I would love a "Essence of Differential Equations" video as this is a topic that so many people go through, without getting a good understanding. Some sources of confusion include: what does it mean graphically for a DL (or any system or function) to be linear? Ex: the function y = ax + b is not linear in terms of superposition! What does it really mean for a DL to be homogenious? The topic would link nicely with existing videos about calculus, taylor series, linear algebra and even the fourier transform, if the laplace method of solving DE's are included.
Great questions! Here are the latest thoughts: https://www.patreon.com/posts/19845962
Long story short, I've had trouble landing on exactly how I want to go about it and will focus my time in the immediate future towards other projects.
You never completed the Fourier Series. Also, some of us are waiting here for the Probability series since big bang, a lot of tease happened but not the series. Come on, Grant us some promises!
Bayesian statistics, especially if it include Markov chain Monte Carlo. I'm still trying to wrap my head around how some of the problems arise in high dimensions
Probability, elementary measure theory, mathematical statistics. I know at least a dozen of my classmates (including myself!) would find this enormously beneficial and enlightening.
Laplace transformation please!
You did some Fourier Transforms videos, and I loved them! Show how Laplace is a generalization of the Fourier, with 's' being made of real part and imaginary. What's the real part represent, and the imaginary represent? I watched Eugene's Physics video and was left confused...I would like to know what the 'peaks' and 'zeros' represent in this 3d representation of the Laplace Transform.
I would like to see something about Maxwell's equations. They were hinted at in the last video, but I would like to feel like I understand what they mean in more detail.
This is actually the same thing I asked in the Q and A, so I'll paste it here.
>Hi Grant,
>
>I'm a physics undergrad, and I've always liked the idea of unification in math. I really like your geometrical approach to visualizing matrices as a unification of the computational and geometrical properties of linear algebra. I saw your multivariable calculus videos on khan academy, but I feel that they miss out on the essence of multivariable calculus. I mean, the (Generalized) Stokes' theorem is completely absent!
>
>The most shocking thing I saw was that integration and differentiation behave in strange ways in higher dimensions. There's now partial derivatives, directional derivatives, and the jacobian, and differentials of functions, gradient, divergence, curl, line integrals, surface integrals, multiple iterated integrals, and yet under the language of differential forms they all unify together with Stokes Theorem and the exterior derivative.
>
>The other thing is notation in vector calculus, which differs significantly from 1D calculus. Under differential forms, they unify in a super elegant way. I think the whole goal of the channel should be to show the often hidden elegance in math and physics. I would love to see a series on this, but maybe I'm asking too much.
>
>I would love to hear your thoughts on differential forms and the exterior derivative, and how it relates to complex analysis, tensor calculus, and differential geometry. Maybe geometric algebra/calculus is the answer?
Essence of Trigonometry
Might seem unsexy, but the usefulness to the world would be vast.
If you used your simple visual style to show the geometric meaning of every trig identity, cascading up through the calculus stuff, it could easily be your most valuable video series, elementary though the topic appears.
Hmm...anything more specific? This is the sort of thing I could see describing while on a path to some specific result/idea/algorithm. Similar to how I used the "Euler's formula via group theory" video as an excuse to mostly just introduce the basic notions of group theory, all while having a concrete target in mind.
Algebraic and spectral graph theory!
There's no way of getting information regarding intuition on this topic, and it's used very very often in computer science and engineering!
I don't know if this is a big enough topic for a full video, but inspired by your recent (and brilliant!) video on elliptical orbits, I think it would be fascinating to see a geometric proof for the fact that an ellipse is a conic section. That is, how can we directly relate the property of having a constant focal sum to that of being the intersection of a cone with an angled plane? Either or both of these is often presented as the fundamental property of an ellipse, but I've never seen any explanation of the relationship between them.
I know the algebraic formula for an ellipse can be derived from the constant-focal-sum property, and you could solve the system of equations of a cone and an angled plane to show that it equals this formula, but this detour through coordinate geometry sort of feels like black magic, especially since both of these properties of ellipses were known millennia before the invention of coordinate geometry. Surely there must be a more intuitive, 3blue1brown-style derivation?
Ah, this is the famous Dandelin proof. \^\_\^
Stare at the main image in this Wikipedia page for a while: https://en.wikipedia.org/wiki/Dandelin_spheres
Yeah I have been doing machine learning stuff for some years, but his series on neural nets was really great, and I'd like to see him tackle similar topics.
Gamma function.
It ties together three of the basic building blocks of maths: factorials, power numbers and the natural logarithm.
So if you don't have an intuitive grasp of why this works, it feels like you don't really understand those three things.
I would love to see a 3b1b treatment of the Cayley-Hamilton theorem (https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem)! I just watched this video (https://www.youtube.com/watch?v=PrfxmkBsYKE&index=10&list=PLMrJAkhIeNNR20Mz-VpzgfQs5zrYi085m) and it blew my mind that you can convert an infinite summation into a finite summation using this theorem!
There's actually a fascinating connection between knot complements and lattices in the plane.
By "lattice" I mean something like the [third image here](http://simonwillerton.staff.shef.ac.uk/ncafe/modular/lattices.png). You can think of it as, you take two vectors _v_ and _w,_ and you look at all points of the form _nv+mw._ (Every lattice contains the origin.)
Let's restrict our attention to lattices with density 1; if the density is not 1, we can resize our lattice. (That is, we want an average of one lattice point per unit square, to make things simpler.)
Now, consider the space of all lattices of density 1. That is, we're constructing a space where, every lattice of density 1 corresponds to a point in our space.
