
A former professor of mine, [Ravi Vakil](http://virtualmath1.stanford.edu/~vakil/), put together a delightful picturebook explanation for how he likes to schematically visualize exact sequences. It's too delightful not to share, and he kindly let me throw it up here as a guest post ([pdf version here](/content/blog/exact-sequence-picturebook/PuzzlingThroughExactSequences.pdf)).

The target audience probably consists mostly of graduate students in mathematics, or else anyone who already has some familiarity with and reason to care about exact sequences, the snake lemma, cohomology, etc. This piece is not meant to introduce any of those, but in conjunction with traditional texts on the matters, it stands to be a very helpful companion.

Such topics rarely get bedtime stories, or strong visuals, which is part of what makes this such a gem. Enjoy!


<Figure image="PuzzlingThroughExactSequences-01.svg" width="100%"/>

---

<Figure image="PuzzlingThroughExactSequences-02.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-03.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-04.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-05.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-06.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-07.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-08.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-09.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-10.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-11.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-12.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-13.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-14.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-15.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-16.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-17.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-18.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-19.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-20.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-21.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-22.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-23.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-24.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-25.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-26.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-27.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-28.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-29.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-30.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-31.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-32.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-33.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-34.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-35.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-36.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-37.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-38.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-39.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-40.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-41.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-42.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-43.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-44.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-45.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-46.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-47.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-48.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-49.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-50.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-51.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-52.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-53.svg" width="100%" />

---

<Figure image="PuzzlingThroughExactSequences-54.svg" width="100%" />

---



Some graduate student bedtime reading by Ravi Vakil

Puzzling through exact sequences

Reflections on the Summer of Math Exposition

SoME1 results


We want to encourage more people to put out explainers of math online.  This is a competition where anyone can submit a video, blog post, interactive game, or whatever else they dream up, and after the deadline we will select 5 to be featured in a 3blue1brown video.  Winners may also receive gold pi creatures.  I mean, probably not *gold* gold, but it would be something unique and emblematic of your greatness.

For the rest, we will put together a playlist of the video entries, and have a list of submissions on this website[^1].
To trade tips, share partial progress, and otherwise interact with others involved, join the [discord](https://discord.gg/Q3cC2GXN2f).


> _Note from August 22nd, 2021_: Submissions are now closed. We received over 1,300 entries, which is outstanding! Thank you to everyone who participated, we genuinely hope you keep producing online math explainers after this. We plan to announce winners by September 14th.

> _Note from July 19th, 2021_: Since announcing the contest, Brilliant came in to offer a total of $5,000 in cash prizes to be split among top entries. We'll split it evenly, allocating $1,000 each to the five entries selected to be featured in a video

## Why do this?
It's hard to overstate how powerful it is when anyone can learn something they want just by searching online.  In a thousand subtle ways, when these explainers exist, projects go faster, school becomes easier, and math-phobia becomes a little bit less common.

A lot of excellent lessons and intuitions stay confined to isolated classrooms, or worse, confined to some individual's head. It takes effort to make videos, or blog posts, and to put them out there. Here, we're hoping to offer a little bit of activation energy to anyone who has thought about doing something like this, but just never got around to it.


## Who is this for?
Honestly, anyone who might benefit others by sharing their knowledge.  This could include teachers who want to take what they've learned to work well in a classroom and put it online, students who want to deepen their own knowledge by putting together an explainer, mathematicians who want to get involved with outreach, engineers who want to share what mathematical tools they use on the job and how, etc.

Entries from groups or teams are welcomed and encouraged too.

## What are the constraints for entries?

1. **It has to be something new you make this summer**

The spirit of this is to encourage people who've never put stuff online before.  If you want to work on something you sort of started once before, that's probably fine, but it can't be something you already published before this contest. Optimally, you'd use this as a chance to try something new you otherwise might not have.

2. **It has to be about math.**

Here we mean "math" very broadly, and more applied topics like physics or computer science are abundantly welcome.  It just has to be the case that a viewer/reader might come away knowing something mathematical they didn't before.

The topic could be at any level, whether that's basic math for young children or higher-level math.  If you're assuming a certain background level for the target audience, kindly mention it below.  It's hard because we don't want to discourage topics with a very niche target audience, as those lessons can sometimes be the most valuable.  However, if your lesson assumes particular expertise, e.g. a comfort with algebraic geometry, keep in mind that our judges may not fit into this category. So to actually win the contest, it's helpful if the topic is accessible to someone with, say, a background in standard undergrad math topics.

3. **One entry per person/group**

We hope you make more, but we only have the capacity to judge participants based on a single entry.

4. **It has to be available in English**

If you want to put out an explainer in another language, wonderful! Please do! But the judges here will be english speakers, so to be considered for the contest the lesson has to be accessible to them.

5. **It has to be publicly visible and available for people to consume for free**

## When is the deadline?

August 22nd, 11:59pm PST


## How will winners be selected?
Here's what we're looking for:

- **Clarity**: Jargon should be explained, the goals of the lesson should be understandable with minimal background, and the submission should generally display empathy for people unfamiliar with the topic.

- **Motivation**: It should be clear to the reader/viewer within the first 30 seconds why they should care.

- **Novelty**: It doesn't necessarily have to be an original idea or original topic, but it should offer someone an experience they might otherwise not have by searching around online.  Some of the greatest value comes from covering common topics in better ways. Other times there's value in surfacing otherwise obscure ideas which more people should know about.

- **Memorable**: Something should make the piece easy to remember even several months later. Maybe it's the beauty of the presentation, the enthusiasm of the presenter, or the mind-blowingness of an aha moment.

To select the final winners, a small group of judges will look through the entries with the criteria above in mind.  Depending on the number of entries, we may have a step to pre-filter entries before these final judgments.  In all likelihood, to be considered for the competition, we may ask you to peer review some of the other participants' submissions. This would involve looking at five other entries and ranking them 1-5 based on how well they fit the criteria above, at which point we'll use the cumulative scoring which comes from those rankings to determine a final set of a few hundred which is manageable for our judges.

For longer works, judges might not be able to consume the full video/post.  Again, what's hard about this is that sometimes great explainers _are_ longer, such as a full lecture and we don't want to discourage those. Just understand that to select winners, there is only so much time, so the substance of your work should be clearly visible with a 5-10 minute view.

## Advice

[Nicky Case](https://ncase.me/) made a delightful little comic encapsulating three of pieces of advice for math exposition mentioned in the video above.

<Figure image="comic-1.png" />
<Figure image="comic-2.png" />
<Figure image="comic-3.png" />
<Figure image="comic-4.png" />

## Other questions?
We'll update this page with more FAQs as they come in.  Feel free to ask below, or on [Reddit](https://www.reddit.com/r/3Blue1Brown/)

[^1]: This only applies to good faith submissions, anything irrelevant or offensive would not be included


The Summer of Math Exposition


<Figure image="through-the-internet.png" />

Many schools around the world are still planning to run their classes and lectures remotely this fall, or at least to operate under a mix of distance and in-person classes. So all the usual balls that teachers and professors need to juggle, keeping students engaged, being responsive to their misconceptions, hitting all the points of a dense lesson plan, etc., are now supposed to be handled with all the awkwardness and impersonality of a video call.

No one seems to have phrased it this way, but effectively this year the world’s teachers were asked to become live streamers.

