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

Summary

StarTalk Radio: The Language of the Universe with Grant Sanderson (3blue1brown)
Release Date: May 20, 2025

In this intellectually stimulating episode of StarTalk Radio, host Neil deGrasse Tyson teams up with co-host Chuck Nice and special guest Grant Sanderson—the creator behind the popular YouTube channel 3blue1brown—to explore the profound role of mathematics in understanding the cosmos. Titled "The Language of the Universe," this episode delves deep into mathematical concepts, unsolved problems, and the evolving landscape of mathematics that continues to shape our perception of the universe.

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1. The Perception and Importance of Mathematics

The episode kicks off with a light-hearted discussion on the general public's relationship with mathematics. Chuck Nice humorously remarks, "If 7 million people follow you for math, that gives me hope for the future of civilization" (02:53). Grant Sanderson counters the common misconception that people dislike math by stating, "More people like math than people suspect" (03:06), emphasizing that intimidation, not dislike, often deters engagement.

2. Unsolved Mathematical Problems and the Clay Millennium

A significant portion of the conversation revolves around some of the most challenging unsolved problems in mathematics. Grant introduces listeners to the Clay Mathematics Institute's Millennium Prize Problems, which include seven notorious challenges each offering a $1 million reward for a valid solution (04:14). Among these, the Twin Prime Conjecture captures Grant's personal interest. He explains, "Do you think there's infinitely many primes that are just two apart?" (06:21), highlighting its historical roots dating back to Euclid and its enduring mystery in number theory.

3. The Complexity of the Navier-Stokes Equations

Transitioning from pure mathematics to applied mathematics, Grant discusses the Navier-Stokes equations—fundamental to fluid dynamics. He elaborates on the unresolved questions surrounding these equations, such as the potential for infinite energy concentrations, which should theoretically be impossible in the physical world (07:37). This underscores the intricate relationship between mathematical modeling and physical phenomena.

4. From Solvable to Unsolvable: The Evolution of Polynomial Equations

The trio delves into the historical progression of solving polynomial equations. Starting with the quadratic formula, Grant explains how mathematicians successfully devised solutions for polynomials up to the fourth degree. However, attempts to solve fifth-degree (quintic) equations using radicals proved futile. He recounts the pivotal contributions of Évariste Galois, who, before his untimely death at 20, laid the groundwork for abstract algebra and demonstrated the impossibility of a general solution for quintic equations using traditional operations (12:48).

5. The Riemann Hypothesis: Prime Mysteries Unveiled

One of the episode's highlights is the discussion on the Riemann Hypothesis, which Grant identifies as his favorite unsolved problem. He describes it as a beautiful and profound question that connects the distribution of prime numbers to the zeros of the Riemann zeta function. "Grant shares, 'If you understand something about this function, you completely understand the primes' (23:01), emphasizing its significance in number theory and its lingering mystery in mathematics.

6. Demystifying Complex Numbers

A listener's question about why complex numbers are termed "imaginary" opens up an engaging dialogue. Grant critiques the nomenclature, suggesting that terms like "lateral numbers" might have been more appropriate (24:07). He expounds on the practicality of complex numbers in fields like quantum mechanics and electrical engineering, where they adeptly model cyclical phenomena and wave behaviors (25:07). This segment demystifies the concept, making it accessible to a broader audience.

7. Exploring Higher Dimensions and Tensors

The conversation advances to the abstraction of higher dimensions and the role of tensors in mathematics and physics. Grant explains tensors as multi-dimensional arrays essential in areas like general relativity and machine learning. He illustrates their origins as natural extensions when dealing with problems requiring representations beyond three dimensions (30:36), highlighting their indispensable role in modern scientific computations.

8. Chaos Theory and the Three-Body Problem

Addressing the three-body problem, Grant elucidates how chaos theory reveals the inherent unpredictability in such systems. He explains that even with precise initial conditions, the sensitivity to minor errors leads to exponentially diverging outcomes, rendering long-term predictions impossible (54:43). This discussion underscores the limitations and fascinating intricacies of mathematical models in describing real-world phenomena.

