Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics | Lex Fridman Podcast #64

The following is a conversation with Grant Sanderson. He's a math educator and creator of 3blue1brown, a popular YouTube channel that uses programmatically animated visualizations to explain concepts in linear algebra, calculus, and other fields of mathematics. This is the Artificial Intelligence podcast. If you enjoy it, subscribe on YouTube, give it five stars on Apple Podcast, follow on Spotify, support on Patreon, or simply connect with me on Twitter and at Lex Friedman, spelled F R I D M A N. I recently started doing ads at the end of the introduction. I'll do one or two minutes after introducing the episode. And never any ads in the middle that can break the flow of the conversation. I hope that works for you and doesn't hurt the listening experience. This show is presented by Cash App, the number one finance app in the App Store. I personally use Cash App to send money to friends, but you can also use it to buy, sell and deposit Bitcoin in just seconds. Cash App also has an investing feature. You can buy fractions of a stock, say $1's worth, no matter what the stock price is. Brokers services are provided by Cash App Investing, a subsidiary of Square and member sipc. I'm excited to be working with Cash App to support one of my favorite organizations called first. Best known for their First Robotics and Lego competitions, they educate and inspire hundreds of thousands of students in over 110 countries and have a perfect rating on Charity Navigator, which means the donated money is used to maximum effectiveness. When you get Cash App from the App Store, Google Play and use code LEXPodcast, you'll get $10 and Cash App will also donate $10 to first, which again is an organization that I've personally seen inspire girls and boys to dream of engineering and a better world. And now here's my conversation with Grant Sanderson.

If there's intelligent life out there in the universe, do you think their mathematics is different than ours?

Jumping right in? I think it's probably very different. There's an obvious sense the notation is different. Right. I think notation can guide what the math itself is. I think it has everything to do with the form of their existence. Right.

Do you think they have basic arithmetic? Sorry to interrupt.

Yeah. So I think they count, right? I think notions like 1, 2, 3, the natural numbers, that's extremely, well, natural. That's almost why we put that name to it. As soon as you can count, you have a notion of repetition, right? Because you can count by two, two times or three times. And so you have this notion of repeating the idea of counting which brings you additional multiplication. I think the way that we extend to the real numbers, there's a little bit of choice in that. So there's this funny number system called the surreal numbers that it captures the idea of continuity. It's a distinct mathematical object. You could very well model the universe and motion of planets with that as the back end of your math. Right. And you still have kind of the same interface with the front end of the. What physical laws you're trying to or what physical phenomena you're trying to describe with math. And I wonder if the little glimpses that we have of what choices you can make along the way based on what different mathematicians have brought to the table is just scratching the surface of what the different possibilities are. If you have a completely different mode of thought or mode of interacting with.

The universe, and you think notation is a key part of the journey that we've taken through math, I think that's.

The most salient part that you'd notice at first. I think the mode of thought is going to influence things more than the notation itself. But notation actually carries a lot of weight when it comes to how we think about things, more so than we usually give it credit for. I would be comfortable saying, do you.

Have a favorite or least favorite piece of notation in terms of its effectiveness?

Yeah, well, so least favorite one that I've been thinking a lot about that will be a video. I don't know when, but we'll see the number E. We write the function E to the X, this general exponential function with a notation E to the X. That implies you should think about a particular number, this constant of nature. And you repeatedly multiply it by itself. And then you say, oh, what's E to the square root of two? And you're like, oh, well, we've extended the idea of repeated multiplication. That's all nice. That's all nice and well. But very famously, you have e to the PI I, and you're like, well, we're extending the idea of repeated multiplication into the complex numbers. Yeah, you can think about it that way. In reality, I think that it's just the wrong way of notationally representing this function, the exponential function, which itself could be represented a number of different ways. You can think about it in terms of the problem. It solves a certain very simple differential equation, which often yields way more insight than trying to twist the idea of repeated multiplication. Like, take its arm and put it behind its back and throw it on the desk and be like, you will apply to complex numbers. That's Not. I don't think that's pedagogically helpful.

So the repeater multiplication is actually missing the main point, the power of E to the X.