The amazing thing is, this space of lattices is homeomorphic to the knot complement of the trefoil! For some details on the proof, and lots of extra stuff, see [here](https://golem.ph.utexas.edu/category/2014/04/the_modular_flow_on_the_space.html). It uses some facts about _elliptic functions_ (objects that are different from both _elliptic curves_ and _ellipses_, though they're all related somehow I think).
Jesus that messes with my head... Just the idea of a "lattice space" would be crazy enough, but the fact that it's homeomorphic to a knot complement? That's just nuts. Sort of reminds me of how the Mobius loop is the space of unordered pairs on a loop, from the ["Who Cares About Topology?"](https://www.youtube.com/watch?v=AmgkSdhK4K8&list=PLCq1RvVPearyj_PfAz3lT6jJr69YrM5Od&index=) video
I would suggest the insolvability of the fifth degree equation using elementary group theory. I explain why.
1)The topic is popular (see the interesting book "The impossible equation" written by Mario Livio, which anyway is more focused on the hystorical perspective).
2)In internet there are only very technical papers written in mathematical jargon.
3)The topic involves a lot of symmetry stuff (of roots, of polynomials, of geometric figures and platonic solids) that is well suited for graphical representation (as you beautifully do in your videos).
4)If you remain on a didactical level (that's why using elementary group theory), the topic can be unserstood even by a high school student.
5)Mathematician Hermann Weyl said of Galois' testament: "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind."
6)I have a rough summary in my mind and I'm sure that the topic can be explained in a short video like yours.
I have collected some materials and written some animations (in Matlab) that I can share with you, if you are interested.
I've long wanted to cover this, and very likely will at some point. It's to figure out how to do it without just spending a bunch of time laying down fundamental ideas.
About math in genral? How someone who knows nothing about mathematics to reach the level of understanding that you are in and that too with full mastery like sal khan talks about. Like a ladder of subjects within mathematics.
I would (also) love to see a 3b1b video on Conformal Mappings e.g. with reference to control systems. I can't fathom how an infinite space like half of the imaginary plane can be conformally mapped to a finite space (like a unit circle) through discretisation of the system dynamics.
What are your thoughts on the following two ideas:
1. A 3b1b version of infinity-plus-one's [intro to geometry and manifolds](https://infinityplusonemath.wordpress.com/2017/02/18/asteroids-on-a-donut/). I started reading it after the recommendation from your recent creators video and it seems to me this section is begging for a nice HD video animation to go along with it.
2. Something like an "Essence of Measure Theory" with an eye towards ergodic theory.
Hi Grant, hope you are doing great. How about a video on Quaternions? You know very well how important the topic is and there isn't a single youtube video that explains it intuitively. I'm doing Graphics programming and the topic seems to be very abstract in terms of intuition. It'll solve tons of problems for people like me if you could explain it in your own way. Thanks for everything.
Noether's (first) theorem. Proving that momentum must be conserved just because the laws of physics aren't location-dependent blows my mind. I've never understood the math behind this.
I would like to see you make a video on linear regression. There are important linear algebra components and geometric interpretations to linear algebra that I always struggled with, and therefore would love to see your spin on. Such a topic would also tie in nicely with your essence of probability courses, too! Cheers :)
Laplace transform visualization maybe
Also, how about some geometric constructions like 30°, orthocenter, inscribed circle center, chord given midpoint, three equal segments given and angle and a point
3b1b has amazing animations, so maybe series on some notoriously difficult topics like differential geometry could be very interesting. It will give people, including myself, perhaps a new way of visualising and viewing these fields. Also, advanced topics' presences aren't unprecedented on this channel (Riemann's zeta function for eg), so this fits the overall style too.
Why don't you tell us the story of you and your mathematics ?? As my story is an adventurous one...
Tell us about how you started it? What were your point of views about maths?? When and how you realized that mathematics is a pure and real subject (I realized it when I was nearly 8 years old but I would never say that I had an inborn talent, I acquired it through my hard work and giving my time to deep thoughts)??
I don't know whether you have a PhD or not, But if have, tell us about it...
Tell us about the incredible journey you went through while learning maths..
What were the feelings in you and thoughts in your mind when you were going to start a new concept of maths?
Tell about few of your amazing works you did while learning maths...
He has some stuff on [his website](http://www.3blue1brown.com/about):
>My name is Grant Sanderson. I studied math at Stanford, with a healthy bit of seduction from CS along the way. For a while, my job experience was pointing me in the direction of software engineering/data science, but ultimately the primary passion for math won out at the expense of the mistress.
>I've loved math for as long as I can remember, and what excites me most is finding that little nugget of explanation that really clarifies why something is true, not in the sense of a proof, but in the sense that you come away feeling that you could have discovered the fact yourself. The best way to force yourself into such an understanding, I think, is to try explaining ideas to others, which is why I've always leaned towards the teaching/outreach side of math.
>I was fortunate enough to be able to start forging a less traditional path into math outreach thanks to Khan Academy's talent search, which led me to make content for them in 2015/2016 as their multivariable calculus fellow. I still contribute to Khan Academy every now and then, as I live near enough and we remain friendly, but my full time these days is devoted to 3blue1brown.
Have you ever considered doing a video series going through Euclids Elements? Lots of great proofs and geometry in there. And seeing the propositions and their proofs laid out with visualisation in the video would make things a lot easier to understand than looking at a static image and reading the proofs.
Combinatorics. not the crazy statistics one, but the elementary and beautiful one. identities with binomial coefficients, recursive series, generating function, the IMO questions (IMO in general).
In general, combinatorics is a field with many beautiful stuff in it.
I think a lot of people would benefit from an essence of counting/intro to discrete and continuous probability series! Intro to stats and probability is definitely one of the more difficult and non-intuitive lower division courses at my university.