A few weeks ago I collaborated with two friends to do an experimental set of [10 live-streamed math lectures](https://itempool.com/3b1b). Since then a number of teachers and professors have reached out asking about the setup I used, whether the live quizzing software made for those streams is available to others ([it is!](https://itempool.com/)), and for general thoughts on the whole distance learning process, so I wanted to share my answer in one place.

Despite technically being a professional YouTuber, I certainly didn’t know what I was doing when I jumped into this. The first time I downloaded the broadcasting software to see what it was all about, I [accidentally streamed myself publicly for 3 seconds](https://youtu.be/Bdblmebf_QY). For any educators out there who feel as unaware of this stuff as I was, this is for you.

There’s actually a silver lining to every teacher suddenly being thrust into the role of a streamer. To the extent that your institution is comfortable with you recording and uploading your lessons publicly (which if you can, you should absolutely do), this year may end up providing the greatest influx of educational material to the internet that the world has ever seen.

If instead of thinking of remote classes as a more awkward form of a normal class, you view it as chance to practice video creation and turning your lessons into something more lasting and sharable, the benefit of handling this transition well may extend well beyond the education of this year’s students.

## Software
### OBS
Most of the heavy lifting for live streaming is handled by a free broadcasting software called [OBS](https://obsproject.com/). There are other options, but this is a popular choice, and it’s what I used. It lets you organize various input sources, for example, cameras, microphones, or screen captures, and mix and match them to make various scenes. It’s very seamless to use with YouTube or Twitch, but you can also [connect OBS to Zoom](https://streamgeeks.us/how-to-connect-zoom-obs/), which is probably more relevant to most teachers.

For example, in my lessons, I set up 5 basic scenes:

<Figure image="face-cam.png" caption="A simple face camera" />
<Figure image="overhead.png" caption="An overhead shot of pen and paper where I wrote out the lesson." />
<Figure image="screen-capture.png" caption="A screen capture used to show animations, graphs on Desmos, and coding examples" />
<Figure image="audience-questions.png" caption="Questions from the audience" />
<Figure image="live-quiz.png" caption="The live quiz" />

In OBS you can set up keyboard shortcuts, so I set up a few to associate certain keys under the lefthand of my keyboard with different scenes so that I could switch between them easily, potentially even while writing or gesturing with my right hand under the overhead camera. It may not have been quite as seamless as real live TV with a production team orchestrating the feed, but I think it can get abitrarily close with practice. The upshot is that the look and feel of a professional production is more accessible to an individual teacher than you might think.

### Itempool
The tool used for the live quiz is part of [Itempool](https://itempool.com/), put together by two former coworkers of mine from Khan Academy. The way it worked is that I’d queue up a question on the screen, and students could go to a certain link (e.g. from their phones) to follow along with the quiz and answer questions. The statistics of how everyone was answering would show up in real time on the stream, where viewers would see bars showing the relative proportions for each answer. But until I flipped a switch to reveal, it was a mystery which answers corresponded to which bars.

<Figure image="itempool.png" />

To get the bars overlaid on the stream, we set up a webpage that displays the real-time updated bars and a title on a transparent background. That way, making that webpage an input source to OBS and creating a scene that combines the face camera with that page gave the effect we wanted.

At the time, it was a very bespoke tool just for that series, but now things have been set up so that you too can create questions, and turn any set of questions you write into one of these live challenges by clicking "administer live". This sets up the link where you can send students as well as the link to the page you can overlay on the stream. Also, and I'll admit I'm biased because it was built by friends, the act of actually writing math questions on Itempool is a delight, especially given the integration with [MathLive](https://mathlive.io/).

When you’re in a room alone talking to a camera, it’s hard to overstate how nice it is to see some connection with the fact that there are really students there, in that moment, interacting with what you’re providing them. Not to mention the value of seeing when there’s a misconception, and for the students to see that many others had the same misconception.

### Desmos and Geogebra
I expect many math teachers already know about [Desmos](https://www.desmos.com/). It's an online graphing calculator that makes it very simple to whip up surprisingly complicated graphs. I found it abundantly useful for live presentations over a stream, and what's nice is that you can share the links of what you build so that students can play with them. For example, here is a [cobweb diagram](https://www.desmos.com/calculator/nul32eaaa9) I put together for the lesson on power towers. Beyond the graphing calculator, they also have an abundance of classroom activities.

One other very powerful tool for making interactive demos that I made use of was [Geogebra](https://www.geogebra.org/). For example, here is the "[complex slide rule](https://www.geogebra.org/m/mbhbdvkr)" that came up during the lecture on complex number fundamentals.

One potential benefit of classes becoming remote is how much more natural it becomes to use the best tools for interacting with math that the internet has to offer. Sure, in principle there's no reason that these tools couldn't also be used for in-person teaching, but shifting to a setting that is necessarily online already lowers the activation energy to make it happen.

## Hardware
As far as the physical setup goes, let me start by saying most teachers don’t have to be too fancy about this. A simple webcam will do, along with whatever mechanism suits you best to show what you’re writing, whether that’s a tablet, an overhead camera, or a whiteboard behind you. What matters most is the lesson itself and your charismatic presence. To goal of any recording and streaming setup is to give you the flexibility to show whatever you want to show in that lesson with as little friction as possible.

During my time at Khan Academy, I found using a Wacom tablet worked quite nicely, and I’ve also done casual remote lessons over a video call where my “overhead camera” was just a phone sitting on a stack of books.

That said, being a “professional” YouTuber, and not wanting to disgrace my people, I thought I’d try to do it right. Well, okay, phrased more honestly my buddy Ben Eater was kind enough to lend a bunch of equipment and to patiently explain the numerous things I didn't know about live action filming.

Here’s what we used:

- Two GH5 cameras, one rigged up on an overhead mount.
- Two USB to HDMI converter adapter cables, which essentially make the cameras operate like webcams.
- A set of studio lights
- A lav mic (More specifically a Rode Lavalier Lapel Omni-Directional Condenser Microphone)
- USB Audio Interface to take the feed from the mic to the computer (Focusrite Scarlett Solo)
- Two monitors. One to see the OBS dashboard, the other used for screen captures, managing the live quiz, seeing comments from the audience, etc.

<Figure
  image="hardware.png"
  caption="The masking tape on the desk was to know what region is inside the shot of the overhead cam, though even then I’d occasionally write off-screen. The sticky notes on the overhead camera rig let me know where to point in the air to reference links and things on the screen."
/>

<Figure
  image="hardware-2.jpeg"
  caption="The microphone in this shot was just a backup in case the lav mic failed on us for some reason."
/>

As a small technical note, for the first two streams we did, the audio was slightly out of sync with the video. Hooking up the microphone to feed through the same input source as the camera fixed this, but there are also options in OBS to force some kind of offset in the audio to account for any delays on the camera’s end. Just do a simple test recording where you clap or something like that to see if the same issue comes up for you.

I would say, while for the most part being simple and barebones with the setup is fine, it's definitely worth having audio at a higher quality than what built-in computer or headphone microphones provide. I suspect part of the reason that video calls can be so much more exhausting than simply watching a video is that there’s a constant, albeit slight, strain to make out what people are saying.

This advice comes from the category of “do what I say, not what I do”. The earliest videos on my channel have terrible audio, and several of the early streams had some issue with the sound, whether it was being out of sync, being too quiet, or clipping. If you take the time to get it right early on, your students will be grateful.

