When do programmatic visuals help in understanding math? 3b1b, SIGGRAPH 2021 Featured Speaker

Featured speaker at SIGGRAPH 2021, on 11 August 2021. SIGGRAPH is the premier conference for computer graphics and interactive techniques worldwide and is a special interest group within ACM.
SIGGRAPH 2021: s2021.siggraph...
#SIGGRAPH2021 #ComputerGraphics
Double pendulum video:
• The Double Pendulum Fr...
Linear algebra series:
www.3blue1brow...
Quaternion interactive:
eater.net/quat...

[@GrantSanderson]

Hey folks, I did a streamed session at SIGGRAPH this year, which they've kindly given me permission to put up publically here. My apologies for the iffy sound quality, something about the mic setting and zoom recording seemed not to have meshed well.

[@mastershooter64]

oh niceeee

[@yash1152]

1:34 hey grant, the audio/vocals quality is good. just there's a li'l bit of noise in the bg. i guess u can remove that noise in post processing and change the audio of this upload itself. yes, youtube allows changing audios of an already uploaded video iirc.

[@yash1152]

31:29 actually, the noise and stuff is un-noticeable now.... so, yeah, good.

[@riddhiman2926]

Is there any way you can share on KZbin the lecture you gave to the IMO participants?

[@goldnutter412]

Going to guess the extra dimension is because you can move the object, translation as well as manipulation This is all very very cool and revealing :D

[@MasterHigure]

14:00 Printers those days were bad. Really bad. The attendants would often clean up printouts with a scalpel or similar. Mandelbrot had to very specifically instruct them not to do that with his prints because some random speck off to the side might actually be relevant. They didn't always listen.

[@chedzeesheeda1019]

Wholeheartedly agreed on the last point and so excited to hear it put into words. I went to a high school where we used coding for Wolfram to shape our work for multivar every time we started a new topic, and I loved it, then I went to an undergrad program where there was so little done in that way and it was almost scoffed at when I asked about tools like this that it just took the wind out of my sails. It makes me so happy any time I hear people with a math background validate the role that computing can play in developing an understanding of mathematics, especially when it lets you get your hands into the guts of the math and the systems of logic you're working with. This made my day.

[@michael-nef]

Before watching more, I definitely think your linear algebra series has been the most "actually educational" set of videos (at least for me). In my day-to-day I now always think of transformations in the way you teach them, determinants in the way you teach them, etc. For example, it's just visually and intuitively obvious why a matrix with two matching columns would have a determinant of zero (because their basis vectors just land on each other). Your other videos are great too, but I definitely feel there was something particularly good about the linear algebra series.

[@yash1152]

14:44

[@NehaKumari-tl1by]

Really that algebra series about it i always wonder how much i have been confused without knowing them

[@surprisedpikachu3782]

You are so right, one of his other two vieos that made me see math in a new light, was the "what does it feel like to invent math" and "math tattoos"

[@mito._]

That's how I learned linear Algebra too! Now I can easily visualize what I want to do with an equation, and how I need transforms to be oriented in some virtual space. :)

[@ishworshrestha3559]

Ok

[@JayTheYggdrasil]

That's a pretty big deal conference as far as I'm aware, awesome you got to speak at it! Very nice presentation as well :D

[@JoelRosenfeld]

Wow! SIGGRAPH! That’s amazing, Grant! Congratulations.

[@pendragon7600]

I like all your videos, but I'm a bit surprised that the essence of linear algebra has received the "most gratitude". Personally, the series didn't reveal any new concepts / ways of thinking about linear algebra; it just put a nice visual to a bunch of the stuff from a textbook. And as you said, there's nothing particularly special about those animations because they're a bunch of vectors and lines. the truly incredible part of your 3b1b channel in my opinion is the beautiful way you explain concepts as if we were going through the process of discovering them for the first time, coupled with unique/surprising animations.

[@yash1152]

4:28 chaos, linear algebra, quaternions 12:37 islands of stability

[@eccentricOrange]

I've been waiting for something like this, yay!!

[@tlijaniraed1599]

one word i love u grant so much i m a hugeeeee fannn and i love your series so much , i think you and khan are the best teacher in the earth from tunisia

[@johnchessant3012]

I need to know more about the islands of stability in the double pendulum graph!! Is it similar to the logistic map?

[@yash1152]

timestamps?

