The Letter That Changed Mathematics

This episode is brought to you by Cancer Research UK. Imagine this. Inside all of us, billions of cells follow millions of instructions written in microscopic code. And when a new cell grows, it copies those instructions, but the smallest error can lead cancer to develop. Right. And this is the reason why there isn't a single cure for cancer, because, you know, there are more than 200 different types. Each of them have got different distinct characteristics, you know, different challenges, different mysteries. And that means that trying to cure cancer isn't like following a single path. It's like trying to map out an entire forest. That's right. And Cancer Research UK is the world's largest charitable funder of cancer research. I mean, their work spans more than 20 countries, with over 4,000 scientists, doctors and nurses pushing knowledge forward to save and improve lives worldwide. You know, over the last 50 years, the work that this charity has done has helped to double cancer survival in the UK. And you have to think about that is that is more parents at the dinner table, right? That is more friends at their birthday parties. That is more people who are living longer, better lives. For more information about Cancer Research UK, their research breakthroughs and how you can support them, visit cancerresearchuk.org forward slash rest of science. Welcome to the rest is science. This is Field Notes, a kind of expedition diary where Hannah and I track the discoveries, curiosities, riddles and questions that have excited our minds recently. Now, what have you got for me today, Hannah? Okay. Today, I want to talk to you about, and I read this as an exaggeration, one of the most important pieces of paper in the history of mathematics. Oh, good. It's a big deal. I got to go and see it. I haven't got it with me today because I think it's locked behind about 15 doors under continual armed guard. I'm joking. But I'm going to get to what it is in a moment and why it ends up being so important. But the first thing I want to ask you about is, have you heard of a guy called Ramanujan? Yes, mathematician, Indian. Yeah, absolutely. So actually, he is the character that Goodwill Hunting is based on, right? I mean, Goodwill Hunting is like this modern version of the story. But essentially, the essence of that story is there's truth to it. We're talking just after 1900 here. And at the time, there are these great titans of mathematics who are at Trinity College in Cambridge. One of them is called Hardy, and the other is his co-collaborator, a guy called Littlewood. And as lots of mathematicians do, even to this day, they get all of these letters from people coming in and saying, oh, I've discovered something absolutely extraordinary. I've come up with a new formula for infinity, et cetera. Anyway, one day, Hardy opens these letters and he sees this letter that's come through in the post, and it's just a total load of absolute gibberish, right? Like complete load of nonsense. The notation is all over the place. It's like pages and pages of formula. And he's like, this is absolute junk. So he immediately just throws it in the bin and leaves it. And then he leaves. He kind of goes off for lunch and goes for a walk. And then later that evening, he finds himself sitting with his longtime collaborator, Littlewood. And he's like, I just can't get this thing out of my head. Because there was a couple of things in it, even though the notation of it was absolute nonsense. There were like a couple of bits in it that sort of, I feel like I've been struggling to understand myself for a number of years. So him and Littlewood go back into the, to the waste paper basket and they get this letter out of the bin and they start to go through and line by line, I mean, it is full of the absolute biggest load of junk in a lot of ways. In terms of the notation, in terms of there are no proofs anywhere. There's no, it's not rigorous to it. It's like a mess. However, contained within it are these like little ideas, these little like mathematical secrets, some of which Hardy knew that the rest of the world didn't know, and some of which were even beyond what Hardy himself thought that was, was possible. And him and Littlewood were like looking at this and being like, okay, you know what, either this person is crazy, either this is a letter that has come in from a madman, or as is more likely, this has come in from a genius. And they decided that on the balance of probability, that if someone was a madman to send this letter, they wouldn't possibly have the imagination to come up with something quite so crazy. And so the only possible explanation was that this must be the real deal. And I should tell you, some of the like crazy things that appear in this letter are that if you add up one plus two plus three plus four plus five all the way to infinity, you get minus one over 12. Is that where that comes from? That's where that comes. Well, yes, that's one of the places that comes from. Yes, absolutely. I mean, can you imagine, I like trying to, I get letters like this now, right? So I am now a professor at Cambridge University, I get letters like this all the time. I open them and think they are nonsense. And there's a little bit of me that's like, oh no, hang on a second, maybe I could be missing out on an actual genius here. Maybe there could