Grant Sanderson
@3blue1brown
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Pi creature caretaker. Contact/faq:
Joined October 2014
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Product idea: An app/website that's identical to YouTube in every way, except without any shorts.
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Fun fact, this is the number of milliseconds in a day.
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Remember that video about how block collisions can compute the digits of pi? A friend, Adam Brown, just showed that the math underlying this is actually identical to the math behind a very famous quantum search algorithm (Grover's):
Genuinely crazy!
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For the most recent video, I had way too much fun simulating the electric field (or rather, the component of that field responsible for radiation) and how it responds to an accelerating electric charge.
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Food for thought: There exists some smallest whole number which no human will ever think about.
I wonder how big it is.
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The birthday paradox is very famous in probability. If you take 23 people, there's about a 50/50 chance that two of them share a birthday. With 50 people, it's a 97% chance.
We could make many other fun examples to illustrate the same counterintuitive phenomenon (thread).
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I have to say, seeing so much of math/science Twitter come to the defense of this video, and the premise of rewarding inquisitiveness rather than mocking inaccuracies, has really made my evening.
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What is a convolution? What is a "Gaussian Kernel"?
(This was the main animation used at the intro of last week's convolutions lecture).
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Here's a short dynamics-based explanation of the equation e^(πi)=-1
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Polar plots of primes. Wait for it...
Can you guess what's going on? See the new video:
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The Fourier series video out!
In this animation, each vector rotates at a constant integer frequency. They're added together, tip to tail. The _only_ control you have is the starting position of each, and from that alone, they'll draw almost anything.
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With recorded COVID-19 cases (outside china) so eerily matching an exponential, I couldn't resist making a primer on exponential/logistic growth. At least 3 counterintuitive things about this kind of growth seem worth putting into the discussion.
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YouTubers often overestimate how much their followers actively await new videos. I bet most followers aren't even aware when it's been a while.
Still, for those of you who are, I want you to know I really am working quite hard on the next video(s?). All will come in due time.
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New video: Simulating an epidemic.
What happens when people avoid each other for the most part but still go to a common central location like a store?
What if you can track and isolate cases, but 20% slip through the cracks? 50%?
And much more.
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How discovery (and the retrospective telling of discovery) works.
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COVID-19 cases outside mainland china follow an exponential so closely it could be a literal textbook example for exponential growth.
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The next chapter about transformers is up on YouTube, digging into the attention mechanism:
The model works with vectors representing tokens (think words), and this is the mechanism that allows those vectors to take in meaning from context.
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Please be considerate to others and arrange yourself in a tetrahedron when in public.
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Take a computer from 1995 and one from today, which one is "old"?
Take the version of you from 1995 and the version from today, which one is "old"?
Some things age into the past, others into the future.
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Really excited that YouTube gave me the first 90 seconds of this year's Rewind to explain the volume formulas for higher dimensional spheres.
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Pretty clever trick. Celcius is the percentage around a semicircle, Fahrenheit gives the angle (plus 32).
This makes for handy intuitive conversions like
25°C = 32 + 45 = 77°F.
50°C = 32 + 90 = 122°F.
Even handier if you're comfortable rounding 32 to 30.
Between 0°C and 100°C are 180°F. This allows for a nice mnemonic to convert between those units using an angle as a guide for °F.
Made possible via
#manim
, thanks to
@3blue1brown
and
@manim_community
!
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I remember having way too much fun writing this back in 2017 when I made the Bitcoin explainer.
If someone describes a cryptographic protocol as having "256-bit security", what does that really mean?
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New video! Bayes' theorem, and making probability intuitive.
I had fun bringing in some Kahneman and Tversky results to show where human intuition seems to jive with probability, and where it doesn't.
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New video! If the answer here does not blow your mind, I don't know what will. In short, how does this number of collisions grow as you increase the mass ratio? E.g., what is it for 1 trillion to 1?
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I propose we adopt a convention of naming each new variant of covid after an airline. It's only fair.
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Wow, making Fourier series animations turns out to be super fun. Not exactly original at this point, but so lovely! Any shapes/figures you'd like to see show up in a video?
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Can you embed the Möbius strip into 3d space in such a way that its boundary forms a perfect circle?
Yes!
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Early impressions of
@OpenAI
's DALL-E 2. 🧵
All images below were produced by AI, with me feeding it the quoted prompt. I was most curious about how helpful such a tool might be in creative work.