## Questions from the audience
The first stream we did had about 26,000 people watching, so the live chat was...not helpful. Actually, it was actively problematic given that people would just post the answers to the questions in there. To take questions from the audience in later streams, we switched instead to having people Tweet to a certain hashtag, which a buddy of mine then combed through to decide which ones were worth pulling up on the screen. The added barrier to contributing made it possible to comb through and the average thoughtfulness of each question was an order of magnitude higher.

Admittedly this part of the experience my not be as transferable for most teachers, but I do think there’s something generalizable in this. In any classroom environment, whether big, small, remote, or in-person, you want engagement and contributions from the students, but only in the right direction. A class full of ambient chatter is in some sense engaged, but without guidance, it’s unlikely to be in the direction of learning. Handling this remotely becomes a whole new ballgame, and it’s worth finding whatever way works for you to connect with the good questions without there being a distracting onslaught of chatter. If you have someone, perhaps a TA, who can comb through questions and feed you the good ones, put them to good use.

As far as integrating this with the setup, what was easiest was to have a shared google doc, with one of the OBS scenes taking a webpage input that simply pointed to the URL of that doc. To give the question being overlaid on the camera shot, we made the Google doc background pure green and applied a chroma key filter.

## Other tips
Rehearse!

Even for ordinary lectures, doing some kind of dry run always helps to make what’s written in your notes come out as smoothly as you intend. But when you’re using any kind of new technology, this goes triply.

Because I had set up the stream with these five different scenes, face, paper, screen capture, quiz, and audience question, running the lesson meant keeping each of these in mind. Which one am I on? Do I need to switch? And if I’m on the screen capture, what should be in the shot, the browser, an animation, or the terminal? Where are we in the live quiz right now? The goal should be for all of this to be second nature so as to stay most focussed on the lesson itself, and with a little practice, it can be.

Tips for setting up remote lessons


Many people, especially college students, ask what books I recommend for learning math, but it's impossible to give a universal answer to such a question. Aside from the obvious fact that it depends on what exactly you want to learn, it's entirely possible for two students with similar backgrounds and goals to approach the same book, with one having a fantastic experience while the other is left perpetually banging their head against the page in confusion. Likewise, a book might work for you at one time in your life but not another.

I've put together a small list of textbooks I've thoroughly enjoyed at some time in the past, with the added condition that each one stands out in some way from the surrounding body of math texts. Take each recommendation with a grain of salt, precisely because the pairing of a book to a learner can be so personal, and who I was when I read each of these may be fairly different from who you are now.


## Books for college students and beyond

---

<Book
  title="Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach"
  author="John Hubbard and Barbara Burke Hubbard"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0971576688&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/3AkLNU7"
/>

This is what I wish I learned from in my freshman year of college. Honestly, the third of it that is about linear algebra is one of the best linear algebra resources out there.

It doesn't pull any punches for the pure math students, but unlike many books with that target audience it stays grounded in frequent numerical examples and scientific applications. The culmination to the generalized Stokes' theorem, as well as the way to frame Maxwell's equations with differential forms, are particularly satisfying.

---

<Book
  title="An Introduction to Thermal Physics"
  author="Daniel Schroeder"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0201380277&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://www.amazon.com/Introduction-Thermal-Physics-Daniel-Schroeder/dp/0201380277/ref=as_li_ss_il?dchild=1&keywords=thermal+physics&qid=1600278073&s=books&sr=1-2&linkCode=li3&tag=3b1b04-20&linkId=3ffbf07a5e3f5b3544d1d4eb19f22b7d&language=en_US"
/>

Despite my aim to keep this list purely about math books, I cannot help but include this one in the list. Simply put, it's one of the most enjoyable textbooks I've ever read, giving a very satisfying first-principles understanding of what thermodynamic concepts like temperature and entropy really are, and how we can to understand them.

Statistical mechanics is all about taking surprisingly minimal assumptions, and mathematically analyzing the life out of them to explain where the laws of thermodynamics come from. I feel all math students should at least expose themselves to this topic, if for no other reason than to see what powerful results can come from simple combinatorics. There's also value in practicing what it feels like to take a complicated expression, write an approximation that capture the core part that really matters, and use it to make experimental predictions about insanely complicated systems.

---

<Book
  title="Visual Complex Analysis"
  auothors="Tristan Needham"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0198534469&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/3jgBs4B"
/>

As the title suggests, this offers a highly pictorial introduction to complex analysis. I would not use this as the _only_ text to learn about complex analysis, but Needham does a marvelous job of conveying the beauty of the subject, with numerous helpful intuitions for otherwise famously tricky ideas.

---

<Book
  title="Visual Group Theory"
  author="Nathan Carter"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=088385757X&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/3AbxXTT"
/>

Another text with a titular promise to focus on pictures, this one tackles a subject which is more commonly taught purely symbolically. One of the most important things for any student learning about groups to understand is just how many different ways there are to think about a group, and how different views might be helpful in different contexts. Again, I wouldn't use this as the only book to understand the topic, but it offers a refreshingly different perspective on the topic from most others out there to help arm your arsenal of intuitions.

---

<Book
  title="Proofs from the book"
  author="Martin Aigner and Günter Ziegler"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=3662572648&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/2Vw1vwb"
/>

If you like clever proofs, this is the book for you. It's filled with some of the most elegant arguments in math, taken from combinatorics, geometry, graph theory, analysis, set theory and more. I can't think of any other books with a higher density of "aha" moments.

Elegant proofs can often be difficult to understand due to their density, but Aigner and Ziegler skillfully set the stage for each argument to make it more understandable. There are also numerous helpful illustrations in the margin which help elucidate the charm in all the proofs they've collected.

---

<Book
  title="Nonlinear Dynamics and Chaos"
  author="Steven Strogatz"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0813349109&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/2Vixu3n"
/>

Steven Strogatz is one of the most well known math communicators, and for good reason. He has a way of making you love a problem before diving into telling you how it's solved. Chaos is an incredibly fascinating field, which has only emerged relatively recently in the history of math. This is the well-deserved gold standard for understanding what it's all about.

---

<Book
  title="Ordinary Differential Equations"
  author="Vladimir Arnold"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=B0046I9D6E&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/2Vbc1cJ"
/>

If you want to understand differential equations, but more from the pure math side of things, Arnold gives a wonderful picture of what the field is all about, with a mixture of high level intuitions and low level details, with a satisfying focus on visual and physical intuition along the way.

---

<Book
  title="The Art Of Probability"
  author="Richard Hamming"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0201406861&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/3lr7qxC"
/>

Hamming was a practitioner, spending much of his career in Bell Labs, so unlike many books on probability this stays grounded in the kind of intuitions which come up a lot in solving practical problems. It has countless helpful puzzles and problem solving tricks which are organized in such a way as to give a sense for the deeper theory underlying probability and statistics.

---

<Book
  title="Information Theory, Inference and Learning Algorithms"
  author="David MacKay"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0521642981&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/3je4OjY"
/>

Information theory has a way of shaping the way you view many things in math, computer science, physics and beyond. It's one of those fields that takes ideas which you wouldn't think of as being quantifiable or rigorous, and makes them so.

To be honest, my _first_ recommendation for anyone looking to learn about information theory is to read [Shannon's original paper](https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf) on the topic. After that, for a fuller view of what it's become, MacKay's book is fantastic.

