Designing Math ft. Grant Sanderson (3Blue1Brown) I Config 2026

Hello.

Hello Config. Are you ready for day two?

There we go. Let's go. I'm Lorna

Croshawn, Figma's Chief Design Officer, and I'm excited to welcome you to the second day of Config. I'm going to introduce our first speaker of the day.

Now, I know many of you were up late hanging out with friends and playing with motion and shaders.

I've seen all of your work online. Keep

it coming.

So, to start us off today, we went with with a chill subject, math.

Now, math is often misunderstood. I

should know, I went to music school and I didn't get math until it connected with something that I loved.

Uh in my case, first sound, then visuals animation.

When you look at it that way, math becomes a field full of creativity, ideas, choices that fundamentally change how you see the world.

Our next speaker helped millions discover math in this way. He's the

creator of 3Blue1Brown, a YouTube channel that uses animation to bring math to life.

He built the open source library Manim to power it.

He lectured at MIT, contributed to an Emmy Award-winning documentary, and worked with Khan Academy.

And if you've ever watched one of his videos and all of the sudden understood something that you had given up on, well, then you know what he does.

Welcome to the stage, Grant Sanderson.

[music] Thank you.

For the last 11 years or so, I have had an obsession. Um as mentioned, I run a

an obsession. Um as mentioned, I run a YouTube channel. I've been making videos

YouTube channel. I've been making videos about visualizing math. And the

obsession is around finding topics which risk being complicated, but finding the right visual that somehow unlocks their underlying meaning.

And at first it was a little bit of a surprise to me to be invited to an event like this.

On the surface, mathematicians and designers feel like very different species.

But really both are trying to get at the same general idea. In both cases, you find yourself awash in a situation with an abundance of choice and a large potential for complexity. And you're

trying to find a clear path through it.

This is what mathematicians do. This is

what designers do.

And so I want to talk about principles of design that we can apply to the specific case of trying to explain math.

And I want to focus on one specific example, which is something that's come up an almost embarrassing number of times on my channel over the years, which is the equation e to the pi i equals -1.

So the reason this has come up so many times is because depending on what you're trying to answer about it, there are fundamentally different ways to visualize it and to explain it.

And whenever you have an abundance of choice, this is where design is at its most needed.

So what we're going to do today, over the next 20 minutes or so, is answer three distinct questions about this equation.

Each question is going to lend itself to a distinct way that we can visualize what's going on.

And each one ties to a principle of design that I think we can apply to math explanation more broadly.

So are we ready to get into it? Are we

ready to spend our early morning getting into imaginary exponentials and uh listening to me not hold back on you?

Yeah.

All right.

[cheering] [applause] My basic assumption is everyone here is maximally curious. I'm not going to

maximally curious. I'm not going to treat design community any different than I would treat a math community.

Question number one, very simply, what does this mean?

It's actually not clear from the symbols what the expression is even trying to say.

So, let's let's break it down. Let's

take each character one at a time. Pi,

we know pi, we love pi. It's all about circles. Traditionally, you define it as

circles. Traditionally, you define it as the ratio of a circle's circumference to its diameter, but mathematicians actually often like to think about unit circles, circles with a radius one, and in that context, it would give you the distance halfway around the circle. So,

that's the image I want in your mind for pi.

Now, I, this is the one that's shrouded in a little bit of mystery. It's often

called the imaginary constant. It's a

little bit of a silly name. It's

honestly no more imaginary than most real numbers are, but it's defining characteristic is that when you square it, you get -1.

Okay. The image you should have in your mind, though, is really it's all about extending our usual notion of numbers instead of them being on a one-dimensional number line into a two-dimensional plane.

And this, by the way, is part of why it's potentially such a delightful topic for designers is once you gain a certain fluency with this new kind of number, you have a new tool in your belt for how to manipulate two-dimensional space.

We'll get there in about 15 minutes.

But for right now, when you think about um I, you should think of it as one member of a big family of what we call complex numbers, where each one has a real component, that's how you move left and right, and an imaginary component,

how you move up and down.