"A sloth playing a guitar, photograph 35mm lens"
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Here's the gist of a beautiful argument Archimedes gave for the surface area of a sphere.
In the YouTube video I did on this, I also described another way to relate this area more directly to a sphere's shadow, which I'll copy on the thread below as a progression of puzzles.
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New video!
Raising e to a matrix, from the dynamics of love to Shrödinger's equation.
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Ah pi day. When non-Americans everywhere wonder what on earth pi has to do with 14.3.
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Here are the problem-solving tips I went through yesterday. Each may seem simple, but it’s shocking how often they help to get unstuck.
What would you add?
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Next time you find yourself writing the number 6, remember who to thank!
All the following symbols and notations were introduced by Gauss
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I'm helping to teach a course at MIT this semester with Alan Edelman, David Sanders, and James Schloss (
@LeiosOS
) that intends to blend math and scientific computing
Anyone in the world is free to follow along.
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I wonder how many hand-sanitizer-resistant bacteria we're all accelerating the evolution of these days.
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I'd like to tell you about a game/puzzle to help celebrate today.
We'll call it "Death's Dice".
(1/9)
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In math, you can sometimes prove a claim before fully understanding it.
But you cannot fully understand a claim without also being able to prove it.
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It's even more fun in 3D, albeit a bit busy.
For the most recent video, I had way too much fun simulating the electric field (or rather, the component of that field responsible for radiation) and how it responds to an accelerating electric charge.
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It’s here! Quaternions:
“But Grant, 30 minutes seems like a bit much to set aside right now.”
Pssht, don’t give me that, I saw you on Netflix last night.
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"Everyone's overreacting, the flu kills way more people each year. Besides, those of us who are young have a low risk even if we do get it."
Do you...hate the elderly?
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Notice the vertex markers, the beginning and ending triangles do indeed have the same vertices. So where do the two units of area go?
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What happens if you take a grid of 1,000,000 points centered in the complex plane, starting off in a 2π-by-2π box, and repeatedly apply the function z -> exp(z)?
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Me playing with simulations: What if I double the radius of infection?
...oh...oh god.
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Has anyone ever come across this way of visualizing the convolution of two functions f and g: Look at the graph of the two-variable function f(x)g(y), and consider diagonal slices over the line x + y = k.
The area of those slices represents (f * g)(k).
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Follow this man!
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My TEDx talk has now been posted.
Among other things, I share some thoughts on the question "when will I ever use this" as it comes up in math education.
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It begins! Differential equations chapter 1, studying the unsolvable:
If transitioning from the fast-pace of twitter scrolling to a 27-minute video is just not in the cards, please enjoy this strangely calming animation of the three body problem instead.
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New video!
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🟩🟩🟩🟩🟩
It's an exploration for writing a Wordle solver, with the challenge of not using the official list of Wordle answers (except as a test set), which is really just an excuse for an information theory lesson.
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For a brief moment in tomorrow's video, I wanted an ambiently wiggling loop.
Now I can't look away.
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New video!
This is easily the longest one I've ever made. When you find yourself in the mood to settle in with a good puzzle, and a story about two different problem-solving styles, I hope you enjoy it.
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In light of the very sad news about Ron Graham's passing, I thought I'd share an interesting tidbit about his famous constant which I only learned recently (thread).
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I can't quite tell if this is a joke, but I love it either way!
Created a kickstarter to make "From Scribble To Readable" a reality! Check it out and back us if you'd like :)
Kickstarter:
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The group of cube rotations is the same as the group of permutations on four objects (S4).
This feels very surprising (to me at least), but becomes more visible when you think of permuting the diagonals of a cube.
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I would love to see side-by-side graphs of recorded COVID-19 cases and Zoom's server costs.
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Now that the US Open has fully replaced line judges with robots, I propose they take things one step further.
Replace all the ball boys and ball girls with a crew of well-trained golden retrievers. Tennis viewership would go up by a factor of 3, guaranteed.
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New video! Why “probability of 0” does not mean “impossible”.
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I realized recently I didn't know why prisms work. Like, why does light slow down? And why would that depend on color?
It turns out, answering the first question with enough detail automatically answers the second, and it's completely delightful.
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Suppose that over the last few years, there's been an exponential rise in the amount of opium usage. It's especially prevalent among the youth. You've started to notice as you walk through airports and look at how people twiddle away their time, sneaking the occasional glance…
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My next lecture for MIT 18.S191 is up, about the seam carving algorithm. Which is actually quite cool!