9. Listener Engagement and Mathematical Proofs

Throughout the episode, Neil, Chuck, and Grant engage with listeners' mathematical queries, providing lucid explanations on topics like dividing by zero, conformal geometry, and the irrationality of square roots. For instance, in response to a question about why you can't divide by zero, Grant demystifies the concept by linking it to projective geometry and explaining the undefined nature of such operations in a practical context (14:57).

10. Promoting Mathematical Literacy and Resources

As the episode nears its conclusion, Grant directs listeners to his YouTube channel, 3blue1brown, encouraging them to explore more visual and intuitive explanations of complex mathematical concepts (58:57). This invitation fosters further engagement, allowing listeners to deepen their mathematical understanding beyond the podcast.

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Notable Quotes with Timestamps

• Grant Sanderson: "More people like math than people suspect." (03:06)
• Chuck Nice: "If 7 million people follow you for math, that gives me hope for the future of civilization." (02:53)
• Grant Sanderson: "Do you think there's infinitely many primes that are just two apart?" (06:21)
• Grant Sanderson: "If you understand something about this function, you completely understand the primes." (23:01)
• Grant Sanderson: "Everyone loves math. I just think that most people are intimidated by it." (03:12)
• Grant Sanderson: "They thought complex numbers were just a notational trick. But they have these cyclic properties that make them really useful." (27:20)

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Conclusion

This episode of StarTalk Cosmic Queries Edition masterfully intertwines humor, expertise, and captivating storytelling to illuminate the pivotal role of mathematics in deciphering the universe's mysteries. Grant Sanderson's ability to translate complex mathematical theories into digestible concepts makes this episode a treasure trove for both math enthusiasts and curious minds alike. As Neil aptly concludes, "Keep looking up!"—a fitting mantra for those inspired to explore the boundless realms of mathematics and its cosmic significance.

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Timestamps

For a more detailed exploration of specific discussions, refer to the notable quotes above, each accompanied by its corresponding timestamp.

Transcript

[Neil deGrasse Tyson]

This Memorial Day, turn up the heat.

[Grant Sanderson]

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[Neil deGrasse Tyson]

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[Grant Sanderson]

Next level all summer long with the Home Depot.

[Neil deGrasse Tyson]

See homedepot.com delivery for more details. McCrispy strips are now at McDonald's. Tender, juicy and its own sauce. Would you look at that? Well, you can't see it, but trust me, it looks delicious. New McCrispy strips now at McDonald's.

[Chuck Nice]

I think we need to do more episodes that feature math.

[Neil deGrasse Tyson]

Yes. With maybe a different co host because I don't do math. No, I love math.

[Chuck Nice]

You love math.

[Neil deGrasse Tyson]

I do, I do.

[Chuck Nice]

You realize how important, important it is?

[Grant Sanderson]

It.

[Neil deGrasse Tyson]

Well, you know, it's, it's. Yes.

[Chuck Nice]

Without math, we don't know nothing in this universe.

[Neil deGrasse Tyson]

Well, I'm still in that boat, no matter what.

[Chuck Nice]

Coming up, StarTalk cosmic queries, a little bit of math. Welcome to StarTalk, your place in the universe where science and pop culture collide. StarTalk begins right now. This is StarTalk Cosmic Queries Edition. Got Chuck nice with me. Chuck, how you doing, man?

[Neil deGrasse Tyson]

I am doing great.

[Chuck Nice]

Yeah? Yeah. This one is going to be on math.

[Neil deGrasse Tyson]

Oh, I was told there would be no math.

[Chuck Nice]

Nobody told you there was going to be no math.

[Neil deGrasse Tyson]

That's right.

[Chuck Nice]

I know some math. But if we're going to have math as a subject, we got to bring on the big guns, right?

[Neil deGrasse Tyson]

Especially the mathiest of the mathiest.

[Chuck Nice]

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

[Grant Sanderson]

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

[Chuck Nice]

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 with a highly followed channel three blue one brown. Can you get more cryptic than that?

[Neil deGrasse Tyson]

Yes. Sounds like a three card Monty scheme.