And when you think about what does I do, what is its reason for existence, for today, I want you to think about the action of multiplying by it as being something that rotates. So, for example, let's say you you multiply I by a real

number A, what's going to happen is it moves 90° up onto that imaginary axis, and it turns into A * I.

If you multiplied it by a purely imaginary number B * I, that also gets rotated 90° because B * I * I is BI

squared. By definition, I squared is -1.

squared. By definition, I squared is -1.

And so, for a general combination of these, A + BI, if each part of it gets rotated 90°, the number as a whole also gets rotated 90°. So, let that burn in your mind. If you see I, you think a

your mind. If you see I, you think a thing that wants to rotate by 90 degrees.

All right. Now, the the last character on here is that number E, and the idea that we're trying to exponentiate it.

And the thing that's so weird about the equation that we're digging into right now is usually when you learn about exponentiation in school, we learn that it's about repeated multiplication. What

does E to the X mean? Well, E is some number, it's around 2.7, and this should mean multiplying by it X times. But,

this just completely breaks down if we're talking about more exotic kinds of inputs like complex numbers. And in

fact, this is just not what that expression even means anymore. What it

actually means is tied to a piece of calculus. Um there's this very beautiful

calculus. Um there's this very beautiful result around how when you use E as a base um for your usual notion of exponents, you know, repeated multiplication, how you extend that to real numbers and all of that, um it

turns out to equal a certain infinite polynomial. I won't give the full story

polynomial. I won't give the full story on why it does, just going to present this is what it happens to equal.

So, in that case, that's a discovery.

That is a discovery in math that these two are the same.

When we extend the definition though, we start using that as a definition.

We sort of invent a notion of what we want exponentiation to mean when we throw in more exotic inputs by saying take that expression, which actually can make sense when we plug in something more exotic, and use E to the X as

almost like a notational shorthand. It's

a little bit of slight of hand. So, the

actual meaning of the thing that we're exploring today is much much richer than you might think. And we can even step through and try to visualize it. So,

we'll start start going through this polynomial. When you add one, that moves

polynomial. When you add one, that moves you one unit to the right. When you add pi times I, that's going to move you up pi units. The next term in this is going

pi units. The next term in this is going to look like 1/2 pi I squared. Okay, so

I squared, that you've rotated 90 degrees twice, you're now moving in the leftward direction.

If you take the next term, it's got a pi I cubed, that I cubed, you think, okay, I've rotated 90 degrees three different times, we're now moving down.

And something very interesting happens when you keep adding on more and more terms of this.

Each power of I means you're rotating another 90°, but that denominator in the expression grows really rapidly. It

grows so rapidly that all of your terms start to shrink and get really small.

So, your spiraling sum zeros in on one specific point of the plane.

And that point happens to be at -1.

So, this is the claim, right? This is

what the expression we're studying actually means, where we're no longer hiding it behind all of this notational sort of And this brings us to the very first principle of treating math explanation

as a design challenge, which is simply treat it as a visual design challenge. Ask the question, what does

challenge. Ask the question, what does it look like?

I'm not saying every single piece of math has to be visualized. I'm not even necessarily saying visualizing it will always give you the best explanation.

But what's undeniably true is that there is a lot of low-hanging fruit simply asking the question, what does it look like?

What's also true, and I've noticed this over the last 11 years of seeing people respond to the stuff that I've made, is that it makes an incredible difference to students.

Simply asking that question is often the difference between sitting in a room in silence trying to read sheet music versus listening to the song.

For this specific example, though, that question alone is not enough. It's

insufficient. Cuz even if you can see what the expression is claiming, that this spiraling sum lands on -1, I've done nothing to explain why. Why would

it land there? That is a strangely clean value for it to land on given that we've got all these powers of pi and a seemingly chaotic spiraling sum.

So, we have to go at it in a different way.

And before you ever try to prove something, I think it's always a good idea to believe that it's true.

So, rather than just asking, why is this true? I want to try to get at, why does

true? I want to try to get at, why does it want to be true?

For that, we're going to take another look at that function, e to the x.

