The Language of the Universe with Grant Sanderson (3blue1brown)

Why can’t you divide by zero? Neil deGrasse Tyson and Chuck Nice discuss higher dimensions, dividing by zero, and math’s unsolved questions with math YouTuber Grant Sanderson (3blue1brown).

Speaker names marked ? are inferred from speaking time and the show’s known hosts — they are not voice-verified.

• 0:00Neil deGrasse Tyson?

  I think we need to do more episodes that feature math. Yes, with maybe a different co-host because I don't do math.

• 0:11Neil deGrasse Tyson?

  No, I love math. You love math? I do. You realize how important it is? It will, 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, StarTalk Cosmic Queries, a little bit of math.

• 0:27Neil deGrasse Tyson?

  Welcome to StarTalk, your place in the universe where science and pop culture collide. StarTalk begins right now.

• 0:40Neil deGrasse Tyson?

  This is StarTalk Cosmic Queries Edition. Got Chuck Nice with me. Chuck, how you doing, man? I'm doing great. Yeah, yeah, this one is gonna be on math. Oh, I was told there would be no math.

• 0:58Neil deGrasse Tyson?

  Nobody told you there was gonna be no math? That's right. I, I know some math, but if we're gonna have math as a subject, we gotta bring on the big guns. Right. Especially- The Mathiest of the Mathies.

• 1:12Neil deGrasse Tyson?

  We've got with us, not his first time on StarTalk, Grant Sanderson. Grant, welcome back.

• 1:18Speaker 01

  Hey, thanks for having me again. It was fun the first time. Let's see how it goes.

• 1:22Neil deGrasse Tyson?

  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.

• 1:35Speaker 01

  Yeah.

• 1:35Neil deGrasse Tyson?

  With a highly followed channel, Three Blue, One Brown. Mm-hmm. Can you get more cryptic than that? Yes, it sounds, uh, sounds like a three-card Monte scheme.

• 1:47Neil deGrasse Tyson?

  How long have you had the channel?

• 1:48Speaker 01

  Man, it's been around 10 years now, which feels a little wild, yeah.

• 1:52Neil deGrasse Tyson?

  And how many followers, followers you have?

• 1:54Speaker 01

  It's, uh, it's, it's around 7 million. Um, I remember sometime recently across-

• 1:58Neil deGrasse Tyson?

  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.

• 2:19Speaker 01

  In general, more people like math than people suspect, I think. Like, it's a little bit, it's a little bit of an underdog. Everyone thinks people hate it, but like-

• 2:26Neil deGrasse Tyson?

  I think everyone loves math.

• 2:29Chuck Nice?

  I just think that most people are intimidated by it and intimidate and 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.

• 2:43Neil deGrasse Tyson?

  But that's why Grant exists so that he can grow the comfort zone that people feel.

• 2:50Chuck Nice?

  Look at you.

• 2:50Neil deGrasse Tyson?

  When encountering this content.

• 2:52Chuck Nice?

  Yes.

• 2:53Neil deGrasse Tyson?

  So let me pick up some, some, some broad, deep topics in math before we get to our Q and A. We, 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. Right? Is there, is there like a, a, a, a book of the latest unsolved problems in math? And then that's what nerd geeks should, should check out of the library first.

• 3:26Chuck Nice?

  Yeah.

• 3:26Speaker 01

  Yeah. I mean, there's, there's, there's been many. I mean, one of the most famous bodies of unsolved problems. Uh, 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.

• 3:51Speaker 01

  So very classic.

• 3:53Neil deGrasse Tyson?

  You can't even describe the problem of the problem you're supposed to solve.

• 3:56Chuck Nice?

  Right. You gotta solve the problem, but you can't even describe, nobody can tell you what the problem is. Now to me, sir.

• 4:01Neil deGrasse Tyson?

  That's how you keep your million dollars.

• 4:02Chuck Nice?

  Yes. I was gonna say, that sounds a lot like being married.

• 4:09Speaker 01

  But what is the problem even that we're beginning to solve? So, but there, 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, for the public.

• 4:25Chuck Nice?

  But there's, there's no prize money for those, right?

• 4:27Speaker 01

  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 awards, there's like plenty that would come if you could, um, solve one of these problems. But of course that's not what most people care about.

• 4:47Neil deGrasse Tyson?

  Are they arise purely within math or is there some scientific? A pumping of what these problems are or engineering solutions.

• 4:55Speaker 01

  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 initial, 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, the like you, 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.

• 5:34Neil deGrasse Tyson?

  His dad asked him this question, this question when he was 11.

• 5:37Chuck Nice?