[Chuck Nice]

How long have you had the channel, man?

[Grant Sanderson]

It's been around 10 years now, which feels a little wild. Yeah.

[Chuck Nice]

And how many followers followers you have?

[Grant Sanderson]

It's, it's around 7 million. I remember sometime recently across the, you.

[Neil deGrasse Tyson]

Know, that's, that's, that's monumental in terms of accomplishment. No no, I'm just saying 7 million people to follow you for math.

[Chuck Nice]

Math. That's what I'm saying.

[Neil deGrasse Tyson]

Exactly.

[Chuck Nice]

If 7 million people, that gives me hope for the future of civilization.

[Neil deGrasse Tyson]

Yeah, that gives me Hope for those 7 million people, not for the future of the rest.

[Grant Sanderson]

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.

[Neil deGrasse Tyson]

Everyone thinks people hate it, but, like, everyone loves math. I just think that most people are intimidated by it and intimidate 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.

[Chuck Nice]

But that's why Grant exists, so that he can grow the comfort zone that people feel. Look at you when encountering this content.

[Neil deGrasse Tyson]

Yes.

[Chuck Nice]

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

[Neil deGrasse Tyson]

Yeah, yeah.

[Grant Sanderson]

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.

[Chuck Nice]

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

[Neil deGrasse Tyson]

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

[Chuck Nice]

Keep your million dollars.

[Neil deGrasse Tyson]

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

[Grant Sanderson]

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.

[Neil deGrasse Tyson]

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

[Grant Sanderson]

Not in the explicit sense that there is a particular institute that put this check behind it. But 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 if you want financial rewards. There's plenty that would come if you could solve one of these problems, but of course, that's not what most people care about.

[Chuck Nice]

They arise purely within math. Or is there some scientific pumping of what these problems are or engineering solutions?

[Grant Sanderson]

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 pipes over, he's like, hey, Grant, you know prime numbers? I'm like, oh, yeah, prime numbers. 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.

[Chuck Nice]

His dad asked him this question when he was 11.

[Neil deGrasse Tyson]

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

[Grant Sanderson]

So I'm just letting go and think. I'm 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 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 coming spaced out by two? And you know that 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.

[Chuck Nice]

And you got one from science?

[Neil deGrasse Tyson]

Yeah. That's great.

[Grant Sanderson]

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 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 equation.

[Chuck Nice]

Navier Stokes equations.

[Grant Sanderson]

Yeah, yeah. 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 could tell the computer to try to run forward a sim. But the theoretical understanding of these equations is worryingly thin in some respects. Where, 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. 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. And it's actually very, very hard to understand the 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.

[Chuck Nice]

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's unsolvable. Correct.

[Grant Sanderson]

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 Nice]

Chuck, recite the quadratic formula. Do you remember it?

[Neil deGrasse Tyson]

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

[Chuck Nice]

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.

[Neil deGrasse Tyson]

Right.

[Chuck Nice]

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?

[Grant Sanderson]

That's exactly right.

[Chuck Nice]

Okay, okay. I just did that by rote. Not because I had a fricking feeling for it.

[Neil deGrasse Tyson]

You just had to know it so many times.

[Chuck Nice]

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

[Grant Sanderson]

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, and 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. But pure mathematicians for a while were curious. Okay, hey, 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, 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.

[Chuck Nice]

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

[Grant Sanderson]

It's wild to contrast history of science.

[Chuck Nice]

Well, maybe they stopped by the 1500s, the Renaissance was kicking in.

[Neil deGrasse Tyson]

Yeah, exactly.

[Chuck Nice]

I'm thinking maybe like the Dark ages.

[Grant Sanderson]

You go to the right part of the world, one person's contemplating cubics. You can definitely find another part of the world where disemboweling 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. 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.

[Chuck Nice]

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

[Grant Sanderson]

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

[Neil deGrasse Tyson]

Wow.

[Grant Sanderson]

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 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, what's the reason?

[Chuck Nice]

Why does it, why does the symmetry break between 4 and 5?

[Grant Sanderson]

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. 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 took a time.