---

<Book
  title="The Cauchy-Schwarz Master Class"
  author="J. Michael Steele"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=052154677X&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/37hfVD5"
/>

The strength of this book is the central role it places on problems. It's aim is to teach you about inequalities, which are the backbone of analysis. But it's all-too easy to teach the statement of an inequality without conveying the intuition for when and how you'd use it. By focusing on well chosen puzzles, this book does an admirable job circumventing that risk and leaving you with the feeling of having trained a skill, rather than having learned a list of facts.

---

<Book
  title="Fourier Analysis: An Introduction"
  author="Elias Stein and Rami Shakarchi"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=B003V4BQ46&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/2WIv0vo"
/>

Not only is this a wonderful foundation for Fourier analysis, with a surprising and satisfying application to prime numbers given in the last chapter, it's a good starting point for analysis as a whole. Stein and Shakarchi have a series of four excellent books on the foundations of analysis, based on a now-famous series of lectures given at Princeton.

 - [Fourier Analysis](https://amzn.to/2WIv0vo)
 - [Real anlysis](https://amzn.to/3fKQlLF)
 - [Complex analysis](https://amzn.to/37kzg6m)
 - [Functional analysis](https://amzn.to/2Vvfq5W)

Analysis, in all its forms, can be very tricky to wrap your mind around, and very fiddly to work with in trying to form airtight proofs and definitions. This set of books, though, is a good place to start for anyone feeling ambitious.

---

<Book
  title="Primes of the Form x^2 + n y^2"
  author="David A. Cox"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=1118390180&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/3Caxsez"
/>

This might seem like a bizarrely specific question to write a whole book about, but that's actually the charm. It covers deep and general ideas, such as class field theory and elliptic functions, which are usually only taught to graduate students, but Cox approaches it all from an extremely concrete question which Fermat thought about.

The typical culture in modern math texts, especially graduate texts, is to first build up an abstract theory, with indications of what you can do with these constructions appearing as offhanded as exercises or examples later on. In any other book on class field theory, the question of which primes can be expressed as $x^2 + n y^2$ for a particular $n$ would be some exercise sitting in the back of a later chapter, easily ignored. The fact that Cox puts one little example so front-and-center as to be the title reflects his general adherence to strong motivation before any abstraction.

---

<Book
  title="The Princeton Companion to Mathematics"
  author="Timothy Gowers (and many, many others)"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0691118809&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/2WR4foN"
/>

Books can never replace the intuition available if there's a professor down the hall whose door you can knock on to start asking questions. However, this book gets pretty close.

Where it shines is in section IV, which includes many expository introductions to various fields of modern math. If there's a field of research you've heard of, say something like analytic number theory, and you're curious to get a feel for what it's all about, the corresponding essay in that section is likely to do a fantastic job.

---

For any textbooks that you read, try to avoid being passive. Read with a pencil and paper in hand to jot down notes and work on exercises (yes, you should actually do the exercises!). Try to predict what proofs will look like before reading them. Be willing to meditate on what the right way to think about a given object is, and ask what would happen if definitions were tweaked.

Ask yourself if each new construct feels motivated, or if it's out of the blue. If it is out of the blue, it's okay to move forward anyway, just keep note of the fact that there is a lurking question mark.

Also, although it’s not quite a book, you may also enjoy the [expository papers](https://www.math.uconn.edu/~kconrad/blurbs/) written by Keith Conrad, especially if you’re looking to learn group theory or number theory.

## Pre-college

The recommendations above are targeted at those in college or beyond. There are undoubtedly many great books for those in high school and below, but I feel less well-positioned to give recommendations in that direction. It was really only once I got to college that I began learning meaningfully from math books, with more of my learning before then coming from interactions with teachers, poking around online, and getting lost in my own head.

That's not to say I didn't learn from books at all before then, there are a handful of texts such as the original [Art of Problem Solving](https://amzn.to/3xrv5QJ) books which were very influential. And when I was younger I remember a certain fascination with many of [the](https://amzn.to/3yu1UOE) [wooden](https://amzn.to/3lu82Tm) [books](https://amzn.to/3luq33M) my dad gave me. But part of me feels like the best thing for any young learner will be a set of good problems to chew on, more so than a set of good books.


## Popular books

There are countless wonderful works out there about math aimed at the general public. Here's just a very small handful which I’d recommend taking a look at.

<Book
  title="Chaos: Making a New Science"
  author="James Gleick"
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0143113453&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/3jnG4FW"
/>

This is one of my all-time favorite books, closely followed by everything else Gleick has written. It gives an inspiring look at the birth of a new field, and unlike many popular science books it manages to talk substantively about the math which was being discovered.

---

<Book
  title="Change Is the Only Constant: The Wisdom of Calculus in a Madcap World"
  author=""
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0316509086&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/3fLOupV"
/>

If you haven't already come across [math with bad drawings](https://mathwithbaddrawings.com/), you're severely missing out.

I feel like Ben Orlin solved what was previously an unsolved problem with this book: Teach calculus in a way which is as enjoyable to those unfamiliar with the topic as it is to experts who use it every day. I, for one, was laughing from page 1. One of the more delightful aspects of this book is how Orlin doesn't just draw from the usual suspects like physics and economics to get practical insight about calculus, he also looks to history, poetry, literature, and swimming corgis.

---

<Book
  title="Infinite Powers: How Calculus Reveals the Secrets of the Universe"
  author=""
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0358299284&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="(https://amzn.to/2VcGxmB)"
/>

This is another wonderful book aiming to popularize calculus (2019 was a good year for that goal). What I enjoyed about this one is how broad a view Strogatz takes of the origins of calculus, ranging from the clever proofs Archimedes was developing up to Fourier's study of heat.

---

<Book
  title="Humble Pi: A Comedy of Maths Errors"
  author=""
  image="https://ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&ASIN=0141989149&Format=_SL250_&ID=AsinImage&MarketPlace=US&ServiceVersion=20070822&WS=1&tag=3b1b04-20&language=en_US"
  url="https://amzn.to/2VumlMs"
/>

Leave it to Matt Parker to write an entire book about making mistakes in math.


Book recommendations


If there’s one thing I hope 3blue1brown videos do, it’s to inspire people to spend more time playing with and learning about math. In a [separate post](/blog/book-recommendations), I lay out some recommendations for books on math which I've personally enjoyed. Here I thought I'd list some of the other places online you might enjoy exploring, should you find yourself with an itch for more math.[^1]

## Practice
Passive consumption, say by watching videos or reading books, is certainly the lowest friction way to expose yourself to new math. But if you care about long-term retention and deeper understanding, there really is no substitute for practice. So before getting to other YouTube channels and books you might enjoy, here are a few places you can go to learn more actively.

<Figure image="brilliant.png" width="150px" />

The first is [brilliant.org](https://brilliant.org/). I know a few of the people there, and I can vouch for the fact that they are extremely thoughtful about what makes for good learning. It shows in the product. Rather than putting lectures and videos front and center, problems and puzzles take precedence, all presented in a structured progression. The site also benefits from an active and helpful community of users.

<Figure image="aops.png" width="150px" />

Another good reservoir of problems with a helpful community surrounding them is the [Art of Problem Solving](https://artofproblemsolving.com/). In particular, they have an extensive [archive of contest math problems](https://artofproblemsolving.com/community/c13_contests), ranging from more approachable ones like the AMC8 to extremely hard ones like the IMO, all with excellent surrounding discussion from the community.

The original [Art of Problem Solving book](https://artofproblemsolving.com/store/item/aops-vol1) by Richard Rusczyk, by the way, was one of the things that made me fall in love with creative math as a young student.

