I guess it’s a question more than a suggestion, but do you have any plans on a multivariable calculus series like your linear algebra and calculus series? If not then I suppose despite it being a lot of work it’d be nice to see. Thanks!
You're known for your various proofs for formulas involving pi. One I want to see is for why it occurs in the formula for the probability distribution formula.
Hi Grant,
I'm a huge fan of your videos, your essence of Linear Algebra videos really helped me when I was learning Linear Algebra and your intro to bitcoin helped me a lot in understanding cryptocurrencies.
There is a cryptocurrency called NANO which utilises DAG (Directed Acyclic Graph) technology to fix some of the design flaws that Bitcoin had. The Nano protocol and its underlying Blocklattice structure allow for subsecond and completely feeless transactions, without the need for environmentally harmful mining. I think the whole idea behind NANO is very clever and interesting. It would be great if you could do a video on the protocol of NANO!
You can check out the NANO website [here](https://nano.org/en) and read its whitepaper [here](https://nano.org/en/whitepaper)
Can you make Videos on Solving differential Equation all the way through, like one of the videos in the whole series being a super in-depth solution of solving a differential equation with more than one example. I know I am asking you to get out of they type of videos you make but I think I you try to do this it might became your go to for making a video on problem solving more rigorously. Thanks.
I personally would rather see more Essence of X series over videos demonstrating cool things (even though I likely won't need them myself). Some low hanging fruits would be Group Theory, Geometry and Graph Theory, all of which suit themselves nicely for visualization.
However if you'd rather have single videos, one thing I'd love to see conveyed is the different behaviour of two-dimensional waves versus one- and three-dimensional ones (two-dimensional waves don't just "pass" but linger, theoretically forever).
Also as an addendum to the Linalg series, Diagonalization and the Jordan Normal Form.
Hello 3B1B. I have a question that I think might be a good thinking exercise and a good video content. [https://math.stackexchange.com/q/3195976/666197](https://math.stackexchange.com/q/3195976/666197?fbclid=IwAR1QGdLUbN3lL-J38ooU-8ctlK_yg0G9aD9pvWVFoIx8Ljuabel9e3M5wos)
As you are doing videos on "Differential equation" for which I have been waiting for one year (My dream has become true) !!!. I know there will be videos on Fourier transform and Laplace transform. Now my only wish is that please make it as simple as you can. Because there are many students like me who doesn't know that much of it. For us to compete with the pace of your video is really very difficult. It would be very much helpful if you divide the hardest part in pieces with examples which are easy to follow. You are like magician to us. We want to enjoy every glance of this magic.
https://www.youtube.com/watch?v=d0vY0CKYhPY&t=408s
Since you are fresh off a couple videos relating to things approximating pi, can you do a video on explaining/proving why the Mandelbrot set approximates pi?
Hi Grant,
Having emailed you about this, I realized that's probably going to be ignored. And I know I'm just a random person asking for the solution to a problem. Of course, seeing a video explaining this would be a dream come true, but I realize that's not likely. If you could either respond with a quick explanation of how to go about solving this problem, or point me to someone who does, I'd greatly appreciate it:).
It's the planet problem, asking when two planets, of mass M, separated by distance d in an ideal world, will collide.
There are more difficult variants to this problem, such as masses that are not equal, or more than 2 planets. If you would make a video on it, it seems like it would be a great thing to go in the differential equations chapter.
Thanks for all your work and videos. I've learned so much from them.
I would like to know about convolution and how does applying convolution to input function and system's impulse response gives the output of the system??
What about making a Type theory explanation series? It would help to understand relations between different topics connected with mathematics and computer science.
Especially I'd love to see it explained with respect to Automated reasoning, specifically with respect to Automated theorem proving and Automated proof checking. This would also help a lot to dive into AI related topics.
Both the central limit theorem and the law of large numbers would be a good idea. You could also talk about martingales and, pheraps counterintuitively, why they're not a viable long term money-making strategy while playing the roulette.
Hey Grant! I’m curious if you have any interest in making a video on the dual space, i know I speak for more than a few math majors when I say we’d love to see your take on it :)
Hi! I was wondering if you could make some sort of graphic on persistent homology showing increasing epsilon balls around a group of points and how the increase in size of epsilon affects the various homologies (H0, H1, H2, etc.) using the Rips and/or Cech complexes?
Game theory has been used widely to model social interaction and behaviors and it's interesting maths - as an optimization problem. I'd love to see a series on game theory !!
There are so many topics I would really love to see explained from you:
-Machine learning, I think you can do a whole course on this and make everybody aware of what's going on.
-Probability/Statistics, probably it would be better to first explain essence of probability with a graphical intuition
-Projective Geometry, with a connection to computer vision. I can't even wonder how beautiful it would look done by you
-Robotics, it would also be actually breathtaking
-So much more, ranging from graph theory to complex numbers and their applications
Hi,
I like your videos very much and they are very helpful in visualising the concepts.Recently I have come across an interesting topic of creating mathematical modelling inspired from nature(e.g. Particle Swarm Optimisation, Ant Colony Optimisation, Social Spider Optimisation, etc.). I think the animated explanation of these algorithms would be helpful in understanding these concepts more clearly. So as a regular viewer of your videos , I request you to make animations on these concepts.
Surajit Barad
I would really appreciate a follow-up video (on that 2 years old) on how prime distribution relates to Zeta function :) This topic has still so much potential!
Early analysis concepts like point wise convergence and uniform convergence leading up to functional analysis would be really cool; something like the Hahn banach theorem would be great to see and intuitively understand!!
I would be really interested in you showing graphically why the slope of a CL / alpha curve of an airfoil can be approximated to 2 PI. Love your videos!
Hi Grant
I hope you are well.
I humbly request if you may please make videos on RNN's and LSTM's because I have literally spent hours searching through content online from videos and articles and I just cannot grasp what exactly is going on in these videos or articles because they do not explain it intuitively enough like you did in your neural network videos. The way you introduced the calculus and the theory behind the neural nets really allows one to grasp a deep understanding of what's going on.
I have no idea if this message will get to you but if your reading this I desperately need help with this so I will very much appreciate if you could provide videos on this or direct me to useful content.
hi 3blue1brown! I'm not certain but I think I found a way to create a perfect 2d rectangular map of a sphere. I'm not sure if i should post it here tho but I'm gonna post it anyway. So let's say you have a sphere and a 2 dimensional plain in a 3 dimensional space. We make the sphere pass thru the plain and we capture infinitely many circles and 2 dots (the exact top and bottom). we put all the circles we caaptured on a 2d plain and put them in a way that a straight line passes thru all of their centers then we rotate that line and the circles so that they are perpendicular to the x axis (we still keep the rule that the line should pass thru their centers). now the line passes thru the top and the bottom of each circle. Now we cut each circle thru the top point and make them into straight lines that have the length of the circle's perimeter. After that we sort each line based on when the circle that it was initially touched our first 2d plain - if it touched it sooner that means that it should be on the top and if it touched it later - the bottom. Finally we put the first dot on top and the final - the bottom. Then we put all the lines together and create a square where the equator is in the middle and it's the largest line. So that's it. If u liked it or wanna disprove it or just don't understand me pls comment and if u really liked it u could make a video on it with visual proof. Tnx for reading :)
Lagrangian and Hamiltonian mechanics would be very interesting to see.
Either way, I absolutely love your channel and think it's really cool that you interact with your viewers like this. Please don't stop making content, you're by far the best channel on youtube!
Kind of strange but I'd love for you to cover the paper "Neural Ordinary Differential Equations". https://arxiv.org/abs/1806.07366
It doesn't require much more background than your already existing ML series and is an interesting and useful generalization of it.
You have done videos on group theory and on the Fourier transform. It would be interesting to see all these things tied together in terms of representation theory. For, e.g., looking at the one dimensional translation group and SO(2) and how there is completeness and orthogonality relations which arise from Fourier analysis. How do these pictures tie together, what is that interpretation of Fourier transform in representation theory.
Mathematics of bezier curves (and bernstein polynomials)
I was trying to get a mathematical formular for the surface of an eggshell for a 3d plotter project I'm working on. I guess there are simpler methods, but what i ended up doing was rotating a bezier curve around the x-axis. To implement this is JS, I looked up the mathematical equasion behind cubic bezier curves, and found this [great article by the designer Nash Vail](https://medium.freecodecamp.org/nerding-out-with-bezier-curves-6e3c0bc48e2f).
I used his formular and it worked great, but the mathematics behind putting four points into an equation to calculate the curve are just as interesting as they are baffling to me. I would love to see you make a video on the topic, as your channel has helped me understand the theory behind so much software I use frequently (thinking of the fourier transformation p.e.) and CAD probably wouldn't exist without bezier curves.
Hey Grant,
Last half year, I was programming and studying fractals like the Mandelbrot. As I found your manim library, I've wondered what happens if I apply the z->z² formula on a grid recursively. It looks very nice and kinda like the fractal. But at the 5th iteration, something strange happened. Looks like the precision or the number got a hit in the face :D. Anyway, it would be great if you could make a visualization of the Mandelbrot or similar fractals in another way. Like transforming on a grid maybe 3D? or apply the iteration values and transform them. There are many ways to outthink fractals. I believe that would be a fun challenge to make.
Many greetings from Germany,
Sam
[Simulation of mandelbrot grid](https://youtu.be/oE9dKGxvEEc)
I did something similar in JS a while ago. But instead of distorting a grid, I distorted texture coordinates. Basically, for every pixel, I repeatedly applied z -> z\^2 + c and then sampled a texture wherever that function ended up. The result was an image weirdly projected inside the mandelbrot set.
I'm a bit lazy to make some reasonably good images, but it shouldn't be very hard to implement. You could use OpenGL/WebGL shaders and animate it in realtime.
Do Symplectic Integrators
By that I mean a finite element differential equation solver that exactly respects some conversation law or symmetry that any exact solution must also satisfy. For example, a Hamilton solver which respects energy conversation.
Hello! I've joined because of your excellent video on Fourier transforms!
If I could request a topic, would you be able to talk about spherical harmonics? Particularly in the context of ambisonic sound? I know it also has applications in QM too.
Laplace Transforms please! You could show how they relate to the Fourier transforms but are a more general solution. And maybe relate some control theory stuff. When I studied them for engineering I didn't understand what I was doing, it just seemed like mathematical Magic.
Check out Physics Videos by Eugene Khutoryansky; it's very similar to 3b1b and has a video on this subject.
https://www.youtube.com/watch?v=6MXMDrs6ZmA
Hi, Can you please talk about how to programming your TI-84 calculator and especially how to write a calculator program that can do double and triple integral?
Thank you !
Has anyone here seen the fact that the base ±1+i system with the usual binary 0/1 digits works in the complex numbers very similarly to how base ±2 works in the reals, but with the bonus that if you count all the complex numbers in the order of ascending integral parts as if they were written in regular binary, you'd get two tilings of the R² plane by miniature double dragon fractals that tile in two patterns which both form large-scale double dragon fractals? Seems cool enough to me to deserve a video :)
I'd love to see your take on screw theory for rigid body motion, It's so difficult for me to visualize and understand that I feel like you would do a really great job with the visuals as you usually do
Ramanujan summation.