[@sibiakkash8229]

@@yash1152 12:00

[@etepeteseat7424]

They're regions surrounding points of equilibria, where the initial state is (for whatever specific reason in each separate case) near enough to a local point of stability that the evolving systems are kept near to the state of the true point of equilibrium as it evolves. At least, that's my suspicion, without having looked at the topic in detail. Likely, the values of the relevant equations there converge, or else diverge much more slowly, than the values for the surrounding chaotic regions. So in that sense, it is analogous behavior to the regions of stability in the logistic map, at the level of systems theory. A lot of chaotic and complex systems have such localized points of equilibria. An analogous concept that I've always found helpful in grasping that behavior is Pareto optimality, which you can get a good enough baseline understanding of for this purpose just from its Wikipedia article; though, as always, the topic can always be investigated further.

[@kennymclaren7900]

Building off what @Etepeteseat 7 said, intuitively, the equilibrium points are where the pendulum is directly vertical. So I'd guess the pockets are where the pendulum is nearly straight up and taking a while to fall and start behaving chaotically and the stripey part in the center (where both initial angles are small) is where the pendulum is nearly straight down and only swinging back and forth with a small amplitude.

[@PhDHugo]

I think this video was what I needed to start my learning in programmatic visuals.

[@yash1152]

56:08 yeah, throughout the whole part of double pendulum here, i was thinking about this video itself. it was really nice video.

[@kagan770]

How did you generate the graphics? What program used? Thanks a lot in advance.

[@ibrahimabushawish2839]

Are you still wondering?

[@imranq9241]

So in the end the person who learned the most math from 3b1b videos was ... 3b1b . It's the perfect crime

[@alwysrite]

Grant is just brilliant !

[@Aidiakapi]

@32:04 Video games usually use quaternions for most rotations, especially for animations. However, this can be caused by many other things, such as multiple constraints fighting to avoid "the wrong positions", but it obviously isn't working properly :).

[@Holobrine]

Quaternions make more sense as the even sub algebra of 3 space geometric algebra tbh

[@jdcrunchman999]

Fonts are very very small, In your next videos can you please use a larger font? I’ve only got an iPad which don’t allow zooming.

[@sophiabrown8573]

A picture is a thousand words only if you can draw it ~ Herb Gross

[@douglasalexander3218]

Weirdly, I found the yellow background on this video a bit distracting.

[@norude]

Are octonians useful to describe a 4d rotation?

[@rikhalder5708]

Can you tell what is best book for quarternion

[@r75shell]

how about another topic: when do programmatic visuals hurts in understanding math.

[@quocanhhbui8271]

The US education department should invest in Grant's works and pay for what he has contributed.

[@danfg7215]

The US education department should cease to exist, education would progress a lot more freely, competitively and cost-effectively without their self-serving agenda and massive waste of resources.

[@BrianAmedee]

Do you think that programmatic visuals will help with understanding subjects outside of maths/physics? Subjects such as chemistry, psychology, or aviation.

[@yash1152]

35:32 the quaternion part is reallly REALLY interesting to me

[@AlanWil2]

Cheers!!!

[@jeff9nomathematicsclass330]

My Gee I wish to chat up with you

[@arunoruto]

Will we get a face reveal at 5M subs? xD

[@VariantNode]

*takes bong hit* *nods along, pretending to understand* *gets multiple choice question right by accident* what

[@sebastiansimon7557]

Some timestamps, notes, and quotes: 00:00 Introduction of Grant Sanderson 01:03 Introduction to the Quiz 02:56 Grant’s KZbin channel Programmatic animations 04:28 Applications of programmatic animations Three examples: 04:52 1. Chaos Double pendulums, Sam Maksimovich 08:27 Quiz: Evolution of double pendulums What happens to the colored pixels in the grid in the double pendulum simulation? 11:18 Epitome of chaos What the evolved colored pixel grid shows is what chaos means: if you change the initial conditions slightly, you’ll get vastly different results 12:00 Islands of stability Surprising: Why are there patches that are less chaotic? 12:52 Case 1: Unexpected result This illustrates an important feature about programmatic visualizations: Something about the result is unexpected. No one would’ve thought to ask the question how e.g. the Mandelbrot set looked like before it was printed out. 14:20 2. Linear Algebra 16:31 Quiz: Matrix multiplication What effect does multiplying by the matrix [ [ 0, -1 ], [ 1, 0 ] ] have on a vector? 18:33 Linearity Why does it follow from the rotation behavior of two unit vectors, that the entire grid rotates? 21:38 Rotation 23:18 Case 2: Medium of explanation 24:33 3. Quaternions 24:55 Quiz: What is your level of familiarity / comfort with quaternions? 27:22 Blog with explorable videos Collaboration with Ben Eater. 28:39 Quaternions vs. Linear Algebra (Euler Angles) Quaternions offer an alternate approach of describing orientation in 3D space. 35:07 How something 4-dimensional is related to 3 dimensions 38:32 Similarity to complex numbers: rotation Interactive tool. 46:09 Quiz: Quaternion rotation Suppose we want to choose a quaternion q so that the function f(p) = q · p · q⁻¹ takes on the following values: f(i) = j; f(j) = k; f(k) = i. What should q be? 49:23 Projections of higher dimensions 51:34 Case 3: Testing your own hypotheses 51:55 Quiz: Most necessary for understanding math For which of these three examples do you think computer graphics are most necessary for understanding math? 54:45 How computer graphics help in understanding linear algebra How does computer graphics help in teaching linear algebra? - It provides a context where you’re required to use and understand it. 55:14 Visualizations help most when learner writes the code Example: WebGL version of double pendulum 57:08 Outro

