I think we need to do more episodes that feature math. Yes, with maybe a different co-host because I don't do math. No, I love math. You love math? I do. You realize how important it is. Well, you know, it's, it's, yes. Without math, we don't know nothing in this universe. Well, I'm still in that boat no matter what. Coming up. Dark Talk, Cosmic Queries, a little bit of math. Welcome to StarTalk, your place in the universe where science and pop culture collide.
This is StarTalk, Cosmic Queries Edition. Got Chuck Nice with me. Chuck, how you doing, man? I am doing great. Yeah, yeah, this one is going to be on math. Oh, I was told there would be no math. Nobody told you there was going to be no math. That's right. I know some math, but if we're going to have math as a subject, we've got to bring on the big guns. Right. Especially the mathiest of the mathiest. We've got with us, and not his first time on StarTalk, Grant Sanderson. Grant, welcome back.
Hey, thanks for having me again. It was fun the first time. Let's see how it goes. Excellent, excellent. So you have sort of academic chops in math and in computer science. All right, those are related on some levels. And you took it to the road. I mean you took it to YouTube. Yeah. With a highly followed channel, three blue, one brown. Can you get more cryptic than that? Yes. Sounds like a three-card Monty's game. That's right. How long have you had the channel?
Man, it's been around 10 years now, which feels a little wild. And how many followers? powers you have? It's, it's, it's around 7 million. I remember some time recently across the, you know, that's, That's monumental in terms of accomplishment. No, no, I'm just saying. You got 7 million people to follow you for math. Math, that's what I'm saying. Exactly. If 7 million people, that gives us.
gives me hope for the future of civilization. Yeah, that gives me hope for those 7 million people. But not for the future of the rest. In general, more people like math than people suspect, I think. It's a little bit of an underdog. Everyone thinks people hate it. I think everyone loves math. I just think that most people are intimidated by it and nobody wants to feel stupid.
I like feeling dumb. That's why I'm on this show. I'm always the dumbest person on this show. But that's why Grant exists. So that... grow the comfort zone that people feel when encountering this content. Let me pick up some broad, deep topics in math before we get to our Q&A. We hear occasionally about problems in math. And no, they're not talking about what's eight times seven. No, they're talking about problems that the deepest thinkers in the field have attempted to solve over the centuries.
Is there like a book? of the latest unsolved problems in math and then that's what Nerd geeks should. Yeah, should check out of the library first. Yeah. Yeah, I mean, there's been many. I mean, one of the most famous bodies of unsolved problems in the year 2000, the Clay Math Institute put out these seven problems that they offered a $1 million prize for. And so these ones are kind of the celebrities among unsolved problems. They're called like the Clay Millennium Math Problems.
They are typically very hard to even describe what the problem is stating. So very classic. You can't even describe the problem. the problem you're supposed to solve right you gotta solve the problem but you can't even describe nobody can tell you what the problem is not to me sir that's how you keep your million dollars yes i was gonna say that sounds a lot like being married But what is the problem even that we're beginning to solve?
But there's also these celebrity unsolved problems that are in some sense less important, but they're easier to state. And as a result, they're a lot more fun to just engage with for the public. But there's no prize money for those, right?
like not in the explicit sense that there is a particular institute that like put this check behind it but like absolutely if you solved any of these problems you gain a certain fame within the math world probably you're doing it as an academic where it really bolsters your career like there's there's plenty
if you want financial rewards there's like plenty that would come if you could solve one of these problems but of course that's not what most people care about are they arising purely within math or is there some scientific pumping of what these problems are or engineering solutions. Yeah, so I'll give you one that's purely within math and then some that are more like
come from the outside world. And this one, it's near and dear to me because I remember when I was, I don't know, maybe 11 or so, my dad from across the kitchen, he like pipes over, he's like, hey Grant. you know, prime numbers. I'm like, oh yeah, prime numbers, like numbers that you can't divide into two smaller pieces, like six equals two times three, but seven, you can't break it up into two smaller pieces. He's like, yeah. Do you think there's infinitely many of them that are just two apart?
So like 11 and 13 are primes that are two apart, or 29 and 31 are a pair of primes that are two apart. And he was asking this because he was reading some news article that mentioned, hey, this is an unsolved problem. His dad asked him this question when he was 11. Oh wow.
So either you were really smart or you had a really bad dad. So I was like, oh, is this true? And eventually, it didn't take too long for him to say, oh, this was a thing I was reading about in this science news magazine that mentioned this.
and this is a problem no one in the world knows how to answer. And that was fascinating to me. Here's a question you can ask. You teach kids about prime numbers. Maybe not everyone remembers them when they grow older, but it's a common topic. All you're asking is, hey,
Are there infinitely many that are two apart? Do they ever stop becoming spaced out by two? And you know the primes get a little sparser. Like as you get much bigger, there's fewer of them. But you might wonder, do they stop clumping up in that way? Euclid asked this. over two millennia ago, we still don't know the answer. And it cuts pretty deep to the understanding of primes that we have and the lack of understanding to be able to answer questions like this.
So that's pure math. It's just a puzzle. Yeah, that's great. Yeah, I mean, so one that's... I'm not going to be able to state the exact nature of this question, but I can give you like the high level overview, which I know, Neil, you're going to know this, but.
There's a set of equations that describe fluid flow. They're very famous. They're called the Navier equations. And so if you're just modeling some fluid and you understand certain aspects of its pressure and viscosity and things like this, There's something, for example, you can tell the computer to try to run forward a simulation. But the theoretical understanding of these equations is... Like, worryingly thin in some respects.