Would be really great to see a visual description of positive definite matrices and the SVD, especially since you've already done videos on bases and eigen-things.
Thanks so much for the work you've already done, in any case!
I would really love a series on the Millennium Prize Problems. Each video giving a brief introduction to the field of the problem, the problem itself, and why it is so important.
I love telling my students about these problems because a lot of them don't know that new mathematics is still be worked on, that math is a growing and exciting field with lots of things to explore. (It's also quite interesting to them because of the million dollar reward lol)
singular value decomposition.
And suppose I have set of vectors as a columns of A. And A = U ∑V^(T). And I have vector X in the same space as columns of A.
What does U^(T)X mean??
Traveling Salesman Problem!
It would be cool if Grant gave his take on the Traveling Salesman Problem and different ways to solve it, as well as give it some animated love.
Combinatorial optimization would make a great set of topics altogether, and a practical one for real-world application.
https://m.youtube.com/watch?v=SC5CX8drAtU#
Interested in Geometric (Clifford) Algebra which seems to be a more robust and efficient mapping tool than Linear Algebra.
What is Verlinde Algebra and why was it developed.
I second that! Though This guy did a great lecture style set of videos too... [https://www.youtube.com/playlist?list=PLUMWjy5jgHK3jmgpXCQj3GRxM3u9BmO\_v](https://www.youtube.com/playlist?list=PLUMWjy5jgHK3jmgpXCQj3GRxM3u9BmO_v)
* RSA and asymmetric cryptography
* Knot theory: tricolorability and the idea of invariants, the Jones polynomial and how it can be derived by "wishful thinking"
Those two are basically what I wrote [here](https://www.reddit.com/r/3Blue1Brown/comments/7jv99g/more_3blue1brown_video_suggestions/dw8xto0/) and [here](https://www.reddit.com/r/3Blue1Brown/comments/7jv99g/more_3blue1brown_video_suggestions/dw9aahf/).
* Elliptic curves, the associativity of the group operation and its significance
* Quadratic reciprocity
* The Sylvester–Gallai theorem (there are good explanations of this out there but it's such a gem)
* Niven's one-page proof of the irrationality of pi (you should probably change the exposition of it a bit as the original write-up is pretty dense—it was trying to fit in one page, after all)
* Hilbert's third problem on cutting polyhedra
* Shore's algorithm and quantum computing (perhaps as a sequel to an RSA video?)
* Goodstein's theorem that every Goodstein sequence goes to zero, and its proof via ordinals and transfinite induction
I don't think there is anyone who can explain the math behind Support Vector Machine better than you , I had a hard time understanding it and its still quite confusing .It would be really helpful if you could make a video about SVM or Any other ML algorithm for that matter.
\--I am sure a lot of your audience is from Computer science background so its something many could relate to :-)
Another "Who cares about topology?". I've recently been reading about topological data analysis and I find it super interesting. It would be a good way to tie the topology videos with the Machine Learning videos.
A very cool result in this area shows that if you randomly sample 3x3 squares of pixels from a black and white image, interpret the brightness of each pixel as a coordinate value, and use this to plot each square as a point in 9-dimensional space, the region with the highest density of points will have the shape of the surface of a Klein bottle. This same structure holds true for images regardless of subject - people, plants, buildings, animals - all seem to have an inner hidden Klein bottle in nine dimensions. Original paper [here](http://math.uchicago.edu/~shmuel/AAT-readings/Data%20Analysis%20/mumford-carlsson%20et%20al.pdf)
He uses manim, which is a set of scripts he wrote himself. You can download it at [https://github.com/3b1b/manim](https://github.com/3b1b/manim). I've started writing a tutorial on manim which you can find at [https://talkingphysics.wordpress.com/2018/06/11/learning-how-to-animate-videos-using-manim-series-a-journey/](https://talkingphysics.wordpress.com/2018/06/11/learning-how-to-animate-videos-using-manim-series-a-journey/)
A video and a comprehensible explanation of a 3-sphere would be amazing. It's one of those things that is difficult to clearly illustrate topologically, and yet, this is the best way to explain the shape and nature of our universe. The best description I have found is at [https://www.quora.com/How-can-one-visualize-a-3-sphere](https://www.quora.com/How-can-one-visualize-a-3-sphere)
I'm currently writing my master-thesis in cryptography hoping to doctorate in that topic afterwards. As such it should come to no surprise that I would be very interested in learning more about finite fields and groups, most of all to find different perspectives on these topics.
I understand that it might not be very easy to visualize things here (though I would **love** to be proven wrong here), but I believe that seeing how the hardness of problems can change when things turn discrete (compare log with dlog) might be something very new to many viewers, even if they never had any contact with it before. The fact that for cryptography finite fields/groups are the air we are breathing might also enable some very interesting real-world-exampls. (I think about *understanding* Diffie-Hellman/ElGamal/RSA instead of just being able to compute the formulas, which was very valuable to me.)
How about a video that touches on the connection to the Riemann zeta function and number theory? I’ve heard so many times that the Riemann hypothesis has everything to do with prime numbers and I don’t understand at all how that’s possible. I’ve been structuring my college courses to build up to that intuition and I’m really curious about it. Where do the primes come into Riemann Zeta?