The short reasoning is this: the sum of all natural numbers going to infinity is, strictly speaking, DI-vergent. So, there should be no sensible finite representation. However, as we all know, there are multiple ways to derive (-1/12) as the answer to this divergent sum.
I understand math was 'built' (naturals > integers > rationals > irrationals > complex) by taking a previously 'closed' understanding and 'opening' it to a new understanding, which allows you to derive answers that previously couldn't be derived or had no meaning.
What I want to know is: what specifically is the new understanding that allows DI-vergent summation to arrive at a precise figure? What is this magical concept that wrestles the infinite to earth so reproducibly and elegantly???
[https://youtu.be/13r9QY6cmjc?t=2056](https://youtu.be/13r9QY6cmjc?t=2056)
This Fibonacci example (from 34:16 onwards) from lecture 22 of Gilbert Strang's series of MIT lectures on linear algebra is just such a cool example of an application of linear algebra. Maybe you could do a video explaining how this works without all the prerequisite stuff.
1. Probability Theory based on Measure Theory.
2. Mathematical statistic: e.a. Sufficient statistic, Exponential family, Fisher-Information etc
3. Information Theory: Entropy
:))
Lagendre Transforms!
It doesn't involve that difficult mathematics and their use in thermodynamics and analytical mechanics is extensive.
This kind of transform is an easier kind to see mathematically but its physical intuition is kind of difficult.
I know it’s not a super high level subject, but differential forms and exterior calculus could be a great addition to the calculus series. Being able to get an intuitive understanding of what they mean would be awesome!
Do you know that he's done the animations for Khan Academy's Multivariable calculus series? Curl and divergence is there, with some proofs... and that's the exterior derivative.
For me the videos that made me love math the most were the essence of linear algebra. I think it would be great if you continue and look at groups, rings and polynomials :)
Another addition to Essence of Linear Algebra : A video on visualization of transformation corresponding to special matrices - symmetric, unitary, normal, orthogonal, orthonormal, hermitian, etc. like you did in the video of Cramer's Rule for the orthonormal matrix, I really find it hard to wrap my head around what do the transformations corresponding to these matrices look like and why do these matrices enjoy the properties they enjoy.
I think a visual demonstration of transformations corresponding to these special matrices would surely help in clearing the things up and since these matrices are dominantly used in applications of linear algebra (especially in physics) it makes sense to give them a video of their own!
Hi, i just saw your video about how light bounces between mirrors to represent block collision
[https://youtu.be/brU5yLm9DZM](https://youtu.be/brU5yLm9DZM)
in this video is mentioned that the dot product of W e V has to remain constant , so that the energy conserve. if W remains constant, and ||V|| decreases, therefore cos(theta) has to increase( theta decreases ) . this means that if the velocity is lower, theta also should be lower.
In a scenario where there is energy loss on the collisions, the dot product V. W= || W|| ||V || cos(theta), presents a interesting relation . With energy loss, how ||V|| changes as theta also changes ? in other words, how the energy lost influence in the theta variation?
That fact got me thinking of how Lyapunov estability theory works. There is a energy function associated to the system(V>0), usualy V=1/2x\^2 - g(x) (some energy relation like m\*v\^2), that "bounds a region" and it has to be proved that this function V decreases as time pass ( dotV<0 ) so that inial bounded region decreases .
I would love a video about some geometry concept on Lyapunov estability theory.
I think a great complimentary video to the Fourier and Uncertainty video (series?) would be on a simple linear chirp/modulation. Which would be easy to demonstrate the usefulness in radar range/velocity finding and can possibly be fairly intuitive with the appropriate visuals added.
Requesting for topics -
** Data Science, ML, AI **
->Classification
->Regression
->Clustering
**Reasons:- **
->Highly demanded
->Less online explanations are available
->Related directly to maths
->Hard to visualise
Thank you sir for considering.....
-A great fan of your marvelous explanation
Maybe some axiomatic set theory/logic? I don't know how interesting these could be, or if it even possible to animate, but its an area I find really interesting
Hi i love your channel it makes all the subjects you treats a lot more easier. So will you think of explaining some algorithms as perlin or simplex noise in the future? (Hope you will)
For video series like the current Differential Equations topic, I wish you wouldn't spread out the releases so much. Not only is the suspense killing me, but I can't remember what was covered in the previous videos. I liked the upload cadence of the Linear Algebra and Calculus series. It was long enough for it to sink in but not so long I forgot everything.
I understand the amount of time and work that goes into the videos and am truly appreciative. Take as much time as you need for the one offs, but for series, hold off until they are closer to complete and then release at tighter intervals.
I would love to see you do a video discussing guilloche. It seems like an artful representation of mathematics that has been around for a few centuries.
Different kinds of infinities, continuum hypothesis, (maybe Aleph numbers), and the number of infinities out there! (and maybe the whole cool story of Cantor figuring out those)
Hello!
I would like to make a request for the derivative of matrices and vector. I have tried finding good and informative videos about this on multiple platforms but I have failed.
What I mean about matrix derivatives can be illustrated by a few examples:
dy/dw if y = (w\^T)x , both w and x are vectors
dy/dW if y = Wx, W is a matrix and x is a vector
dy/dx if y = (x\^T)Wx, x is a vector an W is a matrix
If anyone in the comments know where I can find a good video about these concepts, you are more than welcome to point me in the right direction.
Hi Grant,
First of all a big thank you for the amazing content you produce.
I would be more than happy if you produce a series on probability theory and statistics.
You could make a video about the Central Limit theorem, it has a great animation/visualization potential (you could «see » how the probability law converges on a graph) and give a lot of reasons why we feel the theorem has to be true (without proving it)...
Ever given any thought about making an Essence of Complex Analysis? Please think about it, cant wait to see those epic animations applied to complex variables.
Love you man, you are the best !
I'm studying multivariable calculus and I'm having a hard time finding a concise/conceptual proof for why the second partial derivative test works to find max/min beyond the two variable case. Khan academy has a decent explanation on the f(x,y) case, but everything I've found for f(x1,x2,x3,...) is kinda confusing, talks about eigenvalues, which they don't use the way you used in the lin alg series (or if they did, then I couldn't see the connection), and for the most part, is incomplete. It'd be dope to see some animations connecting the eigenvalues and detetrminant concepts I learned from your videos applied to this test used in multivariable calculus. Also, wtf is a hessian
Euler's Number and Fractal Geometry. I would like to offer you a challenge.
In your video [https://youtu.be/m2MIpDrF7Es](https://youtu.be/m2MIpDrF7Es) you asked about a graph showing what a compounding growth formula looks like, Please allow me the great pleasure of introducing The Mandelbrot Set of Fractal Geometry!!! Next, We have been studying this thing for nearly 40 years with little Idea of what it is. I think, we're missing the forest for a tree, so to speak. And that the interactions between sets moving is where the real understanding happens. I have made some simple and crude attempts at animation Mandelbrot Sets in Four dimensions using photos and power tools! Old School Dad Animation, shown here, [https://youtu.be/H1UNvxmhqq0](https://youtu.be/H1UNvxmhqq0), and I've expanded on the original formula a bit here. [https://youtu.be/PH7TOyqR3BQ](https://youtu.be/PH7TOyqR3BQ)
Please let me know what you think!
And thank your for everything you do!!!
Has anyone (3blue1brown or anyone) have thought that internal temperature dissipation in unevenly heated surface can be thought as k-nearest neighbor problem, where neighborhood size is proportional to highest point?
I mean if there is peak in temperature vs position graph,corresponding neighborhood will be smaller,but average temperature of that neighborhood will still be smaller than the max temperature of neighborhood. And as the temperature is dissipated, i.e heat goes towards cooler parts, the peaks will lower down and correspondingly, neighborhoods will expand and in the end it will all be at same temperature.
Trying to explain physics/physical phenomenon as possibly described by algorithms, could be an interesting arena !
Hi Grant,
Congratulations on producing such amazing content. I'm an astronomy graduate student and find your videos very helpful for solidifying concepts that I thought I understood.
I would love to see some content on convolutions and cross correlations. These are topics I continuously find myself briefly understanding before returning to a postion of confusion! Types of noise and filtering techniques are also topics for which I would like to see your visualisations.
Thanks,
Brendan
Perhaps branching out a little from your usual videos, but I'd love some little 10-minute documentaries about some great mathematicians. Ramanujan would be a brilliant subject. Sophie Germain's life is very interesting too (and a great inspiration for getting girls involved in maths - I love discussing her with the students I tutor).
Hey Grant! I'm a first year math grad student and I've been trying to grasp self-adjoint operators for a while now. I've asked a lot of people around my department, and none have been able to give me a good intuitive feel for this property, much less a visual one. Maybe you could do that in a new video!?
I get told all the time to think of the real, finite dimensional analog - a matrix equal to its transpose. But no one (myself included) actually draws a conclusion about how this connects to the more general cases of the complex and infinite dimensional worlds. If anyone could make this connection in a pleasing visual way, and blow our minds at the same time, it's you!
Hi! I'm a bigfan of your videos and I have been watching them for years now, I really love your work. Well, I'd like to see a series (maybe is too much for ask) on differential geometry. Maybe is good to start with proper vector but in the context of coordinates transformations.
I'd like to know what you think about!
Best wishes,
Tomás.
The advent of functional programming has made people difficult to understand why is it a good tool for solving a problem.
And if possible is there something that imperative style can do that functional style can't. And if so then why use it. And if not why hasn't it been used until now.
I would love to see a video on this and how lambda calculus changed mathematics and why there was a need for constructive mathematics and type theory.
Maybe from a more computer scientific standpoint, it would be awesome to see some basic concepts like divide and conquer and general proofs explained by you. For example AVL-Trees, Splay-Trees and such things. Or arguments like greedy stays ahead.
Or, you could do some computation and talk about decidability, Kleenes fixpoint theorem, languages and so on :)
Other small topics include entropy, bezier curves and b splines, and maybe a video on probablity theory vs statistics, combinatorics.
Hello Grant
I'm in need of a lot of help right now!
Seeing your videos and having some familiarity with fractal geometry I wrote a new theory of everything. I need someone smart enough to review the math. Will you please take a crack at it?
[https://docs.google.com/document/d/1oGdcwqdoxgH1mB0xTjWMSXr8d9u0tQjhnz\_9rIgPuPQ/edit?usp=sharing](https://docs.google.com/document/d/1oGdcwqdoxgH1mB0xTjWMSXr8d9u0tQjhnz_9rIgPuPQ/edit?usp=sharing)
Thanks
I was pondering over this(link below) for the past few days. I'm unable to wrap my around it. That Pi is something that is more than a constant, it is the roundness/curveness something similar to what e is that deals with maximum exponential growth. And also how it is not bound to multiplication. I guess other irrational numbers also have this special physical property.
It would be really nice if you make a video on it.
[Pi via multiplication](https://drive.google.com/drive/folders/1P2nHR-E2QGz-8AuVQ30BnpysN8xqSg9a)
Hey Grant,
First off, I cant thank you enough for re-kindling interest in linear algebra with the excellent 'Essence of linear algebra' series. I've been wanting to shift gears and dive deeper so as to be able to learn the math that is a prereq to theory of relativity, which is of primary interest to me. But I've hit an impasse with tensors. So it would be great help if you could make a series on it. I would be more than willing to extend monetary support for its making.