  Oh, wow. So either you were really smart or you had a really bad dad.

• 5:44Speaker 01

  So I was like, go and think, I'm like, oh, like, is this true? And eventually it didn't take too long for him to say like, oh, this was a thing I was reading about in this like science news magazine that like mentioned this is a problem. Nowhere, no one in the world knows how to answer. And that was fascinating to me. Like, here's, here's a question you can ask, like, you teach kids about prime numbers. Maybe not everyone remembers them when they grow older, but like, it's a common topic. All you're asking is, hey, are there infinitely many that are two apart? Do they ever stop coming 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? You could ask this over two millennia ago. We still don't know the answer and it's, it cuts pretty deep, like to the understanding

• 6:27Speaker 01

  of primes that we have and the lack of understanding to be able to answer questions like this. So that's pure math.

• 6:32Neil deGrasse Tyson?

  Interesting.

• 6:32Speaker 01

  It's just a puzzle.

• 6:33Neil deGrasse Tyson?

  And you got one from science?

• 6:34Speaker 01

  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-Stokes equations.

• 6:50Neil deGrasse Tyson?

  Equations, yeah.

• 6:51Speaker 01

  Yeah. And so if you're, if you're just modeling some fluid and you understand certain aspects of its, you know, pressure and viscosity and things like this, there's something, for example, you could 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 like you get an infinite concentration of energy at some point. You don't like, clearly that shouldn't happen. You don't think that would happen in the in the physical world. But 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.

• 7:32Speaker 01

  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.

• 7:52Neil deGrasse Tyson?

  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?

• 8:04Speaker 01

  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 like 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.

• 8:33Neil deGrasse Tyson?

  Chuck recite the quadratic formula. Do you remember it?

• 8:35Chuck Nice?

  Here's the quadratic formula. What you get for number 13?

• 8:41Neil deGrasse Tyson?

  Looking over the shoulder. But I mean, it's it was I remember getting drilled into us on a level where it prevented me from appreciating what it was actually doing.

• 8:50Chuck Nice?

  Right?

• 8:50Neil deGrasse Tyson?

  It was just a memorizable formula. It was like negative b plus or minus the square root of b squared minus four a c over two a. Did I get that right?

• 8:59Speaker 01

  That's exactly right.

• 8:59Neil deGrasse Tyson?

  Okay.

• 9:00Chuck Nice?

  Okay.

• 9:01Neil deGrasse Tyson?

  I just did that by rote, not because I had any freaking feeling for it.

• 9:05Chuck Nice?

  You just had to know it so many times.

• 9:06Neil deGrasse Tyson?

  Yeah, I got it from eighth grade. It came in. Okay.

• 9:09Speaker 01

  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 would be it takes a much longer song to do it. And if you 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.

• 9:31Speaker 01

  But like pure mathematicians for a while were curious, okay, hey, how far can we take this? You know, 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, it's an even more monstrous formula. If you tried to write it down in full, it would just fill 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 five 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.

• 10:05Neil deGrasse Tyson?

  Wait, are you telling me people had these thoughts in the 1500s while others were disemboweling heretics?

• 10:12Speaker 01

  Yeah.

• 10:13Neil deGrasse Tyson?

  These two things coexisted.

• 10:14Speaker 01

  It's wild to contrast history of science.

• 10:17Neil deGrasse Tyson?

  Well, maybe they stopped by the 1500s. The Renaissance was. Yeah, exactly. I'm thinking maybe like the Dark Ages. Mm hmm.

• 10:24Speaker 01

  Well, you go to the right part of the world. You know, 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 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.

• 10:59Neil deGrasse Tyson?

  It works up to fourth order polynomial, but not fifth.

• 11:02Speaker 01

  And then suddenly five, it stops. Yeah, it stops working.

• 11:05Neil deGrasse Tyson?

  Wow.

• 11:06Speaker 01

  And there's a deep reason why. And the, 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 were 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.

• 11:32Neil deGrasse Tyson?

  What's the reason? Why does it, why does the symmetry break between four and five?

• 11:37Speaker 01

  Neil, I would love so much to be able to like, give you the pithy answer that's like, here's what's true at four, that's not true at five. And 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?

• 11:54Neil deGrasse Tyson?

  I want a full YouTube video on my desk Monday morning of you explaining this.

• 12:00Speaker 01

  It would, it would make me and a lot of other people very happy. Okay, I'm gonna give, I'm gonna give an answer that will, will make no sense, but I promise that there's some sense in which this is true. It's something called Galois theory. And the, one of the young mathematicians who came up with the arguments that led to explaining this, his name was Everest Galois. He died in a duel at the age of 20. This is his big headline fact.