[@yudoball]

Wow so detailed. Thx for the effort

[@littlenarwhal3914]

Choas

[@johnchessant3012]

That FIFA video is hilarious! Quaternions are awesome though so that was the only time I'll ever be happy that most people don't know about them

[@yash1152]

timestamps?

[@yash1152]

32:05

[@Kram1032]

32:33 so I'm not *dead sure* about what's going on there either, but couldn't this still be quaternions? Just accidentally going "the long way round"?

[@GrantSanderson]

I think you're right here, actually, that's a good correction. I'm trying to think of any tell tale signs about which of the two it might be, e.g. if it seems like part of the interpolation involves more rotation about one axis than another, but it's difficult to tell in this blurry image.

[@sethrenshaw8792]

@@GrantSanderson It looks like quaternion rotation to me. There doesn't seem to be the kind of squash and stretch you'd get from arbitrarily bad matrix or euler-angle interpolation. There's a talk by Casey Muratori from a few years ago, where he specifically talks about using the double-covering of quaternions to coerce quaternion lerping into always going the right way. It looks like they didn't do that here. I'm not a graphics expert per-se, but I am in the video-game space and I've spent some time looking into animation systems.

[@SimGunther]

25:41 Quaternions could be represented as a special case of rotors according to Marc ten Bosch

[@LoganCTanner]

Hey I convinced flammable maths to do a sultry calendar with stem people for a tbd stem fundraiser(s), will u plz dress up as a group of sexy Santas with me and Andrew Dotson for charity plz and thank u

[@MathTutor1]

Excellent work. Thank you.

[@Wouter10123]

10:10 I think you may have biased the answers there a little bit by introducing the topic as "Chaos".

[@JWentu]

3:21 "Without Fourier Transform, humankind would cease to exist" ... did I succeed? :)

[@imacds]

There is an artificial symmetry along the line a=b to the double pendulum visualization because the two pendulums are interchangeable. In a mathematical proof I am really tempted to write "Without loss of generality, let a be the lesser angled pendulum and b be the greater angled pendulum." In the visualization, I am tempted to cut the diagram in half along the line a=b and keep a half of the graph.

[@sunorcio3901]

I entered this video thinking it would be about visualizing maths, as in intuitively understanding maths in your mind, reiterating the logical pieces to create fractals and have a grasp of the possible ramifications of systems and its functions. Thought i would learn about my bias toward maths and programming. Anyway this is still an interesting video past my wishful thinking.

[@kagan770]

What is the discord server I can join to see the #programming-visualization... channel?

[@mdal-amin_ku]

Which software do you use for visualization?

[@fredg8328]

When you use quaternions everything magically works... Except when it doesn't work... I'm pretty sure that they use quaternions in FIFA. All modern game engines do. Everybody keeps saying that Euler angles are bad because of gimbal lock and that quaternions are perfect. But in reality quaternions have a lot of problems too.

[@LaplacianFourier]

This guy Grant is a treasure of humanity. Must protec at all costs.

[@adityakhedekar9669]

Hi, can you please create video on Singular Value decomposition for the linear algebra series

[@galenseilis5971]

The form qpq^-1 reminds me of the form of an inner automorphism.

[@yash1152]

timestamps?

[@98danielray]

thats just conjugation

[@subpopulations]

37:23 I think

[@ammyvl1]

what a time to be alive!

[@silverpharoah388]

Can u please make a video about all of ur books?

[@StormyDoru]

Really informative video. Thanks for giving this talk!

[@TheStarDreamer]

Wowww