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!

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.

3b1b Talent

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.

Reminder of how prizes work this year

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.

Part of why I’m so egregiously late in giving a final selection here is that there is no perfect and objective way to do this. I say this every year, but for those who were not selected, please take the word “winner” with a heavy grain of salt. I spent countless hours watching and reading many, many pieces that I thoroughly enjoyed, and which I see as positive contributions to the internet, constituting a significant superset of the ones featured here.

Final selection

As I reviewed each entry, I asked, “If I were teaching a course on this subject, how eager would I be to share with my students?”

It was very rare that I’d imagine simply assigning a video or article wholesale to be consumed by a student on their own time. Teaching is best when it’s dynamic and personal, so instead I’d feel drawn towards picking out bits and pieces, perhaps borrowing examples or pulling out interactive elements that could be used in the classroom. I’d then weigh this together with the teacher feedback that we had gathered.

In no particular order, these were the five final selections:

The fight over fairness that revolutionized math. This offers a nice motivation for expected value and adjacent topics in probability. I think it’s easiest to remember a piece of math if you feel like you invented it yourself, and the examples laid out by the creator here offer a nice scaffold that any probability teacher could use in a class of their own to guide students to reinvent core concepts in probability for themselves.

Relativity in Desmos. For students in any physics class introducing special relativity, this would be most relevant after having built up some of the fundamentals. The creator has offered an easily shareable set of demos to highlight some key properties of Lorenz transforms. Engaging with these stands to sharpen a student’s intuition for thinking about spacetime diagrams, and I can easily envision a teacher picking and choosing ones to fit their particular lesson’s needs.

A trick for analyzing cubics, by Dr PK Math. This highlights a nice property of cubic functions useful for problem-solving. I appreciate the spirit represented by the video, which is that of a teacher sharing something obscure-but-fun with their fellow educators. Tricks like this are often very specific, but the acts of both using and proving it offer students a great chance to practice fundamental ideas for working with polynomials.

The world’s oldest algorithm. As the creator points out, Euclid’s Algorithm is shown in essentially all elementary number theory textbooks when they talk about unique factorization. I appreciated the framing of the lesson here, and the underlying motivation, and think anyone teaching an introductory number theory class would find it a useful structure to draw from.

The science of complexity. This is a beautiful article featuring a few examples of studying interesting macroscopic behavior from simple microscopic rules. This would be a nice warmup into any stat mech class, and many of the demos could be pulled out by teachers to fit their specific lessons.

A few other personal favorites I’d like to share

As I said, the peer ranking gives a great starting place for anyone looking to discover new creators/explainers, with or without added commentary from me. Nevertheless, since I looked through quite a few entries, I can’t help but highlight a few I particularly enjoyed.

• Sphere is a loop of loops

• What is temperature

• The hidden law of crossing paths

• Drawing with perspective

This is a very small subset of those I could have brought up. Please do explore the others if you find yourself hungry for more.

• 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.

Looking back 10 years, looking forward 10 more

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.

And I love it!

It’s amazing how great it can feel to start a day with nothing on the calendar, no team to manage, no customers to interface with, no applications to review, and instead to become fully absorbed in writing, animating, and editing a video. The production is undoubtedly slower for it, but I’ve never minded that too much. Quantity has never been the goal.

I’m not naive, though, and I’ve been aware of how such a solo mode of creation grates against two important facts. 1) There is value in specialized labor and leveraging oneself. 2) The best ideas are often those born out of collaboration.

Growing a team is the “obvious” move. But in 10 years of making videos, I’ve seen many cases around me where a channel grows, and its production capacity grows, but something ineffable is lost. It’s something in that space between the old days when a typical YouTube channel was just some person making videos, and the modern diaspora of thousands of small media companies.

For all its potential inefficiencies, crafting a project from start to finish has always brought me great joy. My hope has been that each video feels like a personal project. It’s me speaking to you through the screen, not mediated by a production studio in between.

There’s nothing inherently wrong with high-output production studios; many of the greatest videos come from them. However, as a viewer, I find myself drawn to the channels that have remained very personal to the creator, and as a friend of many YouTubers, I’ve noticed how many of my colleagues in the space end up unfortunately stressed and exhausted by their jobs.

This is all to say, despite plenty of opportunities to grow the business and to grow a team over the years, it’s something I’ve very deliberately declined out of a clear view of how I like to spend my own working hours, and what kind of channel I want to produce.

That said, sometimes the “obvious” move is, well, the obvious move.

Reflecting on the last 10 years, and looking forward to the next 10, there is a lot I feel inspired to do. I would love to make many more “Essence of ____” style series that genuinely help students build intuition for their course. I have hundreds of topics that have accumulated on my wish list that I know would make fun and inspiring videos. I receive hundreds (thousands?) more via requests each year. Moreover, the character of videos has plenty of room to broaden in ways that would benefit from complementary skillsets, e.g. incorporating more of the human stories behind these topics with hand-drawn illustration.

I’m aware of my own limitations here, and I’d like the next chapter to involve putting together a small team to help chip away at all of this. I think I’ll always see the channel as a personal project, more than a company, and I’d like it to feel that way to viewers, too. With care, though, that’s not incompatible with adding a few more rowers in the boat.