Instead of thinking x, I actually want you to think t, where t is going to be time. And the defining characteristic,

time. And the defining characteristic, the the thing that brings E to life, is the idea of a function whose rate of change is equal to itself. So, here's

what I mean by that. I want you to think of this expression E to the T as a position. It's going to be a position

position. It's going to be a position that changes with time. And we're going to think of it living on the real number line. I'm going to draw a little arrow

line. I'm going to draw a little arrow there. We know that the position starts

there. We know that the position starts at one, because if you take anything to the zeroth power, that lands you at one.

So, we have an initial condition.

And this defining equation for our expression, sort of the the meaning of life for the number E, is that its rate of change, what I have written up on the left there, um is always going to be

equal to the expression itself. So, you

imagine a velocity vector whose rule is it always has to be locked to equal that position. So, right off the bat, as

position. So, right off the bat, as we're going to play time forward, you know what's going to happen, because the velocity is pointing you to the rate, that position is going to have to grow.

But, as it starts to grow, because the position is bigger, the velocity has to get bigger. But, as the velocity gets

get bigger. But, as the velocity gets bigger, it means you're growing even faster. But, the faster you grow, the

faster. But, the faster you grow, the bigger that position gets. And you've

got this positive feedback loop that gets out of hand in a kind of delightful way. And this is the sensation of

way. And this is the sensation of exponential growth.

And more specifically, the idea that both these quantities are always locked to be the same is the sensation of exponential growth when the base is E.

This is really the the meaning of that number E.

And there's just one other little sprinkle of calculus that I need to give you to tell the full story here, which is how you think about it if you put a little constant in front of that time.

So, for example, let's say we put a constant two in front of that time.

Well, intuitively, this is kind of like you're playing time twice as fast. Um

so, you might expect that velocity vector to get twice as big. And that's

exactly what happens. Um if you've taken some calculus, you'll know something known as the chain rule, um where as a formula, what this would look like is our rate of change is still it's not quite equal to the expression itself,

but it's equal to two times that expression. You basically take that

expression. You basically take that constant, and now that's a a multiplying factor. But to visualize what's

factor. But to visualize what's happening, we still think of the idea of whatever that position is, the velocity is locked to be dependent on it, but this time the velocity is always two times that position. So, you still have

your runaway growth, where the bigger the position gets, the bigger that velocity gets, and the bigger the velocity gets, the position has to grow all the faster. But now your runaway growth is running away all the more quickly.

We could also think about a negative constant sitting up there.

So, this is a little bit like if you are playing time backwards.

But as a as a formula, if you follow what the calculus is telling you to do, it says, "Okay, your rate of change looks like the expression itself, but this time we're multiplying by" I've

chosen the example -0.5.

So, visually you'd imagine that negative is kind of rotating everything 180°, and the 0.5 means that it's squished down.

So, now, if I'm going to play time forward, the very first thing that's going to happen is the position is moving to the left. It's going toward zero.

But instead of running away, the smaller the position gets as it approaches zero, that velocity has to get correspondingly smaller. So, as it approaches zero, it

smaller. So, as it approaches zero, it gets smaller, the velocity gets smaller, it slows down, and it approaches but never quite reaches it. And this is that visceral sensation of exponential decay.

And once you have this in mind, we can start to ask the more interesting question of what should it mean, even if we haven't defined it, what should it mean if we put that imaginary number I

sitting in front of that T.

Now, on the one hand, imaginary time feels like a you know, a very woo-woo sort of concept you'd find in a quantum crystal shop, but it's actually very sensible if we remember what's the reason for existence for I. What does I

want to be doing?

Remember, multiplying by I wants to rotate things 90°.

So, if we still think of this expression as a position, and that position is going to change over time, this meaning of existence for E combined with the meaning of existence

for I, is actually giving us a really visceral and visual rule. It's saying

whatever that position is, the velocity vector is going to be a 90° rotation of it.

So, right off the bat, this lends itself to the idea that we should be thinking not confined to the real number line, cuz at the moment zero, our position is actually being pulled to move off of that real number line. It's being pulled

to move into the broader two-dimensional complex plane.

And this rule is also telling us even if we don't know exactly where it's going to be at any given point in time, we know that wherever that position ends up at any given point in time, the corresponding velocity is locked to be a

90° rotation of it.

Naturally, there's one and only one motion where this is going to be followed, and that's if starting from zero, you rotate in a circle.