<Figure image="mathigon.png" width="150px" />

[Mathigon](https://mathigon.org/) is a very beautifully done set of interactive articles on all sorts of fun topics in math. The experience sits somewhere between reading and playing a game, with a really wonderful blend of artwork, technology and pedagogical clarity. One small feature, but which adds a ton in my opinion, is that you have to answer questions in order to continue reading. It's a nice way to force the reader to be honest with themselves about whether they're being passive or not.

<Figure image="khan.png" width="150px" />

For building up your fundamentals, say starting with algebra, take a look at [Khan Academy](https://www.khanacademy.org/). They are probably most famous for the videos Sal does, but arguably the site is most valuable as a source of problems you can drill on, starting from the very basics if necessary. I may be biased, though, given that I used to work there!

## Other YouTube channels

---

<Portrait image="/content/blog/recommendations/mathologer.jpeg" />

[Mathologer](https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg), by Burkard Polster. Half of the appeal here comes from the delightful bits of math this channel covers, often with novel ways of presenting otherwise extremely complicated topics to make them easier. For example, he has a video showing [why $\pi$ is transcendental](https://www.youtube.com/watch?v=WyoH_vgiqXM), which is a famously difficult topic, but which he skillfully boils down into an approachable step-by-step presentation. The other half of the appeal is watching Burkard delightedly laugh at his own jokes every 2 minutes or so.

---

<Portrait image="/content/blog/recommendations/numberphile.jpeg" />

[Numberphile](https://www.youtube.com/user/numberphile), by Brady Haran. I’m guessing if you’re here, you already know about Numberphile. If somehow you don’t, it’s a great channel full of interviews with mathematicians. As a result, you see a wide array of the types of topics which different mathematicians find exciting to showcase or teach, and a little more of the human element behind this subject. There is also an [associated podcast](https://www.numberphile.com/podcast), less about the math, and more about those human stories.

---

<Portrait image="/content/blog/recommendations/stand-up-maths.jpeg" />

[Standup Maths](https://www.youtube.com/user/standupmaths), by Matt Parker. This channel is more whimsical and comedic, but that’s not to say there isn’t a lot of serious math it covers.

---

<Portrait image="/content/blog/recommendations/looking-glass-universe.jpeg" />

[Looking Glass Universe](https://www.youtube.com/user/LookingGlassUniverse), by Mithuna Yoganathan. She covers ideas from physics and math explained simply and unusually honestly. Start with the [one on spin](https://www.youtube.com/watch?v=cd2Ua9dKEl8), it’s great. She also places an emphasis on doing practice, often assigning “homework” at the end of each video.

Also, can we just step back to appreciate that the four channels above are all run by people either from or based in Australia? What’s going on over there? And that’s not even counting [Tibees](https://www.youtube.com/user/tibees), [Eddy Woo](https://www.youtube.com/user/misterwootube), [James Tanton](https://www.youtube.com/channel/UCib_J32VI8rQI_LCFXn1XAA) and countless others.

---

<Portrait image="/content/blog/recommendations/welch-labs.jpeg" />

[Welch Labs](https://www.youtube.com/user/Taylorns34), by Stephen Welch. Perhaps best known for his [series on imaginary numbers](https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF), this channel is full of high production quality and well-written videos. There are also several series on machine learning, the topics underlying self-driving cars, so this is certainly one of the more applied math channels out there.

---

<Portrait image="/content/blog/recommendations/bprp.jpeg" />

[Black pen red pen](https://www.youtube.com/channel/UC_SvYP0k05UKiJ_2ndB02IA), by Steve Chow. If you like seeing how to work out integrals in all sorts of crazy ways, boy is this the channel for you. Steven is a math teacher, and the videos have a feeling of being in office hours, honestly working through problems with a very skillful and enthusiastic teacher. Many of the problems covered are the kind you might come across in school, and many or just for fun.

---

<Portrait image="/content/blog/recommendations/patrickjmt.jpeg" />

[Patrick JMT](https://www.youtube.com/user/patrickJMT), by… well… Patrick. If you want good, honest worked examples of practice problems, especially for calculus, it’s hard to find a deeper well of examples than what this channel provides.

<br />

---

Needless to say, there are countless others. But often less is more when it comes to recommendations, so let's leave things here for the time being.


[^1]: Some of the companies listed above have sponsored 3blue1brown videos in the past, but the recommendations here are not sponsored or advertisements in any way.

Recommendations


#### (Note from many years later)

For whatever reason, when I first started 3blue1brown, I also found myself jotting down silly math poems on the side. Math is often most naturally expressed with symbols or pictures, and words are a more tricky medium, so something about the added constraint of making those words adhere to certain meters or rhyming structures felt like a fun puzzle, albeit an unabashedly cheesy one.

After I made the first couple of 3blue1brown videos, and put together a website with some vague sense that that's what I was supposed to do, I tossed up the poems to that site while I was at it. Even if they seem a bit strange now, I still find a certain charm in leaving them here as a remnant of that early-2015 mindset associated with trying something wacky and throwing it up online, which was the same mindset which led to the channel in the first place.


## Moser's Circle Problem


<Portrait image="/content/blog/poems/circ2.png" />

Take two points on a circle,  
and draw a line straight through.  
The space that was encircled  
is divided into two.  

<Portrait image="/content/blog/poems/circ4.png" />

To these points add a third one,  
which gives us two more chords.  
The space through which these lines run  
has been fissured into four.  

<Portrait image="/content/blog/poems/circ8.png" />

Continue with a fourth point,  
and three more lines drawn straight.  
The new number of disjoint  
regions sums, in all, to eight.  

<Portrait image="/content/blog/poems/circ16.png" />

A fifth point and its four lines  
support this pattern gleaned.  
Counting sections, one divines  
that there are now sixteen.  

<Portrait image="/content/blog/poems/circ31.png" />

This pattern here of doubling  
does seems a sturdy one.  
But one more step is troubling  
as the sixth gives thirty-one.  

## Primes
The primes,  
through times,  

mystified  
those who pried.  

One fact answers why  
they’re simple yet sly:  

Layers of abstraction yield  
complex forms when pierced and peeled.  

Addition lies under multiplication,  
defining him as repeated summation.  

Then he defines primes as the atoms of integers,  
for when multiplied, they give numbers their signatures.  

But when we breach the layer between these two distinct operations,  
asking about how primes add and subtract, there are endless frustrations.  

Even innocuous questions, “what are all their sums?”, or “how often do they differ by two?”,  
stump everyone who has ever lived, with progress made only quite recently by just a few.  

However, to recruit for and progress math we need to have such questions which can be phrased simply and remain unsolved.  
What child does not hear such conjectures and dream, if only for a moment, that they will be the one to see them resolved?  

For otherwise the once vigorous curiosity of a child towards math’s patterns, as they grow older, tends to grow tame,  
just as the rhyme and rhythm of the primes seems to fade as numbers grow, though in both the underlying patterns remain the same.  

## All Numbers are Interesting
Is any number that we count banal,  
and lacking in a feature one might note?  
Suppose some are, and gather up them all.  
The smallest one has quite the cause to gloat.  
It is precisely how far one must count  
before the numbers get uninteresting.  
But this would be a curious amount,  
which contradicts how we defined the thing!  
So every number has something to show,  
including ones whom we will never know.  