Yes. I mean, what it addresses is things where the rate at which something changes depends on its own value. But more specifically, it depends on it linearly. So, for example, if you have a population that's growing and the rate at which it grows depends on how many members of the population are already there, it looks like this nice exponential curve. It makes sense to talk about repeated multiplication because you say how much is there after one year, two years, three years? You're multiplying by something. The relationship can be a little bit different sometimes. Where let's say you've got a ball on a string, like a game of tetherball going around a rope, and you say its velocity is always perpendicular to its position. That's another way of describing its rate of change as being related to where it is. But it's a different operation. You're not scaling it. It's a rotation. It's this 90 degree rotation. That's what the whole idea of complex exponentiation is trying to capture. But it's obfuscated in the notation when what it's actually saying. If you really parse something like E to the PI I, what it's saying is choose an origin, always move perpendicular to the vector from that origin to U. Then when you walk PI times that radius, you'll be halfway around. That's what it's saying. It's kind of the. You turn 90 degrees and you walk, you'll be going in a circle. That's the phenomenon that it's describing. But trying to twist the idea of repeatedly multiplying a constant into that. I can't even think of the number of human hours, of intelligent human hours that have been wasted trying to parse that to their own liking and desire among scientists or electrical engineers or students, have we. Which, if the notation were a little different or the way that this whole function was introduced from the get go, were framed differently, I think could have been avoided. Right.

And you're talking about the most beautiful equation in mathematics. But it's still pretty mysterious, isn't it? Like you're making it seem like it's a notational.

It's not mysterious. I think the notation makes it mysterious. I don't think it's. I think the fact that it represents. It's pretty. It's not like the most beautiful thing in the world, but it's quite pretty. The idea that if you take the linear operation of a 90 degree rotation and then you do this general exponentiation thing to it, that what you get are all the other kinds of rotation, which is basically to say if your velocity vector is perpendicular to your position vector, you walk in a circle. That's pretty. It's not the most beautiful thing in the world, but it's quite pretty.

The beauty of it, I think comes from perhaps the awkwardness of the notation somehow still nevertheless coming together nicely, because you have several disciplines coming together in a single equation. Well, in a sense, historically speaking, that's true.

So the number E is significant. It shows up in probability all the time. It shows up in calculus all the time. It is significant. You're seeing it mated with PI, this geometric constant. And I like the imaginary number and such. I think what's really happening there is the way that E shows up is when you have things like exponential growth and decay, right? It's when this relation that something's rate of change has to itself is, Is a simple scaling. A similar law also describes circular motion. Because we have bad notation, we use the residue of how it shows up in the context of self reinforcing growth, like a population growing or compound interest. The constant associated with that is awkwardly placed into the context of how rotation comes about because they both come from pretty similar equations. And so what we see is the E and the PI juxtaposed a little bit closer than they would be with a purely natural representation, I would think. Here's how I would describe the relation between the two. You've got a very important function we might call exp. That's like the exponential function. When you plug in one, you get this nice constant called E that shows up in probability and calculus. If you try to move in the imaginary direction, it's periodic and the period is tau. So those are these two constants associated with the same central function, but for kind of unrelated reasons. Not unrelated, but orthogonal reasons. One of them is what happens when you're moving in the real direction. One's what happens when you move in the imaginary direction. And yeah, those are related. They're not as related as the famous equation seems to think it is. It's sort of putting all of the children in one bed and they'd kind of like to sleep in separate beds if they had the choice. But you see them all there and there is a family resemblance, but it's not that close.

So actually think of it as a function is the better idea. And that's a notational idea.

Yeah. And here's the thing, the constant E sort of stands as this numerical representative of calculus, right? Calculus is the study of change. So at the very least there's a little cognitive dissonance using a constant to represent the science of change.

Never thought of it that way.

Yeah, right, yeah. It makes sense why the notation came about that way, because this is the first way that we saw it. In the context of things like population growth or compound interest, it is nicer to think about as repeated multiplication. That's definitely nicer. But it's more that that's the first application of what turned out to be a much more general function that maybe the intelligent life your initial question asked about would have come to recognize as being much more significant than this single use case which lends itself to repeated multiplication notation.

But let me jump back for a second to aliens and the nature of our universe.

Okay.

Do you think math is discovered or invented? So we're talking about the different kind of mathematics that could be developed by the alien species. The implied question is, is. Yeah, is math discovered or invented? Is, you know, is fundamentally everybody going to discover the same principles of mathematics?

So the way I think about it, and everyone thinks about it differently, but here's my take. I think there's a cycle at play where you discover things about the universe that tell you what math will be useful and that math itself is invented in a sense. But of all the possible maths that you could have invented, it's discoveries about the world that tell you which ones are. So a good example here is the Pythagorean theorem. When you look at this, do you think of that as a definition or do you think of that as a discovery?