Or, for example, it's not entirely known if it would imply that you get an infinite concentration of energy at some point. Clearly, that shouldn't happen. You don't think that would happen in the physical world with the mathematical model being used. in the pure landscape of what are called differential equations. It's got these properties where people aren't sure whether it falls one way or another.
And it's actually very, very hard to understand the specific type of differential equation. And so again, I won't phrase the specific nature of the question that's unsolved, but broadly speaking, it's... Some basic questions about do these equations behave in the way that you would hope they would behave, no one actually knows. And this is clearly motivated by modeling physics and modeling the world. Now you also have a category of problems that are unsolvable.
Yet you have people who think that they solved it, right? So you can prove that something is unsolvable, correct? Everyone might have seen in school the quadratic equation. So this is something where if you have an expression that looks like x squared plus some constant times x plus some constant equals zero, and you want to solve it. It's a systematic way to do it. This is an equation that comes up all the time for engineers.
all the times in computer graphics programming, just solving equations like this left and right. And so there's a formula. A lot of kids memorize it in school. It's called the quadratic formula. This is a formula that has been known for a really long time. Chuck, recite the quadratic formula. Do you remember? Here's the quadratic formula.
What'd you get for number 13? Looking over the shoulder. But I mean, it was, I remember getting drilled into us on a level where it prevented me from appreciating what it was actually doing. Right. It was just a memorizable formula. It was like negative b plus or minus the square root of b squared minus 4ac over 2a. Did I get that right? That's exactly right. Okay. Okay. I just did that by rote, not because I had a freaking...
I'm feeling for it. You just had to know it so many times. Yeah, I got it from eighth grade that came in. Okay. Here's something you should be thankful for. So there exists a formula to solve any cubic equation, but our teachers never made us memorize it. It's much longer. Yeah.
And it takes a much longer song to do it. And if they think the quadratic formula is this thing that would cause a rote engagement with math instead of a substantive one, forcing kids to memorize the cubic would be even worse. But like pure mathematicians for a while were curious, okay, how far can we take this? Can we have a formula that solves any equation where the highest power of x is x to the fourth?
And you can. It's an even more monstrous formula. If you tried to write it down in full, it would just fill out like an entire page, this formula that solves degree four. And for a long time, the natural question is, okay, can we solve any degree 5 equation?
And between the 1500s, when this was kind of initially being explored with the cubic and everything, up to around the 1800s, people tried, but no one could find a way to answer that. Wait a minute, Tommy. People have these thoughts in the 1500s. while others were disemboweling heretics. These two things coexisted. It's wild to contrast history of science. Well, maybe they stopped by the 1500s. The Renaissance was... Yeah, exactly. I'm thinking maybe the Dark Ages. Mm-hmm.
You go to the right part of the world. One person's contemplating cubics, you can definitely find another part of the world where disembowelings happen. Just skipping to the punchline here, and I'll tell a little bit more story behind it. The answer ended up being, it's literally impossible.
In the appropriate sense, like if you're trying to write a formula using the usual symbols that we do, plus minus times divide, maybe you allow for roots and things like that, if these are the operations at your disposal. you will literally never be able to write down a formula for degree five or higher. And two very young mathematicians were the ones to make the initial steps in this. It works up to a fourth quarter, but not fifth.
And then suddenly five, it stops. Yeah, it stops working. Wow. And there's a deep reason why. And the discovery of this was one of the things that gave birth to a huge part of modern math, which is called abstract algebra. Like most people don't know that this exists, that there was this revolution in math in the 19th century that kind of changed the way that we think about math. But the beginnings of it.
We're born out of this question of trying to solve polynomial equations and realizing, hey, maybe it's impossible once you get above a certain point. What's the reason? Why does the symmetry break between 4 and 5? Neil, I would love so much to be able to give you the pithy answer that's like, here's what's true at 4 that's not true at 5. End.
I have struggled for years to think, is this something I could do in even like a 50-minute video or something that like compellingly describes it to the general public? Okay, I want a full... YouTube video on my desk Monday morning. It would make me and a lot of other people very happy. Okay, I'm gonna give an answer that will make no sense, but I promise that there's some sense in which this is true. It's something called Galois theory.
And one of the young mathematicians who came up with the arguments that led to explaining this, his name was Evariste Gawa. He died in a duel at the age of 20. This is his big headline fact. He knew he was going to lose that duel. So the night before the duel he like does like a brain dump on On the page.
Oh. This is the story. This is the classic story that we all tell in the lecture halls. That's insane. Let the man explain. Go ahead. What you're saying is what everyone hears when they're learning physics and learning math. Oh, that's not true? So he was compiling stuff that he had tried to get published three or four times before.
So he had written the stuff down before. He had gotten it in front of very famous mathematicians like Fourier or Cauchy and Poisson. They had seen the beginnings of his work before. So it wasn't like the first time he's ever jotting this down.
