3Blue1Brown

In April, the next item on my roster of video plans was to explain what cross-entropy is and why it’s used for pre-training language models. Fast-forward two months, and this has turned into a little 3-part mini-series on the foundations of information theory as they pertain to the phrase “compression is intelligence.”

It’s not uncommon for me to get excited by a topic and have it grow; it’s something I love about this work. However, it feels especially funny to me in this case because cross-entropy as a feature of LLM training amounts to just one line of code, and a relatively simple one at that. Of course, what’s interesting is not the line of code, but the question of what the term means and why it’s a principled loss function to use. Given that the term has its origins in studying compression, I don’t think any writer could resist being pulled into the gravity well of that provocative phrase, “compression is intelligence”.

Also, I’ve been itching to cover information theory properly for a while now, and this was a welcome excuse. This first part begins with the fundamentals, explaining where the formulas for information and entropy come from, in the context of leading a viewer to (hopefully) rediscover the core idea behind Shannon’s Noiseless Coding Theorem.

Enjoy, Grant

This may be one of the most involved videos I’ve ever done for the channel, but it was a blast to produce. M.C. Escher has countless pieces that are fascinating for some mathematical reason, but his Print Gallery may top them all (I would say in close contention with his Circle Limit series).

Aside from analyzing the piece, offering both an artistic description based on how Escher approached and a more mathematical one based on work by the mathematicians de Smit and Lenstra, I wanted to use this as a motivation to introduce certain ideas from complex analysis, such as conformal maps and the complex logarithm. Enjoy!

This talk centers around deriving the general formula for the volumes of spheres in higher dimensions. More broadly, it makes an attempt to take various initially counterintuitive facts about higher dimensions and make them (hopefully) a bit more intuitive.

Thanks to UC Santa Cruz for letting me film there, and to all its students for being such a lovely audience. Actually, the same thanks are owed to the International House at Berkeley, to the University of Wisconsin, and to Stanford. Not entirely intentionally, I ended up with a small de facto tour, giving this a couple of times.

New video! This one was a joy to animate.

I foreshadowed this in the last post, but this year I’m launching a new thing this year, which I’m thinking of as a kind of virtual career fair. If you’re looking for a job and want a team that values mathematical and technical curiosity, check out 3b1b.co/talent.

For the 2025 Summer of Math Exposition, I promised a set of supplementary prizes for entries that were helpful to teachers. Although long overdue now, I’m happy to announce the final selection.

In past years, I chose a set of five “general” winners. These were entries I felt enthusiastic about recommending to this audience, as judged by novelty, clarity, and memorability.

For 2025, I decided to do it differently. We already have a peer review system that I think does a wonderful job of surfacing the generally great content that this audience will enjoy. Indeed, we announced its results for SoME4 a few months ago, and if you explore the top entries, I’m confident you’ll find lessons you enjoy.

This year, I wanted to use prizes to reward something not necessarily already rewarded by the peer review and the social internet, which is the content that shows strong promise to be helpful to teachers. Sometimes this aligns with what peer reviewers will enjoy, but not always.

We solicited teacher feedback on all the results and compiled them. My job was then to review that feedback, and to watch/read many of the most promising entries myself, and select five of them to receive a prize of $1,000 each.

If you run a company or are part of a team that may be interested in recruiting technical talent from the 3blue1brown audience, please fill out this form, and I’ll follow up with you.

I have a job posting for an operations role. If someone you know is a good fit, please pass it along, and if you’re interested, feel free to apply.

Shockingly, I’ve been making videos for over a decade now.

For most of that time, I’ve been happily making videos more or less on my own. Well, that’s not exactly true or fair. There have been (delightful) contractors I’ve worked with along the way, most notably for music and hand-drawn illustrations, and for SoME. Several videos have been more collaborative and co-authored; the one on holograms or the one on the Basel problem both come to mind. However, by and large, the writing, animating, and editing of a typical 3bluebrown video is a one-man shop.

The next part in the sequence on Laplace Transforms is now up. In case you missed it, here is the previous video introducing this transform, and here is the one before laying down groundwork and intuition about complex exponents.

In this new part, we dig into a concrete example to see why this transform is useful both to understand the qualitative dynamics of a system and to find a final analytic solution.

For a while, ever since I made a video about Fourier Transforms, one of the most requested topics on the channel has been its close cousin, the Laplace Transform.

I’ve been working hard on a mini-series about this topic, to be inserted into the differential equation series, and the main part is now out.

This chapter visualizes what this transform is, how it’s defined, and how it exposes the exponential pieces lurking inside a function.

Creating these visuals was a real joy. In particular, one of the steps to understanding what it’s really doing is to understand what it means to integrate a complex-valued function, and building up a machine to do that piece by piece and watching what it does is, to me at least, extremely satisfying.

The previous chapter, for those who missed it, talked about how to interpret complex exponentials, from a physical point of view, and why those functions are, in a certain sense, the “atoms of calculus”.

This is the first in a new trilogy of videos I’m making to help understand the Laplace transform. In this chapter, I want to make sure everyone is up to speed on how to think about complex exponents and why they naturally arise in the study of differential equations.

For any of you who are Patreon supporters, you can find an early draft of the next video on the Laplace Transform itself here.

In this fifth and final guest video, Ben Syversen discusses a question anyone who has done ruler and compass constructions for a geometry class may have wondered: What’s the point?

There is a lot about Euclid’s Elements that is easily misunderstood. Arguments that seem to have logical gaps, some constructions that seem pointless, others that seem needlessly convoluted. But each of these actually provides a window into how the ancient Greeks thought about math, and the philosophical role that geometry played.