[Chuck Nice]

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

[Grant Sanderson]

It would make me and a lot of other people very happy. Okay, I'm going to 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 Everiste Galois. He died in a duel at the age of 20. This is his big headline fact.

[Chuck Nice]

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

[Grant Sanderson]

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

[Neil deGrasse Tyson]

That's insane. Yeah.

[Chuck Nice]

Wait, let the man explain.

[Neil deGrasse Tyson]

Go ahead.

[Grant Sanderson]

No, no. So, well, what you're saying is what everyone hears when they're learning physics and learning math. Math and whatnot. 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 was in the crazed pre dual state. But it's such a good story to tell it that way. It's so good to be like, he knew he's going to die, 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 trisecting the angle using a straight edge and Compass to take any angle and divide it into three pieces. His theories, his 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.

[Chuck Nice]

Wow.

[Neil deGrasse Tyson]

Cool, man.

[Chuck Nice]

Well, math is the language of the universe.

[Grant Sanderson]

It certainly is.

[Chuck Nice]

So we shouldn't be surprised if it's some spillage into cosmic discovery.

[Neil deGrasse Tyson]

Yeah, makes sense.

[Chuck Nice]

So can we bring questions your way from our audience?

[Grant Sanderson]

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

[Neil deGrasse Tyson]

Here's the first question from Buck Rice, who says, why can't we divide by zero? Why?

[Chuck Nice]

I love that.

[Grant Sanderson]

That's a great question.

[Chuck Nice]

I've carried that with me my whole life. Because the answer is, well, it's undefined. And so my response is, well, then define it.

[Neil deGrasse Tyson]

Right.

[Chuck Nice]

Get off your duff and define it.

[Grant Sanderson]

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 one divided by zero. 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 one divided by zero. But if you're not doing that and you want to say, why isn't it defined? It depends on what you're doing with division. If what you want to say with division is like, I have one cupcake, and I'm dividing it among three people, how many does each of them get? You're 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.

[Neil deGrasse Tyson]

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. That's the answer. And that's the answer Chuck eats. The answer is, I eat the cupcake.

[Chuck Nice]

Now it's defined.

[Neil deGrasse Tyson]

Now it's defined. All right.

[Andy Richter]

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[Chuck Nice]

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[Grant Sanderson]

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[Chuck Nice]

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[Grant Sanderson]

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[Chuck Nice]

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[Grant Sanderson]

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[Neil deGrasse Tyson]

Hello, I'm Finky Brooke Allen and I support StarTalk on Patreon. This is StarTalk with Nailed Grass 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 gonna read both their questions successively so you can answer them. Okay? Hi, this is Kira from Georgia in the U.S. 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 universe. And then part B. What is your personal favorite unsolved?

[Chuck Nice]

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

[Neil deGrasse Tyson]

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

[Grant Sanderson]

Tell you what, I'm going to punt off part one to Neil here because I. I'll have some humility here. I'm not sure what the important mathematical problems of cosmology are. Not much of a cosmology person, so I'm curious what you say.

[Chuck Nice]

We have singularity problems in the universe. All of our equations tell us that at the center of a black hole, nature is dividing by zero. Okay. Everything goes. The denominator goes to zero. 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.

[Grant Sanderson]

Sense.

[Neil deGrasse Tyson]

No, it doesn't.

[Chuck Nice]

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.

[Neil deGrasse Tyson]

Gotcha.

[Chuck Nice]

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 did. 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?

[Grant Sanderson]

So Riemann is the big one there.

[Chuck Nice]

So Riemann. And when was that, like 1800 sometimes?

[Grant Sanderson]

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

[Chuck Nice]

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. And he 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.

[Grant Sanderson]

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.

[Chuck Nice]

There'S part one and then part B.

[Neil deGrasse Tyson]

Yes, part one and part B.

[Grant Sanderson]

So Riemann, 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 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 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's a very, very beautiful question because it's kind of asking 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.

[Chuck Nice]

Is that your favorite unsolved problem?

[Grant Sanderson]

I think it's my favorite, yeah.

[Neil deGrasse Tyson]

Wow.