• 12:22Neil deGrasse Tyson?

  He knew he was gonna lose that duel. And so the night before the duel, he like, does like a brain dump on, on the page.

• 12:30Speaker 01

  Oh, this is the, this is the story. This is the classic story that we all tell in the lecture halls.

• 12:35Chuck Nice?

  That's insane.

• 12:36Speaker 01

  Yeah, okay, wait, let the man explain.

• 12:38Chuck Nice?

  Go ahead.

• 12:39Speaker 01

  No, no, so what you're saying is what everyone hears when they're learning physics and learning math and whatnot.

• 12:43Chuck Nice?

  Oh, that's not true?

• 12:44Speaker 01

  He wrote down all the, so he did, he like was compiling stuff that he had tried to get published like three or four times before. So the stuff, he had written the stuff down before. He had gotten it in front of very famous mathematicians like Fourier or Cauchy and Poisson. Like they had seen the beginnings of his work before. So it wasn't like the first time he's ever jotting this down was in the like 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 gotta get these ideas out in some way. He's got this classic story behind him in terms of the duel 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 five or higher. It also showed other impossibilities. There's this Classic problem about trisecting the angle,

• 13:28Speaker 01

  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. 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.

• 13:46Chuck Nice?

  Wow, cool, man.

• 13:48Neil deGrasse Tyson?

  Well, math is the language of the universe.

• 13:49Chuck Nice?

  It certainly is.

• 13:50Neil deGrasse Tyson?

  We shouldn't be surprised if it spillage into cosmic discovery.

• 13:55Chuck Nice?

  Yeah, it makes sense.

• 13:56Neil deGrasse Tyson?

  So can we bring questions your way from our audience?

• 13:58Speaker 01

  Yeah, yeah, yeah, let's do it.

• 14:00Chuck Nice?

  Here's the first question from Buck Rice, who says, "Why can't we divide by zero? Why?" I love that.

• 14:11Speaker 01

  That's a great question.

• 14:12Neil deGrasse Tyson?

  I've carried that with me my whole life.

• 14:13Chuck Nice?

  Right.

• 14:14Neil deGrasse Tyson?

  Because the answer is, well, it's undefined. And so my response is, well, define it. Get off your duff and define it.

• 14:25Speaker 01

  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,

• 15:10Speaker 01

  it's like, it's an incoherent question. Like 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.

• 15:17Chuck Nice?

  See, but that's, 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.

• 15:31Neil deGrasse Tyson?

  That's the answer.

• 15:31Chuck Nice?

  And that's the answer.

• 15:32Neil deGrasse Tyson?

  Chuck eats the cupcake.

• 15:34Chuck Nice?

  The answer is, I eat the cupcake.

• 15:38Neil deGrasse Tyson?

  Now it's defined.

• 15:39Chuck Nice?

  Now it's defined. All right.

• 15:54Speaker 03

  Hello, I'm Vinky Marok Allen and I support StarTalk on Patreon. This is StarTalk with Neill Grass Tyson.

• 16:13Chuck Nice?

  This is Kira. Actually, I'm gonna combine two questions in one because Kira and Gavin Bamber actually are similar, but I'm gonna 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 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, fundamentally change the way we view the universe. And then, part B, what is your personal favorite unsolvable?

• 16:54Neil deGrasse Tyson?

  Wait, you can't say one and then part B. It's either one and two, or A and B.

• 16:58Chuck Nice?

  No. See, what I am doing is a new kind of math.

• 17:04Speaker 01

  Tell you what, I'm gonna punt off part one to Neil here because I 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.

• 17:17Neil deGrasse Tyson?

  We have singularity problems in the universe.

• 17:19Chuck Nice?

  Right.

• 17:20Neil deGrasse Tyson?

  What, all of our equations tell us that the center of a black hole, nature is dividing by zero.

• 17:25Chuck Nice?

  Right.

• 17:26Neil deGrasse Tyson?

  Okay, everything goes, the denominator goes to zero.

• 17:31Chuck Nice?

  Right.

• 17:31Neil deGrasse Tyson?

  What happens to the value of the whole thing?

• 17:33Chuck Nice?

  The value of everything else.

• 17:34Neil deGrasse Tyson?

  Okay. We say there's infinite density and infinite this, and it doesn't even make any sense.

• 17:37Chuck Nice?

  No, it doesn't.

• 17:38Neil deGrasse Tyson?

  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.

• 17:47Chuck Nice?

  Right.

• 17:48Neil deGrasse Tyson?

  Because we're not the first to blame the math. I'm just saying.

• 17:51Chuck Nice?

  Gotcha.