And you can say something more specific than that, too. You know that you're going to be rotating at a specific rate, because that velocity vector is locked to equal the position vector in length.

The rate at which you're going around has to be one unit per second.

So, in particular, think about what it would mean to let time run forward for exactly pi seconds.

That would mean if you're traversing one unit per second, and you go pi units around the circle, you've gone exactly halfway around.

And so, this is not just the the meaning of what this expression is saying, it's also showing you why it wants to be true.

It's not just a coincidence that that expression that we had was spiraling in and landing on -1. Uh instead, it feels like it must be that way. How could it have ever been otherwise?

And this gets us to a second principle when it comes to designing a math explanation.

Really, the substance behind any explanation is understanding not just what's true, but why it's true.

And I think whenever you want to understand why something's true, take each component of what you're trying to explain. So, in this case, we have an

explain. So, in this case, we have an equation, but it could be each part of a theorem, it could be each piece of anything else. Take each component, and

anything else. Take each component, and And sure each one has a very clearly defined reason for existence, the thing that it wants to do.

If you were writing a novel, it's never going to work unless each one of the characters has a clearly defined motivation.

The same is true in math. Nothing is

different in math. Give your characters motivation.

Now, while we're here, uh it's actually very fun to see what this motion around the circle looks like in the context of our previous visual. I had shown that spiraling sum for one input, pi * I, but

if instead I make it I * t and I let that number t change, um it's quite fun to see how this spiraling sum does indeed trace out a circle. But from this visual, it's actually very mysterious

why that would be the case.

It's not obvious that it would end up on a circle.

So, we needed a different way of viewing it to understand why it's true, but then the first way of of viewing it shows us what the real claim is. And combining

those, I think, is where you you see that sense of both mystery and a sense of satisfying explanation combined into one.

All right, so these are two questions we could ask. Each one of them lends itself

could ask. Each one of them lends itself to a different principle.

But arguably, there's a much more important question before we even start.

Education 101 is that before anybody is going to learn from you, they have to want to learn. You have to motivate the subject. And often, the best motivation

subject. And often, the best motivation is some form of application. How do you actually use it?

If we were at a conference full of electrical engineers, we could talk all about the many uses of imaginary exponents for signal processing.

If this was a conference full of physicists, we could talk all about the abundant uses of it in the context of quantum mechanics.

But because we're at a conference full of designers, I actually want to talk a much a much more left-field application that you wouldn't think about, which is how it's relevant to art and to understanding one very specific art

piece.

So, for this, I want you to imagine you are in a print gallery and you're in this kind of strange warped world and you're looking at this picture of a boat.

And as [snorts] you look up and say, "What's in that picture?" you see in the picture, the boat is in a harbor and that harbor has a town.

That town has a lot of buildings.

And as you keep looking around, you realize one of those buildings is actually this open-air print gallery.

And you scan through the hall of that print gallery.

And as you keep looking around, you find yourself sitting there, looking at a picture of a boat.

So, zoomed out, this is a piece called the print gallery. It's by M.C. Escher.

So, M.C. Escher is a favorite artist to many, especially a favorite artist to mathematicians, because he had a real knack for finding paradoxical ideas to

flirt with the concept of infinity and seeming contradiction with a two-dimensional concept. Now, he has a

two-dimensional concept. Now, he has a whole repertoire of delightful paradoxical imagery. But, in a letter

paradoxical imagery. But, in a letter that he wrote to his son, he described this piece as, quote, "The most peculiar thing I have ever created." Which, for him, is saying a lot.

And there's a very interesting story, actually, around the math underlying this image that actually himself knew nothing about. One of the most recent

nothing about. One of the most recent projects that I did on YouTube, really digs into this. It's actually one of my favorite pieces, I think, I've done over the years. The full story takes maybe 45

the years. The full story takes maybe 45 minutes to say, so we don't have time here. But, I do at least want to give

here. But, I do at least want to give you a little bit of a glimpse and the way that it's relevant to our story today.

So, the concept that Escher was working with starts this way.