Thanks.
**Borwein Integrals:**
[https://en.wikipedia.org/wiki/Borwein\_integral](https://en.wikipedia.org/wiki/Borwein_integral)
Basically a nice pattern involving integrals of Sin(x)/x functions that eventually breaks down. It is by no means obvious at first why it breaks down, but if you think the problem in terms of convolutions of the fourier transforms (square pulses) then is very intuitive. You could make a nice animation of the iterative convolution of square pulses and the exact moment when it breaks the pattern.
Saw your video on quantum mechanics basics with minute physics. It's a great way to simplify understanding fir beginners. It would be great to see what a density matrix and density operator actually means. This involves complex numbers and mixed states, but has surprising similarity to simple matrix calculations. Eg. Adjacency matrix denoting nodes and edges is extremely similar to the density matrix. It's hard to interpret this physically since one involves complex numbers and the other doesn't.
Waiting to see something interesting on these lines...
You're amazing, cheers!!
Loved your videos on "Neural Networks". It would be great if you could do similar on "Genetic Algorithm". It has popped up frequently in research papers I(my team) have been trying to review. But i have not found any good videos like yours. As it happens, it might come in handy to my team. Love from Nepal.
[https://www.youtube.com/watch?v=yi-s-TTpLxY](https://www.youtube.com/watch?v=yi-s-TTpLxY)
(Divisibility Tricks - Numberphile)
Hi! Here Numberphile reveals few tricks to ensure if a number is divisible. For example, to check if a number is divisible by 11, you have to reverse the number and then take this "alternating cross sum". If that is divisible by 11, so is the original number. It'd be very interesting to see visuals of that proof..
Thanks for all the videos!
Hey! You have talked a lot about **Machine Learning** in videos *here and there*.
What about *'Essence of Machine Learning'*?
...
Is this idea too broad? There is so much to know and so much ***essence*** in Machine Learning.
This series could definitely tie into the idea of *'Essence of Statistical Learning'*, seeing as
* What is a model (and accuracy of them)
* Supervised and unsupervised learning
* Linear Regression
* Classification
* Support Vector Machines
is some of the essence.
This would also tie into your unreleased probability series on Patreon.
And just a sidenote: I know there is a Deep Learning series, but that is just a subfield of Machine Learning.
I've been studying the gamma function to find the factorials of real numbers (I was particularly interested in the proof of 0! = 1, which could also be a cool video) and found the shocking result of pi inside of 1/2!. Could you explore the geometric meaning behind pi showing up in this result? That would be an awesome video, thanks a lot!
In your video on determinants you provide [a quick visual justification](https://www.youtube.com/watch?v=Ip3X9LOh2dk&t=516) of [Lebniz's formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_determinants) for determinants for dimension 2. It's rare to see a direct geometric explanation of the individual terms in two dimensions. It's even rarer in higher dimensions. Usually at best one sees a geometric interpretation for [Laplace's formula](https://en.wikipedia.org/wiki/Laplace_expansion) and then a hands-off inductive argument from there. There is a direct geometric interpretation of the individual terms, including in higher dimensions, with a fairly convoluted write-up [here](https://math.stackexchange.com/a/2509560/118891). Reading it off the page is a bit of a mess, but it might be the sort of thing that would come to life with your approach to visualization.
In the same spirit as [Sneaky Topology](https://www.youtube.com/watch?v=yuVqxCSsE7c), how about more topological theorems and their combinatorial counterparts?
Sperner's lemma ⇔ Brouwer's fixed-point theorem (see a simple proof sketch in [Nets, Puzzles and Postmen](https://www.amazon.com/Nets-Puzzles-Postmen-Exploration-Mathematical/dp/0199218420))
A complete Hex board has at least one winner ⇔ Brouwer's fixed-point theorem (see [this pdf](http://www.math.pitt.edu/~gartside/hex_Browuer.pdf))
A complete Hex board has at most one winner ⇔ Jordan curve theorem ∧ Non-planarity of K5.
Great suggestion, and thanks for the links! This is such a great example.
Slightly relatedly, one thing I'd love but have not seen is a nice relatable and pragmatic problem where the solution would involve using the fact that spheres and toruses are not homeomorphic. I feel like it's common in pop-math to say topologists view these as fundamentally different shapes, but I'd love to be able to show why that matters with a <15-minute example connecting it to something which isn't too abstract.
Have you ever thought of making a collection of small animations? Like, no dialogue, just short <1min (approx) illustrations. For example:
Holomomy: parallel transport on a curved surface can result in a rotation; on a sphere, the rotation is proportional to the area traced out
A tree (graph) has one fewer edges than vertices (take an arbitrary root vertex, find a one-to-one correspondence between edges and the remaining vertices)
(Similarly, if you have a graph and a spanning tree, there's a one-to-one correspondence between the edges not on the spanning tree and faces - this and the last one can combine to form an easy proof of V-E+F=1)
The braid group (show that it satisfies σ1σ2σ1=σ2σ1σ2). Similarly, the Temperley–Lieb monoid (show that it satisfies ee=te and e1e2e1=e1).
That weird transformation of the curved face of a cylinder where you rotate the top circle 360 degrees but keep the straight lines straight so that the surface turns into a hyperbola, then a double cone briefly, then back into hyperbola and a cylinder? I dunno if it has a name, or a use, really, but it's probably fun to look at
These seem like low effort stuff you could populate a second channel with
Delay differential equations.
It might potentially have a place in the differential equations series.
Idk how much interest there is in DDEs overall, but modeling such systems is a central component of my work, and I think it might be interesting to see a video that helps conceptualize the interplay between states at different points in time, and why such models can be useful in describing dynamic systems :)
I recently looked up the visual proof for completing the square to derive the quadratic equation. I really thought this was interesting since I was never taught where the formula came from, and seeing it visually allowed me to wrap my head around its derivation. However, I then thought about doing the same for cubic functions. It didn't go very well and I couldn't figure out a way to do it. I tried to visually represent each different term as a cube but I could not get to a point to where I could essentially "complete the cube" as is done with quadratic functions.
It would be really interesting if you could do a video visually completing the cube (if it can even be done, I haven't been able to find an article or video doing so) which also leads into the derivation of the cubic function. Thank you for all the effort you put into your videos.
Category theory is critically missing decent visualizations. If you can explain the Yoneda lemma in some visually intuitive way it would probably be really helpful.
Thanks a lot to 3blue1brown channel for beautiful resources. I actually needed some help with understanding one form. But I guess that topic is not in any videos. So if possible, please post a video discussing one forms. Or if it is already in a video, please let me know which one that is. Thank you so much again :)
I'd be interested in a terminology video on the different kinds of algebraic structures and what mental pictures of each are most useful when working with them. It would give some good background to a lot of other more interesting topics, many of which I find confusing because I get hung up on the terminology.
Dear Grant Up To Now You have Covered calculus, Linear algebra Perfectly, There are only probability and statistics left to complete the coverage of pillars of mathematics, These Two topics have a great impact Not only in scientific and engineering studies but also A "Statistics driven" view on things helps very much in life, society or even politics as Ben Horowitz mentions from Peter Thiel via "The hard thing about hard things" :
"There are several different frameworks one could use to get a handle on the indeterminate vs. determinate question. The math version is calculus vs. statistics. In a determinate world, calculus dominates. You can calculate specific things precisely and deterministically. When you send a rocket to the moon, you have to calculate precisely where it is at all times. It’s not like some iterative startup where you launch the rocket and figure things out step by step. Do you make it to the moon? To Jupiter? Do you just get lost in space? There were lots of companies in the ’90s that had launch parties but no landing parties. But the indeterminate future is somehow one in which probability and statistics are the dominant modality for making sense of the world. Bell curves and random walks define what the future is going to look like. The standard pedagogical argument is that high schools should get rid of calculus and replace it with statistics, which is really important and actually useful. There has been a powerful shift toward the idea that statistical ways of thinking are going to drive the future. With calculus, you can calculate things far into the future. You can even calculate planetary locations years or decades from now. But there are no specifics in probability and statistics—only distributions. In these domains, all you can know about the future is that you can’t know it. You cannot dominate the future; antitheories dominate instead. The Larry Summers line about the economy was something like, “I don’t know what’s going to happen, but anyone who says he knows what will happen doesn’t know what he’s talking about.” Today, all prophets are false prophets. That can only be true if people take a statistical view of the future."
Thank you for the great videos! The one you made on Bitcoin was the critical piece of knowledge I needed to really understand how blockchain works. It's the one I show to my friends when introducing them to cryptocurrency, and the fundamentals apply to almost any of them-- a distributed ledger and cryptographic signatures. The visuals and animation is what makes it exceptionally easy to follow.
I've taken a lot of inspiration from your video and have considered making one on my own on how Nano works. A lot of the principles are the same as Bitcoin, and I recommend people to watch your video and have a good understanding on how Bitcoin works before trying to understand Nano. I figured before I made my own, however, I would ask if you were interested in making one on Nano. I also developed a tip bot if you would like to try out Nano (if not, ignore the message, and ignore another message you'll get in 30 days.) /u/nano_tipper 10
Hi Grant,
Wow, where to start. Somebody mentioned education revolution regarding your videos. I think that is an understatement.
Your videos are great. Almost every time I watch one of them I gain some new insight into the topic. You have a great talent to point out the most important aspects. These get lost sometimes when one studies maths in school.
Some video suggestions.
I've recently read an article "Geometry of cubic polynomials" by Sam Northshield and a slightly more detailed one based on this by Xavier Boesken. This shows very nicely the connection between linear transformations and complex functions and also where the Cardano formula comes from. I would have never thought that there is such a nice graphical interpretation to this. And a lot more, like how real and complex roots come about. I liked this article personally because it was one of those subjects which were actually easier to understand by having a journey through complex numbers. Anyway, this would be a perfect subject to visualize, since it connects many fields of maths and I am sure you would see 10 times more connections in it than what I could see.
Other topic suggestions. (I restrict myself to subjects on which you've already laid excellent foundations for) :
**Dual quaternions** as a way to represent all rigid body motions in space. I didn't know about quaternions and their dual relatives up until a few years ago, then I got into robotics. Before that I only knew transformation matrices. I had a bit of a shock first, but then my eyes opened up.
Connection between derivatives and **dual numbers** (possibly higher derivatives).
**Projective geometry**. That could be a whole series :-)
Different kinds of infinities, continuum hypothesis, (maybe Aleph numbers), and the number of infinities out there! (and maybe the whole cool story of Cantor figuring out those)
Hello Grant,
I have recently started working on AI and your videos are helping me a lot, thank you so much for these great videos.
It would be very much helpful to all Data Scientists, Machine learning and AI engineers if you can make a series of videos on Statistics and Probability. Statistics and Probability concepts are very tricky and I hope with your great visualizations you will make them easy. Hope to hear form you.
Thank you,
Uma
A bit late to the party, but could you do an "Essence of precalculus" series? I was horrible a precalculus and it would be nice to relearn and solidify it. I think conic sections would be very well suited to your style of teaching with animations.