be a real genius. But can you imagine getting a letter from someone saying, if you add one, two, three, four, all the way up to infinity equals minus one over 12, you would just dismiss it out of hand, surely? Yeah, you think it was a joke. And you'd be concerned for the person. But you're right. To have the imagination to come up with that, but then also the confidence to send it to leading mathematicians would make me take a closer look. Right, absolutely. So that's exactly what Hardy does. He finds a way to contact the person who wrote in the letter, a person called Ramanujan. And he organizes for him to come over from India and become a resident in Trinity College in Cambridge. And when he does so, he realizes that the person in front of him, who is untrained, this kind of rough diamond as it were, has the most extraordinary natural ability of basically any mathematician who has ever walked this earth. Not only that, with his only training being this textbook that's used for high school students, right? Like, if there's no advanced mathematical ideas in it at all, he had managed by himself to advance the field of number theory further than the best, most well-funded minds across Cambridge and the rest of the world. Right, on his own, he had managed to do things that had evaded other people. So the thing that I got to see when I was first talking to Cambridge about moving over there as a professor, they organized this day for me where I got to go and see all of the really exciting things around Cambridge, right? Got to go meet the Vice Chancellor, got to see the fancy halls and stuff. And there was one day, one afternoon, there was one bit where we got to go into the library and dig out their treasures. And they had the original Newton copy of Principia Mathematica, the textbook that he wrote, his handwritten annotations in it. Absolutely amazing. That's sort of the gem of their collection that everyone gets really excited about. And of course, they found that exciting. But then they were like, oh, and there's some stuff by some other mathematicians that we've dug out over there in the corner. And we went over, had no idea that this is what we were going to find. But there was a folder with the original letter, the one that had been thrown in the bin by Hardy and the subsequent follow-ups. And I basically, I've never been more excited by anything in my entire life. What kind of mathematics was it? What was being argued or shown? So this is its number theory, essentially. So you're looking at different ways that you can partition series numbers. So I have sent over a couple of pictures that I took on the day. Oh, for today from Hannah Fry. There you go. Oh, wow. Okay. Right. So for starters, look how nice his handwriting is, obviously. That's the first thing I was going to say. This is not a madman. This is an artist. It's so readable. It's so readable. Dear sir, I am very much qualified on perusing your letter of the 8th February. I mean, I'm not good at reading cursive, but I love this. This is readable, right? But this is someone who notices actually how crazy he sounds, because there's a section here where it says, I told him that the sum of an infinite number of terms of the series 1 plus 2 plus 3 plus 4 plus plus plus plus plus plus equals minus 1 over 12 under my theory. If I tell you this, you will at once point out to me the lunatic asylum as my goal. Like he knew it sounded crazy. Yeah. Knew it sounded crazy. And yet it turns out it's not crazy. It's absolutely correct. And not only that, but it turns out to be phenomenally useful in the field of string theory. It's an actual result that actually gets used by actual physicists. Yeah. And Numberphile has a fantastic video on it. It sounds like a joke. And yet there are ways of looking at the sum of all the integers and you go, well, how can it not be negative 112? Yeah. That proof from Numberphile is, I think it's pretty controversial. I think that's probably fair to say. Well, it's controversial, but it really forces you to say, well, what is mathematics and what are we doing here? Because there are different kinds of intuitions and different sorts of behaviors and things to focus on that really can make something like this true. Yeah. Absolutely. The other thing about this Ramanujan story is that I should add is that actually the winter in Cambridge, it really didn't suit him. This is also 1914, 1915. Right at the very beginning of the First World War, he was vegetarian and it was almost impossible for him to get the nourishment that he needed while there were rations and so on in place. There was nowhere in Cambridge that sold vegetarian food around that time, I believe. But he really didn't do very well. Around three years later, he ended up passing away. He contracted tuberculosis and passed away. And there's this really beautiful story about Hardy, who I have to tell you, Hardy is the most upright of people. If you imagine a really traditional English gentleman, that's Hardy all over. This is a man who had five things that he was most proud of in the world. And number five was that he didn't have any children. He used to boast about how happy he was that none of his maths was any use whatsoever to anybody, which is a bit of a shame because it ended up being the foundation of modern cryptography. But he's like this very, very upright man, loves cricket and mathematics