I mean, this is going to be a straightened-out version of the scene we just looked at. We have a man looking at a picture of a boat. Um but, in this case, instead of looping around, we can zoom in. That boat is in a harbor. That

zoom in. That boat is in a harbor. That

harbor is next to a town with many buildings. One of those buildings is

buildings. One of those buildings is this open-air print gallery. And as you get closer in, you see a copy of where we started.

And so, this concept, it's also fun.

It's not quite paradoxical. It's just an image that's nested with I inside itself. It's very deeply nested inside

itself. It's very deeply nested inside itself in this case. Uh but, this isn't this isn't enough to be the Escher style genius. The simple idea of a picture

genius. The simple idea of a picture nested deeply within itself.

Escher somehow intuitively realized that there must be a way to take this concept and kind of pull out that inner version of that world nested within itself and pull it out to connect with the outer

version.

I know that animation I'm actually very proud of. Uh,

proud of. Uh, I think yeah, thank you. Thank you.

It's [applause] I don't deserve the credit. I'll give

credit where credit is due in a moment.

I made it, but but the math behind it comes from elsewhere. I think this would have blown Escher's mind because he went about creating this piece in a very different way. It was this very

different way. It was this very intuitive kind of folk artist, very tactile, physical approach. But what

underlies that animation I just showed, which I'm just going to play it again because I want to I just want to see what it looks like.

Let's see.

Um, now we got we already have our zoom.

We already have our zoom.

Let's see it. Let's appreciate it. All

right, so we're pulling out that inner world.

Oh, yeah. Oh, that's nice.

And it doesn't line up and it doesn't line up and then at one point it just perfectly lines up. It's so good. It's

so good.

Okay, so the what's underlying this is a a set of complex functions. A set of functions treating each pixel on the plane of that image like they're complex numbers. Um, and then there's a very

numbers. Um, and then there's a very specific manipulation that pulls off um, this this move that Escher was doing.

So like I said, I don't deserve the credit. This comes from two

credit. This comes from two mathematicians named de Smit and Lens trap who around 50 years after Escher made that piece provide this really nice analysis on how actually there's a lot

of hidden math related to complex numbers um, underlying what's really going on with the piece.

Now as I said, I wish I had the time to tell you the full story right now. I

don't. But what I will do is try to give you that 10,000 foot view glimpse of the core parts of the story and the way that it relates to our our exploration today of designing explanations. Specifically

with respect to that e to the z function.

See here instead of talking about e to the t or e to the x, I'm saying z cuz z is often something we use to describe complex numbers. And the new way that

complex numbers. And the new way that I'm going to have us visualize it is going to be to think more globally, where we want to say, "What does this do to the entire plane?" Cuz here we're using it for an artistic goal, where we want to manipulate space and do

something new with space.

So, just to review, we um we already talked about what happened if we plug in a purely imaginary number, some multiple of i, which is as you walk up that line, the output kind of walks around a circle.

And what ends up happening is if you walk up a different line, where maybe it's shifted one unit to the right, you walk around a bigger circle, specifically one scaled up by e.

If you walk up a different vertical line, maybe this time two units to the right, you walk around an even bigger circle, this one scaled up by e squared.

And in general, the thought to have in your mind is how all of these vertical lines are turning into circles.

And what I want to try to visualize is what does this function do to the entirety of the input plane, the entirety of all possible inputs? How can

we hold that in our head? I think

there's a really nice way to visualize it, where we think of all these vertical lines, so I'm going to highlight them, and think of the plane that exists in between them.

And what I like to imagine is sort of rolling up that plane into a tube, so that we've turned all those vertical lines into circles, and then specifically it's circles with radius one to start, so every kind of 2 pi

units, that's when you're repeating.

And then you take that whole tube, and we know each one of those circles needs to be centered in the output plane, so we kind of move that tube, we center it above the output plane that we're going to have, and then we squish it all down.

Squish it on so that all those circles are centered around the origin, and they grow at this exponential rate, where each one is e times bigger than the last.

So, this is our new way of thinking about the exponential function. It's a

third distinct way of visualizing it based on the fact that we have a new goal, which is to do something artistic, which is to understand manipulations of space.

