3blue1brown

To epsilon-delta or not to epsilon-delta?

Alright, so I'm putting together the video on limits, and I'm trying to decide if it makes sense to talk about the epsilon-delta definition.  On the one hand, there's something nice about showing the full rigor.  But at the same time, epsilons and deltas are largely beside the point for understanding what calculus is all about.  After all, the modern formulation of limits in these terms came much later in history than the advent of calculus itself.   That is, maybe its a...

New Calc video! Derivatives of exponentials

Hello Patrons,

Originally this was just going to be a footnote video, but it scope-creeped its way into a fully fledged chapter.  This stands to reason, given just how important exponential functions are to calculus.  So this is now going in as chapter 4, after chapter 3 on derivative formulas with geometry, and before what is now chapter 5 on the product rule, chain rule, etc.

This topic is a little less visual than others.  My approach was to give some intuition about ex...

New calc video! Implicit differentiation, what's going on here?

Hello Patrons,

Happy pi day everyone!  I have for you chapter 5 of Essence of calculus, about implicit differentiation.  I am sincere in saying at the start of the video that this was a topic that confused me as a student, and ultimately I think the way to convey it sensibly is to essentially treat it as a little preview of multivariable calculus.

I allude at some point here to a footnote video where I'll talk about the derivative of exponentials.  If you don't remember se...

New video! Euler's formula with introductory group theory

Hey Patrons,

In light of the fact that it's been 2 years since uploading my first video to this channel, I decided to revisit the idea of that video.  Mostly, this an excuse to teach a little bit of group theory, with an emphasis on putting group actions front and center.

-Grant

Some group animations for the fours

This Friday's video will involve a bit of group theory, here are a few preview animations.

Removing ads for each new video's first month, as thanks

Hello 3blue1brown patrons,

Several times I've mentioned just how blown away I've been by your willingness to support what I'm trying to do with this channel, and how thankful I am for that.  But actions speak louder than words, so I want a better way to show you what your support means to me, doing what I can to offer you more enjoyable videos.

I work hard to craft an experience for viewers moment-by-moment from a video's start to its end.  But the fact is, with monetization on...

New Calc video! "Chain rule, product rule, etc. | Essence of calculus, chapter 4"

Appreciated patrons,

This is the fourth instalment of the "Essence of calculus" series, covering the various combination rules for derivatives.  That is, how differentiating plays with sums, products and function composition.

I think the more I get into this,  the more I realize what I actually hope to accomplish with the series itself.  While there are countless resources to learn about and practice calculus online, I'd like this series to be a canonical place to go for t...

New video! Who (else) cares about topology

Dearest patrons,

This is a beutiful piece of math, a platium-level proof if ever there was one.  I'm a bit of a sucker for anything that connects two disparate ideas in math (who isn't?), so when I first came accross this it immediately jumped towards the top of my video list.

Much like the first "Who cares about topology" video, a strong motivation in making this is to show how ideas from this field can actually be used to solve problems.  Sure, any time you squish a sphere on...

Some preview animations

Hello $4+ patrons,

Here are a few animations that will be included in this weeks video.

-Grant

$8 Patrons name animations!

Hello $8+ patrons,

I've made a playlist included all of your names being transformed in the complex plane by the function f(z) = z^2.  It shows your name first moving about a bit in the pre-transformed plane, then being transformed, then it shows what the image of that previous motion would be in the post-transformed space.

Here is the playlist: https://www.youtube.com/playli...

Essence of calculus chapter 3: Derivative formulas through geometry

Hello Patrons,

To finish off January, the early version for chapter 3 in the calculus series is ready for you.  This involves numerous visuals which I wish I was shown as a student learning calculus.  I don't know about you, but my early relationship with taking derivatives was one of some memorization mixed with graphical intuition for the loose shape of things.  

As I say in the video, I think it's important to think directly about the "tiny nudges" underlying derivative...

A few calculus chapter 3 animations

Hello $4+ patrons,

Here are two animations from the upcoming chapter in the calculus series.  Hope you enjoy!

-Grant

New video! Fractals are not self-similar

Hey folks!

This is one I've been excited to make for a while.  I've always been fascinated by fractals (who isn't?), but ever since reading Mandelbrot's "The Fractal Geometry of Nature", I've been fascinated in a completely different way.  The concept of fractal dimension is at first so very strange, but the more you start thinking about it and seeing it in the world, the more you wonder why you never saw things in this way all along.

I hope you enjoy it.

-Grant

A few more fractal animations...

Hello $4+ patrons!

The fractal video will go out this Friday.  Here are a few more animations, some from the video, some that I might include in a supplmental "Fractal charm" short.

-Grant

More preliminary fractal dimension animations

Hello fours!

Here are a few animations I'll use to introduce self-similarity dimension in the upcoming video.

-Grant

Fractal animations for the fours!