From the historical perspective, it's a discovery because they were. But that's probably because they were using physical object to build their intuition. And from that intuition came the mathematics. So the mathematics wasn't in some abstract world, detached from the physics. But I think more and more math has become detached from. You know, when you, when you even look at modern physics, from, from string theory to even general relativity, I mean, all math behind the 20th and 21st century physics kind of takes a brisk walk outside of what our mind can actually even comprehend in multiple dimensions. For example, anything beyond three dimensions, maybe four dimensions.

No, no, no, no. Higher dimensions can be highly, highly applicable. I think this is a common misinterpretation that if you're asking questions about a five dimensional manifold, that the only way that that's connected to the physical world is if the physical world is itself a five dimensional manifold or includes them.

Wait, wait, wait a minute, wait a minute. You're telling me you can imagine a five dimensional manifold?

No, no, that's not what I said. I would make the claim that it is useful to a three dimensional physical universe, despite itself not being three dimensional.

So it's useful meaning to even understand a three dimensional world. It'd be useful to have five dimensional manifolds.

Yeah, absolutely. Because of state spaces.

But you're saying there, in some, in some deep way, for us humans, it does, it does always come back to that three dimensional world for the useful, usefulness at the dimensional world. And therefore it starts with a discovery. But then we invent the mathematics that helps us make sense of the discovery.

In a sense, yes. I mean, just to jump off of the Pythagorean theorem example, it feels like a discovery. You've got these beautiful geometric proofs where you've got squares and you're modifying the areas. It feels like a discovery. If you look at how we formalize the idea of 2D space as being R2, all pairs of real numbers, and how we define a metric on it and define distance, you're like, hang on a second, we've defined distance so that the Pythagorean theorem is true. So suddenly it doesn't feel that great. But I think what's going on is the thing that informed us what metric to put on R2, to put on our abstract representation of 2D space came from physical observations. And the thing is, there's other metrics you could have put on it. We could have consistent math with other notions of distance. It's just that those pieces of math wouldn't be applicable to the physical world that we study because they're not the ones where the Pythagorean theorem holds. So we have a discovery, a genuine bona fide discovery that informed the invention, the invention of an abstract representation of 2D space that we call R2 and things like that. And then from there you just study R2 as an abstract thing that brings about more ideas and inventions and mysteries which themselves might yield discoveries. Those discoveries might give you insight as to what else would be useful to invent. And it kind of feeds on itself that way. That's how I think about it. So it's not an either or. It's not that math is one of these or it's one of the others. At different times, it's playing a different role.

So then let me ask the Richard Feynman question then along that thread is what do you think is the difference between physics and math? There's a giant overlap. There's a kind of intuition that physicists have about the world that's perhaps outside of mathematics. It's this mysterious art that they seem to possess, we humans generally possess. And then there's the beautiful rigor of mathematics that allows you to, I mean, just like, as we were saying, invent frameworks of understanding our physical world. So what do you think is the difference there and how big is it?

Well, I think of math as being the study of, like, abstractions over patterns and pure patterns in logic. And then physics is obviously grounded in a desire to understand the world that we live in. I think you're going to get very different answers when you talk to different mathematicians, because there's a wide diversity in types of mathematicians. There are some who are motivated very much by pure puzzles. They might be turned on by things like combinatorics. And they just love the idea of building up a set of problem solving tools, applying to pure patterns. There are some who are very physically motivated who try to invent new math or discover math in veins that they know will have applications to physics or sometimes computer science, and that's what drives them. Chaos theory is a good example of something that's pure math. It's purely mathematical, a lot of the statements being made, but it's heavily motivated by specific applications to largely physics. And then you have a type of mathematician who just loves abstraction. They just love pulling into the more and more abstract things, the things that feel powerful. These are the ones that initially invented topology and then later on get really into category theory and go on about infinite categories and whatnot. These are the ones that love to have a system that can describe truths about as many things as possible. People from those three different veins of motivation into math are going to give you very different answers about what the relation at play here is, because someone like Vladimir Arnold, he's written a lot of great books, many about differential equations and such, he would say math is a branch of physics. That's how he would think about it. And of course, he was studying differential equations, related things, because that is the motivator behind the study of PDEs and things like that. But you'll have others who, especially the category theorists, who aren't really thinking about physics necessarily. It's all about abstraction and the power of generality. And it's more of a happy coincidence that that ends up being useful for understanding the world we live in. And then you can get into, why is that? The case. It's sort of surprising that that which is about pure puzzles and abstraction also happens to describe the very fundamentals of quarks and everything else.