• 17:51Neil deGrasse Tyson?

  Because math is badass, and we're not. Okay, so, we're gonna 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 did. And so this is the curvature of spacetime. It's not flat and Euclidean geometry is flat. And who came up with curved geometry?

• 18:25Speaker 01

  So Riemann is the big one there.

• 18:27Neil deGrasse Tyson?

  So Riemann, and when was that, like 1800s sometime?

• 18:29Speaker 01

  Yeah, it was in the 1800s, like maybe 50s, let's say.

• 18:33Neil deGrasse Tyson?

  Okay, all right, so 19th century, we have the tools to think about curved geometry. That was immediately uptook by the cosmologists to think about what could be the geometry of the universe.

• 18:49Speaker 01

  Right.

• 18:49Neil deGrasse Tyson?

  They needed a way to talk about. So that's all I got here, but it's the math leading us, not us finding a math problem that's not solved.

• 18:56Speaker 01

  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.

• 19:04Neil deGrasse Tyson?

  Oh.

• 19:04Speaker 01

  For one of my favorite unsolved problems.

• 19:06Neil deGrasse Tyson?

  It was part one and then part B.

• 19:07Chuck Nice?

  Yeah, it's part one and part B.

• 19:09Speaker 01

  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.

• 19:54Speaker 01

  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.

• 20:30Neil deGrasse Tyson?

  Is that your favorite unsolved problem?

• 20:32Speaker 01

  I think it's my favorite. Yeah.

• 20:33Neil deGrasse Tyson?

  Oh, wow. Okay.

• 20:35Chuck Nice?

  It sounded like it was too. You make it sound very elegant as a.

• 20:39Neil deGrasse Tyson?

  He sounded a little bit excited as he was talking about it.

• 20:41Chuck Nice?

  I love it. Yeah. Okay. All right, this is Tony Isaacs and Tony says, G'day Astro Nail and Lord Nice. This is Tony here from Melbourne, Australia.

• 20:50Neil deGrasse Tyson?

  Melbourne?

• 20:50Chuck Nice?

  Melbourne. Um, he says, love the show. Go on Chuck and do an RC accident.

• 20:59Chuck Nice?

  Okay, that's somebody who's been listening to the show for a while.

• 21:02Neil deGrasse Tyson?

  Listening too long.

• 21:03Chuck Nice?

  Yeah, yeah. It totally telegraphs 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.

• 21:27Neil deGrasse Tyson?

  Hmm.

• 21:28Chuck Nice?

  Yeah, so.

• 21:29Neil deGrasse Tyson?

  So Grant, I'm going to lead off by saying, why did you label those numbers imaginary?

• 21:36Speaker 01

  My God, it's the worst name in all of math.

• 21:38Neil deGrasse Tyson?

  Worst name ever.

• 21:40Speaker 01

  Gauss proposed calling them lateral numbers, which would have been, you know.

• 21:43Neil deGrasse Tyson?

  A little better.

• 21:44Speaker 01

  Just a little better.

• 21:45Neil deGrasse Tyson?

  And then, then you add another aspect to the imaginary number to get the complex number.

• 21:49Chuck Nice?

  Right.

• 21:50Neil deGrasse Tyson?

  And now you call it a complex? These are words that are complete turn offs and I blame you.

• 21:54Chuck Nice?

  Well, I mean, if that's.

• 21:56Neil deGrasse Tyson?

  I blame your people.

• 21:57Chuck Nice?

  I was going to say, 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?

• 22:05Speaker 01

  Man, so many people struggle with this, right? Because you label it as imaginary. You, you start by pitching it by saying, you know, square roots of negatives don't exist, but pretend like they do and run forward. And like, you could, 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, uh, there's a natural number system to try to describe that. Call that number system what you want.

• 22:35Chuck Nice?

  Oh, so you can think of them as like clocks then?

• 22:38Speaker 01

  Think of them as clock numbers, right? Anytime you're doing clock stuff, these numbers are gonna be great. So we use the clock numbers. Why are they relevant in quantum mechanics? You 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.

• 23:04Neil deGrasse Tyson?

  So the engineers adopted the complex plane, 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.

• 23:21Speaker 01

  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.

• 23:42Neil deGrasse Tyson?

  I don't thank God for that.

• 23:46Neil deGrasse Tyson?

  About what I thank God for.

• 23:47Chuck Nice?

  Thank math.

• 23:49Speaker 01

  Thank the education system. But like if you would ask someone, hey, solve the equation x squared plus one equals zero, they would have said, there is no solution. Like, obviously, there's no, 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 like square roots of negatives. But when you use the formula, you can find that real valued 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 like kind of works.

