up to it. So that super cold water just is desperate to be ice. Yeah, yeah. Yeah, it would much rather be ice, but it has this kinetic barrier. This barrier has to get over to get there. We made ice! Why the branches, though? What is it about the branches? So if you imagine you have this sort of very perfect hexagonal plate, as Christoph was saying, as a crystal forms, you get heat being produced by that crystal. That will heat up the supercooled liquid around that plate. And what that means, in effect, is that you're slowing down the crystal growth. If you get some sort of slight irregularity in the surface of that crystal, you're going to get a little bit of the crystal poking further out into that heat. So that'll actually be exposed to slightly colder water. And then, of course, that will grow quicker. So, in effect, if you have a sort of a slight irregularity that allows part of the crystal to access this colder water, that will grow faster and faster and faster. And hence you get a needle shooting out away from the initial ice crystal. Yeah, it's this sort of interplay between tiny imperfections in something that, you know, this is a naturally occurring phenomenon. It's not going to be absolutely mathematically perfect, but these tiny things actually are helping to make the shape because we're taking the path of least resistance, essentially, and you get this all sorts of places in natural phenomena. It's why lightning makes these kind of shapes as well. It's just sort of slightly easier to go this way and this way, so that's what happens. And then you kind of get a positive feedback loop, which, as Tom was saying, you know, you get... if a crystal's already going out in one direction, it's getting to the colder liquid, and so then it's more likely to carry on going in that direction, and so then you get these sort of shooting forms if the conditions are right. You've been, I'm actually looking at Hannah here, dying for the word fractal. I'm so excited. As all the mathematicians are thrilled by the word fractal. Can you explain what a fractal is and how that apply in this context? Yeah. So essentially they're kind of shapes or curves that have self-similarity, that they repeat themselves more or less on different scales, smaller and smaller and smaller and smaller. Now, in nature, we don't go to infinity. You know, mathematicians like to just carry on processes forever and we glibly say, yes, dot, dot, dot, to infinity. Natural processes, of course, have finite limits, but we see structures appearing, ice is one of them, trees, roots, river deltas, where you get branching-like structures that look the same on big scales and small scales, roughly speaking. And that's what we kind of understand by a fractal structure, that you get this self-similarity. At every scale, unless you have another clue for what scale you're at, the scale, it looks the same more or less every scale. You could zoom in on this and look the same. Yeah. So mathematically speaking, we can create shapes that look rather like this with very simple rules. That's another reason why we see them in nature and in animals and plants. it's very simple often to to set up what's going to happen so with branching you know you could you could write an instruction that just says okay you start off with a line okay now let's just have a branch point and then after a little while those branches themselves branch and branch branch branch so we get this in kind of fern-like shapes in plants it's so so simple a great advantage if you're a plant for had to have this kind of branching structure is that you can start being that shape right from the beginning. So what do plants want to do? They want to get loads of sunlight and CO2 and they want to start doing that as soon as they're alive, right? As soon as they start to grow. So it was all very well to say, well, I'll just, I'll grow a massive trunk and then a load of leaves. Well, you can't do that because you haven't got the energy to grow a massive trunk and a load of leaves. It's much more efficient, makes sense, to evolve a shape that can grow and still stay basically the same. I like the idea that the tree AGM, have a vote on how are we going to grow come on guys there's a lot of difficulty with the issue of choice and you know mathematicians I don't want to tar us all with the same brush we are basically all lazy because I know but we like to if something works and we can find a way that why do we do proofs we want to be sure this thing's always true right we don't have to test every single triangle in the universe it's impossible we just find a little argument that works. So this laziness is also what plants and animals and nature like to do. An efficient method that is simple to do over and over again. Mathematicians are of nature. Yeah, that's often been said. The most natural of all the scientists. Because this is a relatively new relatively new branch We known about ferns a lot longer than we known about factiles We have we have And one of the reasons for that is if you have an idea and you think what would happen if I replicated this a thousand billion times It's kind of hard to do that unless you have a computer. In fact, the most famous that we now call fractals were discovered a long, long time ago. For example, relevant to this discussion, the Koch snowflake curve, so-called now. I believe that was first looked at in 1904 but you can make one of those on a piece of paper I haven't got these brilliant wonderful experiments I'm a mathematician so I've got