So why do you think the fundamentals of quarks and the nature of reality is so compressible into clean, beautiful equations that are for the most part simple, relatively speaking, a lot simpler than they could be. So you have we mentioned somebody like Stephen Wolfram who thinks that sort of. There's incredibly simple rules underlying our reality, but it can create arbitrary complexity. But there is simple equations. What I'm asking a million questions that nobody knows the answer to, but no.

Idea why is it simple? It could be the case that there's a filtration at play. The only things that physicists find interesting are the ones that are simple enough they could describe it mathematically, but as soon as it's a sufficiently complex system, like that's outside the realm of physics, that's biology or whatever have you. And of course that's true. Maybe there's something where it's like, of course there will always be some thing that is simple. When you wash away non important parts of whatever it is that you're studying. Just from an information theory standpoint, there might be some. You get to the lowest information component of it. But I don't know, maybe I'm just having a really hard time conceiving of what it would even mean for the fundamental laws to be intrinsically complicated. Like some set of equations that you can't decouple from each other.

Well, no, it could be that we take for granted that the, the laws of physics, for example, are for the most part the same everywhere or something like that. Right. As opposed to the sort of an alternative could be that the rules under which the world operates is different everywhere. It's like a, like a deeply distributed system where just everything is just chaos. Like not, not in a strict definition of cast, but meaning like just. It's impossible for equations to capture for, to explicitly model the world as cleanly as the physical does. I mean, we almost take it for granted that we can describe, we can have an equation for gravity, for action at a distance. We can have equations for some of these basic ways the planets move and just the low level at the atomic scale, how the materials operate at the high scale, how black holes operate, but it doesn't. It seems like it could be there's infinite other possibilities where none of it could be compressible into such equations. It just seems beautiful. It's also weird, probably to the point you're making that it's very pleasant that this is true for our minds.

Right.

So it might be that our minds are biased to just be looking at the parts of the universe that are compressed, compressible, and then we can publish papers on and have nice E equals empty squared equations.

Right. Well, I wonder, would such a world with uncompressible laws allow for the kind of beings that can think about the kind of questions that you're asking?

That's true.

Right. Like an anthropic principle coming into play at some weird way here. I don't know. I don't know what I'm talking about at all.

Maybe the universe is actually not, not so compressible. But the way our brain, the. The way our brain evolved, we're only able to perceive the compressible parts. I mean, we are. So this is a sort of Chomsky argument. We are just descendants of apes over like really limited biological systems. So it totally makes sense that we're really limited little computers, calculators that are able to perceive certain kinds of things. And the actual world is much more complicated.

Well, but we can do pretty awesome things. We can fly spaceships. And we have to have some connection of reality to be able to take our potentially oversimplified models of the world, but then actually twist the world to our will based on it. So we have certain reality checks that physics isn't too far afield simply based on what we can do.

Yeah, the fact that we can fly is pretty good.

It's great. It's a proof of concept that the laws we're working with are, are working well.

So I mentioned to the Internet that I'm talking to you. And so the Internet gave some questions, so I apologize for these. But do you think we're living in a simulation that the universe is a computer or the universe is a computation running on a computer?

It's conceivable. What I don't buy is, you know, you'll have the argument that, well, let's say that it was the case that you can have simulations, then the simulated world would itself eventually get to a point where it's running simulations, and then the second layer down would create a third layer down and on and on and on. So probabilistically, you just throw a dart at one of those layers. We're probably in one of the simulated layers. I think if there's some sort of limitations on the information processing of whatever the physical world is, it quickly becomes the case that you have a limit to the layers that could exist there. Because the resources necessary to simulate a universe like ours clearly is a lot just in terms of the number of bits at play. And so then you can ask, well, what's more plausible? That there's an unbounded capacity of information processing in whatever the highest up level universe is, or that there's some bound to that capacity which then limits the number of levels available. How do you play some kind of probability distribution on what the information capacity is? I have no idea, but I don't immediately people almost assume a certain uniform probability over all of those meta layers that could conceivably exist when it's a little bit like a Pascal's wager on like you're not giving a low enough prior to the mere existence of that infinite set of layers.

Yeah, that's true. But it's also very difficult to contextualize the amount of information processing power required to simulate our universe seems like amazingly huge.

But you can always raise two to the power of that. Yeah, like numbers get big and we're.