• 24:32Speaker 01

  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. Imaginary was kind of derogatory. It wasn't like, hey, we want to teach kids this. What should we call it?

• 24:49Chuck Nice?

  It was, it was like imaginary girlfriend.

• 24:54Speaker 01

  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 that involve or physics that involves waves and cycles. And so that's what we're stuck with.

• 25:06Chuck Nice?

  Hello, Dr. Tyson, Lord Nice, Mr. Brown. This is Brandon from New Jersey. Is there any relation between the conformal geometry 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 the line after curving it. Is this at work in conformal geometry as well? I'm completely confused, but wow. What, what, 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.

• 25:52Chuck Nice?

  Stop showing off. This is not the place. All right, go ahead.

• 25:55Speaker 01

  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. 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, but.

• 26:11Chuck Nice?

  All right, can you do it in 20 seconds as a challenge? And if we understand it, who cares? If not, then we now we have something to go look up, which is even more fun. Go ahead. All right, go.

• 26:21Speaker 01

  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 do 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. Some of the. 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.

• 27:07Chuck Nice?

  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.

• 27:16Speaker 01

  Yeah, I gotta give actual examples for that to carry teeth, but you know, a little too long.

• 27:20Neil deGrasse Tyson?

  He was still showing off though.

• 27:21Chuck Nice?

  Yeah, he's showing off without a doubt. Brandon was 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 it's now it's fodder for look up, which is great. Ethan step and Ethan says, hello, Dr. Tyson and Mr. Grant. And, and Lord Nice, if you're there, hello. I love these people, man. 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?

• 28:05Speaker 01

  Okay, I think the easiest way to describe it, it 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 like computer scientist way to answer what a tensor is. But then in physics, the, 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

• 28:46Speaker 01

  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 to answer the question like, where did this come from? Or who would come up with it? I think usually it's, 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 like 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.

• 29:15Neil deGrasse Tyson?

  But somebody's got to be clever enough to see that need and then come up with it, right? Not everyone is that clever. They would be a prisoner of the known math of the And you need someone who can step out of that and say, I have a new way to think about this problem.

• 29:30Speaker 01

  I mean, some physicists will joke that one of Einstein's greatest contributions to physics was his notation for tensors. And like, 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.

• 29:54Neil deGrasse Tyson?

  This is not T I J, is that?

• 29:56Speaker 01

  Yeah, this is when you're like the Einstein summation notation where you wanna...

• 30:00Neil deGrasse Tyson?

  Yeah, it's a simplified thing. It makes the equation look way simpler and more tame than what's actually going on.

• 30:07Chuck Nice?

  Well, I'm gonna 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 gonna be honest. I'm sitting here, and it's rare that I am lost within a conversation, but this one is like out there, man, which is very cool. So he...

• 30:27Neil deGrasse Tyson?

  Well, plus he's got seven million followers on his YouTube, so he's doing something right.

• 30:30Chuck Nice?

  He's doing something 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...

• 30:54Neil deGrasse Tyson?

  No, no.

• 30:55Chuck Nice?

  I'm making this part up. I'm like, thanks, Akia, 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.

• 31:24Neil deGrasse Tyson?

  It sounds like a diss. Where's the next branch of math?

• 31:27Chuck Nice?

  Right.

• 31:28Neil deGrasse Tyson?

  What's up with that?

• 31:30Chuck Nice?

  You guys have done nothing.

• 31:33Speaker 01

  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?

• 31:59Speaker 01

  One of those fields that tends to take that perspective, kind of stepping back and saying, hey, what if, what if prime numbers and like 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 like way of thinking and it's just quietly happening among these circles, not in a way that's very popularized. It's never going to show up in your high school calculus class, but it's absolutely

• 32:44Speaker 01

  like a new thing.

• 32:45Neil deGrasse Tyson?

  Okay, but you're saying it exists within math, but none that have shown up in our K through 12 textbook?

• 32:51Speaker 01

  Nor should they. Like, I don't think you should shove category theory into a K through 12 textbook.

• 32:55Chuck Nice?

  I gotta 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?

• 33:12Neil deGrasse Tyson?

  Only three people can learn it in the whole world.

• 33:14Chuck Nice?

  That's it, baby.

• 33:16Speaker 01

  No, I know you're joking, I know you're joking, but I hate when this is kind of how things come across as 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 come into 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, like, we shouldn't, we shouldn't put it into K12. And there's a ton of stuff that's too complex to describe that's vocational, right? If you want to understand like 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

• 34:00Speaker 01

  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.

• 34:13Chuck Nice?