pieces of paper It's the foreground of the mind Exactly, the mind So you can imagine an equilateral triangle Now we're going to make a fracture with this All we do is every line in this picture we replace it we just put a little smaller equilateral triangle halfway along and then we get a star Very cool And then if we do exactly the same thing again, every little line here, we replace it with a line with a little extra triangle, smaller. And then we say dot, dot, dot, which is a famous thing that mathematicians do. And if you repeat that process infinitely many times, where in my case infinity equals six, you get about as big as I can count. You get this lovely shape being produced, which is called the snowflake curve because it looks a bit snowflakey. Now, you can get an idea for what this is, but you can literally draw it by hand. Because this is, I mean, reminiscent of coastlines. It's reminiscent of the inside of a lung. It's like you do find these patterns all over the place. Yeah. And then the reason for that, the inside of a lung, why is a fractal form so useful there? Well, fractals have this technical term wiggliness. that it's really, really good for getting a huge amount of surface area for a finite amount of volume. And the reason for that is that fractals have this curious property. They are kind of between dimensions. Take a line, right? If you take a line and you make it, the scale it up by a factor of three, it's three times as long. If you take a square and scale everything up by a factor of three, it would go up by a factor of nine. It would be nine times the area. If you take this snowflake curve and scale it up by a factor of three, it turns out that when you do that, you've actually made a thing that's four times as long. Because of it. I know. Because you took a line and then you put a little wiggle in it. Right. So so when you do that, you've got something that it's it's not a line, not line like it's not one dimensional. It's not going up by a factor of nine. It's not two dimensional. It's somewhere in between. and you can actually put a mathematical, a detailed definition of this, this fractal is somewhere around 1.26 dimensional. So it's kind of more, fills more space in a line and less than a square. Oh, Dara, this is my favourite thing we've ever covered. I know, I love this stuff, but I also know we were going, no, no, no, no, no, what happened? We get to this? What happened? But the point of all this is to say you've got something so like in a lung, its surface area has this sort of fractal branching and branching and branching thing. It's like its surface area is increasing more like volume than area should. So it's filling up more of the space. And even in a finite, you know, our bodies are only a finite size, even after Christmas. We haven't got the room to have just a whole load of lung laid out, 100,000 square metres. But we need to get to the surface area. If we squiggle the surface of the lung in a sufficiently interesting way, we can fit a lot of surface onto the same amount of volume. I think the main conclusion here is, Jane, what has happened to your windshield is that it has broken the dimensions that it existed in. It is no longer. It is no longer confined. Either two or two dimensions, yes. Because in this situation, just to check if we can bring it back to this for a second, the fractal in this situation is the shape of the crystals of the ice. it is making choices as it grows from the nucleation point this massive event where a microscopic piece of dust fell onto this super cool water and then from that point it goes slowly as we've seen spreading spreading spreading spreading spreading making choices about whether to go left or right left or right left or right like lightning moving through the atmosphere absolutely and then but it does in a way that makes the pattern look like it's repeating repeating repeating and that gives it that fern like impression that you have here it's probably what do you reckon 2.4 dimensions I reckon yeah we can quibble over that yeah I mean I don't I think we'll all live with an exact number 2.71 your record not too many decimal places yeah but for this simple picture that Jane just wanted to know what was happening has now even Jane's kind of going I'm fine I'm great I'm great I got the gist I got the gist some time ago now before we went into you know infinite surface areas for a finite volume or whatever we have answered the question haven't we We very much answered the question. Well, I guess we'll leave it there. Thank you so much to our guests, Sarah Hart, Christoph Saltzman and Thomas Whale. What a wonderful Christmas gift that was for Jane. To not just find out how her car got iced over, but also really deep dive to physics. Hey, it's a gift that keeps on giving. Isn't it? On a fractal level. And keeps on. And keeps on. And keeps on. And smaller and smaller amounts. Ad infinitum. Yes. Subscribe to Curious Cases on BBC Sounds and make sure you've got push notifications turned on and we'll let you know as soon as new episodes are available. Nature Bang. Hello. Hello. And welcome to Nature Bang. I'm Becky Ripley. I'm Emily Knight. And in this series from BBC Radio 4, we look to the natural world to answer some of life's big questions. Like, how can a brainless slime mould help us solve complex mapping problems? And what can an octopus teach us about the relationship between mind and body? It really stretches your understanding of consciousness. 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