Easily humbled by basically everything around us. So it's very difficult to, to kind of make sense of anything. Actually when you look up at the sky and look at the stars and the immensity of it all, to make sense of us, the smallness of us, the unlikeliness of everything that's on this earth coming to be, then you could basically anything could be. All laws of probability go out the window to me because I guess because the amount of information under which we're operating is very low, we basically know nothing about the world around us, relatively speaking. And so when I think about the simulation hypothesis, I think it's just fun to think about. But it's also, I think there is a thought experiment kind of interesting to think of the power of computation, whether the limits of a touring machine, sort of the limits of our current computers. When you start to think about artificial intelligence, how far can we get with computers? And that's kind of where the simulation hypothesis is useful to me as a thought experiment. Is, is the universe just a computer? Is it just a computation? Is all of this just a computation and sort of the same kind of tools we apply to analyzing algorithms? Can that be applied? If we scale further and further and further, Will the arbitrary power of those systems start to create some interesting aspects that we see in our universe? Or is something fundamentally different needs to be created?

Well, it's interesting that in our universe it's not arbitrarily large. The power that you can place limits on, for example, how Many bits of information can be stored per unit area. All of the physical laws. You've got general relativity and quantum coming together to give you a certain limit on how many bits you can store within a given range before it collapses into a black hole. The idea that there even exists such a limit is, at the very least thought provoking when naively you might assume, oh, well, technology could always get better and better. We could get cleverer and cleverer, and you could just cram as much information as you want into, like, a small unit of space. That makes me think it's at least plausible that whatever the highest level of existence is doesn't admit too many simulations or ones that are at the scale of complexity that we're looking at. Obviously, it's just as conceivable that they do and that there are many. But I guess what I'm channeling is the surprise that I felt upon learning that fact that there are. That information is physical in this way.

That there's a finiteness to it. Okay, let me just even go off on that. From a mathematics perspective and the psychology perspective, how do you mix? Are you psychologically comfortable with the concept of infinity?

I think so.

Are you okay with it?

I'm pretty okay, yeah. Are you okay?

No, not really. It doesn't make any sense to me.

I don't know, like, how many. How many words. How many possible words do you think could exist that are just, like, strings of letters?

That's a sort of mathematical statement. That's beautiful. And we use infinity in basically everything we do. Everything we do in science, math, and engineering. Yes, but you said exist. The question is, you said letters of words.

I said words.

Words. To bring words into existence to me, you have to start saying them or writing them or listing them.

That's an instantiation. How many abstract words exist?

Well, the idea of abstract. The idea of abstract notions and ideas.

I think we should be clear on terminology. I mean, you think about intelligence a lot like artificial intelligence. Would you not say that what it's doing is a kind of abstraction? Abstraction is key to conceptualizing the universe. You get this raw sensory data. I need something that every time you move your face a little bit and they're not pixels, but analog of pixels on my retina change entirely, that I can still have some coherent notion of. This is Lex. I'm talking to Lex. Right? What that requires is you have a disparate set of possible images hitting me that are unified in a notion of Lex. That's a kind of abstraction. It's a Thing that could apply to a lot of different images that I see. And it represents it in a much more compressed way and one that's much more resilient to that. I think in the same way, if I'm talking about infinity as an abstraction, I don't mean non physical, woo woo, ineffable or something. What I mean is it's something that can apply to a multiplicity of situations that share a certain common attribute in the same way that the images of your face on my retina share enough common attributes that I can put this single notion to it. In that way, infinity is an abstraction and it's very powerful. And it's only through such abstractions that we can actually understand the world and logic and things. And in the case of infinity, the way I think about it, the key entity is the property of always being able to add one more. No matter how many words you can list, you just throw an A at the end of one and you have another conceivable word. You don't have to think of all the words at once. It's that property, the. Oh, I could always add one more. That gives it this nature of infinity ness in the same way that there are certain properties of your face that give it the lex ness. So infinity should be no more worrying than the I can always add one more sentiment.

That's a really elegant, much more elegant way than I could put it. So thank you for doing that as yet another abstraction. And yes, indeed, that's what our brain does. That's what intelligent systems do, that's what programming does, that's what science does, is build abstraction on top of each other. And yet there is at a certain point abstractions that go into the quote, woo. Right. Sort of. And because we're now, it's like, it's like we built this stack of, you know, the. The only thing that's true is the stuff that's on the ground. Everything else is useful for interpreting this. And at a certain point, you might start floating into ideas that are surreal and difficult and take us into areas that are disconnected from reality in a way that we can never get back.