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Dear aspiring YouTubers: Never, *ever*, name your channel after a piece of your own physiology.
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There's a famous proof for why π shows up in the Gaussian distribution, originally due to Poisson. Despite it being covered well on the internet, I found the prospect of animating it too fun to resist.
But! I wanted to go deeper.
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The puzzle below is a famous cautionary tale in math. But there's more to it than just coincidence, and on the YouTube version, we dig into what's really going on.
Full explanation:
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@waitbutwhy
Give a man a fire, and he'll be warm for the night.
Set a man on fire, and he'll be warm for the rest of his life.
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Need to create a fake chart that looks like a stock market graph? Happy to go with an overkill solution? Try taking one coordinate from some Brownian motion!
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I'm looking for an intern! See the application below.
This may be a little late in the season, but we're flexible for part-time if you have a main gig and still want to help.
Please share it with any undergrads who might be interested.
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A little while ago this integral pattern was making the Twitter rounds. It arose in a paper by Jonathan and David Borwein, whose colleague "concluded that there must be a bug in the software" when evaluating.
I made a video to explain what's going on
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I'm doing a TEDx talk at Berkeley this February. If you want to come, they gave one of those fancy discount codes for my followers (i.e. you): SPEAKER35
Hope to see you there!
Our first speaker is the incredible Grant Sanderson, founder of
@3blue1brown
, a math-based education Youtube channel with over two million subscribers and over 85 million total views! Sanderson is shaping education in today’s technological age, learn more at TEDxBerkeley 2020! 😍
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Yet another unexpected way that colliding blocks compute pi.
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After watching the "Simulating an Epidemic" video, a viewer (Prajwal DSouza) made this interactive, which is absolutely wonderful.
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Emails are funny.
"Dear 3blue1brown,
....<intro>...
I understand that you do not do sponsored videos. Would you be interested in doing a 30-second mention of our AI / Machine Learning / Data Science course?"
Do you understand? I don't think you do.
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I was playing around with shading a surface using multiple textures today and thought I'd share this.
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New problem-solving video coming out on Monday.
It's a long one!
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Pro tip: When
@nattyover
writes something, read it. You'll thank yourself later.
This article on Gödel's incompleteness theorem(s) is one of the most accessible coverages of the topic I've seen that actually captures the substance of the result.
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Thanks to the help of some interns, many of my videos have now been adapted into a written/interactive form. The clip below shows an example of one interactive elements made for the neural networks series by
@PullJosh
.
If you have a moment, take a look.
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"What's great about the heat equation is how well it behaves with sinusoidal func...oh wait...dear god..."
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Thing I love: This picture of 10-year-old Terence Tao with Paul Erdős.
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It seems so unnatural to graph periodic functions on a plane, rather than a cylinder since a circular input automatically forces the periodicity. So I made some Fourier series socks, graphing things *properly*.
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What are your favorite (potentially long) mathematician quotes? Bonus points if it's not just a pithy restatement of "math is beautiful", but something which changes how you think.
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New video!
Learn how asking the right question about Newton's method leads to a hidden Mandelbrot set, in the context of a general primer on holomorphic dynamics.
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Unexpected YouTube growth hack: Have ChatGPT analyze your thumbnail incorrectly and tweet about it.
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One neat difference between ChatGPT and previous iterations is its willingness to acknowledge uncertainty or absurdity (e.g. these are less of an issue now )
A funny quirk is it may be _too_ confident in its uncertainty, as in its take on Alice here.
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About a month ago when I published this video, recorded cases outside China were ~21k. In it, I mentioned that if you naively projected out the current trend, it'd mean hitting 1,000,000 cases by April 5th.
Just saying.
With recorded COVID-19 cases (outside china) so eerily matching an exponential, I couldn't resist making a primer on exponential/logistic growth. At least 3 counterintuitive things about this kind of growth seem worth putting into the discussion.
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I thought it might be fun to do a quick distraction project on the mathematically optimal way to play wordle.
Wrote a little program yesterday which uses a bank of ~13,000 words along with their frequencies in English, tried it today. Not bad!
🟨🟨⬜🟨⬜
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🟩🟩🟩🟩🟩
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"An army's effectiveness depends on its size, training, experience, and morale, and morale is worth more than any of the other factors combined."
- Napoleon
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