  Akia, guess what? It's there, but it's not necessary for, you're on a need to know basis. You're on a need to know basis, Akia. And that's, there we go. All right. This is Gina. She says, hello, smarty pants. I was

• 34:32Chuck Nice?

  I love it. She says, Gina Martin says, I was recently learning about circles and pi. I learned that the circumference of a circle can never be a whole rational number. I am having such a hard time wrapping my head around this, pun intended, wrapping my head around. Okay. Uh, could you please explain this a little better for me? Thanks, Gina from North Carolina.

• 34:59Speaker 01

  Wait, wait, wait, wait. Is she correct? Can't you have the diameter be an irrational number? And end up with the circumference rational? Yeah. So to be clear, if you want the circumference, the circumference could be anything you want. It could be the number five. Um, I think what, 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, 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, ugly. It becomes irrational, becomes hard to write down.

• 35:42Speaker 01

  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 one, 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. Now, that is irrational.

• 36:24Speaker 01

  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 you can, we can, 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. Right, like maybe you think, oh, maybe square root of 2 is gonna 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.

• 37:06Speaker 01

  So if, 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 like multiply everything out by the bottom, that q squared, p squared is gonna equal 2 times q squared. So if this, if you could 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 2 times something. P must be an even number.

• 37:48Speaker 01

  Yeah. And so you're like, okay, I don't know, let's, let's call p, you know, 2 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 gonna 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.

• 38:30Speaker 01

  But if you reduce it down, now both of those have to be even and you'll never get to a coherent answer.

• 38:35Neil deGrasse Tyson?

  It's just a little weird that to get this irrational number, you have to take the ratio of two numbers. That's a weird fact.

• 38:43Speaker 01

  So this is a common mathematician tool. They say, oh, you want to prove that something's impossible? Right. They're like, man, this problem's really hard. I think it might be impossible. Like, 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.

• 39:22Chuck Nice?

  I love that, what you just did. I love starting with the square. That's very cool.

• 39:49Chuck Nice?

  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.

• 40:11Neil deGrasse Tyson?

  Tell us about Flatland. I happen to have a copy right here on my desk.

• 40:15Speaker 01

  Oh, really? That's great. Yeah, I've got one over on the shelf over here. So this is, this is a very classic book where the author, had you imagined a world that's just two dimensional? So here in three dimensions, you can look 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 you 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, we are, it is as hard to describe to us as it is for us to describe to flatlanders.

• 40:58Speaker 01

  But to the, to the question on what shape would I be in flatland?

• 41:04Speaker 01

  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, are the other patron boy is only going to appreciate. But that's probably all I got.

• 41:19Neil deGrasse Tyson?

  As I remember the story, the more sides you had, the more aristocratic you were. Ooh, look at you. Right. 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. Right. So I think I'd be a hexagon. Okay. I'd be hexagon.

• 41:41Speaker 01

  You want to tessellate the plane?

• 41:43Neil deGrasse Tyson?

  Yes. Ah, look at that. Yeah. Although, any shape can tessellate, right?

• 41:49Speaker 01

  Well, not any shape. I mean, you got your squares, you got your triangles.

• 41:53Neil deGrasse Tyson?

  No, no, no. I mean, if you look at, if you look at Escher paintings, he is, isn't that tessellation? Where you have two, two shapes intersect. Right. So the, the difference is, I'm a, what they call 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?

• 42:15Speaker 01

  Well, not any shape can tessellate. So you're right that Esher shows you've got this infinite family that can tessellate.

• 42:20Neil deGrasse Tyson?

  I agree, but he has like angels and devils tessellating.

• 42:23Speaker 01

  Yeah, yeah, yeah, yeah.

• 42:24Neil deGrasse Tyson?

  But that you can test, that's called tessellation, isn't it?

• 42:26Speaker 01

  Yeah, 100%.

• 42:28Neil deGrasse Tyson?

  Right, right. So now, if you're going to restrict yourself to a, to a shape, to a polygon, what's called a regular polygon, then we're limited, right? But so I want to be a hexagon, because you can tile a floor with a hexagon.

• 42:41Chuck Nice?

  Yes, you can, yeah.

• 42:42Neil deGrasse Tyson?

  And get other people to tile with you, and you can snuggle, and everything fits.

• 42:46Chuck Nice?

  Very, very snugly together.

• 42:47Neil deGrasse Tyson?

  Snuggly.

• 42:48Chuck Nice?

  Very cool.

• 42:49Neil deGrasse Tyson?

  So what shape would you be?

• 42:51Speaker 01

  Now that we're talking about tiling, there was a whole tile that was discovered a couple of 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.