and nothing else, not interested in another single thing. But he described his encounter with Ramanujan as the most romantic encounter of his life. Or the only romantic encounter of his life. Yeah, which is quite sad really in a lot of ways. But he really thought that this man was absolutely extraordinary, like had been sent by God. And when Ramanujan was in the hospital, when he was extremely sick and about to pass away, Hardy got a taxi down. He was in Putney, in a hospital in Putney. So Hardy got a taxi all the way over to go and see him and sit with his friend. And he didn't know what to say, right? He was rubbish at small talk, sort of like a very awkward guy. And he sat there and said, Oh, the number of the taxi that I got here was 1729, like an extremely unremarkable number. And 1729 has now become a sort of inside joke for mathematicians because Ramanujan without even pausing, this is a man who's on his deathbed, right? Like one of the most gifted mathematicians to have ever lived, immediately responds, 1729 isn't uninteresting at all. It's the smallest number that can be comprised of two cube numbers added together in two different ways, which I mean, I don't know about you, Michael, but that would take me about an afternoon to work that out. Yeah. And on my deathbed, I'll probably say something like, nothing that becomes a mathematical meme for the centuries. No, absolutely not. It's just a really lovely story. And I mean, I think it's sort of like, I think the kind of the idea of there being these geniuses who walk the earth is there's something so appealing about it, there's something like really magnetic about it. I also think it's sort of problematic because almost all science and maths and advancing of humanity is not done by, you know, the absolute like one in a squillions, it's done by big teams of people who just are normal people who work really hard, right? Yeah, right. Yeah. I mean, the great man theory of history and culture and discovery, it forgets the fact that like there needed to be doctors to help Ramanujan, there needed to be like literally the messengers who would send the correspondences back and forth so that these mathematical conversations could happen. Like you need everything, everyone. There's also, I think this idea that, you know, not everyone is Usain Bolt, but that doesn't mean that you can't have extreme value in sort of going for a jog around your park on a Sunday afternoon, you know, you sort of don't have to be a genius. And we all stand on the shoulders of giants for every Ramanujan, there's a high school math textbook that set him on that path. Yeah, absolutely. You do get these things popping up there in math sometimes. You do get people coming from nowhere with like no formal training, no connection. First of all, I'll say that not only do they just kind of pop up, they're often quite young. Yeah. I'm sure you've heard a lot about this like sweet spot for mathematicians and their productivity. And it's like in the early 20s. What are your thoughts on that? Do you feel like you've got some mathematical discovery that you're going to, you're going to stumble upon in the next couple of decades and you're going to throw the average off? Oh, no, I'm way too old, Michael. Way too old, yeah. Way too old. Also, I'm just enjoying my time spent with you far too much to be thinking deeply about mathematical truths and reality. Oh, now you're blaming me for keeping you from changing the world mathematically. I'll take it, I'll take that responsibility. Okay, that's good. Yeah, it is. It does seem to be our young persons game. I mean, actually, they always used to say it was a young man's game. And I think that's sort of a real emphasis there. Only one woman has ever won the Fields medal. This is a bit like the Nobel Prize for mathematics, except it's way harder because you can only receive it if you're under 40 and they only give it out every four years. Wow. But yeah, only one woman ever has won it. So, yeah. Oh my gosh, I've got two months left to win that. Uh-oh. Come on, let's do this. Do you have to be under 40 or does your discovery just have to have been made before you were 40? No, you received the medal before you're 40. When's this next ceremony? 2026, that's the next Fields medal. Oh, but what month? Why, when's your birthday? January 23rd. It's really soon in the year, but it only happens every four years. 2030, 2034, 2038. Let's see, my daughter is going to be, she won't be 40 until 2059. She's got time. She's got a chance. All right, so I'm looking at these photos. I still have them up on my computer and there's one where you're actually standing next to a bound book, a hard cover bound book. And is that what contains these letters? No, so that is Newton's original copy of Brinkipia Mathematica. Oh, no kidding. I love, but it's on a pillow. I know. It just needs to be tucked in. It's so precious. The thing is, is that even though it's on a pillow, if you like, flick through these, there's one point clearly where it's got moldy. Oh no. It's like these pages are just covered in black mold. I love that. You know, Richard Feynman once said something like, if you think about something for a really long time, you can write down all of your thoughts in a book and then put it in on a shelf in a library full of basically what happens when people think about things. And it was just such