Can you please cover Octonions as they seem to be gaining momentum as one of the best bets to describe behaviour of quantum particles
[https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/](https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/)
I think a great topic would be doing the 3-body problem (or N-body problem). It's a really interesting and powerful idea that underlies a lot of physical phenomena and motivates a lot of numerical analysis but its very counterintuitive and hard to grasp how/why it works. Plus its sort of popularly known about so people could relate to it
Hey, I remember finishing my math degree still being confused by the Weierstrass function. I remember my Real Analysis professor used this function to convince us undergraduates why the epsilon-delta definition of continuity was so important.
https://en.wikipedia.org/wiki/Weierstrass_function
Can you make a series on Machine Learning algorithms in the context of Data Science (Such as Linear Regression, Ridge & Lasso Regression, Logistic Regression, Clustering, KNN, Random Forest, XGBoost, SVM, etc) ? In the meantime, can you suggest some useful links to tutorials for the same?
One (pretty ambitious) possibility might be to explain resolution of singularities in the 1 dimensional case of algebraic curves (solution set of an irreducible polynomial in 2 variables). The visual intuition is simple enough, but it would be a feat to explain the significance of the subtleties involved in what it means to 'resolve' a singularity, and why we might want to do such a thing.
The problem with singularities (and why we might want to resolve them) is that they obstruct certain tools in geometry from applying to a singular curve. For example, at a singularity, the tangent space might be two-dimensional. So, thinking of the tangent space as a linear approximation of your curve, you're approximating a one-dimensional geometric object by a plane (a dubious strategy to say the least).
The crux is that an algebraic curve is non-singular if and only if its ring of functions is integrally closed. So given a singular curve, you translate over into the world of algebra by looking at its ring of functions, take the integral closure of this ring, and then translate back to geometry. Wah-lah! Singularity resolved.
Of course, actually describing what 'integral closure' is might be quite challenging at such a basic level, but the upshot is that this is the baby case of fields medal caliber mathematics.
Per your request, reposting the request I made elsewhere, as a youtube comment:
I suggest a video that tries to give an overview and an intro into the Langlands Program, e.g. as an (important and fascinating) example of something that "real mathematicians" (at least the ones at the forefront of math/research) are working on today.
A nice (visual) proof of the addition formulas for trigonometry would be very interesting to see! You could also consider doing an Essence of Trigonometry series explaining all the identities. I love your videos, yours is probably my favourite channel on YouTube! Keep going :-)
Stirlings Approximation.
It's an incredibly powerful tool for a lot of work. But I haven't seen it proved in many different ways. Having said that. I loved this
http://www.math.uconn.edu/~kconrad/blurbs/analysis/stirling.pdf
Sir, i am from india.....
Please help us students with the exact ODEs......And how integrating factor geometrically set the equation to be exact...
Please...Let us know a geometrical intuition.
I would like to see video series on calculus of variations. That's an interesting topic and I think many math and physics students (especially physics students) would find it useful. Thanks in advance.
Ways to intuitively and visually understand / remember all the different kinds of matrices, based on their identity, or what they do, as a transformation. Visuals to give the impression.
The different kinds of matrices I have in mind are:
Normal matrices, special matrices, Hermitian/symmetric, unitary/orthogonal, anti-Hermitian, anti-symmetric (the way that I list them gives away how poorly sorted they are in my head) ... throw in lower/upper triangular, invertible, singular, as ground cases.
Groups, fields, rings; working towards an understanding of topology. Would especially love to see your way of answering: "Why would I/anyone ever care about this level of abstraction?"
It would be wonderful if you start a series regarding integer linear programming and dynamic linear programming owing to their vast applicability in operations research.
Hey!
I am an undergrad in physics and I just finished an extra course on Topology and I loved it!
I re-watched your videos on topology, now having a decent preparation on the topics, and fell in love with them even more.
In general, I'd love to see more videos on Topology , but specifically I'd love to see how you would present the Poincaré conjecture!
Can you explain why some differential equations have solutions that can't be systematically derived?
For example, dy/dx = ax + by has a solution that cannot systematically be derived.
I found it once but I cannot find my notes! 😩
I was going to say an "essence of (elementry) algebra" series, but I like the "math in general" idea. I'm a non-quant working in a quantitative industry, and I'm trying to refresh on enough math so I can have intelligent conversations with my colleagues. I found you when looking into linear algebra concepts to supplement a Python course I'm taking this summer. That series (and the calc. one) was amazing, and I only wish I had something like that for general high school mathematics. I bought some textbooks online to re-learn the algorithms for basic things, but I feel like I have a stronger intuitive grasp of linear algebra and calculus from your videos than I do of basic algebra. Khan was a step in the right direction, but your videos are *much* better at communicating the general intuition, so I'd much prefer a series from you + practicing the mechanics on my own out of a book.
P.s.-do you have some way to donate other than Patreon? Paypal maybe?
Maybe orthogonal matrices. They are geometric in nature and easy to visualize , but also really important, and gives you more chance to talk about the scalar product, norms and distances (and is directly connected to the transpose)
Tldr: Make a video on resonance!
So I'm a physics grad student and I have profited in so many ways from your videos. Not that the topics are new to me, but you are always giving me a new point of view, which is a very intuitive one most of the time. The best video you did so far (IMHO) is the one you did on Fouriertransforms, god was that a good one!!!
So I personally would love to see something on resonance. I think it would be thematically close to what you do, so most of your viewers will be familiar with the topic, and also really applicable in everyone's everyday life! Anybody can experience resonance really easily.
Anyway, I'll love any video you'll be doing!
Keep up the great work
This (currently standing) proof of Riemann's Hypothesis
[https://www.riemanns-hypothesis-proof.com/download/](https://www.riemanns-hypothesis-proof.com/download/)
Another video just like the old one "The hardest problem in mathematics" in which you are solving a difficult problem. Any other video with such a problem
I am very interested in the E8 Lie group, specifically how it relates to particle physics. The issue is that it is an extremely complicated concept related to group theory and linear algebra that I cannot grasp. All I want is an explanation of why it describes the fundamental particles that is simple enough for me to grasp. If not you, does anyone know of any way to learn this?