• 43:02Neil deGrasse Tyson?

  I heard about that. But, but is that a regular polygon, or is it just some other shape?

• 43:07Speaker 01

  No, no, no, it's a wild, it's not that weird a shape. It looks like a hat, kind of, people call it. But what's cool is it was discovered by an amateur. So it was this, people didn't know if such a, if there was a shape that tiles things non-periodically, or that tiles things non-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 of months back then.

• 43:34Neil deGrasse Tyson?

  So what's the difference between a periodic and a non-periodic pattern?

• 43:37Speaker 01

  Great. So most of the patterns you can think of are periodic, like.

• 43:40Neil deGrasse Tyson?

  It's kind of what we mean by pattern.

• 43:42Speaker 01

  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, 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, it's a, 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.

• 44:16Neil deGrasse Tyson?

  So there's no way to shift it to be the same as what it once was?

• 44:20Speaker 01

  Yeah. It's kind of like how the digits of pi are these irrational numbers. They, 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.

• 44:30Chuck Nice?

  Okay. Very cool. This is William Walker. And William Walker says, hello, gentlemen, from Florida, from the Florida Panhandle.

• 44:39Chuck Nice?

  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?

• 44:54Neil deGrasse Tyson?

  Yeah, how do you get there?

• 44:55Speaker 01

  Great, great, great, great. I think this is one of the big misconceptions when mathematicians talk about higher dimensions. People are like, assume that they are talking about something that should be physically realized. But 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, I don't know, little particle moving around and you describe its like 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.

• 45:32Neil deGrasse Tyson?

  Uniquely.

• 45:33Speaker 01

  But the math.

• 45:33Neil deGrasse Tyson?

  Yeah.

• 45:34Speaker 01

  Uniquely, yeah.

• 45:34Neil deGrasse Tyson?

  Yeah.

• 45:35Speaker 01

  And mathematicians and physicists 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 like 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, right? When a, when a, 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

• 46:19Speaker 01

  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.

• 46:33Neil deGrasse Tyson?

  I have on my shelf here.

• 46:34Chuck Nice?

  So these dimensions are placeholders.

• 46:36Neil deGrasse Tyson?

  What, what? I got it on my shelf here. A Klein bottle, bottle opener.

• 46:43Speaker 01

  I love it.

• 46:45Neil deGrasse Tyson?

  So this is an attempt to represent a four dimensional object in three dimensions.

• 46:51Chuck Nice?

  Yeah.

• 46:51Speaker 01

  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.

• 47:06Neil deGrasse Tyson?

  So the Klein bottle is a bottle that has no inside.

• 47:08Speaker 01

  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, I had some friends in college who got in trouble for trespassing in a certain building and they, they're 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. Like 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.

• 47:37Neil deGrasse Tyson?

  Were these bored Stanford students?

• 47:41Speaker 01

  Yeah, I don't know how bored they were, but they were, they were creative Stanford students.

• 47:46Chuck Nice?

  Yeah, not creative enough not to go to jail.

• 47:52Chuck Nice?

  Cause you're still going down.

• 47:54Neil deGrasse Tyson?

  But that's clever, you have to flip the fence, but then reattach it.

• 47:56Chuck Nice?

  Reattach it.

• 47:57Neil deGrasse Tyson?

  Yeah.

• 47:57Speaker 01

  Yeah, exactly. So in the middle of the night, you know, while you're doing whatever you're doing trespassing there, just make sure that as you leave, you like, you cut the fence, you twist the fence, you reattach it so that it's a Mobius fence and then your defense is solid.

• 48:09Neil deGrasse Tyson?

  So Klein bottle is a four dimensional version of a Mobius strip.

• 48:14Speaker 01

  Kind of, yeah, I, I don't, I don't love that description. I mean, it's what, 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, but they're different, like a Klein bottle is a closed shape. It doesn't have an edge. Mobius strip has an edge. So like topologically, they're pretty different animals, but they, they like swim in the same waters.

• 48:43Neil deGrasse Tyson?

  Well, 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.

• 48:57Speaker 01

  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 gonna like go to the edge where all the water is falling off, right? If it was a flat disk, 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 like peer off the edge at some point. If you're walking around the Klein bottle, you never hit an edge in that way. It's, and like mathematically, we call it a closed surface in this way. So they're both surfaces, they're both 2D.

• 49:25Neil deGrasse Tyson?

  Okay, so there's an important distinction.

• 49:27Chuck Nice?

  Closed surface, that's the key.

• 49:28Neil deGrasse Tyson?

  Time for just one more question.

• 49:30Chuck Nice?

  Okay, all right. Let's go to our old friend, Kevin the Sommelier.