Crazed pre dual state, but it's such a good story to tell it that way, right? It's so good to be like he knew he's gonna die You know, so he's got to get these ideas out in some way He's got this classic story behind him in terms of the dual and all that but the math that he was doing
One of the things that it showed is that it's impossible to solve equations degree 5 or higher. It also showed other impossibilities. There's this classic problem about um trisecting the angle using a straight edge and compass to take any angle and divide it into three pieces his theories his like new math can prove that that's not possible um it's simply too hard to describe right here over a podcast
But it gave birth to a whole field of math that is both central to math and also particle physics these days. Wow. Cool, man. Well, math is the language of the universe. It certainly is. So we shouldn't be surprised if it doesn't spillage into cosmic discovery. Yeah, it makes sense. So can we bring questions your way from our audience? Yeah, yeah, yeah. Let's do it. Here's the first question from Buck Rice.
who says, why can't we divide by zero? Why? I love that. That's a great question. I've carried that with me my whole life. Because the answer is, Well it's undefined. And so my response is, well, define it. Get off your duff and define it. Yeah, exactly, define it. And you know who does? The mathematicians, actually. There are parts of math, one in particular called projective geometry, where one of the objects in there is essentially what we want to get at by the idea of 1 divided by 0.
And the thought is, if you have a number line and you walk infinitely far, either to the left or to the right, there's this unified point that you're approaching called the point at infinity.
And you can do useful math by defining that. And that's kind of what you're getting at when you have this notion of 1 divided by 0. But if you're not doing that and you want to say, like, why isn't it defined? It depends on what you're doing with division, right? If what you want to say with division is, like, I have...
You know, I have one cupcake and I'm dividing it among three people. How many does each of them get? Like a third of a cupcake. If I have one cupcake and I'm dividing among zero people. It's an incoherent question. The cupcake's got to go somewhere. The fact that that question is incoherent is maybe what we mean by saying that it's undefined. See, but that's my point. In practical terms, I have one cupcake. And I want to divide it between zero people.
We got to go back to the beginning of the statement. I have one cupcake. And that's the answer. The answer is I ate the cupcake. Now it's defined. Now it's defined. All right. Hello, I'm Vicki Brooke Allen and I support StarTalk on Patreon. This is StarTalk with Neil deGrasse Tyson. This is Kira. Actually, I'm going to combine two questions in one because Kira and Gavin Bamber
actually are similar, but I'm going to read both their questions successively so you can answer them. Okay. Hi, this is Kira from Georgia in the US. In your opinion, what is the most fascinating unsolved mathematical problem in cosmology that if understood could fundamentally change? how we view the work universe. Hold that. Gavin Bamber says, hey, Gavin here from North Vancouver. Please visit, Neil. What's your favorite unsolved math question and how would you illustrate it? So, one.
uh fundamentally change the way we view the universe and then part b what is your personal favorite unsolved. You can't say one and then part B. It's either one and two or A and B. No. See, what I am doing is a new kind of math. I'm going to punt off part one to Neil here because I'll have some humility here. I'm not sure what the most important mathematical problems of cosmology are. I'm not much of a cosmology person, so I'm curious what you say.
We have singularity problems in the universe. All of our equations tell us that at the center of a black hole The nature is dividing by zero. Right. Okay? Everything goes the denominator goes to zero right what happens to the value of everything else we say there's infinite density and infinite this and that doesn't even make any sense no it doesn't so what we don't know but we suspect is that
That's a limit to the application of our theory of the universe, not a limit to the invocation of the math. Right. Because we're not the first to blame the math. I'm just saying. Gotcha. Because math is badass. And we're not, okay? So we're going to take the blame first. But... It is true that certain discoveries in math have led the discoveries in astrophysics. We had no need for non-Euclidean geometry. Until we dead.
And so this is the curvature of space-time. It's not flat, and Euclidean geometry is flat. And who came up with curved geometry? So Riemann is the big one there. So Riemann, and when was that, like 1800 sometime? Yeah, it was in the 1800s, like maybe 50s, let's say. Okay, all right, so 19th century, we have the tools to think about curve geometry that was immediately uptook by the cosmologists.
to think about what could be the geometry of the universe. So that's all I got here. It's the math leading us, not us finding a math problem that's not solved. Well, I mean, you bring up Riemann talking about non-Riemannian geometry, but he's also the source of how I was going to answer the part B of that question for one of my favorite unsolved problems. There's part one and then part B. Yes, part one and part B.
So Riemann, you know, he did a lot of geometry stuff. He also was one of the fathers for complex analysis, basically using complex numbers to solve other problems within math. And he had one paper on number theory. So that's the, I described prime numbers earlier, like this twin prime conjecture.
He has this one paper that he puts out, I think it's 1857, about prime numbers. Otherwise, he doesn't do any number theory. And it completely changed the whole field because he basically said, hey, here's this. continuous function. It doesn't feel like it's about primes that are all discrete. It's very like continuous.
It's got complex numbers. That makes it very different. And if you understand this function, you completely understand the primes. These days we call it the Riemann zeta function because he used the Greek letter zeta. And he basically said, hey, we can really, really well understand how the primes are distributed if we understand something about this function. And he put this conjecture up about where all the, if you want to solve when this function equals zero.
He didn't know how to solve it. He had a guess for where those solutions are. And this is called the Riemann hypothesis. He was hypothesizing it. It's one of those million dollar problems. And it clearly, it's a very, very beautiful question because it's kind of asking like if the prime numbers form a chord in a certain sense because it studies them based on frequency information.
And nobody knows how to answer it, but the more you dig into this question, it paints a really, really beautiful picture. Is that your favorite unsolved problem? I think it's my favorite. Oh, wow. Okay. It sounded like it was, too. You make it sound very elegant. And you sound a little bit excited, as you were talking about it. I love it. Okay. All right, this is Tony Isaacs, and Tony says, G'day, Astro-Neil and Lord Nice. This is Tony here from Melbourne, Australia. Melbourne? Melbourne.