a humbling reminder that like, the mold doesn't care what's on the page. No, no, absolutely not. Or whose hand had turned the pages. There are some nice things in there though. There's like a couple of bits where Newton has done a couple of little doodles. So there's like, I mean, he's got much messier handwriting than Ramanujan, that's what I will say. But there's like a few little graphs that he's drawn there, like little tangents and stuff. There's a bit where he's sort of correcting the print, being like, no, no, no, no, I don't like this at all. I want to change it completely. I mean, my Latin is not great. So I'm assuming that's what it says. But there's one bit which I quite like, little Newton doodle. A Newtonian emoji. You know, in the Royal Society, where they elect fellows, like really great people from science and mathematics, they elect them as fellows. And it was the institution, it's been sort of the longest running scientific institution in kind of continuous existence in the entire world, been going since the 1600s. Newton was like a very prominent early president of this place. Anyway, when you are elected as a fellow, which is very, very hard to do, you get one black ball and you're out. When you get elected, you get to sign your name in this original book, which has everybody in it all the way back to the 1600s. And the thing that's really nice is that it's obviously Newton's signature is in there. But the person underneath Newton's signature is completely rubbed out because for centuries, people have flicked backwards, put their finger on the page, you're going, oh, look, it's Newton. And they've pointed at Newton's smudging the name below him. Oh, I love that. It raised, literally a raise from history. This does happen every now and then. Someone pops up from nowhere who is sort of outside of the like usual institutions, but has done something really extraordinary. This happened a few years ago, there was a guy called Gregori Pelman. I don't know if you've come across this guy. Yes, I've seen like photos on Reddit of him, like taken candidly from his back while he's waiting in an intersection and they're like, we found him. And it's just such a mystery. He is so fascinating. Okay, so in the year 2000, the Claymaths Institute, they decided that they would write down the seven most important unsolved problems in mathematics, right? Like the biggest mysteries and they were like, we'll give a million dollars to anyone who could solve one of these. And all of them almost have stood unsolved for a quarter of a century. But from nowhere a few years ago, this proof landed on the internet by Gregori Pelman, a Russian mathematician. I mean, I'm sort of saying that, but like he wasn't associated with anywhere else. He solved this like very important theorem called the Poincaré conjecture. And people who initially were like very skeptical, where's this guy come from? You know, you don't just sort of, I mean, you may as well like, you may as well put it up on Twitter, you know, like, you don't expect sort of a random Twitter poster to have solved it. But people very carefully went through and checked and it was absolutely correct. Like this whole thing was totally, totally spot on. And so, you know, people were like, well done you have all of these prizes, have the medal, have the, have the million dollars, right? But the thing is, is the million dollar prize, it was a ceremony that was going to be held in Spain. And Gregori Pelman, he replied that to go to Spain would mean one day traveling, one day to collect the prize, one day to return home. And that would be three days when he wouldn't be able to do mathematics. So he turned it down. Wow, says something, doesn't it? Yeah. Yeah. But he hides from everybody. He's like, he doesn't want to give talks, he doesn't want to be invited to things. He just wants to do his own thing. So yeah, hence people stalking him and putting photos of him on the internet. Living his best life alone. Well, we don't know if he's alone. He might have like a bunch of friends. So he did live with his mum in an apartment in Moscow for when the prize was suggested. Okay. Which is like, you know, actually I think a million dollars in Moscow is like, it's enough money to be noticed, but not enough money to protect yourself. So I think that might be one of the reasons. Oh, okay. Yeah, that makes sense. Or maybe he just doesn't want his mom to know he's a nerd. I think it might have leaked out on the side. She's like, what's that letter you received? And he's like, oh, junk mail. I gotta go lift some weights. Is that what you were like in your teenage years, Michael? No, no, I was very proud to be as nerdy as possible. Me too, absolutely. Shall we go to a break? This episode is brought to you by NordVPN. VPNs are an incredible invention. Short for virtual private network or VPN. It works to shield your IP address. It virtually connects you to any network in the world and protects your information from prying eyes. Yeah, no more worrying that that public network you joined that like didn't even require a password will end in disaster. No, just fire up NordVPN and its encryption software will keep your data safe. And NordVPN's unique threat protection pro. It blocks malicious links. It scans downloads for viruses for up to 10 devices for your whole house when attached to your router. NordVPN even protects