[Chuck Nice]

Okay.

[Neil deGrasse Tyson]

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

[Chuck Nice]

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

[Neil deGrasse Tyson]

I love it. Yeah. Okay. All right. This is Tony Isaacs, and Tony says, g' day, Astroneil and Lord. Nice. This is Tony here from Melbourne, Australia. Melbourne, Melbourne. He says, love the show. Go on, Chuck, and do Aussie accent. Okay. That's somebody who's been listening to the show for a while.

[Chuck Nice]

Listening too long.

[Neil deGrasse Tyson]

Yeah, it totally telegraphed me. All right. He says, I have a complex problem. 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.

[Chuck Nice]

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

[Grant Sanderson]

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

[Chuck Nice]

Worst name ever.

[Grant Sanderson]

Gauss proposed calling them lateral numbers, which.

[Chuck Nice]

Would have been, you know, a little better. And then, and then, then you add another aspect to the imaginary number to get the complex number right. And now you call it a complex. These are words that are complete turnoffs. And I blame you.

[Neil deGrasse Tyson]

Well, I mean, if that's.

[Chuck Nice]

I blame your people.

[Neil deGrasse Tyson]

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?

[Grant Sanderson]

So many people struggle with this, right? Because you label it as imaginary. You start by pitching it by saying square roots of negatives don't exist, but pretend like they do and run forward. And 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.

[Neil deGrasse Tyson]

So you can think of them as like clocks.

[Grant Sanderson]

Then think of them as clock numbers, right? 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 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.

[Chuck Nice]

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.

[Grant Sanderson]

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 there's no such thing as a square root of negative, but pretend like there is. And 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.

[Chuck Nice]

I don't thank God for that. I'm very picky about what I thank God for.

[Grant Sanderson]

Thank math think that the education system. But if you had asked someone, hey, solve the equation x squared plus 1 equals 0, they would have said, there is no solution. 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 valued answer. And 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 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.

[Neil deGrasse Tyson]

What should we call was like, imaginary girlfriend.

[Grant Sanderson]

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.

[Neil deGrasse Tyson]

Hello, Dr. Tyson. Lord, nice. 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 the line after curving it. Is this at work in conformal geometry as well?

[Chuck Nice]

I'm completely confused, but, wow, what do you get out of that?

[Neil deGrasse Tyson]

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 place. All right, go ahead.

[Grant Sanderson]

If this were a lecture hall, this would be the time. I'm like, brandon, let's talk after class. It's gonna be great. You and me at a blackboard. The lesson will be better for everyone if we don't engage with what you wanna engage with right now. If you wanted, we can try describing what conformal geometry and circle inversion are.

[Neil deGrasse Tyson]

But 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.

[Chuck Nice]

All right, go.

[Grant Sanderson]

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.

[Neil deGrasse Tyson]

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.

[Grant Sanderson]

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

[Chuck Nice]

He was still showing off though.

[Neil deGrasse Tyson]

Yeah, he's showing off. Without a doubt. Brandon was showing off. Okay. But guess what, I'm glad. I'm, 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 is 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.

[Chuck Nice]

I freaking love these people.

[Neil deGrasse Tyson]

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?

[Grant Sanderson]

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 of. 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 a three dimensional grid of numbers like this and to answer the question, 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.

[Chuck Nice]

But somebody's got to be clever enough to see that need and then come up with it. Not everyone is that clever. There 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.

[Grant Sanderson]

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 represent in a. A useful and manipulable way.

[Chuck Nice]

This is not tij, is that.

[Grant Sanderson]

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

[Chuck Nice]

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

[Neil deGrasse Tyson]

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 in a conversation, but this one is, like, out there, man, which is very cool. So he.

[Chuck Nice]

Well, plus he's got 7 million followers on his YouTube, so he's doing something right.

[Neil deGrasse Tyson]

He's doing something right. All right. Yeah. Hi, Grant. Hello, Dr. Tyson. I'm Akiya, 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.

[Chuck Nice]

Nice.