He says, love the show. Go on Chuck and do an Aussie accent. Okay, that's somebody who's been listening to the show for a while. Listening too long. Yeah, yeah. Totally telegraphed me. All right. he says i have a complex problem um it's the complex number i it's the square root of negative one which doesn't exist, but it is used in so much math.
that predicts things accurately, including quantum. It fries my brain. I hope Neil and Grant can help me out. Thanks. Yeah, so... So, Grant, I'm going to lead off by saying... Why did you label those numbers imaginary? My god, it's the worst name in all of that. Worst name ever! Gauss proposed calling them lateral numbers, which would have been... A little better. And then you add another aspect to the imaginary number to get the complex number. And now you call it a complex?
These are words that are complete turn-offs, and I blame you. Well, I mean, if that's the case, then we need to go all the way back to the beginning because we call them math problems. And who wants to deal with problems?
so many people struggle with this right because you label it as imaginary you start by pitching it by saying you know square roots of negatives don't exist but pretend like they do and run forward You could teach the whole topic completely differently where you start off by talking about processes that cycle and trying to model processes that cycle and using our normal number systems for that. and anything that has cyclic kind of behavior.
there's a natural number system to try to describe that. Call that number system what you want. Oh, so you can think of them as clocks then? Think of them as clock numbers. Anytime you're doing clock stuff, these numbers are going to be great. So we use the clock numbers. Why are they relevant in quantum mechanics? You've got a bunch of waves, right? There's things that are cycling and there's frequencies that are relevant. You've got like E equals HF type stuff.
when there's frequencies, you should suspect that complex numbers are there. It's useful in electrical engineering. Why? Because you're dealing with a bunch of waves. You've got a bunch of cyclical processes and frequencies, and so it's natural to model them with these kinds of numbers. So the engineers... adopted the complex
playing the complex numbers after the fact, right? They didn't say, gee, we need a way to do this. Let's invent it. They said, we don't know how to do this. And a mathematician comes up, here's a way.
I mean, is that, of course, I'm exaggerating that. I mean, do you want to know where complex numbers came from originally? Because I think it's sort of a lie that we tell in schools where we say, you know, there's no such thing as a square root of negative, but like pretend like there is and like mathematicians just love pretending things.
It actually cuts to something we were talking about earlier, which is when people were solving cubic equations and they wrote down a cubic formula that, thank God, neither of us had to memorize. I don't thank God for that. Everybody's thinking about what I thank God for. Thank math. Thank the education system. But if you would ask someone, hey, solve the equation x squared plus 1 equals 0,
they would have said, there is no solution. Like, obviously, we're not going to make up a solution. That won't do anything. But there were certain cubic equations where when you tried to use the formula, You had a real number answer. So real numbers in, real numbers out. Never any whiff of the square roots of negatives.
But when you use the formula, you can find that real value to answer if you take seriously the idea that somewhere inside that formula, there's a square root of a negative and it all cancels out at some point. But like while you're working it out, you're engaging with these square roots of negatives. So for a while, for mathematicians, they're like, oh, it's this one weird trick that kind of works. I don't know what it means, but it seems to work for solving cubics.
And I think one of the reasons they were called imaginary is because it took a long time for people to take them seriously. They just thought it was this notational trick. And the word imaginary was kind of derogatory. It wasn't like, hey, we want to teach kids this, what should we call it? It was like imaginary girlfriend. But then it took much longer for people to realize how they're useful and the idea that they have these like cyclic properties that...
make them really useful for any kind of math or physics that involves waves and cycles. And so that's what we're stuck with. Hello, Dr. Tyson, Lord and Ice, Mr. Brown. This is Brandon from New Jersey. Is there any relation between the conformal geometry and the Pythagorean theorem? Recently, I've learned about some circle inversions, and it seems to me that these inversions are leveraging the Pythagorean theorem. to maintain the symmetry of points across a line after curving it.
Is this at work in conformal geometry as well? I'm completely confused. Wow. What do you get out of that? Look at that, man. Let me just answer Brandon for a second. All right. Stop showing off, man! Stop showing off. This is not the plan. If this were a lecture hall, this would be the time I'm like, Brandon, let's talk after class. It's going to be great for you and me at a blackboard. The lesson will be better for everyone if we don't engage with what you want to engage with right now.
If you wanted, we can try describing what conformal geometry and circle inversion are. Can you do it in 20 seconds as a challenge? And if we understand it, who cares? If not, then now we have something to go look up, which is even more fun. Go ahead. When you look at yourself in the mirror, you see a reflection of yourself. It's like a different version of yourself.
Sometimes you can use that to solve problems like solving your hair and things like that. In mathematics, there's a different kind of mirror that sometimes they use where they pretend like a circle is a mirror and reflects everything from the inside to the outside and the outside to the inside. It's the special transformation of space.
And there's certain geometry problems where they look hard, but then when you reflect it through a circle like this, which is a really weird mind-warping motion, it turns the hard problem into an easy problem. That's called circle inversion. aspects about that i can't describe conformal geometry but that's what circle inversion that's the vibe of it and we won't describe specific problems there but if you're curious on the vibe it's treating a circle as a mirror Cool.
That was fascinating. I mean. I don't. I get with the circle inversion what you're talking about. I don't know what it's used for, and I don't understand. Yeah, I've got to give actual examples for that to carry teeth, but, you know, a little too long. He was still showing off, though. Yeah, he's showing off. Without a doubt, Brandon, we're showing off, okay? But... guess what i'm glad i'm glad you know i never even heard of conformal geometry until just this moment
So I'm happy that Brandon was showing off because now it's fodder for Look Up, which is great. Ethan step and Ethan says, hello, Dr. Tyson and Mr. Grant. And Lord Nice, if you're there, hello. I love these people, man. I freaking love these people. He says, my name is Ethan from North Carolina. I love math, but sometimes I wonder how these things get figured out. My question is, how on earth did someone come up with Tensor products?