And really those two mathematicians, when they were studying Escher, they were like artists who were fluent with a new kind of paintbrush, which is how to use functions of complex numbers to manipulate things in just the precise

ways that you want.

And this exponential function was one of their paintbrushes, and the only other one they really needed was the reverse of this, where you go the other way around. So, the name we have for that

around. So, the name we have for that inverse function is the natural log. Um

so, in this case, exponentiating gets you one direction, natural logs take you another. And the thing to hold in your

another. And the thing to hold in your mind is what would that do? It takes the circles and kind of unwraps them into lines. And this is very fun to think

lines. And this is very fun to think about if instead of just general lines and numbers, we actually put a picture on one of those planes, and we say, "What would it mean to take the logarithm of a picture?"

In this case, what it would mean is that you look at various circles in that picture, and each one of those circles of pixels gets kind of straightened out into a line.

And more specifically, it's in such a way where as you zoom in, every time you zoom in by a factor of E, your line has shifted one unit to the right.

And for the Escher concept, this means if ever you have a picture which is nested within itself, and you've got this kind of infinitely repeating pattern as you zoom in, this unwrapping method turns that into a

tiling pattern that turns out to be kind of easier to work with and manipulate.

So, here's the high-level view of what those mathematicians did. At a high level, they started by uh looking at a logarithm of the straightened out concept of a a man looking at a picture of a boat with that nested deeply inside

itself. You get this wild and bizarre

itself. You get this wild and bizarre image when you take that logarithm.

Um what they understood is if you take that and then you rotate it in just the right way, you have to be very careful, you find just the right way to rotate it, if then you kind of translate back to the normal

world where you exponentiate, that's what gets you this this wild spiral.

So, I have not explained why that's the case. I simply want to show you how it

case. I simply want to show you how it when someone uses these functions of exponential numbers as a kind of new paintbrush, they get enough fluency to be able to rediscover what's really a not obvious concept that I should as I

stumbled on.

This also, by the way, explains how I made that animation that I was showing you earlier. Where really what was going

you earlier. Where really what was going on under the hood is I was working in the log space, and if you rotate things in that log space, and you just present to screen what you get after you exponentiate, that's what gets this

transition between the straightened out version and the Escher version.

And so, this cuts us to the third principle for treating math explanation as a design challenge. So, we all know that we want to motivate our subjects. I

think that's very obvious. We all know that applications are one of the best sources of motivation. But, usually we think of math application as being in the service of science or in the service of technology. And it is, and that's

of technology. And it is, and that's great. There's lots of good stories to

great. There's lots of good stories to be told. But, the third principle here

be told. But, the third principle here is that sometimes the most memorable application is the most surprising one.

It's not obvious that studying imaginary exponentials and complex functions lets you achieve an artistic aim. But, I

think when you see how it does, it burns in your brain just a little bit longer than if I was just talking about applications to technology.

So, here we spent time looking at three distinct principles. One of them, simply

distinct principles. One of them, simply treat it as a visual design challenge.

Simply ask the question, what does it look like?

The other one is that if you want to understand why things are true, ask what are the motivations of your characters.

That motivation is often the difference between a proof and an explanation.

And then finally, the most memorable applications are the ones that are the most surprising.

Now, these are just the ones that I come up with if we situate ourselves for 20 minutes talking about things. I think if we sat here for another hour, we could easily come up with another seven more principles about math and explanation

and design. And what I want to leave you

and design. And what I want to leave you with is the idea that everyone in this room really is a teacher in some form.

Maybe very explicitly your job is related to explanation. Um, but if not, all forms of design are in some form trying to communicate as clearly as you can. And at minimum, everyone is a

can. And at minimum, everyone is a teacher, cuz everyone has the role of trying to explain what they do and why it matters to others.

And I want to encourage you to step back and spend 30 minutes sometime today to ask, what are your own principles around teaching? And how do those principles

teaching? And how do those principles overlap with design?

Cuz in just the same way that math can be in the service of design, you know, offering these different paint brushes for different manipulations of 2D space, design is absolutely in the service of explanation and teaching.

And the more that we try to understand the intersection between those two, the more successfully we'll achieve the ends of both.

And with that, I'll leave you. Thank you

very much.

[applause]