Let’s see if this joke proof applies as well  
to real numbers; do they all stand out?  
Suppose some don’t, and find one who rebels  
by being meaningful beyond a doubt.  
Before it was the smallest who amused,  
but real sets might lack a minimum.  
Will there be a special one to choose  
from subsets of the whole continuum?  
It isn’t clear where such a choice comes from,  
yet some treat “choice” like it’s an axiom.  

## Invention vs Discovery
A lurking question, old as Greece, does ask:  
Is math invention or discovery?  
The answer’s both, but here the harder task  
is knowing how it switches.  So you see,  
most truths are found with constructs, which in turn  
are built from truths like chickens and their eggs.  
One needs to know, should this fact raise concern,  
that constructs can be daft; just logic’s dregs.  
The truths discovered tell how to define  
the structures that make math and world align.  

Pythagoras’s theorem about lengths,  
is one such truth that he (and others) found.  
Observed, discovered, and through mental strength,  
it was then proved with pictures that astound.  
Much later mathematicians, minds alit,  
would formalize, as sets, both “length” and “space”.  
These constructs were defined so that by writ  
Pythagoras’s truth remains the case.  
That is, this “theorem” now seems just defined.  
As such, one might then lose their peace of mind.  

So in this world where "space" is formalized,  
do all those pretty proofs now lose their charm?  
Of course not!  We all ought to realize  
how “length” could change its meaning without harm  
to mathematical consistency  
in all the theory giving facts of space.  
However, then these “facts” would hardly be  
related to the nature we embrace.  
And hence, a truth has told how to define  
a structure that makes math and world align.  

## 2π
Fixed poorly in notation with that two,  
you shine so loud that you deserve a name.  
Late though we are, to make a change it's true,  
We can extol you ‘til you have pi's fame.  
One might object, "Conventions matter not!  
Great formulae cast truths transcending names."  
I've noticed, though, how language molds my thoughts;  
the natural terms make heart and head the same.  
So lose the two inside your autograph,  
then guide our thoughts without you "better" half.  

Wonders math imparts become so neat  
when phrased with you, and pi remains off-screen.  
Sine and exp both cycle to your beat.  
Jive with Fourier, and forms are clean.  
“Wait! Area of circles”, pi would say,  
“sticks oddly to one half when tau’s preferred.”  
More to you then!  For write it in this way,  
then links to triangles can be inferred.  
Nix pi, then all within geometry  
shines clean and clear, as if by poetry.  

## 1-1+1-1+...
When one takes one from one  
plus one from one plus one  
and on and on but ends  
anon then starts again,  
then some sums sum to one,  
to zero other ones.  
One wonders who'd have won  
had stopping not been done;  
had he summed every bit  
until the infinite.  

Lest you should think that such  
less well-known sums are much  
ado about nonsense  
I do give these two cents:  
The universe has got  
an answer which is not  
what most would first surmise,  
it is a compromise,  
and though it seems a laugh  
the universe gives “half”.  

## Defining Math
In math, we strive and stickle with a pride,  
to wash away all ambiguity.  
And yet, definers cannot all decide  
what “math” itself means.  What an irony!  
Perhaps a science, solving mysteries,  
but happy to pursue what seems absurd.  
Perhaps a brother of philosophies,  
but with a rigid rigour in each word.  
Computer science might be suitable,  
but most of math is not computable.  

Perhaps it is a language of its own,  
which, as to english, has the most false friends.  
Perhaps religion, where the truth alone  
is worshiped, giving means as well as ends.  
Perhaps a sister of creative arts,  
but one arousing reason, not a sense.  
Indeed it shares a number of its parts  
with almost every study in some sense.  
But it’s an orphan, floating in abstract,  
and it alone encloses what’s exact  

## Euler's Formula
Famously  
start with e,  
raise to π  
with an i.  
we've been taught  
by a lot  
that you've got  
minus one.  

Can we glean  
what it means?  
For such words  
are absurd.  
How to treat  
the repeat  
of a feat  
πi times?  

This is bound  
to confound  
'til your mind  
redefines  
these amounts  
one can't count  
which surmount  
our friend e.  

Numbers act  
as abstract  
functions which  
slide the rich  
2d space  
in its place  
with a grace  
when they sum.  

Multiplied,  
they don’t slide,  
acting a  
second way.  
They rotate,  
and dilate,  
but keep straight  
that same plane.  

Now what we  
write as e  
to the x  
won’t perplex  
when you know  
it’s for show  
that “x" goes  
up and right.  

It does not,  
as you thought,  
repeat e  
product e.  
It functions  
with gumption  
on functions  
of the plane.  

It turns slides  
side to side  
into growths  
and shrinks both.  
Up and downs  
come around  
as turns round,  
which is key!  

This is why  
π times i,  
which slides north  
is brought forth  
and returned,  
we have learned,  
as a turn  
halfway round.  

Minus one,  
matched by none,  
turns this way,  
hence we’re done.  

## Ode To Infinity
Philosophers have claimed you are not "real",  
but you are no less true than 3, in fact.  
For what is "three"?  Some thinking will reveal,  
a term that points to triplets in abstract.  
And so are you, a pointer in this light,  
but aimed instead at families without size.  
From here it took great Cantor's mental sight  
to see these sets disjoin before his eyes.  
For that's the tricky thing about you, friend,  
what you are is something without end!  

Aside from all the masks you wear as sets,  
you permeate all math like pi and e.  
Without you series, sequences and nets,  
would vanish, as would all geometry.  
To cook the real numbers all it takes,  
is you and all the fractions to react.  
When added as a simple point you make  
both arguments and spaces more compact.  
When patterns, lists and odes drag on and on,  
just say "infinity", and woes are gone.  

## Groups
"What is a group?"  
    “A symmetry of things.”  

“Of things from where?”  
    “Oh, those from any place.”  

“But where in math?”  
    “In math?  A vector space.”  

“And what is that?”  
    “It’s what a field brings.”  

“And what are fields?”  
    “They’re Symmetries of groups.”  

“A group’s own group?”  
    “Why yes, when both are one.”  

“So now we’re back?”  
    “Indeed, where we’d begun.”  

“So where to start?”  
    “Where you start any loop.”  

“And where is that?”  
    “From anywhere within.”  

“And if you're out?”  
    “Then why would you begin?”  

## Abstraction
The crux of thought,  
computer science,  
language, and how math arose,  
abstraction brought  
a key alliance  
syncing facts who seem opposed.  

The first is how  
our minds are tuned to  
focus on at most one thought.  
Though well endowed,  
they’re not immune to  
puzzlement when overwrought.  

But secondly  
the facts of nature  
interweave dependently,  
so how do we  
conceive this nature  
in its multiplicity?  

We first extract  
a common trait  
of things too complex to be known,  
then to “abstract”,  
reformulate  
this trait as *something in its own*.  

As such it’s pulled  
from all distractions,  
single, hence conceivable.  
But when minds hold  
this dense abstraction,  
They enclose a system whole.  

For “things” and their “traits”,  
though separately named,  
do more than relate:  
They’re one and the same.  