Again, I think the easiest way to describe it that doesn't quite capture what it's about, but like a vector, we have a list of numbers that you might write as like a column of numbers. A matrix, you've got this two-dimensional grid of numbers used in computer science all the time. It's how machine learning works.
but sometimes you want a three-dimensional grid of numbers just as the way to hold your data. Imagine like a three-dimensional grid, each cell in that grid has a number, you might call that a tensor. That's the computer scientist way to answer what a tensor is. But then in physics, the use of certain objects which can be represented with tensors like this is relevant for general relativity and describing the curvature of space.
There's also other corners of math where you have objects that could be described. like you could give them a coordinate system where you want to have like a three-dimensional grid of numbers like this and answer the question like where did this come from or who would come up with it I think usually it's once there's a very specific problem that you're dealing with where you realize
The data that represents this is something that naturally organizes itself into a three-dimensional grid or a higher-dimensional grid of numbers. It's just a natural way to try to even hold that idea in your head. But somebody's got to be clever enough to see that need and then come up with it. Not everyone is that clever. They would be a prisoner of the known math of the day. And you need someone who can step out of that and say, I have a new way to think about this problem.
I mean, some physicists will joke that one of Einstein's greatest contributions to physics was his notation for tensors. And he had a really nice notation for how to even write it down, which lets you think about it clearly on a blackboard and such. which is a joke, of course, but it does cut to the fact that they're a central object that it takes a clever mind to even, like, represent in a useful and manipulable way. This is not T-I-J, is that...
Yeah, this is when you're like the Einstein summation notation where you want to, yeah. Yeah, it's a simplified thing. It makes the equation look way simpler and more tame than what's actually going on. Well, I'm going to tell you, you guys just made it look way simpler to me because I have no idea what you're talking about. Okay. I'm just going to be honest. Uh, I'm sitting here, and it's rare that I am lost.
we're in a conversation but this one is like out there man which is very cool so he plus he's got seven million followers on his youtube so he's doing something right he's doing something right all right yeah Hi Grant. Hello Dr. Tyson. I'm Akia and I'm originally from India but I live in San Francisco. I heard the last podcast when Grant was on and I was on my way to Death Valley for a dark sky festival.
for stargazing, and it was a delightful experience. We stopped along the way at a cafe and had a lovely cup of tea. I had Darjeeling. My friend Mary had... No, no. I'm making this part up. I'm like, thanks, okay, for including... All of your information about your trip. My question to you Grant is... Why is there no new revolutionary paradigm in mathematics like calculus and algebra that is uncovered today as opposed to the last millennium?
when many fewer people were working on mathematical problems. Thank you so much for both of you and for popularizing science. It sounds like a diss. Where is the next branch of math? Right. What's up with that? You guys have done nothing. I mean, there's definitely been a ton of development in math and new fields developed. Most of the math that exists.
So, okay, algebraic geometry, there's this very interesting phenomenon that's been happening in the last century or so, but even the latter half of that If someone tried to think of like a grand unified theory of math, something where you're trying to understand what are these weird connections that come up in seemingly very disparate parts,
One of those fields that tends to take that perspective, kind of stepping back and saying, hey, what if prime numbers and functions were really living in the same kind of world and the facts that we know about one tell us facts about another? A lot of the people doing algebraic geometry, that kind of fits in there. You also have a pretty big revolution in the way that people think about math, where There is a thing that's called category theory, which is extremely hard to explain, and it's...
kind of like a language. It's sort of like a new language with which mathematicians think about their work that didn't exist 100 years ago. It's very much a different way of thinking, and it's just quietly happening among these circles, not in a way that's very popular. It's never going to show up in your high school calculus class. But it's absolutely like a new thing. Okay, but you're saying it exists within math, but none that have shown up in our K-12 textbook.
Nor should they. I don't think you should shove category theory into a K-12 textbook. I got to say what Grant's doing right now is he's throwing shade at all those mathematicians back there that Akia is talking about. He's like, our stuff is so complex right now. that Yeah, you can't even learn it, okay? i know you're joking i know you're joking but i hate when this is kind of how things come across it's like oh this is so complex it's more like
So you have certain people doing a certain job, like a research mathematician trying to find proofs. This is tools for that job. It's not a great tool for other jobs, like maybe writing programs and using mathematical modeling for... the like simulation that you're running not as good a tool for that job but for their job of writing proofs it's like here's this new tool
It's coming to invention. Once you want to pursue that job, hey, we can make it as approachable as we want. But because I don't think everyone should do that job, we shouldn't put it into K-12. And there's a ton of stuff that's too complex to describe that's vocational.
If you want to understand the exact way that injection molding works or something like that, it might be something that's inappropriate for a podcast, not because it's like highfalutin, you know, super math brain. It's just because, hey, things that are very peculiar to one job. tend to involve a lot of assumed jargon and a lot of assumed context from the people learning it. And it's just not meant to be. Akia, guess what?