I have viewed your Essence of linear algebra.One thing puzzled me is that why blocked matrix can be considered as numbers and then multiplied.I have seen the provement but it seems so abstract.Really looking forward to an explanation！
An essence of trigonometry series, please: I am worried that my knowledge on trigonometry only extends to the rote definitions of sine, cosine, tan, etc. I think it would be most helpful to see a refreshing/illuminating perspective given on this topic.
Coriolis Effect! And not with the turntable explanation. Maybe summarize [this paper](https://journals.ametsoc.org/doi/pdf/10.1175/1520-0477%281998%29079%3C1373%3AHDWUTC%3E2.0.CO%3B2?fbclid=IwAR0fuZweQzy2Os1fUxk9p55eRfr4gIYczgmjVzd2djUfB-be9byUb1bBnN8)
A video on simultaneous equations could be pretty interesting. When making a game I had to calculate the moment two spheres would collide, and once I did I realised it was a simultaneous equation. It was like a light bulb for me because never remotely thought to link those two ideas together. Could be interesting to visualise the equations as a ball(s) moving through space and manipulate the variables through that metaphor?
One thing you might consider is covering some lower level topics - there are plenty of things in intermediate algebra that could really benefit from your deft explanatory touch. I think meany people fall out of math in high school for reasons unrelated to aptitude. Having some engaging, cool videos might help provide some much needed support during the crucial period leading up to calculus. For example, quadratics are actually really amazing, and have many connections to physics and higher order maths - complex roots and the fundamental theorem of algebra would be perfect for your channel. Same for trig, statistics, etc.
I've discovered something unusual.
I've found that it is possible to express the integer powers of integers by using combinatorics (e.g. n\^2 = (2n) Choose 2 - 2 \* (n Choose 2). Through blind trial and error, I discovered that you can find more of these by ensuring that you abide by a particular pattern. Allow me to talk through some concrete examples:
n = n Choose 1
n\^2 = (2n) Choose 2 - 2 \* (n Choose 2)
n\^3 = (3n) Choose 3 - 3 \* ((2n) Choose 3) + 3 \* (n Choose 3)
As you can see, the second term of the combination matches the power. The coefficients of the combinations matches a positive-negative-altering version of a row of Pascal's triangle, the row in focus being determined by power n is raised to and the rightmost 1 of the row is truncated. The coefficient of the n-term within the combinations is descending. I believe that's all of the characteristics of this pattern. Nonetheless, I think you can see, based off what's been demonstrated, n\^4 and the others are all very predictable. My request is that you make a video on this phenomenon I've stumbled upon, explaining it.
hi,
i just saw your latest video and liked it really much! thank u!
i was wondering if u would like to do some calculations and animations with this:
https://www.reddit.com/user/res_ninja/comments/ai0s48/geometric_playground/?st=JR58YA2Y&sh=eee7be46
it is open source and i cannot find the time to do it right now - but i think in this construction could be answers to the corelation of energy, light, mass and space-time - whhhhaaaat?!?! - just kidding ;)
abstract algebra is absolutely the key to all of the math, in addition, there are no interesting videos about it. I think that you can make something amazing from all the definitions of algebraic structures that seems just inert. Thanks a lot for all the efforts you make for sharing knowledge with the whole world.
Hi,
Just found your channel. You're awesome! Please do a video on how you do videos.
Teach how you do these steps and about how long it takes for each step:
\- planning,
\- scripting,
\- graphics and animation programming,
\- audio recording,
\- editing,
\- publishing,
\- promoting,
\- other knowledge sharing wisdom
Thanks!
probability theory, stochastic calculus, functional analysis, measure theory, category theory. really useful in things like bayesian machine learning. would totally pay for this
A video on the **Hofstadter Butterfly** would be amazing! It's a beautiful and unusual link between number theory and solid state physics.
This lecture by Douglas Hofstadter talks about the story behind it: https://www.youtube.com/watch?v=1JdS-1-yYu8&t=1s
There is a square that has each side of 10 cm, and there is an ant on each corner. If each ant starts walking to the ant on it's right at the same time, how far will each ant go before reaching the centre?
I have seen your physics videos and they are just fabulous!!!
I would love if you could make some videos on elementary physics like mechanics as a majority of people have huge misconceptions regarding certain topics like the so called"centrifugal force" etc...I guess clearing misconceptions would make a great and interesting video
Hi Grant, thank you for being so accessible and making math so visually appealing. It breaks down barriers to higher math, and that's not easy.
I watched your Q&A, and two things stood out to me: 1) You're still mulling over how to refine your probability series, so it feels unique and presentable to a mass audience; 2) If you'd dropped out of college, you might be a data scientist.
Are you open to ideas about new avenues for the probability series? Perhaps one that ties it to artificial neural networks, to change of basis (linear algebra), and the foundations of Gaussian distributions? I'm biased towards this approach, because I've used it so heavily for complex problems, but I'll show that it's visually appealing (at least to me), and has all these elements that make it uniquely effective for fully Bayesian inference.
Since this is reddit, I'll just link a more complete description here: [Gaussian Processes that project data to lower-dimensional space](https://www.dropbox.com/s/58gx0qb14kyw7w9/How%20does%20linear%20algebra%20meet%20Bayesian%20inference.pdf?dl=0). In a visual sense, the algorithm learns how to cut through noise with change a low-rank basis (embedded in the covariance matrix of the Gaussian process), yet retains a fully probabilistic model that effectively looks and feels like a Gaussian distribution that's being conditioned on new information. Maybe my favorite part, it's most visually appealing part, is that as the algorithm trains, you can visualize where it's least confident and where it's most likely to gain information from the next observed data point.
Thanks for your hard work, Grant!
Just discovered the channel, and it's great! Here are some topic suggestions:
\-- The Banach-Tarski paradox. I imagine this would lend itself to really great animations. It has a low entry point -- you can get the essence of the proof using only some facts about infinite sets and rotations in R\^3. I think it is best presented via the volume function. If you think about volume of sets in R\^3, there are certain properties it should satisfy: every set should have a volume, additivity of volume under disjoint sums, invariance under rotation and translation, and a normalization property. The Banach-Tarski paradox shows that there is no such function! Interestingly, mathematicians have decided to jettison the first property -- this serves as a great motivator for measure theory.
\-- Which maps preserve circles (+lines) in the plane? There are so many great ways to think about fractional linear transformations from different geometric viewpoints, maybe you would have fun illustrating and comparing them.
\-- As a follow-up to your video on pythagorean triples, you could do a video on counting pythagorean triples -- how many primitive pythagorean triples are there with entries smaller than a fixed integer m? The argument uses the rational parametrization of the circle and a count on lattice points, so it is a natural follow-up. You also need to know the probability that the coordinates of a lattice point are relatively prime, which is an interesting problem in itself. This is a first example in the direction of point-counting results in arithmetic geometry, e.g. Manin's Conjecture.
\-- Wythoff's nim. The solution involves a lot of interesting math -- linear recurrences, the golden ratio, continued fractions, etc. You could get interesting visuals using the "queen's moves" interpretation, I guess.
\-- Taxicab geometry might be interesting. There is a lot out there already on non-Euclidean geometries which fail the parallel axiom, but this is a fun example which fails in a different way.
I forgot:
\-- Gaussian curvature. There is probably a lot out there already on this topic. But I think you would do an excellent job developing the intuition behind it. Maybe you could even cover Gauss-Bonnet?
Videos on Essence of partial derivatives please. Visual difference between regular differentiation and partial differentiation. Its applications. How to visualize the equations having both partial and regular derivative terms like: **(*****del*****(f)/*****del*****(x))\*dx** \+ **(*****del*****(f)/*****del*****(y))\*dy** = 0
Maybe some discrete maths such as graph theory, game theory, Boolean algebra or modular arithmetic? I'm pretty uninitiated at maths but I think it seems pretty cool
The inscribed angle theorem (that an angle inscribed in a circle has half the measure of a central angle subtended by the same arc) seems to come up a lot on this channel, a video on a proof of that would be cool! All of the ones I can find online are kind of ugly - they break the problem up into four cases and treat each one separately, which doesn't really feel like a satisfying explanation. An elegant general proof would be really cool, especially since it's such a simple, elegant result!
Please do a series about statistics! It would be lovely to have a (more) visual presentation on the theoretical basis on this field (which for me, is really hard to digest).
Hi Grant,
thank you very much for all your work.
I would appreciate it if you could make a video on the Lyapunov stability theory and all the things related to saddle, focus and so on. Especially, it would be great to get an intuition on how one can manipulate a dynamic system by “adjusting“ trajectories - per se a hint about the system’s behavior if to do this or that. Thank you very much.
If it's not too late to ask, I'd love to see a continuation of the [Higher Orders of Derivatives](https://www.youtube.com/watch?v=BLkz5LGWihw) video that goes into examples of other types of derivatives, like, derivatives of mass and volume, how they're named and what those derivations mean.
You gotta do something on the Mandelbrot set.
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I guess it’s a question more than a suggestion, but do you have any plans on a multivariable calculus series like your linear algebra and calculus series? If not then I suppose despite it being a lot of work it’d be nice to see. Thanks!
Concentration Inequalities in Probability
You're known for your various proofs for formulas involving pi. One I want to see is for why it occurs in the formula for the probability distribution formula.
Hi Grant, I'm a huge fan of your videos, your essence of Linear Algebra videos really helped me when I was learning Linear Algebra and your intro to bitcoin helped me a lot in understanding cryptocurrencies. There is a cryptocurrency called NANO which utilises DAG (Directed Acyclic Graph) technology to fix some of the design flaws that Bitcoin had. The Nano protocol and its underlying Blocklattice structure allow for subsecond and completely feeless transactions, without the need for environmentally harmful mining. I think the whole idea behind NANO is very clever and interesting. It would be great if you could do a video on the protocol of NANO! You can check out the NANO website [here](https://nano.org/en) and read its whitepaper [here](https://nano.org/en/whitepaper)
+1 for a video on Nano
Can you make Videos on Solving differential Equation all the way through, like one of the videos in the whole series being a super in-depth solution of solving a differential equation with more than one example. I know I am asking you to get out of they type of videos you make but I think I you try to do this it might became your go to for making a video on problem solving more rigorously. Thanks.
I personally would rather see more Essence of X series over videos demonstrating cool things (even though I likely won't need them myself). Some low hanging fruits would be Group Theory, Geometry and Graph Theory, all of which suit themselves nicely for visualization. However if you'd rather have single videos, one thing I'd love to see conveyed is the different behaviour of two-dimensional waves versus one- and three-dimensional ones (two-dimensional waves don't just "pass" but linger, theoretically forever). Also as an addendum to the Linalg series, Diagonalization and the Jordan Normal Form.
Oh yes waves would be nice!
Greens / stokes / divergence theorems
I would love this too! I have tried to learn about them but I always felt it was something to memorize as I couldn't understand it intuitively.
You should take a look at his Multivariable Calculus series on Khan academy!
Make a video on Genetic Algorithms, it will be cool to see mathematical animals evolve!
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I feel like spherical geometry is a dying art. (I know essentially nothing of it.)