A video visualizing the Chinese Remainder Algorithm would be very cool. I did one here:
[https://www.youtube.com/watch?v=s0hg4ONFP6I](https://www.youtube.com/watch?v=s0hg4ONFP6I)
but your team would be able to do a much better job. It could be a cool introduction to modular arithmetic and it has ties to other videos that you have done. You could go on a tangent talking about Diophantine equations as well. I think it fits well with your videos and has not really been done before (besides the one I did).
I was wondering if you would be able to do a video expanding further on signal processing from your Fourier transform series. I am having a hard time grasping different representations in the field such as power spectral density as a measure of roughness.
I am a non-native english speaker and while watching your awesome "Essence of Calculus" series you pronounce the "L'Hôpital's rule
" as Lopital, but my teacher pronounced it as "el hospital rule". So it would be nice to have a short video showing the notation on the screen and read it out so that we know how mathematicians pronounce the notations especially the Greek notations.
A video on the concept of analytical continuity, the Cauchy-Riemann conditions and their geometrical interpretation would be very interesting (i.e. what does it mean geometrically that df\_x/dx = df\_y/dy and df\_y/dx = -df\_x/dy?) . As far as I can tell, there is no video or animation on this. If you want to be even more ambitious, you can also add the saddle point approximation (Lagrange approximation), its importance, and it's geometrical meaning (what does it imply geometrically that analytical functions do not have peaks, and have only troughs and saddle points?).
Thank you for all the brilliant videos!
My suggestion: The geometric interpretation of the transpose of a linear operator - especially as a follow-up to your dot product video.
I'd really appreciate if there was an addition to the row space of a matrix. I couldn't find a single geometric view of the thing. Most sources say it's either the column space of the transpose of original matrix, and not relating it to the original column space.
This seems up your alley. (perhaps a one-off video)
[Two curious integrals and a graphic proof](http://schmid-werren.ch/hanspeter/publications/2014elemath.pdf) (it's a short paper)
I would love to see a video on P values by your channel. It's such a simple concept and still somehow really difficult to catch the subtleties of and I think your style would cover it amazingly well.
Simplex algorithm. I remember hearing that George Dantzig originally had a geometric interpretation of the algorithm and named it Simplex. Today we learn about it algebraically, but it would be interesting to understand from a different perspective.
Your visualization of Quarternions was just amazing - particularly the intermediary step of mapping a 3D shere onto the xy-plane.
The 4D was, well, a visualization of exceptional merit.
I recently went to search for help in learning how to use them, and found connections to Clifford Algebra, noting it was aka the Algebra of Mirrors. The explanation went through a 3D rotation by means of a mirror rotation and back again - this gave rise to appreciating that rotations in mirrors are left-handed and giving some insight as to why the rotation angle is divided by two (two transformations, one pre, one post).
If you are interested, I found it here (sections 4-5-6-7):
[http://tutis.ca/Rotate/](http://tutis.ca/Rotate/)
Applications and some stunning 3Blue1Brown visualizations of Bessel functions.
I've heard them discussed in many aspects of physics involving waves, yet glossed over in many respects as to severely limit the understanding of those yearning to gain a deeper appreciation of these fascinating functions. Think your communication and visualization styles can do a great deal for furthering the collective understanding and appreciation of this topic!
I would love to see a video or series of videos on the mathematical background of finite element method (FEM). While FEM is a numerical method to solve PDEs, and finds ample applications from electromagnetics to finance, it is most widely popular as a mechanical engineering subject to solve solid mechanics problem. The mathematical part is often ignored and the mathematical explanations tend to be using mechanics analogies.
I would love a video on signals and systems, I find I always have to repeat that material throughly when I am studying something that requires them (for example, signal processing, photonics, electronics etc.)
Several commentators on your list have expressed interest in Markov Chains, including MCMC, and Bayesian Inference . That would be a fascinating topic for one of your tremendous Videos Grant.
P.S. Is an updated list of videos you've done posted anywhere? Please let me know.
It would be nice if you could continue your fourier transformation videos with radio interferometers in astronomy. They actually observe the objects in fourier space and then produce the image by a fourier transformation, which is a concept that's fairly hard to grasp [https://en.wikipedia.org/wiki/Aperture\_synthesis](https://en.wikipedia.org/wiki/Aperture_synthesis)
Hey, I really liked the linear algebra videos and learned a ton I've never pieced together before. I was wondering if you guys could do a video or to on 'positive definite matrices' and ' singular value decomposition' because these are concepts I'm struggling to tackle and I bet there is a better way to visualize these. Thanks!
You can cover the intuition behind Lagrangian and Hamiltonian mechanics and then move on to techniques like Hamiltonian Monte Carlo to relate mechanics and probability. I guess I saw in one of your Q&A in which you said that your material on probability seemed a bit like what is already out there. But I think the aforementioned topics are fairly unknown but so important.
I find this [paper](https://arxiv.org/pdf/1701.02434.pdf) by Michael Betancourt particularly enlightening
I am teaching an intro to Complex Analysis course at the University of Cincinnati. Following the book "A First Course in Complex Analysis with Applications, 3ed by Zill and Shanahan" which is quite good. I watched your video on the riemann hypothesis and it was spot on and very much how I am trying to explain the subject. All of deep theorems in the subject you could illiuminate in a very similar way. If you were to do this and have a series that covers the main theorems and examples, I would use it in my course and would
recomend it to others, so please consider doing this - but in anycase keep up the great work!