What if instead of calling these abstract, how different would it be in your mind if we called them general? And the phenomenon that you're describing is overgeneralization. When you try to have a concept or an idea that's so general as to apply to nothing in particular in a useful way, does that map to what you're thinking of when you think of.

First of all, I'm Playing little just for the fun of it. Devil's advocate. I think our cognition, our mind is unable to visualize. So you do some incredible work with visualization and video. I think infinity is very difficult to visualize for our mind. We can dilute ourselves into thinking we can visualize it, but we can't. I don't. I mean, I would venture to say it's very difficult. And so there's some concepts in mathematics, like maybe multiple dimensions we could sort of talk about that are important, possible for us to truly intuit. And it just feels dangerous to me to use these as part of our toolbox of abstractions.

On behalf of your listeners, I almost fear we're getting too philosophical, right? No. Heck no.

Heck no.

I think to that point, for any particular idea like this, there's multiple angles of attack. I think when we do visualize infinity, what we're actually doing, you write 1, 2, 3, 4. Those are symbols on the page that are insinuating a certain infinity. What you're capturing with a little bit of design there is the. I can always add one more property.

Right.

I think I'm just as uncomfortable with. You are if you try to concretize it so much that you have a bag of infinitely many things that I actually think of. No, not 1, 2, 3, 4. 1, 2, 3, 4, 5, 6, 7, 8. I try to get them all in my head. And you realize, oh, your brain would literally collapse into a black hole. All of that. And I honestly feel this with a lot of math that I try to read where I don't think of myself as particularly good at math. In some ways, I get very confused often when I am going through some of these texts. And often what I'm feeling in my head is like, this is just so damn abstract. I just can't wrap my head around it. I just want to put something concrete to it that makes me understand. And I think a lot of the motivation for the channel is channeling that sentiment of, yeah, a lot of the things that you're trying to read out there. It's just so hard to connect to anything that you spend an hour banging your head against a couple of pages and you come out not really knowing anything more other than some definitions maybe and a certain sense of self defeat. One of the reasons I focus so much on visualizations is that I'm a big believer in. I'm sorry, I'm just really hampering on this idea of abstraction. Being clear about your layers of abstraction. It's always tempting to start an Explanation. From the top to the bottom, you give the definition of a new theorem. This is the definition of a vector space, for example. That's how we'll start a course. These are the properties of a vector space first. From these properties we will derive what we need in order to do the math of linear algebra or whatever it might be. I don't think that's how understanding works at all. I think how understanding works is you start at the lowest level you can get at, where rather than thinking about a vector space, you might think of concrete vectors that are just lists of numbers or picturing it as an arrow that you draw, which is itself even less abstract than numbers because you're looking at quantities like the distance of the x coordinate, the distance of the Y coordinate. It's as concrete as you could possibly get. And it has to be. If you're putting it in a visual right.

It's an actual arrow, it's an actual vector.

You're not talking about a quote unquote vector that could apply to any possible thing. You have to choose one if you're illustrating it. And I think this is the power of being in a medium like video or if you're writing a textbook and you force yourself to put a lot of images, is with every image you're making a choice. With each choice, you're showing a concrete example. With each concrete example, you're aiding someone's path to understanding.

I'm sorry to interrupt you, but you just made me realize that that's exactly right. So the visualizations you're creating, while you're sometimes talking about abstractions, the actual visualization is an explicit low level example. Yes. So there's an actual. Like in the code, you have to say what the vector is, what's the direction of the arrow, what's the magnitude? Yeah. So that's you're going. The visualization itself is actually going to the bottom.

And I think that's very important. I also think about this a lot in writing scripts where even before you get to the visuals, the first instinct is to. I don't know why, I just always do. I say the abstract thing, I say the general definition, the powerful thing, and then I fill it in with examples later. Always it will be more compelling and easier to understand. When you flip that and instead you let someone's brain do the pattern recognition, you just show them a bunch of examples, the brain is going to feel a certain similarity between them. Then by the time you bring in the definition or by the time you bring in the formula, it's articulating a thing that's already in the brain. That was built off of looking at a bunch of examples with a certain kind of similarity. And what the formula does is articulate what that kind of similarity is, rather than being a high cognitive load set of symbols that needs to be populated with examples later on. Assuming someone's still with you.

What is the most beautiful or awe inspiring idea you've come across in mathematics?

I don't know man.

Maybe it's an idea you've explored in your videos, maybe not.