your wallet. How? Great question. You see, some online retailers will charge you a different price depending on where you are. That's a great way to ruin Christmas. Fortunately, using NordVPN means you will never be subjected to algorithmic pricing again. To get the best discount off your NordVPN plan, go to NordVPN.com forward slash rest is science and our link will also give you four extra months on their two year plan. There is no risk either with Nord's 30 day money back guarantee. The link is in the podcast episode description box. This episode is brought to you by Cancer Research UK. Cancer drugs aren't developed overnight. They start as ideas in the lab, then move into testing to check their safe and work effectively. In the late 1990s, Cancer Research UK scientists began exploring a bold idea. Could the antibodies that normally trigger allergic reactions be used to treat cancer? The lab results were promising, but allergic reactions carry real risks. After years of work, an early stage trial showed these antibodies could be used safely. And for one person on the trial, their tumor shrank. Research is ongoing, but this careful process is how treatments move from the lab into hospitals. Cancer Research UK backs innovative ideas. And thanks to decades of support over eight in 10 people in the UK, receiving cancer drugs are using one developed by or with Cancer Research UK scientists. For more information about Cancer Research UK, their research breakthroughs and how you can support them, visit cancerresearchuk.org forward slash the rest is science. This episode is brought to you by Thriver. 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This is your latest idea. It's unique. It's game changing. It's huge. But you can go even bigger with AI-powered PDF spaces in Acrobat Studio, turning your files and links into actionable insights and content, plus share projects and collaborate seamlessly while keeping everything private and secure. So your excellent idea stays yours. Do that with Acrobat. Learn more and try it out on Adobe.com. Welcome back from the break. We are refueled and we are ready for some questions. You survived. Congratulations. We've got people who desperately need answers. For example, Becky asks, why do I sneeze when I first have a mentor, a piece of chewing gum? Well, okay. First things first, Becky. I think you need a lozenge for that sore throat. That's what I was thinking. It's ridiculous, Becky. Chewing gum is not the thing that you should be consuming right now. Okay. Basically, there are these nerves in your face, right? There's one called the trigeminal nerve, like this map. It's a monster. It's like an absolute beast sitting behind your face. It's responsible for all of the sensation that you get and also the movement of your main muscles, your chewing muscles. That nerve essentially holds the power to hit the sneeze button. That's the nerve that gets triggered. If a little bit of dust or a microbe or something ends up in your nose and you're like, okay, I need to eject this immediately, let's sneeze. What can happen is when you eat something like mint or for some people, when you look at a bright light, right? This is also having, it's sort of like, that nerve, it gets a slight sensation that it gets confused by and it just hits the eject button instead. Instead of being like, it's cool, don't worry about it, it's a bit of mint. The looking at bright light one is because the optic nerve runs really close to the other nerve and the insulation between the two of them can be a bit dodgy. You can get a mixed message, quite literally like a faulty wiring of looking at a bright light and this nerve being like, something sneeze. How cool. Yeah, the optic nerve, any nerve near the trigeminal nerve, overstimulated, it can leak in and then it, yeah, the trigeminal nerve says there's something that needs to get out of here, sneeze. Have a sneezing fit. It would be quite bad if you worked in the mint factory, wouldn't it? Well, yeah, I've never had that kind of effect. I don't sneeze when I see bright lights, but I've heard people talk about it and I've been like, wow, you've got leaky nerves, dude. You are miswired. The only way actually that I can make myself sneeze is, which everyone can do, right, is to get something really small. Don't do this, but like a feather and like tickle the inside of your nose. That will work. If you ever want to sneeze, some people quite like sneezing, I think, don't they? Oh yeah, some people really like it. Like it's their bag, baby. Well, like in a kinky way. Yeah, because it represents like a moment where you lose control. It's surrendering yourself to the sneeze. Wow, you better believe I'm going to be searching some of the dark corners of Reddit later. Oh yeah, I can show you my friends in the forums, you know, but that's a good time to move on to Harry B. It probably is. Wait, let me ask you this one. Let me ask you this one. What is your favorite and least favorite element and why? My favorite and least favorite element. My favorite element is sulfur because my father was a sulfur engineer. He was a chemical engineer who worked especially in sulfur recovery from refineries. They've got a lot of sulfur that they don't need. And he loved sulfur. He had jars of sulfur around the house's decoration and he always wanted to get a tattoo of a sulfur molecule. My least favorite, I