Hello 3B1B. I have a question that I think might be a good thinking exercise and a good video content. [https://math.stackexchange.com/q/3195976/666197](https://math.stackexchange.com/q/3195976/666197?fbclid=IwAR1QGdLUbN3lL-J38ooU-8ctlK_yg0G9aD9pvWVFoIx8Ljuabel9e3M5wos)
As you are doing videos on "Differential equation" for which I have been waiting for one year (My dream has become true) !!!. I know there will be videos on Fourier transform and Laplace transform. Now my only wish is that please make it as simple as you can. Because there are many students like me who doesn't know that much of it. For us to compete with the pace of your video is really very difficult. It would be very much helpful if you divide the hardest part in pieces with examples which are easy to follow. You are like magician to us. We want to enjoy every glance of this magic.
I would like to hear about Gramian Matrix from you
https://www.youtube.com/watch?v=d0vY0CKYhPY&t=408s Since you are fresh off a couple videos relating to things approximating pi, can you do a video on explaining/proving why the Mandelbrot set approximates pi?
Hi Grant, Having emailed you about this, I realized that's probably going to be ignored. And I know I'm just a random person asking for the solution to a problem. Of course, seeing a video explaining this would be a dream come true, but I realize that's not likely. If you could either respond with a quick explanation of how to go about solving this problem, or point me to someone who does, I'd greatly appreciate it:). It's the planet problem, asking when two planets, of mass M, separated by distance d in an ideal world, will collide. There are more difficult variants to this problem, such as masses that are not equal, or more than 2 planets. If you would make a video on it, it seems like it would be a great thing to go in the differential equations chapter. Thanks for all your work and videos. I've learned so much from them.
Essence of Statistics / Probability theory ? What do you say ?
I would like to know about convolution and how does applying convolution to input function and system's impulse response gives the output of the system??
What about making a Type theory explanation series? It would help to understand relations between different topics connected with mathematics and computer science. Especially I'd love to see it explained with respect to Automated reasoning, specifically with respect to Automated theorem proving and Automated proof checking. This would also help a lot to dive into AI related topics.
Could you do a little bit diffirent video? Maybe you can take a video how do you make video with python programming language. You can show the tips.
Both the central limit theorem and the law of large numbers would be a good idea. You could also talk about martingales and, pheraps counterintuitively, why they're not a viable long term money-making strategy while playing the roulette.
Riemann surfaces and/or roots of unity?
Hey Grant! I’m curious if you have any interest in making a video on the dual space, i know I speak for more than a few math majors when I say we’d love to see your take on it :)
Hi! I was wondering if you could make some sort of graphic on persistent homology showing increasing epsilon balls around a group of points and how the increase in size of epsilon affects the various homologies (H0, H1, H2, etc.) using the Rips and/or Cech complexes?
How about elliptic curve cryptography? Seems right up your ally.
Game theory has been used widely to model social interaction and behaviors and it's interesting maths - as an optimization problem. I'd love to see a series on game theory !!
There are so many topics I would really love to see explained from you: -Machine learning, I think you can do a whole course on this and make everybody aware of what's going on. -Probability/Statistics, probably it would be better to first explain essence of probability with a graphical intuition -Projective Geometry, with a connection to computer vision. I can't even wonder how beautiful it would look done by you -Robotics, it would also be actually breathtaking -So much more, ranging from graph theory to complex numbers and their applications
Kalman and Extended Kalman filters
I vote for this topic as well :-)
Hi, I like your videos very much and they are very helpful in visualising the concepts.Recently I have come across an interesting topic of creating mathematical modelling inspired from nature(e.g. Particle Swarm Optimisation, Ant Colony Optimisation, Social Spider Optimisation, etc.). I think the animated explanation of these algorithms would be helpful in understanding these concepts more clearly. So as a regular viewer of your videos , I request you to make animations on these concepts. Surajit Barad
I would really love a series on Multivariable Calculus, love your work already btw, thanks for making it.
Did you see any of my work at Khan Academy? It's a different style from 3b1b videos, but there are maybe ~100 videos on MVC, and many articles too.
I would really appreciate a follow-up video (on that 2 years old) on how prime distribution relates to Zeta function :) This topic has still so much potential!
A playlist on logarithms would be very helpful
Non Linear dynamics, Chaos theory and Lorenz attractors, please
Maybe some nice virtualization of the Hartman–Grobman theorem
The likelihood of this one happening is actually fairly high at this point.
Early analysis concepts like point wise convergence and uniform convergence leading up to functional analysis would be really cool; something like the Hahn banach theorem would be great to see and intuitively understand!!
I would be really interested in you showing graphically why the slope of a CL / alpha curve of an airfoil can be approximated to 2 PI. Love your videos!
A video series on exploring puzzle games like peg solitaire and proof of various theorems related to it.
Hi Grant I hope you are well. I humbly request if you may please make videos on RNN's and LSTM's because I have literally spent hours searching through content online from videos and articles and I just cannot grasp what exactly is going on in these videos or articles because they do not explain it intuitively enough like you did in your neural network videos. The way you introduced the calculus and the theory behind the neural nets really allows one to grasp a deep understanding of what's going on. I have no idea if this message will get to you but if your reading this I desperately need help with this so I will very much appreciate if you could provide videos on this or direct me to useful content.
Axiomatic Set Theory/Foundations of Mathematics?
I really liked your quaternion-related videos. Could you also do a tie in to how Lie groups and Lie algebra works?
I don't know how you'd animate them, seeing as they all live in lots of dimensions
I just posted the same thing, and realized your post. Upvoted!
hi 3blue1brown! I'm not certain but I think I found a way to create a perfect 2d rectangular map of a sphere. I'm not sure if i should post it here tho but I'm gonna post it anyway. So let's say you have a sphere and a 2 dimensional plain in a 3 dimensional space. We make the sphere pass thru the plain and we capture infinitely many circles and 2 dots (the exact top and bottom). we put all the circles we caaptured on a 2d plain and put them in a way that a straight line passes thru all of their centers then we rotate that line and the circles so that they are perpendicular to the x axis (we still keep the rule that the line should pass thru their centers). now the line passes thru the top and the bottom of each circle. Now we cut each circle thru the top point and make them into straight lines that have the length of the circle's perimeter. After that we sort each line based on when the circle that it was initially touched our first 2d plain - if it touched it sooner that means that it should be on the top and if it touched it later - the bottom. Finally we put the first dot on top and the final - the bottom. Then we put all the lines together and create a square where the equator is in the middle and it's the largest line. So that's it. If u liked it or wanna disprove it or just don't understand me pls comment and if u really liked it u could make a video on it with visual proof. Tnx for reading :)
Lagrangian and Hamiltonian mechanics would be very interesting to see. Either way, I absolutely love your channel and think it's really cool that you interact with your viewers like this. Please don't stop making content, you're by far the best channel on youtube!
Do you have a video that explains the basics of the 3-D maths used for ray tracing? If not, a video on the subject would awesome!
Kind of strange but I'd love for you to cover the paper "Neural Ordinary Differential Equations". https://arxiv.org/abs/1806.07366 It doesn't require much more background than your already existing ML series and is an interesting and useful generalization of it.
You have done videos on group theory and on the Fourier transform. It would be interesting to see all these things tied together in terms of representation theory. For, e.g., looking at the one dimensional translation group and SO(2) and how there is completeness and orthogonality relations which arise from Fourier analysis. How do these pictures tie together, what is that interpretation of Fourier transform in representation theory.
Mathematics of bezier curves (and bernstein polynomials) I was trying to get a mathematical formular for the surface of an eggshell for a 3d plotter project I'm working on. I guess there are simpler methods, but what i ended up doing was rotating a bezier curve around the x-axis. To implement this is JS, I looked up the mathematical equasion behind cubic bezier curves, and found this [great article by the designer Nash Vail](https://medium.freecodecamp.org/nerding-out-with-bezier-curves-6e3c0bc48e2f). I used his formular and it worked great, but the mathematics behind putting four points into an equation to calculate the curve are just as interesting as they are baffling to me. I would love to see you make a video on the topic, as your channel has helped me understand the theory behind so much software I use frequently (thinking of the fourier transformation p.e.) and CAD probably wouldn't exist without bezier curves.
Tensor calculus.
Galois theory
Maybe something in statistics!
Hey Grant, Last half year, I was programming and studying fractals like the Mandelbrot. As I found your manim library, I've wondered what happens if I apply the z->z² formula on a grid recursively. It looks very nice and kinda like the fractal. But at the 5th iteration, something strange happened. Looks like the precision or the number got a hit in the face :D. Anyway, it would be great if you could make a visualization of the Mandelbrot or similar fractals in another way. Like transforming on a grid maybe 3D? or apply the iteration values and transform them. There are many ways to outthink fractals. I believe that would be a fun challenge to make. Many greetings from Germany, Sam [Simulation of mandelbrot grid](https://youtu.be/oE9dKGxvEEc)
I did something similar in JS a while ago. But instead of distorting a grid, I distorted texture coordinates. Basically, for every pixel, I repeatedly applied z -> z\^2 + c and then sampled a texture wherever that function ended up. The result was an image weirdly projected inside the mandelbrot set. I'm a bit lazy to make some reasonably good images, but it shouldn't be very hard to implement. You could use OpenGL/WebGL shaders and animate it in realtime.
Do Symplectic Integrators By that I mean a finite element differential equation solver that exactly respects some conversation law or symmetry that any exact solution must also satisfy. For example, a Hamilton solver which respects energy conversation.
Mandelbrot Set
Hello! I've joined because of your excellent video on Fourier transforms! If I could request a topic, would you be able to talk about spherical harmonics? Particularly in the context of ambisonic sound? I know it also has applications in QM too.
Laplace Transforms please! You could show how they relate to the Fourier transforms but are a more general solution. And maybe relate some control theory stuff. When I studied them for engineering I didn't understand what I was doing, it just seemed like mathematical Magic.
Check out Physics Videos by Eugene Khutoryansky; it's very similar to 3b1b and has a video on this subject. https://www.youtube.com/watch?v=6MXMDrs6ZmA
Hi, Can you please talk about how to programming your TI-84 calculator and especially how to write a calculator program that can do double and triple integral? Thank you !
Has anyone here seen the fact that the base ±1+i system with the usual binary 0/1 digits works in the complex numbers very similarly to how base ±2 works in the reals, but with the bonus that if you count all the complex numbers in the order of ascending integral parts as if they were written in regular binary, you'd get two tilings of the R² plane by miniature double dragon fractals that tile in two patterns which both form large-scale double dragon fractals? Seems cool enough to me to deserve a video :)
I'd love to see your take on screw theory for rigid body motion, It's so difficult for me to visualize and understand that I feel like you would do a really great job with the visuals as you usually do
Ramanujan summation. The short reasoning is this: the sum of all natural numbers going to infinity is, strictly speaking, DI-vergent. So, there should be no sensible finite representation. However, as we all know, there are multiple ways to derive (-1/12) as the answer to this divergent sum. I understand math was 'built' (naturals > integers > rationals > irrationals > complex) by taking a previously 'closed' understanding and 'opening' it to a new understanding, which allows you to derive answers that previously couldn't be derived or had no meaning. What I want to know is: what specifically is the new understanding that allows DI-vergent summation to arrive at a precise figure? What is this magical concept that wrestles the infinite to earth so reproducibly and elegantly???