[Neil deGrasse Tyson]

And it was a delightful experience. We stopped along the way at a cafe and had a little lovely cup of tea. I had jar jealing. My friend Mary had. 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 math, 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.

[Chuck Nice]

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

[Neil deGrasse Tyson]

Right?

[Chuck Nice]

What's up with that?

[Neil deGrasse Tyson]

You guys have done nothing.

[Grant Sanderson]

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 popularized. It's never going to show up in your high school calculus class, but it's absolutely like a new thing.

[Chuck Nice]

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

[Grant Sanderson]

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

[Neil deGrasse Tyson]

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.

[Chuck Nice]

Only three people can learn it in the whole world.

[Andy Richter]

That's.

[Neil deGrasse Tyson]

That's it, baby.

[Grant Sanderson]

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 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, we shouldn't put it into K12. 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 highfalutin 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.

[Neil deGrasse Tyson]

Akiya. Guess what? It's there, but it's not necessary for you on a need to know basis. You'll need to know basis, Akiya. And that's. There we go.

[Grant Sanderson]

All right.

[Neil deGrasse Tyson]

This is Gina. She says, hello, smarty pants. I was.

[Chuck Nice]

Okay.

[Neil deGrasse Tyson]

I love it. She says. Tina Martin says, I was recently learning about Smith Circles and Pie. 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, could you please explain this a little better for me? Thanks, Gina from North Carolina.

[Chuck Nice]

Wait, wait, wait. Is she correct? Can't you have the diameter be an irrational number and end up with the circumference rational?

[Grant Sanderson]

Yeah. So to be clear, if you want the circumference, the circumference could 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 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 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 two. Now, that is irrational. It's also much, much easier to prove to you why it must be irrational. And if you indulge me, I think it's possible to do this in 45 seconds, and we can see how this goes. All right, you ready? Here we go. Proof the 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. 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 factors. 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 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 P2 times K or something, right? It's some even number. If you then write this down algebraically and you replace it with 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 of, 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.

[Chuck Nice]

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.

[Grant Sanderson]

So great. So this is the common mathematician tool. They say, oh, you want to prove that something's impossible, right? They're like, man, this problem is 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. That's a very common mathematician thing.

[Neil deGrasse Tyson]

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

[Andy Richter]

Hi there, it's Andy Richter, and I'm here to tell you about my podcast, the Three Questions with Andy Richter. Each week I invite friends, comedians, actors and musicians to discuss these three where do you come from, where are you going, and what have you learned? New episodes are out every Tuesday with guests like Julie Bowe and Ted Danson, Tig Notaro, Will Arnett, Phoebe Bridgers, and more. You can also tune in for my weekly Andy Richter Call in show episodes where me and a special guest invite callers to weigh in on topics like dating, disasters, bad teachers, and lots more. Listen to the three Questions with Andy Richter wherever you get your podcasts. Ryan Reynolds here from Mint Mobile with.

[Chuck Nice]

A message for everyone paying Big Wireless way too much. Please, for the love of everything good in this world, stop with Mint. You can get premium wireless for just $15 a month. Of course, if you enjoy overpaying, no.

[Grant Sanderson]

Judgments but that's weird.

[Chuck Nice]

Okay, one judgment anyway. Give it a [e-mail] Switch upfront payment.

[Grant Sanderson]

Of $45 per three month plan, equivalent to $15 per month required. Intro rate first three months only, then.

[Chuck Nice]

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[Grant Sanderson]

See full [e-mail] every day is a.

[Chuck Nice]

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[Neil deGrasse Tyson]

I'm Tien, from Vietnam. If your two 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.

[Chuck Nice]

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

[Grant Sanderson]

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 imagine 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, wouldn't it be really hard to describe three dimensional shapes to them? If you have someone who lives in Flatland and you want to describe what a cube is or a sphere or 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 the other patron boy is certainly going to appreciate. But that's probably all I got.

[Chuck Nice]

As I remember the story, the more sides you had, the more aristocratic you were.

[Neil deGrasse Tyson]

Oh, look at you.

[Chuck Nice]

Right. So a triangle would be like the lowest, the scum of the earth.