Math Poetry

<Section dark={true}>
  <HomepageFeaturedContent title="Animated math" show_latest_video={true}>
    <HomepageFeaturedItem
      lesson="neural-networks"
      caption="Learn how neural networks work"
      interactive_src="lessons/2017/neural-networks/neural-network-interactive/index"
    />
    <HomepageFeaturedItem
      lesson="vectors"
      caption="Learn how to visualize linear algebra"
      video_src="eola-preview.mp4"
    />
    <HomepageFeaturedItem
      lesson="essence-of-calculus"
      caption="Learn the essence of calculus"
      video_src="../content/lessons/2017/derivatives-power-rule/figures/5.20-6.20-trimmed.mp4#t=0.001"
    />
    <HomepageFeaturedItem
      lesson="clacks"
      caption="Learn how sliding blocks compute an unexpected constant"
      video_src="clacks.mp4"
    />
    <HomepageFeaturedItem
      lesson="fourier-series"
      caption="Learn how this animation relates to the physics of heat"
      video_src="fourier-pi.mp4"
    />
    <HomepageFeaturedItem
      lesson="bitcoin"
      caption="Learn how bitcoin works"
      interactive_src="lessons/2017/bitcoin/live-transactions-interactive/index"
    />
    <HomepageFeaturedItem
      lesson="prime-spirals"
      caption="Learn why polar plots of primes form these spirals"
      video_src="prime-spirals.mp4"
    />
    <HomepageFeaturedItem
      lesson="inscribed-rectangle-problem"
      caption="Learn how topology is used to solve problems"
      video_src="sudanese-band-loop.mp4"
    />
    <HomepageFeaturedItem
      lesson="sphere-area"
      caption="Learn how a sphere's surface area relates to its shadow"
      video_src="sphere-area.mp4"
    />
    <HomepageFeaturedItem
      lesson=""
      caption=""
      image_src="/featured-content/pi-comment.svg"
    />
  </HomepageFeaturedContent>
</Section>

<Announcement 
  id="0"
  headline="Summer of Math Exposition #2"
  description="Create an online lesson, submit it by August 15th"
  link="https://summerofmathexposition.substack.com/p/the-summer-of-math-exposition-is?s=r"
/>

<Section id="lessons">

<LessonGallery show="topic" />

</Section>

<Section>

## New notebooks

<figure>
  <div>
    <a href="https://store.dftba.com/collections/3blue1brown/products/mathematical-quotebook-notebook">
      <img
        src="/images/home/notebook.jpeg"
        width="500px"
        alt="Image of the new 3blue1brown quotebook notebook available on the DFTBA store.
      "/>
    </a>
  </div>
</figure>

</Section>

<Section width="narrow">

<Portrait image="images/home/patreon-logo.svg" />

If these lessons add value to your life, consider joining the contributing members who help make them possible.

Benefits range from early access to new videos, to having your name in the video credits.
This model allows for the lessons to remain free from brand integrations and sponsor messages, without introducing paywalls.

<Center>
  <Clickable
    link="https://www.patreon.com/3blue1brown"
    icon="fab fa-patreon"
    text="Memberships"
    design="rounded"
  />

  <Clickable
    link="/thanks"
    icon="fas fa-heart"
    text="Special thanks"
    design="rounded"
  />
</Center>

</Section>

<Section>

<iframe src="https://3blue1brown.substack.com/embed" width="480" height="180" />

</Section>

Three false proofs and what lessons they teach.

How to Lie With Visual Proofs

Generating functions, as applied to a hard puzzle used for IMO training.

Olympiad level counting

Following up on the previous video about Wordle and information theory

Oh, wait, actually the best Wordle opener is not “crane”…

A lesson on information theory and Shannon entropy, told through an exploration of writing a Wordle solver.

Solving Wordle using information theory

A tale of two problem solvers, looking for the average area of a cube's shadow

Alice, Bob, and the average shadow of a cube

An introduction to holomorphic dynamics, which studies iterative processes in the complex plane.

How a Mandelbrot set arises from Newton’s work

A surprising fractal that arises from Newton's method for finding roots of polynomials.

Newton's Fractal (which Newton knew nothing about)

A quick way to compute eigenvalues of a 2x2 matrix

A quick trick for computing eigenvalues

Exponentiating matrices, and the kinds of linear differential equations this solves.

How (and why) to raise e to the power of a matrix

The medical test paradox: Can redesigning Bayes rule help?

Hamming codes part 2, the elegance of it all

A discovery-oriented introduction to error correction codes.

Hamming codes and error correction

An introduction to symmetry, group theory, isomorphisms, and the monster group.

Group theory, abstraction, and the 196,883-dimensional monster

An information puzzle with an interesting twist

The impossible chessboard puzzle

Tips to be a better problem solver

Intuition for i to the power i

An overview of a simplified version of the DP-3T algorithm for privacy-first contact-tracing

The DP-3T algorithm for contact tracing (via Nicky Case)

The power tower puzzle

What makes the natural log "natural"?

Logarithm Fundamentals

Imaginary interest rates

What is Euler's formula actually saying?

Complex number fundamentals

Trigonometry fundamentals

The simpler quadratic formula

Lockdown math announcement

Introduction to probability density functions.

Why “probability of 0” does not mean “impossible” | Probabilities of probabilities, part 2

SIR models for epidemics, showing how tweakign behavior can change an outbreak.

Simulating an epidemic

The binomial distribution, introduced as setup to talk about the beta distribution

Binomial distributions | Probabilities of probabilities, part 1

A primer on exponential and logistic growth, with epidemics as a central example

Exponential growth and epidemics

A short explanation of why Bayes' theorem is true, together with discussion on a common misconception in probability

The quick proof of Bayes' theorem

A visual way to think about Bayes' theorem, and strategies for making probability more intuitive.

Bayes' theorem

Q&A with Grant (3blue1brown), windy walk edition

A game of darts yields an interesting connection to higher dimensional geometry.

Darts in Higher Dimensions

A curious pattern in polar plots with prime numbers, together with discussion of Dirichlet's theorem

Why do prime numbers make these spirals?

Problem 2 from the 2011 IMO, a seemingly easy question that turned out to be ridiculously difficult

The unexpectedly hard windmill question (2011 IMO, Q2)

A quick explanation of e^(pi i) in terms of motion and differential equations

e^(iπ) in 3.14 minutes, using dynamics

A montage of "fourier series" drawings, in which the sum of many rotated vectors traces an image

Pure Fourier series animation montage

How Fourier series arose from studying heat flow, and how they can be thought of as decomposing any drawing in 2d into a sum of rotations.

But what is a Fourier series? From heat flow to circle drawings

Solving the heat equation

The heat equation, as an introductory PDE.

But what is a partial differential equation?

What is a differential equation, the pendulum equation, and some basic numerical methods

Differential equations, studying the unsolvable

What Cramer's rule is, and a geometric reason it's true

Cramer's rule, explained geometrically

The third and final part of the block collision sequence.

How colliding blocks act like a beam of light...to compute pi.

A solution to the puzzle involving two blocks, sliding fricionlessly, where the number of collisions mysteriously computes pi

Why do colliding blocks compute pi?

A puzzle involving colliding blocks where the number pi, vey unexpectedly, shows up.

The most unexpected answer to a counting puzzle

Two proofs for the surface area of a sphere

But why is a sphere's surface area four times its shadow?

Solving a discrete math puzzle, namely the stolen necklace problem, using topology, namely the Borsuk Ulam theorem

Sneaky Topology | The Borsuk-Ulam theorem and stolen necklaces

A look at what turbulence is (in fluid flow), and a result by Kolmogorov regarding the energy cascade of turbulence.

Why 5/3 is a fundamental constant for turbulence

An introduction to an interactive experience on why quaternions describe 3d rotations

Quaternions and 3d rotation, explained interactively

How to visualize quaternions, a 4d number system, in our 3d world

Visualizing quaternions (4d numbers) with stereographic projection

Q&A with Grant Sanderson (3blue1brown)

A beautiful proof of why slicing a cone gives an ellipse.