Hello $4+ patrons!

This next video will be full of fun animations, since it's about fractals.  More specifically, it's about fractal dimension.  I've added three such animations to the "for the fours" playlist, and will be adding more through this week.

-Grant

Essence of calculus, chapter 2. "The paradox of the derivative"

Hey folks!

Chapter 2 is ready for your viewing and reviewing.  As always, I welcome any feedback you have, even though I might wait a month or two to go through it all and incorporate improvements.  I'm especially curious to hear from the calculus teachers in the audience; the more anecdotes about a specific student's learning experience, the better.

There are some aspects of the sound that are slightly wonky, e.g. it seems like I hit the microphone once or twice while recordin...

A few calc animations for the 4's

Hello 4+ patrons!

Here are a few animations from the imminent chapter 2 of "Essence of calculus".  

https://www.youtube.com/watch?v=SXdT3kDG-s8&index=9&list=PLKXWoWb0qgQ_ATdosn-ZTvN4j90MYIbar

http...

New video (no charge on this one)

Hey folks!

I put out kind of a different video today, about tattoos on math.  Even though it came out around 8 minutes, I still consider a "short", at least in spirit, and it's typical example of the kind of video that I would not charge patrons for.   While the spirit of this Patreon is to support more in-depth videos visualzing math, this one is more just for fun, targetting a slightly different audience.  Of course, I do try to sneak in a mini-trig lesson in the middle, ...

For the 2's

Hello $2+ patrons!

Back in snowy Park City for the holidays, I thought I'd take the chance to send you a thank you photo with a few of the physical manifestations of your contributions.   Many of you decided to make that 100% jump from 1 to 2, and I thank you for that.

2 is a pretty great number, don't you think?  It's the first prime, the protagonist throughout CS theory, and the heart of many beautiful parity-based proofs.  Not to mention, it's much more fun to give...

Some gratitude

Hi Everyone,

I just have to say, you all continue to blow me away with your willingness and enthusiasm to support this channel.  It's only been about 1.5 months since I launched this campaign, and things are well beyond the highest hopes I had at that time.   As the holiday season approaches, and the year comes to a close, it's a good excuse to take a moment to remember who has made this a great year for me, and for 3blue1brown, and you all sit right there at the top of the list...

New video! Visualizing the Riemann zeta function and analytic continuation

Hey Folks,

I think you're really going to like this one.  About two weeks ago, before jumping into the next Essence of Calculus video, I was playing around with visualizing various complex functions.  When I put in the zeta function, it struck me as so pretty that I just had to make it the next project.  This was especially true when I realized it could give a nice way to motivate the idea of analytic continuation.   Hope you enjoy!

-Grant

Zeta on right half of complex plane

What's going on here should make much more sense when I'm narrating over it (especially because I will insert a few animations in the middle here to talk about complex exponentiation), but I thought I'd share it with you guys because the spiral converging sums were kind of fun.

Visualizing Riemann's zeta function

Hey folks,

Usually I just share sneak-peak animations with >$4 patrons, but I thought I'd share these with everyone.  I was playing around with visualizing various complex functions as transformations, and when I tried it out with the Riemann zeta function, the result was so beautiful I decided that must be the next video I do.

In particular, I think there's a really nice visual intuition behind the idea of analytic continuation to be had here, which, given a proper explanation, ...

Thoughts on derivative video?

Beloved patrons,

So I'm about to start animating chapter 2 of essence of calculus, in which I'll go more into the derivative.  Before I jump into it, I wanted to hear thoughts that y'all might have on giving intuitions about derivatives.  I have the basic script for what I'd like to say, but I sometimes like to do a rewrite before jumping into the animations.  I'm especially curious to hear from the calculus teachers among you, and specific anecdotes involving specific students ...

New video(s): Binary, Hanoi and Sierpinski

Hey folks!  

The new videos are done.  This is full of relations that surprised me: That binary counting can solve a classic puzzle, that ternary counting solves a variant of the puzzle, and that these solutions can be interpreted as a way to walk through a certain graph structure reminiscent of Sierpinski's triangle.   And what's more, it features a phenomenal computer science lecturer from Stanford, Keith Schwarz.

This pair of videos will be charged like a single vi...

Walking the graph

Towers of Hanoi graph structure

Okay, ignore the picture I posted for y'all (which is slightly wrong anyway).  This is the animation overwhich I'll be narrating while introducing the connection between Sierpinski's traingle and the Towers of Hanoi puzzle.

Structure of Hanoi towers

Not an animation, like I usually pass along to you $4+ folk, but nevertheless it's a preview to a structure that the coming video(s) will include.  This particular image is just a zoomed out, very high resolution version of the structure, so the stroke witdths are all super thin, but you should be able to zoom in to see what's going on.

Solve Towers of Hanoi by counting

Hey fours!  See if you can figure out what this represents, and why it works.