• 49:34Neil deGrasse Tyson?

  Oh, okay.

• 49:35Chuck Nice?

  And Kevin says-

• 49:36Neil deGrasse Tyson?

  That's his last name, The Sommelier. His middle name is The, the last name is Sommelier.

• 49:41Chuck Nice?

  He says, 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 Charelló.

• 50:08Neil deGrasse Tyson?

  I'm gonna find that wine.

• 50:09Chuck Nice?

  Yeah.

• 50:10Neil deGrasse Tyson?

  So Principia is his greatest work. It's the Mathematical Principles of Natural Philosophy. Sensibly abbreviated Principia.

• 50:19Chuck Nice?

  Principia.

• 50:20Neil deGrasse Tyson?

  Yeah, but it's Principia Mathematica if you want to give the full thing.

• 50:23Chuck Nice?

  Oh, okay.

• 50:23Neil deGrasse Tyson?

  This is his greatest work. If there's a wine with that name, I'm gonna find it. Thank you.

• 50:27Chuck Nice?

  Hey, there is a wine with that name.

• 50:28Neil deGrasse Tyson?

  Sommelier. 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?

• 50:43Speaker 01

  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 mathematic model 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.

• 51:02Neil deGrasse Tyson?

  A branch of mathematics that you're not inventing today. Just thought I'd rub that in again. Okay.

• 51:07Speaker 01

  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,

• 51:26Speaker 01

  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

• 52:10Speaker 01

  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.

• 52:24Speaker 01

  So 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.

• 52:36Neil deGrasse Tyson?

  So that's not the type of problem lending itself to this issue. Right.

• 52:41Speaker 01

  Yeah.

• 52:41Neil deGrasse Tyson?

  We just need a smarter person to come along.

• 52:43Chuck Nice?

  No, because the idea is this. The answer is it is unknowable.

• 52:49Neil deGrasse Tyson?

  That is.

• 52:49Chuck Nice?

  That is the actual solution.

• 52:51Neil deGrasse Tyson?

  That's the answer.

• 52:52Chuck Nice?

  That's the answer.

• 52:53Neil deGrasse Tyson?

  It's not that we can't solve it.

• 52:54Chuck Nice?

  We did solve it. We did solve it. And the solution is.

• 52:58Neil deGrasse Tyson?

  Excellent.

• 52:58Chuck Nice?

  This is unknowable.

• 52:59Neil deGrasse Tyson?

  You agree with that. That's brilliant.

• 53:01Speaker 01

  Yeah, that's a great summary. I think that's a great summary of chaos.

• 53:04Chuck Nice?

  I've earned my keep here. I can go home now.

• 53:08Neil deGrasse Tyson?

  Just barely.

• 53:09Chuck Nice?

  I know what time it is. Really, Chuck, on the last question?

• 53:15Neil deGrasse Tyson?

  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 as 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.

• 53:58Neil deGrasse Tyson?

  And with that power, I thrive on thinking about higher dimensions, a topic we talked about. What do things look like in four dimensions, five dimensions, six dimensions? You can't picture that in your head. No. Our brains evolved on the plains 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 that way 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?

• 54:40Speaker 01

  Yeah, so the YouTube's channel, it's named Three Blue One Brown, 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.

• 54:51Neil deGrasse Tyson?

  Excellent. And how else do we find you on social media?

• 54:55Speaker 01

  Three Blue and Brown on whatever your favorite social app is these days.

• 54:59Neil deGrasse Tyson?

  Okay, now the three and the one are numerals and the blue and the brown are words.

• 55:05Speaker 01

  Very hard to describe. The numerals mixed with the numbers, yeah. Or just 3B1B, if you search that, you can usually land on it.

• 55:12Neil deGrasse Tyson?

  Okay. Well, just congratulations for your success and, as I say, doing God's work.

• 55:17Chuck Nice?

  Yeah.

• 55:18Neil deGrasse Tyson?

  Because God can't divide by zero.

• 55:20Chuck Nice?

  Nope.

• 55:21Neil deGrasse Tyson?

  So we hope to see more of you. Keep up the work and we need more math fluency in this world.

• 55:26Chuck Nice?

  Yes, we do. God, please, just infect the whole country, can you please?

• 55:31Speaker 01

  Do my best.

• 55:32Neil deGrasse Tyson?

  All right, Chuck, always good to have you, man.

• 55:33Chuck Nice?

  Always a pleasure.

• 55:34Neil deGrasse Tyson?

  All right. This has been StarTalk Cosmic Queries, Mathematics Edition. Neil deGrasse Tyson, keep looking up.