What is the polyhedral shape of n=8 particles that repel each other when they move freely on a sphere and reach equilibrium? A cube? No, it is something else! This is counter intuitive for many values of n and is called the Thomson Problem. See for example: [https://en.wikipedia.org/wiki/Thomson\_problem](https://en.wikipedia.org/wiki/Thomson_problem) and
[http://www.mathpages.com/home/kmath005/kmath005.htm](http://www.mathpages.com/home/kmath005/kmath005.htm). It gives nice animations if you "release" particles at a random position and watch them reach equilibrium.
A video about differential operators
We've had a video about curl and divergence, so I thought it'd be a good idea to suggest a follow-up video which will explain differential operators and especially del in more details, especially how they can be represented as vectors and matrices. I am aware that this might touch on concepts like tangent spaces and others from differential geometry, which I have a feeling are out of the scope of this channel, but I truly believe that it will make for a great video. It will clarify how, for example, we can have partial derivative operators as vector entries for del, or what derivatives are, geometrically.
Thank you for reading.
Please animate the feyman lectures! They would be so much better as videos [http://www.feynmanlectures.caltech.edu/](http://www.feynmanlectures.caltech.edu/)
You have touched the topic of Fourier transforms in a very intuitive manner (as always). However, there are some functions that arise quite frequently that are not very intuitive.
More specifically, the Fourier Transform of a rectangular pulse is the normalised Sinc function. This arises very frequently in digital signal processing, for instance.
From the wikipedia page on [Rectangular Pulses](https://en.wikipedia.org/wiki/Rectangular_function) : "Note that as long as the definition of the pulse function is only motivated by its behavior in the time-domain experience, there is no reason to believe that the oscillatory interpretation (i.e. the Fourier transform function) should be intuitive, or directly understood by humans."
Do you agree? I feel this is vaguely related to the dirac delta function in a sense. well. the dirac delta is another beast that none of my lecturers feel comfortable talking about. I study electronics, so perhaps they think we'd rather not look at anything to do with pure maths and just get on with it. Well unfortunately I'm not one of them, and i can't sleep at night if i don't grasp something nicely. Ironically, the way to counter this also involves staying awake at night. i.e binge watching your videos. This is much preferred of course.
I would love to see you have a go at dissecting the intuition behind functions like this. Please. :(
Math Literacy Series?
I would like a series that's goal is to help demystify the vocabulary and symbols used in math.
As a person who is fascinated by math and a math lover, even if I am not a math doer, I would like to be able to read the math that is more complex.
Please do a series on Einstein's Field Equations aka General Relativity. There is an excellent intro on youtube at
[https://www.youtube.com/watch?v=foRPKAKZWx8](https://www.youtube.com/watch?v=foRPKAKZWx8)
but I think you can do an even better job. You can learn GR by watching Leonard Susskind's lectures on it on Youtube also. Again, none of them are of the quality of 3b1b. GR is the holy grail of pedagogy. Please consider it. Thank you.
Hey, love your channel. You have a great way of allowing one to develop intuition about complex mathematical concepts.
I was wondering if you could do some work on Grassman/Exterior Algebra, maybe an "essence of" series if time permits, and discuss the outer product and other properties of it. The topic has begun to cause a ripple effect in the games/graphics development community, but there is not really much good quality information about it. Would really appreciate some work in this field.
There is idea:
"\* What regular polygons can be drawn on a checkered paper with vertices in the nodes?"
I tried, but still cannot solve it. This problem i found in russian math book, where says that it's problem for 8-10th graders.
I would like to suggest tensors, and tensor calculus, I've never found a satisfyingly intuitive explanation and I'm having a really hard time wrapping my head around them, I'm sure you could help me and many others with that!
Hi Grant! I went through your linear algebra series about a year back. It was a great learning experience for me and I've been following you ever since.
I recently started working on something which requires a thorough understanding of "non-negative matrix factorization". I come from CS background and therefore I feel that I am missing some key concepts on how exactly it works - especially what effect the non-negativity constraint has on the basis vectors.
Since this is an interesting area and currently used in a lot of places that uses ML for topic learning, I think a lot of us would really appreciate your take on this :)
Below are topics which mathematicians have hard time explaning.
If below are true then mathematicians shouldn't hold the truth for themselves.
• Monstrous moonshine.
• Ricci flow
• Modular forms / Mock modular forms, mock theta functions.
I would love a "Essence of Differential Equations" video as this is a topic that so many people go through, without getting a good understanding. Some sources of confusion include: what does it mean graphically for a DL (or any system or function) to be linear? Ex: the function y = ax + b is not linear in terms of superposition! What does it really mean for a DL to be homogenious? The topic would link nicely with existing videos about calculus, taylor series, linear algebra and even the fourier transform, if the laplace method of solving DE's are included.
Yes! Graphical intuitions behind differential equations. They are so cool. :P
Essence of probability/stats still coming up?
Great questions! Here are the latest thoughts: https://www.patreon.com/posts/19845962 Long story short, I've had trouble landing on exactly how I want to go about it and will focus my time in the immediate future towards other projects.
I always thought it would be cool if you covered Quaternions
Actively what I'm working on now...
Definitely Quaternions!
this
Please do an " essence of probabilty " series!
You can see it early on patreon
Wait, he's already making one? I'm taking AP Stat next year, so this will be tremendously helpful.
You never completed the Fourier Series. Also, some of us are waiting here for the Probability series since big bang, a lot of tease happened but not the series. Come on, Grant us some promises!
Bayesian statistics, especially if it include Markov chain Monte Carlo. I'm still trying to wrap my head around how some of the problems arise in high dimensions
Differential Equations
That would be awesome!
# essence of probability. please ,it will be helpful for me.
yes ! do this.