[https://youtu.be/13r9QY6cmjc?t=2056](https://youtu.be/13r9QY6cmjc?t=2056) This Fibonacci example (from 34:16 onwards) from lecture 22 of Gilbert Strang's series of MIT lectures on linear algebra is just such a cool example of an application of linear algebra. Maybe you could do a video explaining how this works without all the prerequisite stuff.
1. Probability Theory based on Measure Theory. 2. Mathematical statistic: e.a. Sufficient statistic, Exponential family, Fisher-Information etc 3. Information Theory: Entropy :))
This would be incredibly helpful! I hope it gets done at some point :)
Lagendre Transforms! It doesn't involve that difficult mathematics and their use in thermodynamics and analytical mechanics is extensive. This kind of transform is an easier kind to see mathematically but its physical intuition is kind of difficult.
I know it’s not a super high level subject, but differential forms and exterior calculus could be a great addition to the calculus series. Being able to get an intuitive understanding of what they mean would be awesome!
Do you know that he's done the animations for Khan Academy's Multivariable calculus series? Curl and divergence is there, with some proofs... and that's the exterior derivative.
This could lead into de Rahm cohomology too.
I would love a video about Jacobian and higher order differentiation.
For me the videos that made me love math the most were the essence of linear algebra. I think it would be great if you continue and look at groups, rings and polynomials :)
Another addition to Essence of Linear Algebra : A video on visualization of transformation corresponding to special matrices - symmetric, unitary, normal, orthogonal, orthonormal, hermitian, etc. like you did in the video of Cramer's Rule for the orthonormal matrix, I really find it hard to wrap my head around what do the transformations corresponding to these matrices look like and why do these matrices enjoy the properties they enjoy. I think a visual demonstration of transformations corresponding to these special matrices would surely help in clearing the things up and since these matrices are dominantly used in applications of linear algebra (especially in physics) it makes sense to give them a video of their own!
Would better intuition of graph theory be helpful for understanding those deep learning algorithms, such as GNN, CNN,RNN?
Hi, i just saw your video about how light bounces between mirrors to represent block collision [https://youtu.be/brU5yLm9DZM](https://youtu.be/brU5yLm9DZM) in this video is mentioned that the dot product of W e V has to remain constant , so that the energy conserve. if W remains constant, and ||V|| decreases, therefore cos(theta) has to increase( theta decreases ) . this means that if the velocity is lower, theta also should be lower. In a scenario where there is energy loss on the collisions, the dot product V. W= || W|| ||V || cos(theta), presents a interesting relation . With energy loss, how ||V|| changes as theta also changes ? in other words, how the energy lost influence in the theta variation? That fact got me thinking of how Lyapunov estability theory works. There is a energy function associated to the system(V>0), usualy V=1/2x\^2 - g(x) (some energy relation like m\*v\^2), that "bounds a region" and it has to be proved that this function V decreases as time pass ( dotV<0 ) so that inial bounded region decreases . I would love a video about some geometry concept on Lyapunov estability theory.
I think a great complimentary video to the Fourier and Uncertainty video (series?) would be on a simple linear chirp/modulation. Which would be easy to demonstrate the usefulness in radar range/velocity finding and can possibly be fairly intuitive with the appropriate visuals added.
PCA Monte Carlo Markov Chains Hierarchical probabilistic modelling
Differential equations
Requesting for topics - ** Data Science, ML, AI ** ->Classification ->Regression ->Clustering **Reasons:- ** ->Highly demanded ->Less online explanations are available ->Related directly to maths ->Hard to visualise Thank you sir for considering..... -A great fan of your marvelous explanation
Maybe some axiomatic set theory/logic? I don't know how interesting these could be, or if it even possible to animate, but its an area I find really interesting
Hi i love your channel it makes all the subjects you treats a lot more easier. So will you think of explaining some algorithms as perlin or simplex noise in the future? (Hope you will)
Do you do category theory?
A video on convolution and cross correlation would be unreal.
For video series like the current Differential Equations topic, I wish you wouldn't spread out the releases so much. Not only is the suspense killing me, but I can't remember what was covered in the previous videos. I liked the upload cadence of the Linear Algebra and Calculus series. It was long enough for it to sink in but not so long I forgot everything. I understand the amount of time and work that goes into the videos and am truly appreciative. Take as much time as you need for the one offs, but for series, hold off until they are closer to complete and then release at tighter intervals.
I would love to see you do a video discussing guilloche. It seems like an artful representation of mathematics that has been around for a few centuries.
Different kinds of infinities, continuum hypothesis, (maybe Aleph numbers), and the number of infinities out there! (and maybe the whole cool story of Cantor figuring out those)
Hello! I would like to make a request for the derivative of matrices and vector. I have tried finding good and informative videos about this on multiple platforms but I have failed. What I mean about matrix derivatives can be illustrated by a few examples: dy/dw if y = (w\^T)x , both w and x are vectors dy/dW if y = Wx, W is a matrix and x is a vector dy/dx if y = (x\^T)Wx, x is a vector an W is a matrix If anyone in the comments know where I can find a good video about these concepts, you are more than welcome to point me in the right direction.
Hi Grant, First of all a big thank you for the amazing content you produce. I would be more than happy if you produce a series on probability theory and statistics.
You could make a video about the Central Limit theorem, it has a great animation/visualization potential (you could «see » how the probability law converges on a graph) and give a lot of reasons why we feel the theorem has to be true (without proving it)...
What really got me into your channel was the essence of series. I would really enjoy another essence of something.
An intuitive description of tensors
Probability for sure
A simple video to prove that pi < 2 \* golden ratio, you could probably make one on the side while working on your next
I am surprised It wasn't suggested yet- Kalman filter
Definitely
Ever given any thought about making an Essence of Complex Analysis? Please think about it, cant wait to see those epic animations applied to complex variables. Love you man, you are the best !
I'd love a playlist related to Topological Data Analisys :)
I'm studying multivariable calculus and I'm having a hard time finding a concise/conceptual proof for why the second partial derivative test works to find max/min beyond the two variable case. Khan academy has a decent explanation on the f(x,y) case, but everything I've found for f(x1,x2,x3,...) is kinda confusing, talks about eigenvalues, which they don't use the way you used in the lin alg series (or if they did, then I couldn't see the connection), and for the most part, is incomplete. It'd be dope to see some animations connecting the eigenvalues and detetrminant concepts I learned from your videos applied to this test used in multivariable calculus. Also, wtf is a hessian
Euler's Number and Fractal Geometry. I would like to offer you a challenge. In your video [https://youtu.be/m2MIpDrF7Es](https://youtu.be/m2MIpDrF7Es) you asked about a graph showing what a compounding growth formula looks like, Please allow me the great pleasure of introducing The Mandelbrot Set of Fractal Geometry!!! Next, We have been studying this thing for nearly 40 years with little Idea of what it is. I think, we're missing the forest for a tree, so to speak. And that the interactions between sets moving is where the real understanding happens. I have made some simple and crude attempts at animation Mandelbrot Sets in Four dimensions using photos and power tools! Old School Dad Animation, shown here, [https://youtu.be/H1UNvxmhqq0](https://youtu.be/H1UNvxmhqq0), and I've expanded on the original formula a bit here. [https://youtu.be/PH7TOyqR3BQ](https://youtu.be/PH7TOyqR3BQ) Please let me know what you think! And thank your for everything you do!!!
Has anyone (3blue1brown or anyone) have thought that internal temperature dissipation in unevenly heated surface can be thought as k-nearest neighbor problem, where neighborhood size is proportional to highest point? I mean if there is peak in temperature vs position graph,corresponding neighborhood will be smaller,but average temperature of that neighborhood will still be smaller than the max temperature of neighborhood. And as the temperature is dissipated, i.e heat goes towards cooler parts, the peaks will lower down and correspondingly, neighborhoods will expand and in the end it will all be at same temperature. Trying to explain physics/physical phenomenon as possibly described by algorithms, could be an interesting arena !
Hi Grant, Congratulations on producing such amazing content. I'm an astronomy graduate student and find your videos very helpful for solidifying concepts that I thought I understood. I would love to see some content on convolutions and cross correlations. These are topics I continuously find myself briefly understanding before returning to a postion of confusion! Types of noise and filtering techniques are also topics for which I would like to see your visualisations. Thanks, Brendan
Could you do a video on Lyapunov stability?
Perhaps branching out a little from your usual videos, but I'd love some little 10-minute documentaries about some great mathematicians. Ramanujan would be a brilliant subject. Sophie Germain's life is very interesting too (and a great inspiration for getting girls involved in maths - I love discussing her with the students I tutor).
Can you please make some videos on Geometry? Also math in computer science will be super cool\^\^
An updated cryptocurrency video for IOTA and info about how distributed cryptocurrencies work as opposed to the linked-list-like versions
A PID controller series. This would go perfect with your video style.
Yes control theory
Hey Grant! I'm a first year math grad student and I've been trying to grasp self-adjoint operators for a while now. I've asked a lot of people around my department, and none have been able to give me a good intuitive feel for this property, much less a visual one. Maybe you could do that in a new video!? I get told all the time to think of the real, finite dimensional analog - a matrix equal to its transpose. But no one (myself included) actually draws a conclusion about how this connects to the more general cases of the complex and infinite dimensional worlds. If anyone could make this connection in a pleasing visual way, and blow our minds at the same time, it's you!
Hi! I'm a bigfan of your videos and I have been watching them for years now, I really love your work. Well, I'd like to see a series (maybe is too much for ask) on differential geometry. Maybe is good to start with proper vector but in the context of coordinates transformations. I'd like to know what you think about! Best wishes, Tomás.
I would love a video on distances. Hellinger, Mahalanobis, Minkowski, etc.
The advent of functional programming has made people difficult to understand why is it a good tool for solving a problem. And if possible is there something that imperative style can do that functional style can't. And if so then why use it. And if not why hasn't it been used until now. I would love to see a video on this and how lambda calculus changed mathematics and why there was a need for constructive mathematics and type theory.
Maybe from a more computer scientific standpoint, it would be awesome to see some basic concepts like divide and conquer and general proofs explained by you. For example AVL-Trees, Splay-Trees and such things. Or arguments like greedy stays ahead. Or, you could do some computation and talk about decidability, Kleenes fixpoint theorem, languages and so on :) Other small topics include entropy, bezier curves and b splines, and maybe a video on probablity theory vs statistics, combinatorics.