Why slicing a cone gives an ellipse

This video recounts a lecture by Richard Feynman giving an elementary demonstration of why planets orbit in ellipses.  See the excellent book by Judith and David Goodstein, "Feynman's lost lecture”, for the full story behind this lecture, and a deeper dive into its content.

Feynman's Lost Lecture (ft. 3Blue1Brown)

Divergence, curl, and their relation to fluid flow and electromagnetism

Divergence and curl:  The language of Maxwell's equations, fluid flow, and more

A visual for derivatives which generalizes more nicely to topics beyond calculus. Thinking of a function as a transformation, the derivative measure how much that function locally stretches or squishes a given region.

The other way to visualize derivatives

A proof of the Wallis product for pi, together with some neat tricks using complex numbers to analyze circle geometry.

The Wallis product for pi, proved geometrically

An algorithm for solving continuous 2d equations using winding numbers.

Winding numbers and domain coloring

A bit of the history behind how we came to use the symbol "pi" to represent what it does today, and how Euler used it to refer to several different circle constants.

How pi was almost 6.283185...

A beautiful solution to the Basel Problem (1+1/4+1/9+1/16+...) using Euclidian geometry.  Unlike many more common proofs, this one makes it very clear why pi is involved in the answer.

Why is pi here?  And why is it squared?  A geometric answer to the Basel problem

The general uncertainty principle, about the concentration of a wave vs the concentration of its fourier transform, applied to two non-quantum examples before showing what it means for the Heisenberg uncertainty principle.

The more general uncertainty principle, beyond quantum

An animated introduction to the Fourier Transform, winding graphs around circles.

But what is the Fourier Transform?  A visual introduction.

A classic puzzle in graph theory, the "Utilities problem", a description of why it is unsolvable on a plane, and how it becomes solvable on surfaces with a different topology.

The three utilities puzzle with math/science YouTubers

A challenging question of geometry and probability on the Putnam.  The shift in perspective used for the solution here carries with it broader loessons for problem-solving.

The hardest problem on the hardest test

The math of backpropagation, the algorithm by which neural networks learn.

Backpropagation calculus

An overview of backpropagation, the algorithm behind how neural networks learn.

What is backpropagation really doing?

An overview of gradient descent in the context of neural networks. This is a method used widely throughout machine learning for optimizing how a computer performs on certain tasks.

Gradient descent, how neural networks learn

Analyzing our neural network

An overview of what a neural network is, introduced in the context of recognizing hand-written digits.

But what is a Neural Network?

An introduction to the quantum behavior of light, specifically the polarization of light.  The emphasis is on how many ideas that seem "quantumly weird" are actually just wave mechanics, applicable in a lot of classical physics.

Some light quantum mechanics (with minutephysics)

A method for thinking about high-dimensional spheres, introduced in the context of a classic example involving a high-dimensional sphere inside a high-dimensional box.

Thinking outside the 10-dimensional box

Drawing curves that fill all of space, and a philosophical take on why mathematics about infinite objects can still be useful in finite contexts.

Hilbert's Curve: Is infinite math useful?

When a piece of cryptography is described as having "256-bit security", what exactly does that mean?  Just how big is the number 2^256?

How secure is 256 bit security?

How does bitcoin work? What is a "block chain"? What problem is this system trying to solve, and how does it use the tools of cryptography to do so?

But how does bitcoin actually work?

There are a few special right triangles many of us learn about in school, like the 3-4-5 triangle or the 5-12-13 triangle.  Is there a way to understand all triplets of numbers (a, b, c) that satisfy a^2 + b^2 = c^2?  There is!  And it uses complex numbers in a clever way.

All possible pythagorean triples, visualized

A beutiful derivation of a formula for pi.  At first, 1-1/3+1/5-1/7+1/9-.... seems unrelated to circles, but in fact there is a circle hiding here, as well as some interesting facts about prime numbers in the context of complex numbers.

Pi hiding in prime regularities

What is the second derivative?  Third derivative?  How do you think about these?

Higher order derivatives

Taylor series are extremely useful in engineering and math, but what are they?  This video shows why they're useful, and how to make sense of the formula.

Taylor series

Derivatives are about slope, and integration is about area.  These ideas seem completely different, so why are they inverses?

What does area have to do with slope?

What is integration?  Why is it computed as the opposite of differentiation?  What is the fundamental theorem of calculus?

Integration and the fundamental theorem of calculus

How does (ε, δ) "epsilon delta" help us formalize what exactly it means for one value to approach another?

(ε, δ) "epsilon delta" definitions of limits

What is L'Hopital's rule and how does it help us evaluate limits?

L'Hôpital's rule

What are limits? How are they defined? How are they used to define the derivative?

Limits and the definition of derivatives

How to think about implicit differentiation in terms of functions with multiple inputs, and tiny nudges to those inputs.

Implicit differentiation, what's going on here?

What is the derivative of a^x?  Why is e^x its own derivative?  This video shows how to think about the rule for differentiating exponential functions.

What's so special about Euler's number e?

The product rule and chain rule in calculus can feel like they were pulled out of thin air, but is there an intuitive way to think about them?

Visualizing the chain rule and product rule

Introduction to the derivatives of polynomial terms thought about geometrically and intuitively. The goal is for these formulas to feel like something the student could have discovered, rather than something to be memorized.

Power Rule through geometry

Introduction to the derivatives trigonometric functions thought about geometrically and intuitively.

Trig Derivatives through geometry

An introduction to what a derivative is, and how it formalizes an otherwise paradoxical idea.

The paradox of the derivative

An overview of what calculus is all about, with an emphasis on making it seem like something students could discover for themselves.  The central example is that of rediscovering the formula for a circle's area, and how this is an isolated instance of the fundamental theorem of calculus

The Essence of Calculus

Euler's formula, e^{pi i} = -1, is one of the most famous expressions in math, but why on earth is this true?  A few perspectives from the field of group theory can make this formula a bit more intuitive.

Euler's formula with introductory group theory

What exactly are fractals?  A common misconception is that they are shapes which look exactly like themselves when you zoom in.  In fact, the definition has something to do with the idea of "fractal dimension".

Fractals are typically not self-similar

After a friend of mine got a tattoo with a representation of the cosecant function, it got me thinking about how there's another sense in which this function is a tattoo on math, so to speak.

Tattoos on Math

What is the Riemann zeta function? What is analytic continuation? This video lays out the complex analysis needed to answer these questions.

Visualizing the Riemann zeta function and analytic continuation

How couting in binary can solve the famous tower's of hanoi problem.

Binary, Hanoi and Sierpinski, part 1

An unsolved problem, a very elegant solution to a weaker version of the problem, and a little bit on what topology is and why people care.

Who cares about topology? (Inscribed rectangle problem)

What is a vector space?  Even though they are initial taught in the context of arrows in space, or with vectors being lists of numbers, the idea is much more general and far-reaching.

Abstract vector spaces

Eigenvalues and eigenvectors are one of the most important ideas in linear algebra, but what on earth are they?

Eigenvectors and eigenvalues

What is a change of basis, and how do you do it?

Change of basis

The formula for the cross product can feel like a mystery, or some kind of crazy coincidence.  But it isn't.  There is a fundamental connection between the cross product and determinants.