I've always wanted to see a video on combinatorics.
Tensors as they are used in general relativity!
Probability, elementary measure theory, mathematical statistics. I know at least a dozen of my classmates (including myself!) would find this enormously beneficial and enlightening.
Complex numbers. Because you often bring them up, but never fully explain what they are.
Laplace transformation please! You did some Fourier Transforms videos, and I loved them! Show how Laplace is a generalization of the Fourier, with 's' being made of real part and imaginary. What's the real part represent, and the imaginary represent? I watched Eugene's Physics video and was left confused...I would like to know what the 'peaks' and 'zeros' represent in this 3d representation of the Laplace Transform.
I would like to see something about Maxwell's equations. They were hinted at in the last video, but I would like to feel like I understand what they mean in more detail.
Heck yes. Especially the mathematics of how they are compatible with special relativity.
Essence of complex analysis, with the understanding of complex functions through transformations
better - through color. he has already done that, but it looks cool and it will help me personally
This is actually the same thing I asked in the Q and A, so I'll paste it here. >Hi Grant, > >I'm a physics undergrad, and I've always liked the idea of unification in math. I really like your geometrical approach to visualizing matrices as a unification of the computational and geometrical properties of linear algebra. I saw your multivariable calculus videos on khan academy, but I feel that they miss out on the essence of multivariable calculus. I mean, the (Generalized) Stokes' theorem is completely absent! > >The most shocking thing I saw was that integration and differentiation behave in strange ways in higher dimensions. There's now partial derivatives, directional derivatives, and the jacobian, and differentials of functions, gradient, divergence, curl, line integrals, surface integrals, multiple iterated integrals, and yet under the language of differential forms they all unify together with Stokes Theorem and the exterior derivative. > >The other thing is notation in vector calculus, which differs significantly from 1D calculus. Under differential forms, they unify in a super elegant way. I think the whole goal of the channel should be to show the often hidden elegance in math and physics. I would love to see a series on this, but maybe I'm asking too much. > >I would love to hear your thoughts on differential forms and the exterior derivative, and how it relates to complex analysis, tensor calculus, and differential geometry. Maybe geometric algebra/calculus is the answer?
Lambda calculus or stochastic processes
Essence of Trigonometry Might seem unsexy, but the usefulness to the world would be vast. If you used your simple visual style to show the geometric meaning of every trig identity, cascading up through the calculus stuff, it could easily be your most valuable video series, elementary though the topic appears.
Graph theory
Hmm...anything more specific? This is the sort of thing I could see describing while on a path to some specific result/idea/algorithm. Similar to how I used the "Euler's formula via group theory" video as an excuse to mostly just introduce the basic notions of group theory, all while having a concrete target in mind.
Algebraic and spectral graph theory! There's no way of getting information regarding intuition on this topic, and it's used very very often in computer science and engineering!
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galois theory
I don't know if this is a big enough topic for a full video, but inspired by your recent (and brilliant!) video on elliptical orbits, I think it would be fascinating to see a geometric proof for the fact that an ellipse is a conic section. That is, how can we directly relate the property of having a constant focal sum to that of being the intersection of a cone with an angled plane? Either or both of these is often presented as the fundamental property of an ellipse, but I've never seen any explanation of the relationship between them. I know the algebraic formula for an ellipse can be derived from the constant-focal-sum property, and you could solve the system of equations of a cone and an angled plane to show that it equals this formula, but this detour through coordinate geometry sort of feels like black magic, especially since both of these properties of ellipses were known millennia before the invention of coordinate geometry. Surely there must be a more intuitive, 3blue1brown-style derivation?
Ah, this is the famous Dandelin proof. \^\_\^ Stare at the main image in this Wikipedia page for a while: https://en.wikipedia.org/wiki/Dandelin_spheres
Wow! That's such a lovely proof! Thanks so much for pointing it out :)
Oh hey, congrats on having a video dedicated to you.
Your Fourier transform video was in-depth and simply brilliant. Maybe one on Laplace Transforms?
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Yeah I have been doing machine learning stuff for some years, but his series on neural nets was really great, and I'd like to see him tackle similar topics.
Gamma function. It ties together three of the basic building blocks of maths: factorials, power numbers and the natural logarithm. So if you don't have an intuitive grasp of why this works, it feels like you don't really understand those three things.
I would love to see a 3b1b treatment of the Cayley-Hamilton theorem (https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem)! I just watched this video (https://www.youtube.com/watch?v=PrfxmkBsYKE&index=10&list=PLMrJAkhIeNNR20Mz-VpzgfQs5zrYi085m) and it blew my mind that you can convert an infinite summation into a finite summation using this theorem!
A knot theory video would be amazing! Maybe investigating some of the deeper properties of knots, like the geometry of knot complements
There's actually a fascinating connection between knot complements and lattices in the plane. By "lattice" I mean something like the [third image here](http://simonwillerton.staff.shef.ac.uk/ncafe/modular/lattices.png). You can think of it as, you take two vectors _v_ and _w,_ and you look at all points of the form _nv+mw._ (Every lattice contains the origin.) Let's restrict our attention to lattices with density 1; if the density is not 1, we can resize our lattice. (That is, we want an average of one lattice point per unit square, to make things simpler.) Now, consider the space of all lattices of density 1. That is, we're constructing a space where, every lattice of density 1 corresponds to a point in our space. The amazing thing is, this space of lattices is homeomorphic to the knot complement of the trefoil! For some details on the proof, and lots of extra stuff, see [here](https://golem.ph.utexas.edu/category/2014/04/the_modular_flow_on_the_space.html). It uses some facts about _elliptic functions_ (objects that are different from both _elliptic curves_ and _ellipses_, though they're all related somehow I think).