“Great!” I hear some of you say, “Where do I apply?” If you’re interested in joining the creative side, e.g., animating, illustrating, or editing, hold your horses for right now. I will put out applications for those roles in a bit, hopefully in a month or two, but I don’t have the capacity this month to address them.

I’m sharing this post now for two reasons. First, there is one role I’m actively hiring for right now.

Hiring for general operations

I’m looking for someone to help manage and own projects within 3blue1brown, especially including those that are not directly part of core video creation. I’m open to a diverse range of backgrounds. The word “operations,” for all its vagueness, seems closest to what I have in mind, though similar roles elsewhere have fallen under names like “producer” or “general manager”. There is also a sprinkling of BD and strategy in the job, though I care less if that’s part of one’s previous work experience.

Here is an application giving more details.

What’s most important is that a candidate is aligned with the mission of 3blue1brown, stays on top of logistics, and can be strategic about shaping the scope and approach of a given project. A background in video production is not strongly necessary, but it helps. Likewise, for a background in math, and any experience coding. One part of the job, especially early on, will be helping to initiate and manage a recruitment-oriented business model for the channel.

And this brings us to my second request.

3b1b Talent

Many companies are interested in recruiting mathematically and technically talented people. The audience of 3blue1brown contains many such people. I know, because I meet them all the time, especially when visiting colleges, and I’m blown away. Some of them are just starting off their careers, or are open to changing jobs, and would find value in being exposed to opportunities with math-enthusiastic teams.

Next year, to support the growth I have in mind, I will experiment with a supplementary business model to help make these connections. The concept I have in mind is to have a simple page on my website that features a selection of partner companies whose teams would be compelling for a 3b1b audience member to join, collectively representing a diversity of job types and industry categories. It stands to be most fun if the partnered companies feature something contentful that audience members may enjoy, like a nerd-sniping technical puzzle representative of the kinds of problem-solving skills their team values.

At the end of the videos, I’d briefly highlight the page and its premise (a virtual career fair, basically) and encourage interested audience members to explore it. It’s important to me that such mentions be relatively light-touch and not influence the video content itself.

If you run a company or are part of a team that may be interested in joining the initial pilot for this concept, please reach out via this form.

It’s worth acknowledging that for many years now, I’ve declined brand integrations and sponsorships on the channel. In fact, a big part of the pitch for Patreon support was this lack of sponsorships, and the qualms I have about it as a business model.

This recruiting concept is, to an extent, a departure from that, and it would feel disingenuous for me not to call that out. Done right, though, I see a chance for a genuine win-win-win that sidesteps many of the misaligned incentives I take issue with.

I could write up a full post on why I think that, and perhaps upon launching it, I will. For now, I’ll spare you the full thought dump and instead simply tell you I’m thinking about it carefully and am receptive to critical feedback. If this helps connect viewers with roles they’d enjoy on teams they’d enjoy working with, and if it helps with some team expansion goals, it would be a shame not to try it on for size.

That’s all for now!
-Grant

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.

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”.

Next up, we’ll delve into the relationship between a derivative of a function and its Laplace Transform, which makes clear why it’s such a useful tool for differential equations. After that, we’ll step back and talk about how you could have reinvented the Laplace Transform for yourself, which walks us down a path exposing its relationship to Fourier transforms, as well as the formula for the inverse Laplace Transform. This will show a completely different way to understand how it breaks down functions as combinations of exponentials.

Stay tuned.

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.

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 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 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 entries, it would help me greatly if you took 60 seconds to fill out this form and let me know: https://forms.gle/jVssKAifNs3kdE9o9

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.

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

Full description

Apply here

For early experiments, we’re budgeting for five languages (Arabic, French, German, Spanish, Hindi). If you’re interested in helping out with other languages, feel free to apply, and if we have the capacity to hire more or coordinate volunteer efforts, we’ll let you know.

Other ways to help

Last year, my friend Ben Eater put together this site to help coordinate text translations for videos. YouTube used to have a feature like this, but it was discontinued. I’m deeply grateful to everyone who has helped submit translations there, it’s not only directly useful for subtitles, but it’s a helpful first step for any dubs.

AI?

In light of the recent AI boom, and the technically impressive results with text-to-voice, a number of companies, including YouTube, have been pushing AI dubs. Many of which will generate a voice that sounds like the original narrator. I went down a rabbit hole of experimenting with them last year. In fact, one of the original goals of the community translation site built with Eater was to take community-curated text translations and use them to quickly create auto-dubs.

My conclusion right now is that once you get past the initial wow factor, AI dubs are simply not that engaging. I would much rather the voice of the channel in Spanish (or any other language) be a real voice that conveys the feeling of a friendly math teacher than have it be something that vaguely sounds like me.

Also, in a world where an increasing percentage of content online is AI-generated, I think the emotional significance of landing on a video whose translated voice is both human and genuine is not to be overlooked.

I believe there’s plenty of room for software to help with the process, which can often get bogged down in the tedium of editing, and one goal is to experiment with tooling to help streamline the process. But for the time being, I’m leaning away from AI dubs themselves.