This is the fourth of five guest videos before I’m back with my own, and I have to say, I absolutely love it. Next time you have 50 minutes to spare, here is the full piece —a wonderful problem-solving saga centered around a piece of modern art and the combinatorial question underlying it.

If you don’t have the time right now, and just want to whet the appetite, here is a short preview.

Unlike the other guest videos, this one does not come from another YouTuber. Instead, it’s from a friend of mine, Paul Dancstep. He’s collaborated with me on a few projects in the last year, including the ones on holograms and cosmic distances, and I think you’ll agree he has a great eye for visual exposition and pedagogical storytelling.

For anyone curious about who he is and the process behind this piece, I recorded a short interview with him.

This is the third of five guest videos while I’m on leave. It comes from a previous winner of the Summer of Math Exposition, Vilas Winstein, currently doing his PhD at Berkeley. In it, he covers some of the fundamental ideas in statistical mechanics, including defining temperature, entropy, and deriving the Boltzmann law, all in the service of creating a simplified and discretized model mirroring the behavior of a fluid transitioning between a liquid and gaseous state. Enjoy!

This is a guest video by Aleph 0, the second in a series of five guest posts this summer while I’m on leave.

This covers AlphaGeometry, a tool for automatically solving most IMO-level geometry problems. But even in the short time since Google DeepMind announced this tool, the abilities of AI in contest and research-level mathematics have come a long way. This year, both DeepMind and OpenAI announced having achieved gold-level performance at the IMO with pure language models.

A project I have in the works for later this year is about the role of AI in modern math. For instance, the loop described in the video above between deductive and intuitive reasoning lends itself very well to math, but it seems the direction the winds are blowing is more towards using the availability of symbolic checkable reasoning on the side of training data generation, more so than during test-time performance.

One aspect of this project will involve interviews with mathematicians about how (if at all) modern reasoning tools are relevant to their work. If you feel you have a story relevant to this, I’d love to hear it, so feel free to fill out this form.

This year, I knew I’d want to take some meaningful time away from the channel, so I reached out to a few creators whose work I respect and asked if they’d be interested in me pointing some of the Patreon funds that come to the channel towards them to commission some guest videos during my absence.

This first installment comes from Stephen Welch, of Welch Labs, a true kindred spirit of 3blue1brown on YouTube, and I couldn’t be happier with the result. He goes into the details of diffusion models, the technology behind modern AI-generated images and videos. There’s much more nuance than the typical “model learns to remove noise” summary given, and he explains and visualizes it beautifully. Enjoy!

If you enjoy this, I’d highly recommend looking at the other content on Welch Labs and considering purchasing his book, too.

We'll be doing it again this year!

One of the great joys of the Summer of Math Exposition is that through the peer review process, we have a flood of activity for a few weeks of the year that helps new creators gain exposure, and allows those of us in the community to see new topics and perspectives we may not have otherwise come across. A risk, however, is that the entries which is most rewarded are lessons that appeal to those already very passionate about math, at least enough so to voluntarily join the peer review.

Given how many students in the world struggle with math, I would love to be able to re-direct the enormous amount of creative energy that goes into all these entries, if only slightly, to encourage people to choose topics not just based on what fellow math-nerds will love, but based on what will be most helpful. To do this, in giving out cash prizes to 5 entries this year, I’ll be placing heavy weight on whether teachers of the relevant subject believe the entry would be helpful to their students.

So, if you’re a math teacher of any kind, I would love to hear what specific topics you think deserve better online coverage. What is especially hard to explain to students? Where would visualizations or better narratives be especially useful? What have you searched for where the results you got left you disappointed? Feel free to discuss on this Reddit post.

Also, if you are a math teacher and you think you might be interested in helping provide feedback to this year’s...

In the most recent video about quantum computing, in response to the section on Grover’s algorithm, I saw many comments expressing a very similar point of confusion. I made a follow-up video, which I hope may help clarify some of the issues, and which also doubles as an excuse to talk a bit more about the central role of linearity in quantum computing.

A common misconception about quantum computers is that they would solve hard problems by trying all possible solutions in parallel. This vaguely gestures at something true, but the reality is more subtle.

What do they do then? This video builds up to Grover’s algorithm, a general method in quantum computing for finding solutions to any NP problem, i.e., anything where you have a quick way to verify solutions, even if finding them in the first place may be hard. To get there, we build up the fundamentals of quantum computing, not with a set of analogies, but as a piece of math.

My original plan for a Pi Day video this year was to describe an analogy between these colliding blocks that compute π and Grover's algorithm, a topic in quantum computing. As I sat down outlining that video though, it was clear that there's a lot of background required to explain a quantum computing algorithm (shocking, I know), and for the comparison to make any sense, you need to have the original block collision stuff all loaded into your head.

Looking back at the original explainer I made for this topic, I didn’t love it. Rather than cramming a summary into the increasingly heavy quantum computing video, I decided to take the opportunity to make a second edition of that block collision explainer. Told right, it’s a story about problem-solving principles, and the contrast between the pragmatic realities of the setup and the idealism required to see the π makes for some good end-of-video pontificating.

I'd love it if you shared it with anyone in your life who likes Pi Day, and stay tuned for the quantum computing follow-on!

This is part 2 of a saga about cosmic distances. Enjoy!

This most recent video is a change of pace for the channel, featuring an interview with Terence Tao on cosmic distances. Enjoy!