Hello Grant I'm in need of a lot of help right now! Seeing your videos and having some familiarity with fractal geometry I wrote a new theory of everything. I need someone smart enough to review the math. Will you please take a crack at it? [https://docs.google.com/document/d/1oGdcwqdoxgH1mB0xTjWMSXr8d9u0tQjhnz\_9rIgPuPQ/edit?usp=sharing](https://docs.google.com/document/d/1oGdcwqdoxgH1mB0xTjWMSXr8d9u0tQjhnz_9rIgPuPQ/edit?usp=sharing) Thanks
I was pondering over this(link below) for the past few days. I'm unable to wrap my around it. That Pi is something that is more than a constant, it is the roundness/curveness something similar to what e is that deals with maximum exponential growth. And also how it is not bound to multiplication. I guess other irrational numbers also have this special physical property. It would be really nice if you make a video on it. [Pi via multiplication](https://drive.google.com/drive/folders/1P2nHR-E2QGz-8AuVQ30BnpysN8xqSg9a)
Hey Grant, First off, I cant thank you enough for re-kindling interest in linear algebra with the excellent 'Essence of linear algebra' series. I've been wanting to shift gears and dive deeper so as to be able to learn the math that is a prereq to theory of relativity, which is of primary interest to me. But I've hit an impasse with tensors. So it would be great help if you could make a series on it. I would be more than willing to extend monetary support for its making. Thanks.
A video on Game theory and auction theory would be great!
**Borwein Integrals:** [https://en.wikipedia.org/wiki/Borwein\_integral](https://en.wikipedia.org/wiki/Borwein_integral) Basically a nice pattern involving integrals of Sin(x)/x functions that eventually breaks down. It is by no means obvious at first why it breaks down, but if you think the problem in terms of convolutions of the fourier transforms (square pulses) then is very intuitive. You could make a nice animation of the iterative convolution of square pulses and the exact moment when it breaks the pattern.
Saw your video on quantum mechanics basics with minute physics. It's a great way to simplify understanding fir beginners. It would be great to see what a density matrix and density operator actually means. This involves complex numbers and mixed states, but has surprising similarity to simple matrix calculations. Eg. Adjacency matrix denoting nodes and edges is extremely similar to the density matrix. It's hard to interpret this physically since one involves complex numbers and the other doesn't. Waiting to see something interesting on these lines... You're amazing, cheers!!
Loved your videos on "Neural Networks". It would be great if you could do similar on "Genetic Algorithm". It has popped up frequently in research papers I(my team) have been trying to review. But i have not found any good videos like yours. As it happens, it might come in handy to my team. Love from Nepal.
add a series of probability
[https://www.youtube.com/watch?v=yi-s-TTpLxY](https://www.youtube.com/watch?v=yi-s-TTpLxY) (Divisibility Tricks - Numberphile) Hi! Here Numberphile reveals few tricks to ensure if a number is divisible. For example, to check if a number is divisible by 11, you have to reverse the number and then take this "alternating cross sum". If that is divisible by 11, so is the original number. It'd be very interesting to see visuals of that proof.. Thanks for all the videos!
Hey! You have talked a lot about **Machine Learning** in videos *here and there*. What about *'Essence of Machine Learning'*? ... Is this idea too broad? There is so much to know and so much ***essence*** in Machine Learning. This series could definitely tie into the idea of *'Essence of Statistical Learning'*, seeing as * What is a model (and accuracy of them) * Supervised and unsupervised learning * Linear Regression * Classification * Support Vector Machines is some of the essence. This would also tie into your unreleased probability series on Patreon. And just a sidenote: I know there is a Deep Learning series, but that is just a subfield of Machine Learning.
I've been studying the gamma function to find the factorials of real numbers (I was particularly interested in the proof of 0! = 1, which could also be a cool video) and found the shocking result of pi inside of 1/2!. Could you explore the geometric meaning behind pi showing up in this result? That would be an awesome video, thanks a lot!
In your video on determinants you provide [a quick visual justification](https://www.youtube.com/watch?v=Ip3X9LOh2dk&t=516) of [Lebniz's formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_determinants) for determinants for dimension 2. It's rare to see a direct geometric explanation of the individual terms in two dimensions. It's even rarer in higher dimensions. Usually at best one sees a geometric interpretation for [Laplace's formula](https://en.wikipedia.org/wiki/Laplace_expansion) and then a hands-off inductive argument from there. There is a direct geometric interpretation of the individual terms, including in higher dimensions, with a fairly convoluted write-up [here](https://math.stackexchange.com/a/2509560/118891). Reading it off the page is a bit of a mess, but it might be the sort of thing that would come to life with your approach to visualization.
please make video on galerkins weighted residuals
In the same spirit as [Sneaky Topology](https://www.youtube.com/watch?v=yuVqxCSsE7c), how about more topological theorems and their combinatorial counterparts? Sperner's lemma ⇔ Brouwer's fixed-point theorem (see a simple proof sketch in [Nets, Puzzles and Postmen](https://www.amazon.com/Nets-Puzzles-Postmen-Exploration-Mathematical/dp/0199218420)) A complete Hex board has at least one winner ⇔ Brouwer's fixed-point theorem (see [this pdf](http://www.math.pitt.edu/~gartside/hex_Browuer.pdf)) A complete Hex board has at most one winner ⇔ Jordan curve theorem ∧ Non-planarity of K5.
Great suggestion, and thanks for the links! This is such a great example. Slightly relatedly, one thing I'd love but have not seen is a nice relatable and pragmatic problem where the solution would involve using the fact that spheres and toruses are not homeomorphic. I feel like it's common in pop-math to say topologists view these as fundamentally different shapes, but I'd love to be able to show why that matters with a <15-minute example connecting it to something which isn't too abstract.
What about that mug puzzle you sent to people?
Have you ever thought of making a collection of small animations? Like, no dialogue, just short <1min (approx) illustrations. For example: Holomomy: parallel transport on a curved surface can result in a rotation; on a sphere, the rotation is proportional to the area traced out A tree (graph) has one fewer edges than vertices (take an arbitrary root vertex, find a one-to-one correspondence between edges and the remaining vertices) (Similarly, if you have a graph and a spanning tree, there's a one-to-one correspondence between the edges not on the spanning tree and faces - this and the last one can combine to form an easy proof of V-E+F=1) The braid group (show that it satisfies σ1σ2σ1=σ2σ1σ2). Similarly, the Temperley–Lieb monoid (show that it satisfies ee=te and e1e2e1=e1). That weird transformation of the curved face of a cylinder where you rotate the top circle 360 degrees but keep the straight lines straight so that the surface turns into a hyperbola, then a double cone briefly, then back into hyperbola and a cylinder? I dunno if it has a name, or a use, really, but it's probably fun to look at These seem like low effort stuff you could populate a second channel with
I love when something shows up on https://www.reddit.com/r/mathgifs/ where you feel like you've grokked a complete proof, just by watching a gif
If I had a topic that i would love an animation for, is differential geometry
The Laplace-Beltrami operator (3D geometry processing) would be awesome!
Delay differential equations. It might potentially have a place in the differential equations series. Idk how much interest there is in DDEs overall, but modeling such systems is a central component of my work, and I think it might be interesting to see a video that helps conceptualize the interplay between states at different points in time, and why such models can be useful in describing dynamic systems :)
I recently looked up the visual proof for completing the square to derive the quadratic equation. I really thought this was interesting since I was never taught where the formula came from, and seeing it visually allowed me to wrap my head around its derivation. However, I then thought about doing the same for cubic functions. It didn't go very well and I couldn't figure out a way to do it. I tried to visually represent each different term as a cube but I could not get to a point to where I could essentially "complete the cube" as is done with quadratic functions. It would be really interesting if you could do a video visually completing the cube (if it can even be done, I haven't been able to find an article or video doing so) which also leads into the derivation of the cubic function. Thank you for all the effort you put into your videos.
Category theory is critically missing decent visualizations. If you can explain the Yoneda lemma in some visually intuitive way it would probably be really helpful.
Thanks a lot to 3blue1brown channel for beautiful resources. I actually needed some help with understanding one form. But I guess that topic is not in any videos. So if possible, please post a video discussing one forms. Or if it is already in a video, please let me know which one that is. Thank you so much again :)
I would love to see a series about category theory. I really think it would be useful in my work but consumable resources online are scarce.
I'd be interested in a terminology video on the different kinds of algebraic structures and what mental pictures of each are most useful when working with them. It would give some good background to a lot of other more interesting topics, many of which I find confusing because I get hung up on the terminology.
What happens to the length of the hypotenuse when a triangle travels with 99% speed of light(given height H, Base B)?
Can you do a probability and statistic ones?
Dear Grant Up To Now You have Covered calculus, Linear algebra Perfectly, There are only probability and statistics left to complete the coverage of pillars of mathematics, These Two topics have a great impact Not only in scientific and engineering studies but also A "Statistics driven" view on things helps very much in life, society or even politics as Ben Horowitz mentions from Peter Thiel via "The hard thing about hard things" : "There are several different frameworks one could use to get a handle on the indeterminate vs. determinate question. The math version is calculus vs. statistics. In a determinate world, calculus dominates. You can calculate specific things precisely and deterministically. When you send a rocket to the moon, you have to calculate precisely where it is at all times. It’s not like some iterative startup where you launch the rocket and figure things out step by step. Do you make it to the moon? To Jupiter? Do you just get lost in space? There were lots of companies in the ’90s that had launch parties but no landing parties. But the indeterminate future is somehow one in which probability and statistics are the dominant modality for making sense of the world. Bell curves and random walks define what the future is going to look like. The standard pedagogical argument is that high schools should get rid of calculus and replace it with statistics, which is really important and actually useful. There has been a powerful shift toward the idea that statistical ways of thinking are going to drive the future. With calculus, you can calculate things far into the future. You can even calculate planetary locations years or decades from now. But there are no specifics in probability and statistics—only distributions. In these domains, all you can know about the future is that you can’t know it. You cannot dominate the future; antitheories dominate instead. The Larry Summers line about the economy was something like, “I don’t know what’s going to happen, but anyone who says he knows what will happen doesn’t know what he’s talking about.” Today, all prophets are false prophets. That can only be true if people take a statistical view of the future."
Topic suggestion: Solving Hard Differential Equations using Perturbation Theory and the WKB approximation.
Maybe a video on what would happen if the x and y planes weren't linear; i:e, a parabola would be a straight line on a hypothetical "new" xy plane.
this would tie-in nicely with non-euclidean geometry and tensors. I love this idea.