It's there, but it's not necessary. You're going to need to know braces. You're going to need to know braces, a kid. This is Gina. She says, hello, smarty pants. I was really... I love it. She says, Gina Martin says, I was recently learning about circles and pie. I learned that the circumference of a circle can never be a hole.
rational number. I am having such a hard time wrapping my head around this. Pun intended. Wrapping my head around. Okay. Could you please explain this a little better for me? Thanks, Gina from North Carolina. Wait, wait, wait. Is she correct? Can't you have the diameter be an irrational number and end up with the circumference? Yeah, so to be clear, the circumference can be anything you want. It could be the number five.
I think the intention of the phrasing was that the ratio of the circumference to the diameter could never be. Or like if the diameter is a whole number, the circumference will never be rational. However you want to phrase it, it's the relationship between those two that's fundamentally irrational.
So it's like a game of whack-a-mole. You make one of them a nice number, the other one looks ugly. You make one of the other one nice, the first one becomes ugly. It becomes irrational. It becomes hard to write down. And so it's a very deep question to try to say, why is pi this ratio between a circle circumference and its diameter? Why is that an irrational number?
There is not a podcastable answer that I can give. This might not be satisfying, but instead of talking about circles, let's talk about squares. where if you have a square and it's got a side length of 1 and you ask how far is it to get from one side length from one corner to the opposite corner, that ends up being the square root of 2.
This is something that follows from the Pythagorean theorem. This is another situation where this geometric length has an irrational relationship with the first length we drew. So the ratio between that diagonal and the square side length is the square root of 2. That is irrational. It's also much, much easier to prove to you why it must be irrational. And if you will indulge me, I think it's possible to do this in like 45 seconds and we can see how this goes. All right, you ready?
Here we go. Proof that square root of 2 is irrational. Assume that you could write it down as a rational number. Maybe you think, oh, maybe square root of 2 is going to be... I don't know, like 5 divided by 3, or maybe something more complicated, like 153 divided by 311. Like, surely I can find big enough numbers that'll make this work.
And I say, whatever you choose, we'll write it down as p over q. We say that's the same thing as the square root of 2. What that would mean, first of all, let's assume that it's fully reduced. So if you wrote something down like... 4 divided by 2. You could reduce that to be 2 divided by 1. So there's no common factor. You can reduce this thing down.
So if that was true, p divided by q is the same as the square root of 2. By definition, you're saying that p squared divided by q squared is equal to 2. So that's what it would mean by definition. So that means when you multiply everything out by the bottom, that q squared, p squared is going to equal 2 times q squared. So if you could have come up with some numbers where it was true, you must admit that p squared is the same as 2 times q squared.
That means that p is an even number because it's two times something. p must be an even number. Yeah, and so you're like, okay, I don't know, let's call P, you know, two times K or something, right? It's some even number. If you then write this down algebraically and you replace it with 2 times k,
you're going to conclude that q also has to be an even number. Because when you take that key equation, p squared equals 2 times q squared, that ends up looking like 4 times k squared equals 2 times q squared. You divide some stuff out, and you say, hey, q also has to be an even number. So you must conclude that P is even. You must conclude that Q is even. But we assumed at the start that it was a reduced fraction. Both of those numbers couldn't be even, otherwise it wouldn't have been reduced.
So there cannot be a way to write it as a fraction, because otherwise you end up in this infinite regress where somehow both of them have to be even, but if you reduce it down, now both of those have to be even, and you'll never get to a coherent answer. It's just a little weird that... to get this irrational number You have to take the ratio of two numbers.
It's a weird fact. So this is a common mathematician tool. They say, oh, you want to prove that something's impossible? They're like, man, this problem's really hard. I think it might be impossible. They have a big ego, and so they want to say, hey, it's not that I can't solve it because I'm dumb. It's because no one can solve it. So they want to prove that it's impossible.
Classic tactic. What you do is you say, I'm going to start by assuming it's possible, like writing some notation to say, what if it was possible? What would follow from that? And then you come to some kind of contradiction. You say, so see, if we assumed it was possible, we land on this thing that could never be. Therefore, our assumption was false. So that's a very common mathematician thing. I love that, what you just did. I love starting with the square. That's very cool.
I'm Tian from Vietnam. If you're... to Journey to Flatland. What shape would you use, I mean would you choose to be and what activity would you like to do there? Thank you, longtime fan of StarTalk and both of you. Tell us about Flatland. I happen to have a copyright here on my desk.
Oh, really? That's great. I've got one over on the shelf over here. So this is a very classic book where the author had you imagine a world that's just two-dimensional. So here in three dimensions, you can left, right, up, down, in, and out. But he said, what if you were just on this two-dimensional world and that's all the world was? And so we have a bunch of creatures there.
And he was making this analogy to say, like, wouldn't it be really hard to describe three-dimensional shapes to them? Like, if you have someone who lives in Flatland and you want to describe what a cube is or a sphere or like a donut, there are these shapes. You just really can't describe it to them. And then the purpose of the book was then to say, hey,
If there's geometric shapes in four dimensions, it is as hard to describe to us as it is for us to describe to Flatlanders. But to the question on what shape would I be in Flatland, I mean, it's kind of basic maybe, but circle seems useful. You can roll around. Everything's nice and symmetric. You can do circle inversion, which, you know, the other patron boy is certainly going to appreciate. probably all i got as i remember the story the more sides you had the more
aristocratic you were. Oh, look at you. So a triangle would be like the lowest, the scum of the earth. Those triangles, I can't believe that. Trying to move in here, they know better. and then the squares, and then pentagons, hexagons. So I think I'd be a hexagon. Okay. I'd be a hexagon. You want to tessellate the plane? Yes. Ah, look at that. Yeah. Although, any shape can tessellate, right?