don't know, it seems so mean to be all like judgmental about just a way matter can be arranged. I guess should I go with something that's dangerous? Let me think. Yeah, go on. What's the most dangerous? You know, like the most dangerous, I'd say like some enriched plutonium would be really bad. Yeah. And yet, plutonium, I think, can also do a lot of good when used in the right way. What about like, is it a phthalium that's like really poisonous? Some of them are very bad for you. Yeah, but they've got to have redeeming qualities. There's a whole part of the periodic table that's full of like the forgettable elements where their properties are all like, yeah, it's a silvery metal that is sometimes used in optics. And you look at its name and the name is like flabubium. And you're like, oh, is that really an element? And you have to look it up and you're like, yeah, it's got to Wikipedia page. Wow, I never hear about that. But I don't want to be mean to them either. Is there an overrated element? I sometimes think carbon gets talked about a lot. It does. It's also, don't if you know this, quite a big deal. You're one of those carbon heads. I see. I see. Maybe it's just that I love sulfur too much. But what about you? What's your least and most favorite element? You're right. Having a least favorite is really difficult. I was going to go for argon as my least favorite just because it's like, wow, I'm boring. Doesn't do anything. It does so much. Ever heard of an incandescent light bulb? I know. But then also I do, the other thing I was thinking about was, because the thing about argon is it does react with anything, right? It's like it just happily runs around on its own. It's like the ultimate loner, basically. But the thing about argon is that actually they use it to detect neutrinos. So there was one point where I got to go to CERN and I just so happened to be there when they were building this giant neutrino detector. These particles that are incredibly difficult to sense that leave almost no discernable imprint on the world around them. But if you have a giant tank of liquid argon, then you can slightly detect these little trails that are left by these neutrinos as they go through. Anyway, this tank, right, that they had was lined with gold on the inside. That was pretty cool. But the liquid argon that they had in it, I was like, that's an absolute beast. How on earth did you get the funding for that? Liquid argon is apparently cheaper than Coca-Cola by the later. No kidding. How cold is it? I actually don't know. We got a question from Michael S. How cold is liquid argon? So I guess argon condensine temp that is negative 302 Fahrenheit, negative 185 Celsius. Not that cold. I mean cold, not like insane cold. Basically, a point is it sounds like argon might be your favorite element. I think it might be. I was also going to go for gold, just because I think actually it's a bit of an easy one, isn't it? A bit of an easy one. No, no, no. Gold has a lot going for it. I think if you're going to say argon is bad for just hanging around and not reacting with stuff, how can then you then say gold is great? Gold famously doesn't react with stuff. It's precious. It doesn't tarnish. The only thing that gets rid of it is aquaresia dissolves it at least. Yeah, which is actually the famous story about the Nobel Prize winner who during the Second World War, because when you get a Nobel Prize, it's like the medallion that they give you is literal gold, right? Like actual full-on gold. And they were concerned about the Nazis coming in and being like, give us that solid gold medallion you've got over there. And so he dissolved it, right? He kind of put it into a solution. Is it a solution or a colloid? I don't know. It's a colloid, yeah. But the point is you look at it, it's just a beaker of gross stuff. Reddish fluids, you've just got no idea what it is. Great way to hide gold. Great way to hide gold, yeah. I think I'm going gold. Also, I just love the idea that it's sort of like created in the, you know, like the collision of neutron stars and came here on an asteroid, which is admittedly true for almost every heavy element. But yeah, yeah. But you know, gold was probably also one of the first elements ever discovered since it can exist in its elemental form. Yeah. You can spot it in the rocks. I think sulfur would have been really early because you also can just find sulfur nuggets, veins of sulfur, whereas carbon, oxygen, it's hard to like see those things if you're just a prehistoric person poking around. Also, I think it's really entangled with human history, right? Like we valued it from time immemorial. Like, you know, the first idea of money was like that you would exchange the snuggle of gold, you know, even before coins were really a thing. So, okay, there you go, listeners of the rest is science. If you want to send in gifts to either me or Michael, Michael, I'll take as much gold as you want to give me, and Michael will take sulfur. I really will. But we'll be back next Thursday with another edition of Field Notes. And until then, keep your eyes open, your pencils sharp, and your minds a little bit wild. And if you have anything you want to send us in, including alternative outros that we can use next week, then please do send them in. The rest is science at gohanger.com. Or join our newsletter by heading to restis.com slash science.