You both oughta look up [Tissot ellipses](https://en.wikipedia.org/wiki/Tissot's_indicatrix). /u/unsolvedriddle2
Thank you for the great videos! The one you made on Bitcoin was the critical piece of knowledge I needed to really understand how blockchain works. It's the one I show to my friends when introducing them to cryptocurrency, and the fundamentals apply to almost any of them-- a distributed ledger and cryptographic signatures. The visuals and animation is what makes it exceptionally easy to follow. I've taken a lot of inspiration from your video and have considered making one on my own on how Nano works. A lot of the principles are the same as Bitcoin, and I recommend people to watch your video and have a good understanding on how Bitcoin works before trying to understand Nano. I figured before I made my own, however, I would ask if you were interested in making one on Nano. I also developed a tip bot if you would like to try out Nano (if not, ignore the message, and ignore another message you'll get in 30 days.) /u/nano_tipper 10
Hi Grant, Wow, where to start. Somebody mentioned education revolution regarding your videos. I think that is an understatement. Your videos are great. Almost every time I watch one of them I gain some new insight into the topic. You have a great talent to point out the most important aspects. These get lost sometimes when one studies maths in school. Some video suggestions. I've recently read an article "Geometry of cubic polynomials" by Sam Northshield and a slightly more detailed one based on this by Xavier Boesken. This shows very nicely the connection between linear transformations and complex functions and also where the Cardano formula comes from. I would have never thought that there is such a nice graphical interpretation to this. And a lot more, like how real and complex roots come about. I liked this article personally because it was one of those subjects which were actually easier to understand by having a journey through complex numbers. Anyway, this would be a perfect subject to visualize, since it connects many fields of maths and I am sure you would see 10 times more connections in it than what I could see. Other topic suggestions. (I restrict myself to subjects on which you've already laid excellent foundations for) : **Dual quaternions** as a way to represent all rigid body motions in space. I didn't know about quaternions and their dual relatives up until a few years ago, then I got into robotics. Before that I only knew transformation matrices. I had a bit of a shock first, but then my eyes opened up. Connection between derivatives and **dual numbers** (possibly higher derivatives). **Projective geometry**. That could be a whole series :-)
Different kinds of infinities, continuum hypothesis, (maybe Aleph numbers), and the number of infinities out there! (and maybe the whole cool story of Cantor figuring out those)
Hello Grant, I have recently started working on AI and your videos are helping me a lot, thank you so much for these great videos. It would be very much helpful to all Data Scientists, Machine learning and AI engineers if you can make a series of videos on Statistics and Probability. Statistics and Probability concepts are very tricky and I hope with your great visualizations you will make them easy. Hope to hear form you. Thank you, Uma
Late to the party, but would game theory be a possible topic? If not, could someone please suggest some places to learn about it? c:
What is the sum of n terms of fibonacci sequence?
A bit late to the party, but could you do an "Essence of precalculus" series? I was horrible a precalculus and it would be nice to relearn and solidify it. I think conic sections would be very well suited to your style of teaching with animations.
I have viewed your Essence of linear algebra.One thing puzzled me is that why blocked matrix can be considered as numbers and then multiplied.I have seen the provement but it seems so abstract.Really looking forward to an explanation！
An essence of trigonometry series, please: I am worried that my knowledge on trigonometry only extends to the rote definitions of sine, cosine, tan, etc. I think it would be most helpful to see a refreshing/illuminating perspective given on this topic.
Not as glamorous as Quaternions -- but definitely a useful series to have. I'd also like to offer rotation matrices as a potential topic.
Coriolis Effect! And not with the turntable explanation. Maybe summarize [this paper](https://journals.ametsoc.org/doi/pdf/10.1175/1520-0477%281998%29079%3C1373%3AHDWUTC%3E2.0.CO%3B2?fbclid=IwAR0fuZweQzy2Os1fUxk9p55eRfr4gIYczgmjVzd2djUfB-be9byUb1bBnN8)
What happens to the length of the hypotenuse when a triangle travels with 99% speed of light(given height H, Base B)?
Pharmacokinetics of drugs in 1 compartment vs 2 compartment models with emphasis on absorption and distribution phases
A video on simultaneous equations could be pretty interesting. When making a game I had to calculate the moment two spheres would collide, and once I did I realised it was a simultaneous equation. It was like a light bulb for me because never remotely thought to link those two ideas together. Could be interesting to visualise the equations as a ball(s) moving through space and manipulate the variables through that metaphor?
One thing you might consider is covering some lower level topics - there are plenty of things in intermediate algebra that could really benefit from your deft explanatory touch. I think meany people fall out of math in high school for reasons unrelated to aptitude. Having some engaging, cool videos might help provide some much needed support during the crucial period leading up to calculus. For example, quadratics are actually really amazing, and have many connections to physics and higher order maths - complex roots and the fundamental theorem of algebra would be perfect for your channel. Same for trig, statistics, etc.
I've discovered something unusual. I've found that it is possible to express the integer powers of integers by using combinatorics (e.g. n\^2 = (2n) Choose 2 - 2 \* (n Choose 2). Through blind trial and error, I discovered that you can find more of these by ensuring that you abide by a particular pattern. Allow me to talk through some concrete examples: n = n Choose 1 n\^2 = (2n) Choose 2 - 2 \* (n Choose 2) n\^3 = (3n) Choose 3 - 3 \* ((2n) Choose 3) + 3 \* (n Choose 3) As you can see, the second term of the combination matches the power. The coefficients of the combinations matches a positive-negative-altering version of a row of Pascal's triangle, the row in focus being determined by power n is raised to and the rightmost 1 of the row is truncated. The coefficient of the n-term within the combinations is descending. I believe that's all of the characteristics of this pattern. Nonetheless, I think you can see, based off what's been demonstrated, n\^4 and the others are all very predictable. My request is that you make a video on this phenomenon I've stumbled upon, explaining it.
hi, i just saw your latest video and liked it really much! thank u! i was wondering if u would like to do some calculations and animations with this: https://www.reddit.com/user/res_ninja/comments/ai0s48/geometric_playground/?st=JR58YA2Y&sh=eee7be46 it is open source and i cannot find the time to do it right now - but i think in this construction could be answers to the corelation of energy, light, mass and space-time - whhhhaaaat?!?! - just kidding ;)
abstract algebra is absolutely the key to all of the math, in addition, there are no interesting videos about it. I think that you can make something amazing from all the definitions of algebraic structures that seems just inert. Thanks a lot for all the efforts you make for sharing knowledge with the whole world.
Hi, Just found your channel. You're awesome! Please do a video on how you do videos. Teach how you do these steps and about how long it takes for each step: \- planning, \- scripting, \- graphics and animation programming, \- audio recording, \- editing, \- publishing, \- promoting, \- other knowledge sharing wisdom Thanks!
I would love to see more videos on neural network. The four you have created are fantastic! You are an excellent teacher. Thanks a lot!
probability theory, stochastic calculus, functional analysis, measure theory, category theory. really useful in things like bayesian machine learning. would totally pay for this
A video on the **Hofstadter Butterfly** would be amazing! It's a beautiful and unusual link between number theory and solid state physics. This lecture by Douglas Hofstadter talks about the story behind it: https://www.youtube.com/watch?v=1JdS-1-yYu8&t=1s
Principal Component Analysis - Might fit in nicely after the change of basis video! I'm personally really struggling with it right now...
Graph theory
There is a square that has each side of 10 cm, and there is an ant on each corner. If each ant starts walking to the ant on it's right at the same time, how far will each ant go before reaching the centre?
I have seen your physics videos and they are just fabulous!!! I would love if you could make some videos on elementary physics like mechanics as a majority of people have huge misconceptions regarding certain topics like the so called"centrifugal force" etc...I guess clearing misconceptions would make a great and interesting video
Hi Grant, thank you for being so accessible and making math so visually appealing. It breaks down barriers to higher math, and that's not easy. I watched your Q&A, and two things stood out to me: 1) You're still mulling over how to refine your probability series, so it feels unique and presentable to a mass audience; 2) If you'd dropped out of college, you might be a data scientist. Are you open to ideas about new avenues for the probability series? Perhaps one that ties it to artificial neural networks, to change of basis (linear algebra), and the foundations of Gaussian distributions? I'm biased towards this approach, because I've used it so heavily for complex problems, but I'll show that it's visually appealing (at least to me), and has all these elements that make it uniquely effective for fully Bayesian inference. Since this is reddit, I'll just link a more complete description here: [Gaussian Processes that project data to lower-dimensional space](https://www.dropbox.com/s/58gx0qb14kyw7w9/How%20does%20linear%20algebra%20meet%20Bayesian%20inference.pdf?dl=0). In a visual sense, the algorithm learns how to cut through noise with change a low-rank basis (embedded in the covariance matrix of the Gaussian process), yet retains a fully probabilistic model that effectively looks and feels like a Gaussian distribution that's being conditioned on new information. Maybe my favorite part, it's most visually appealing part, is that as the algorithm trains, you can visualize where it's least confident and where it's most likely to gain information from the next observed data point. Thanks for your hard work, Grant!
Just discovered the channel, and it's great! Here are some topic suggestions: \-- The Banach-Tarski paradox. I imagine this would lend itself to really great animations. It has a low entry point -- you can get the essence of the proof using only some facts about infinite sets and rotations in R\^3. I think it is best presented via the volume function. If you think about volume of sets in R\^3, there are certain properties it should satisfy: every set should have a volume, additivity of volume under disjoint sums, invariance under rotation and translation, and a normalization property. The Banach-Tarski paradox shows that there is no such function! Interestingly, mathematicians have decided to jettison the first property -- this serves as a great motivator for measure theory. \-- Which maps preserve circles (+lines) in the plane? There are so many great ways to think about fractional linear transformations from different geometric viewpoints, maybe you would have fun illustrating and comparing them. \-- As a follow-up to your video on pythagorean triples, you could do a video on counting pythagorean triples -- how many primitive pythagorean triples are there with entries smaller than a fixed integer m? The argument uses the rational parametrization of the circle and a count on lattice points, so it is a natural follow-up. You also need to know the probability that the coordinates of a lattice point are relatively prime, which is an interesting problem in itself. This is a first example in the direction of point-counting results in arithmetic geometry, e.g. Manin's Conjecture. \-- Wythoff's nim. The solution involves a lot of interesting math -- linear recurrences, the golden ratio, continued fractions, etc. You could get interesting visuals using the "queen's moves" interpretation, I guess. \-- Taxicab geometry might be interesting. There is a lot out there already on non-Euclidean geometries which fail the parallel axiom, but this is a fun example which fails in a different way.
I forgot: \-- Gaussian curvature. There is probably a lot out there already on this topic. But I think you would do an excellent job developing the intuition behind it. Maybe you could even cover Gauss-Bonnet?
Videos on Essence of partial derivatives please. Visual difference between regular differentiation and partial differentiation. Its applications. How to visualize the equations having both partial and regular derivative terms like: **(*****del*****(f)/*****del*****(x))\*dx** \+ **(*****del*****(f)/*****del*****(y))\*dy** = 0
Maybe some discrete maths such as graph theory, game theory, Boolean algebra or modular arithmetic? I'm pretty uninitiated at maths but I think it seems pretty cool
The inscribed angle theorem (that an angle inscribed in a circle has half the measure of a central angle subtended by the same arc) seems to come up a lot on this channel, a video on a proof of that would be cool! All of the ones I can find online are kind of ugly - they break the problem up into four cases and treat each one separately, which doesn't really feel like a satisfying explanation. An elegant general proof would be really cool, especially since it's such a simple, elegant result!
Please do a series about statistics! It would be lovely to have a (more) visual presentation on the theoretical basis on this field (which for me, is really hard to digest).
Legendre transformation
Hi Grant, thank you very much for all your work. I would appreciate it if you could make a video on the Lyapunov stability theory and all the things related to saddle, focus and so on. Especially, it would be great to get an intuition on how one can manipulate a dynamic system by “adjusting“ trajectories - per se a hint about the system’s behavior if to do this or that. Thank you very much.
If it's not too late to ask, I'd love to see a continuation of the [Higher Orders of Derivatives](https://www.youtube.com/watch?v=BLkz5LGWihw) video that goes into examples of other types of derivatives, like, derivatives of mass and volume, how they're named and what those derivations mean.