Well, not any shape. I mean, you got your squares, you got your triangles. No, no, no. I mean, if you look at Escher paintings, he is Isn't that tessellation? Where you have two shapes intersect. Right. So the difference is I'm a regular polygon where all my sides are equal to each other. Then it's only the hexagon. But I think tessellation is all shapes that can do that is called tessellation. Isn't that right?
Well, not any shape can tessellate. So you've got this infinite family that can tessellate. I agree, but you have like angels and devils tessellating. Yeah, yeah, yeah. But that's called tessellation, isn't it? Yeah, 100%. Right, right. So now, if you're going to restrict yourself to a to her shape.
to a polygon, what's called a regular polygon, then we're limited. So I want to be a hexagon because you can tile a floor with a hexagon. Yes, you can, yeah. And get other people to tile with you and you can snuggle and everything fits. It's very, very snugly together. Very cool. So what shape would you be? Now that we're talking about tiling, there was a whole tile that was discovered a couple years ago that's a single tile that tessellates in a non-periodic way.
And it can't tessellate in a periodic, but it only tessellates non-periodically. I heard about that. But is that a regular polygon or just some other shape? No, no, no. It's a wild shape. It's not that weird a shape. It looks like a hat, kind of, people call it. But what was cool is it was discovered by an amateur. So people didn't know if there was a shape that...
tiles things non-periodically or the tiles things on periodically without having any periodic tiling is the like technical question but it's just like an interesting tiling question an amateur found it and it became a little fun celebrity of the math internet for a couple months back then. So what's the difference between a periodic and a non-periodic pattern?
Great. So most of the patterns you can think of are periodic. That's kind of what we mean by pattern. Almost, yeah. Like if you shift the whole picture and it looks identical. So you take your hexagon tiling and then you like shift your view over by one hexagon. It looks identical.
So that's what we would mean by periodic. It wasn't even known that you could have a non-periodic tiling for a while, but Penrose, who is very famous for physics reasons, he found a way to use these two tiles that each look like a rhombus. to have a pattern that fills all of space but it never repeats. So it's a...
describable pattern, you can describe what it should be, but it never repeats. So when you shift your viewing point, it will never look identical. So there's no way to shift it to be the same as what it once was? Yeah. It's kind of like how the digits of pi or these irrational numbers, they don't repeat themselves. They just go on and on in a predictable way, but not that repeats itself. It's the geometric equivalent of that. Okay. Very cool.
This is William Walker. And William Walker says, Hello, gentlemen, from the Florida Panhandle. I've heard it said that mathematically we know properties, some or all, I'm not sure, of dimensions higher than what we observe. Could you please elaborate upon this? What can we say about these dimensions? How do you get there?
Great, great, great, great. I think this is one of the big misconceptions when mathematicians talk about higher dimensions. People assume that they are talking about something that should be physically realized. So ultimately... When you're doing math, sometimes you might have something that can be described by multiple numbers. You have some system like a little particle moving around and you describe its velocity with some list of numbers and its position with some list of numbers.
and you often find it useful to take all your numbers and just list them together. And if you have a list of three numbers, you could think of it as a point in a three-dimensional space. If you have a list of two numbers, you could think of it as a point in a two-dimensional space. Uniquely. Uniquely, yeah. And mathematicians and physicists we realize like, hey, sometimes we're solving a problem and we have a list of like four numbers or five numbers.
And it was really useful to be able to visualize what was going on when it was a list of three numbers by having this unique association between a triplet of numbers and a point in a 3D space. They're like, why can't we do that? Why can't we say there's some abstract four-dimensional space, not in like physical reality, but that's just going to represent.
whatever problem I'm solving where there's a quadruplet of numbers that come up. Or in machine learning these days, when a large language model reads your text. The first thing it does is it turns a given word into a really big list of numbers, like tens of thousands of numbers.
And it's very common for researchers to think of that as a point in an insanely high dimensional space and to use geometric ideas to describe what's happening to it through the model. But of course, we're not saying there's like a 12,000 dimensional space. In physical reality, it's just that it's a nice way to describe lists of numbers. I have on my shelf here. So these dimensions are placeholders. I got on my shelf here. Klein bottle opener. I love it.
So this is an attempt to represent a four-dimensional object in three dimensions. Yeah, so Klein bottles are something that they're most comfortable in four dimensions. This is where they want to live. And if you try to make them live in three dimensions, they have to like unnaturally cross through themselves. There's no way to put it in three dimensions without it crossing through itself. So the coin bottle...
It's a bottle that has no inside. Yeah, I think that's a fair way to say it. There's no, you can't distinguish the inside and the outside. Yeah. I had some friends in college who got in trouble for trespassing in a certain building, and they were like, maybe as part of our defense, we go to the fence outside the building and we apply one twist to it so that the whole fence is a Mobius strip.
and this is another one of those shapes where there's no clear notion of an inside or an outside, then we can argue to the authorities that we couldn't have been inside the trespassing area because there's no coherent notion of the inside of the relevant area. Were these bored Stanford students? I don't know how important they were, but they were creative Stanford students. Not creative enough not to go to jail. Because you're still going down. But that's clever. You have to.
Flip the fence, but then reattach it. Yeah, exactly. While you're doing whatever you're doing, trespassing there, just make sure that as you leave, you cut the fence, you twist the fence, you reattach it so that it's a Mobius fence, and then your defense is solid. so the climb bottom is a four-dimensional version
Of a Mobius strip. Kind of, yeah. I don't love that description. I mean, if you try to take a Mobius strip and you take another Mobius strip and you try to glue their edges together, you'll get a Klein bottle. It's a very mind-warping thing to try to think about.