[Neil deGrasse Tyson]

Those Triangles. I can't believe that trying to move in here. They know better.

[Chuck Nice]

And then the squares and then pentagons, hexagons. Right. So I think I'd be a hexagon.

[Grant Sanderson]

Okay.

[Chuck Nice]

I'd be a hexagon.

[Grant Sanderson]

You want to tessellate the plane.

[Neil deGrasse Tyson]

Yes, I want to look at that.

[Chuck Nice]

Yeah. Although any shape can tessellate, right?

[Grant Sanderson]

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

[Chuck Nice]

No, no, no. I mean, if you look at Escher paintings, he is. Isn't that tessellation where you have two.

[Grant Sanderson]

Yeah, yeah, yeah.

[Chuck Nice]

Two shapes intersect.

[Neil deGrasse Tyson]

Right.

[Chuck Nice]

So the difference is I'm a. What they call 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?

[Grant Sanderson]

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

[Neil deGrasse Tyson]

I agree.

[Chuck Nice]

But he has like angels and devils tessellated.

[Grant Sanderson]

Yeah, yeah, yeah, yeah.

[Chuck Nice]

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

[Grant Sanderson]

Yeah, 100%.

[Neil deGrasse Tyson]

Right.

[Chuck Nice]

Right. So now if you're going to restrict yourself to a shape, to a polygon. Right, what it'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.

[Neil deGrasse Tyson]

Yes, you can, yeah.

[Chuck Nice]

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

[Neil deGrasse Tyson]

Fits very, very snuggly together.

[Chuck Nice]

Snuggly.

[Neil deGrasse Tyson]

Very cool.

[Chuck Nice]

So what shape would you be now.

[Grant Sanderson]

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.

[Chuck Nice]

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

[Grant Sanderson]

Oh, 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 people didn't know if there was a shape that tiles things non periodically or that tiles things on periodically without having any. Periodic tiling is the 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.

[Chuck Nice]

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

[Grant Sanderson]

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

[Chuck Nice]

It's kind of what we mean by pattern almost.

[Grant Sanderson]

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 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.

[Chuck Nice]

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

[Grant Sanderson]

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. That.

[Chuck Nice]

Okay, very cool.

[Neil deGrasse Tyson]

This is William Walker and William Walker says hello gentlemen from Florida, 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?

[Chuck Nice]

Yeah, how do you get there?

[Grant Sanderson]

Great, great, great, great. I think this is one of the big misconceptions. When mathematicians talk about higher dimensions, 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, I don't know, 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.

[Chuck Nice]

Uniquely.

[Grant Sanderson]

Uniquely.

[Chuck Nice]

Yeah.

[Grant Sanderson]

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. 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 a 12,000 dimensional space in physical reality, it's just that it's a nice way to describe lists of numbers.

[Chuck Nice]

I have on my shelf here.

[Neil deGrasse Tyson]

So these dimensions are placeholders.

[Chuck Nice]

I got on my shelf here a Klein bottle bottle opener.

[Grant Sanderson]

I love it.

[Chuck Nice]

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

[Neil deGrasse Tyson]

In three dimensions, yeah, yeah.

[Grant Sanderson]

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 unnaturally cross through themselves. There's no way to put it in three dimensions without it crossing through itself.

[Chuck Nice]

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

[Grant Sanderson]

Yeah, I think that's a fair way to say it. 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.

[Chuck Nice]

Were these bored Stanford students?

[Grant Sanderson]

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

[Neil deGrasse Tyson]

Not creative enough not to go to jail because you still got.

[Chuck Nice]

Well, that's clever. You have to flip the fence, but then reattach it.

[Neil deGrasse Tyson]

Reattach it, yeah, yeah, exactly.

[Grant Sanderson]

So in the middle of the night, 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.

[Chuck Nice]

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

[Grant Sanderson]

Kind of. Yeah. 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 they're different. 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.

[Chuck Nice]

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.

[Grant Sanderson]

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 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 like peer off the edge of. 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.

[Chuck Nice]

Okay. So there's an important distinction.

[Neil deGrasse Tyson]

Closed surface, that's the key.