It's analogous to a Mobius strip in that they both are non-orientable, meaning you have this notion of no clear inside or outside, but they're different. Like a Klein bottle is a closed shape. It doesn't have an edge. Mobius strip has an edge. So topologically, they're pretty different animals, but they swim in the same waters. But it doesn't have a 1D edge, but this has a 2D edge, which would be a surface.
A surface is an edge in four dimensions, isn't it? In the same way the 1D edge of a Mobius strip is an edge in three dimensions. So if you live on the Earth, right, and you try to walk to find the edge of the Earth, it's a sphere. There is no edge. You're never going to, like, go to the edge where all the water is falling off, right?
If it was a flat disc, you could walk to where the edge is. If you're a little ant and you live on a Mobius strip, you can walk to the edge and peer off the edge at some point. If you're walking around the Klein bottle, you never hit an edge in that way. And like mathematically, we call it a closed surface in this way. So they're both surfaces, they're both 2D. Okay, so there's an important distinction, closed surface. That's okay. Time for just one more question. Okay. All right.
Let's go to our old friend. Kevin the sommelier. Oh, okay. That's his last name, the sommelier. His middle name is the sommelier. He says this, hey, Neil has touched on the three-body problem in an explainer episode, but Would there be another branch of mathematics that hasn't been discovered yet that could solve it just like Newton did with motion in honor of Sir Isaac Newton? There is a Spanish wine called Principa. Mathematica, which is a bright white wine made with chariello.
I'm going to find that one. Principia is his greatest work. It's the Mathematical Principles of Natural Philosophy. Sensibly abbreviated Principia. Principia. Yeah, but it's Principia Mathematica, if you want to... give the full. This is the greatest work. If there's a wine with that name, I'm going to find it. Thank you. There is a wine with that name. So I like this question because at what point do you say
It's unsolvable. And at what point do you say the person who's brilliant enough to solve it is yet to be born and to apply their genius to it? Yeah, I would say there's two different ways to think about if a problem's hard to answer. So in the case of Newton modeling the planets,
It wasn't even known what the right math to put to it was, what mathematical you would use to try to make predictions. And his big contribution was to invent the appropriate field of math that you could use to then make predictions. A branch of mathematics that you're not inventing today. Just thought I'd rub that in again, okay? Right. And I mean, the three-body problem feels so different because it's not that it's like we don't know what math should describe it. It's instead saying we know...
It's an intrinsically mathematical question. You say, given this piece of math that's describing it exactly, and it's Newtonian calculus, It's known that you cannot predict what's going to happen if you have a little bit of error in your initial predictions. So this was the big surprise of chaos theory.
where initially you might think, hey, if I know how to solve an equation or if I have some equation, I'm sufficiently smart about it, then if I know the initial state and I just see how the world evolves according to that equation, I can predict the future.
And then chaos theory said there are these situations, including the three body problem, where even if you exactly know what all the solutions are, if you have a little bit of error in your measurement, that error blows up so quickly that subject to that error, the possible states you could end up with after a pretty short amount of time, spans such a wide space of possibilities that effectively the outcome is unpredictable.
so unless you had infinite precision which is just that's not how science or engineering works at all or planets move so it's the result is telling it's not that it's unknown like what the answers will be it's known that the answers are unknowable in a certain way right it's known that it'll be chaotic in the sense that final outcomes are very sensitive
So that's not the type of problem lending itself to this issue, right? Yeah, exactly. We just need a smarter person to come along. No, because the idea is this. The answer is. It is unknowable. That is the actual solution. That's the answer. That's the answer. It's not that we can't solve it. It's not that we can't solve it. We did solve it. And the solution is this is unknowable. You agree with that? That's right.
That's brilliant. Yeah, that's a great summary. I think that's a great summary of chaos. I've earned my keep here. One of the things I like most about mathematics is you get to peer through doorways. in advance of actually stepping there. Because the math is a model of reality. Speaking as a scientist. allows you to explore the world without ever leaving your chair. If the math is a proper model of the physical universe, then you have the power of a god in your hand by making predictions.
If they come true, that gives you that much more confidence that the math and the universe are one. And with that power, I... thrive on thinking about higher dimensions, a topic we talked about. What do things look like and forth? six dimensions. You can't picture that in your head. No. Our brains evolved of the Serengeti trying to not get eaten by lions. We don't have the capacity to think that way, but we have the mind power. To calculate.
and to give us all the information we know about higher dimensions and other places we have yet to visit. And that is a cosmic perspective. So again, tell us your YouTube channel. Yeah, so the YouTube's channel, it's named 3Blue1Brown, an admittedly weird name. You could also search 3B1B. And a lot of topics that we've discussed here, flavors of them show up on that channel. Excellent. And how else do we find you on social media?
Three blue and brown on whatever your favorite social app is these days. Okay, now the three and the one are numerals. Very hard to describe. The numerals mixed with the numbers, yeah. Or just 3B1B. If you search that, you can usually land on it. Okay. Well, just congratulations for your success. And as I say, doing God's work. Yeah. Because God can't divide by zero. Nope. So we're up to see more of you.
keep up the work, and we need more math fluency in this world. Yes, we do. God, please, just infect the whole country, can you, please? I'll do my best. All right, Chuck, always good to have you, man. Always a pleasure. All right, this has been StarTalk Cosmic Queries. Mathematics Edition Neil deGrasse Tyson