[Chuck Nice]

Time for just one more question.

[Neil deGrasse Tyson]

Okay. All right, let's go to our old friend Kevin, the sommelier.

[Chuck Nice]

Oh, okay. And Kevin says that's his last name, the sommelier.

[Neil deGrasse Tyson]

Okay.

[Chuck Nice]

His middle name is the last name of some.

[Neil deGrasse Tyson]

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.

[Chuck Nice]

I'm gonna find that wine.

[Neil deGrasse Tyson]

Yeah.

[Chuck Nice]

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

[Neil deGrasse Tyson]

Principia.

[Chuck Nice]

Yeah. But it's Principia Mathematica. If you want to give the full thing. This is the greatest work. If there's a wine with that name, I'm gonna find it. Thank you.

[Neil deGrasse Tyson]

There is a wine with that name.

[Chuck Nice]

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?

[Grant Sanderson]

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.

[Chuck Nice]

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

[Grant Sanderson]

Right. 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 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 problems, 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 span 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.

[Neil deGrasse Tyson]

At all, or planets move.

[Grant Sanderson]

So the result is telling. It's not that it's unreal 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.

[Chuck Nice]

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

[Neil deGrasse Tyson]

Right.

[Grant Sanderson]

Yeah.

[Chuck Nice]

We just need a smarter person to come along.

[Neil deGrasse Tyson]

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

[Chuck Nice]

That's the answer.

[Neil deGrasse Tyson]

That's the answer.

[Chuck Nice]

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

[Neil deGrasse Tyson]

We did, did solve it, and the solution is excellent. This is unknown.

[Chuck Nice]

You agree with that? That's. That's brilliant.

[Grant Sanderson]

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

[Neil deGrasse Tyson]

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

[Chuck Nice]

Just barely.

[Neil deGrasse Tyson]

What time is it really, Chuck, on the last question.

[Chuck Nice]

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. 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.

[Grant Sanderson]

Yeah, so the YouTube's channel, it's named Three Blue, One Brown. An admittedly weird name. You could also search 3B 1B. And a lot of topics that we've discussed here, flavors of them show up on that channel, so.

[Chuck Nice]

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

[Grant Sanderson]

Three blue and brown on whatever your favorite social app is these days.

[Chuck Nice]

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

[Grant Sanderson]

Very hard to describe the numerals mixed with the numbers. Yeah. Or just 3B, 1B. If you search that, you can usually land on it.

[Chuck Nice]

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

[Neil deGrasse Tyson]

Yeah.

[Chuck Nice]

Because God can't divide by zero. So we hope to see more of you. Keep up the work. And we need more math fluency in this world?

[Neil deGrasse Tyson]

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

[Grant Sanderson]

Do my best.

[Chuck Nice]

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

[Neil deGrasse Tyson]

Always a pleasure.

[Chuck Nice]

All right. This has been StarTalk Cosmic Queries, Mathematics Edition. Yield the grass, Tyson. Keep looking up.

[Andy Richter]

Hi there, it's Andy Richter, and I'm here to tell you about my podcast, the three Questions with Andy Richter. And each week I invite friends, comedians, actors and musicians to discuss these three where do you come from, where are you going, and what have you learned? New episodes are out every Tuesday with guests like Julie Bow and Ted Danson, Tig Notaro, Will Arnett, Phoebe Bridgers, and more. You can also tune in for my weekly Andy Richter call in show episodes where me and a special guest invite callers to weigh in on topics like dating, disasters, bad teachers, and lots more. Listen to the three Questions with Andy Richter wherever you get your podcasts.

[Grant Sanderson]

I've never felt like this before.

[Neil deGrasse Tyson]

It's like you just get me.

[Grant Sanderson]

I feel like my true self with you. Does that sound crazy?

[Neil deGrasse Tyson]

And it doesn't hurt that you're gorgeous.

[Grant Sanderson]

Okay, that's it. I'm done taking you home with me. I mean, you can't find shoes this good just anywhere. Find a shoe for every you from brands you love like Birkenstock, Nike, Adidas, and more at your dsw store or dsw.com.