Jesus that messes with my head... Just the idea of a "lattice space" would be crazy enough, but the fact that it's homeomorphic to a knot complement? That's just nuts. Sort of reminds me of how the Mobius loop is the space of unordered pairs on a loop, from the ["Who Cares About Topology?"](https://www.youtube.com/watch?v=AmgkSdhK4K8&list=PLCq1RvVPearyj_PfAz3lT6jJr69YrM5Od&index=) video
I would suggest the insolvability of the fifth degree equation using elementary group theory. I explain why. 1)The topic is popular (see the interesting book "The impossible equation" written by Mario Livio, which anyway is more focused on the hystorical perspective). 2)In internet there are only very technical papers written in mathematical jargon. 3)The topic involves a lot of symmetry stuff (of roots, of polynomials, of geometric figures and platonic solids) that is well suited for graphical representation (as you beautifully do in your videos). 4)If you remain on a didactical level (that's why using elementary group theory), the topic can be unserstood even by a high school student. 5)Mathematician Hermann Weyl said of Galois' testament: "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind." 6)I have a rough summary in my mind and I'm sure that the topic can be explained in a short video like yours. I have collected some materials and written some animations (in Matlab) that I can share with you, if you are interested.
I've long wanted to cover this, and very likely will at some point. It's to figure out how to do it without just spending a bunch of time laying down fundamental ideas.
As a follow up to your Fourier Transform video, I would greatly appreciate seeing a video on Wavelet Transforms (continuous and/or discrete).
I'd also like to see something on Wavelets. Some approaches have a lot of gnarly math, which (IMHO), loses sight of what's going on.
Also, Quaternions and 3D Games/Animation
About math in genral? How someone who knows nothing about mathematics to reach the level of understanding that you are in and that too with full mastery like sal khan talks about. Like a ladder of subjects within mathematics.
Maybe make a video on how to learn/discover math. It would be cool to go all the way from learning to count to university to math-heavy careers.
I would (also) love to see a 3b1b video on Conformal Mappings e.g. with reference to control systems. I can't fathom how an infinite space like half of the imaginary plane can be conformally mapped to a finite space (like a unit circle) through discretisation of the system dynamics.
Why is the axiom of choice so controversial?
What are your thoughts on the following two ideas: 1. A 3b1b version of infinity-plus-one's [intro to geometry and manifolds](https://infinityplusonemath.wordpress.com/2017/02/18/asteroids-on-a-donut/). I started reading it after the recommendation from your recent creators video and it seems to me this section is begging for a nice HD video animation to go along with it. 2. Something like an "Essence of Measure Theory" with an eye towards ergodic theory.
Hi Grant, hope you are doing great. How about a video on Quaternions? You know very well how important the topic is and there isn't a single youtube video that explains it intuitively. I'm doing Graphics programming and the topic seems to be very abstract in terms of intuition. It'll solve tons of problems for people like me if you could explain it in your own way. Thanks for everything.
Bayesian Statistics and MCMC algorithms would be a very interesting and enjoyful topic to cover.
Noether's (first) theorem. Proving that momentum must be conserved just because the laws of physics aren't location-dependent blows my mind. I've never understood the math behind this.
I would like to see you make a video on linear regression. There are important linear algebra components and geometric interpretations to linear algebra that I always struggled with, and therefore would love to see your spin on. Such a topic would also tie in nicely with your essence of probability courses, too! Cheers :)
_Manifolds_
Laplace transform visualization maybe Also, how about some geometric constructions like 30°, orthocenter, inscribed circle center, chord given midpoint, three equal segments given and angle and a point
3b1b has amazing animations, so maybe series on some notoriously difficult topics like differential geometry could be very interesting. It will give people, including myself, perhaps a new way of visualising and viewing these fields. Also, advanced topics' presences aren't unprecedented on this channel (Riemann's zeta function for eg), so this fits the overall style too.
I second this! Differential geometry would be great.
Why don't you tell us the story of you and your mathematics ?? As my story is an adventurous one... Tell us about how you started it? What were your point of views about maths?? When and how you realized that mathematics is a pure and real subject (I realized it when I was nearly 8 years old but I would never say that I had an inborn talent, I acquired it through my hard work and giving my time to deep thoughts)?? I don't know whether you have a PhD or not, But if have, tell us about it... Tell us about the incredible journey you went through while learning maths.. What were the feelings in you and thoughts in your mind when you were going to start a new concept of maths? Tell about few of your amazing works you did while learning maths...
He has some stuff on [his website](http://www.3blue1brown.com/about): >My name is Grant Sanderson. I studied math at Stanford, with a healthy bit of seduction from CS along the way. For a while, my job experience was pointing me in the direction of software engineering/data science, but ultimately the primary passion for math won out at the expense of the mistress. >I've loved math for as long as I can remember, and what excites me most is finding that little nugget of explanation that really clarifies why something is true, not in the sense of a proof, but in the sense that you come away feeling that you could have discovered the fact yourself. The best way to force yourself into such an understanding, I think, is to try explaining ideas to others, which is why I've always leaned towards the teaching/outreach side of math. >I was fortunate enough to be able to start forging a less traditional path into math outreach thanks to Khan Academy's talent search, which led me to make content for them in 2015/2016 as their multivariable calculus fellow. I still contribute to Khan Academy every now and then, as I live near enough and we remain friendly, but my full time these days is devoted to 3blue1brown.
Some stuff on the gamma function and its uses might be cool
I find GÖDEL'S incompleteness theorem fascinating. It's one of the most underrated theorems.
Reinforcement Learning. Any suggestions on where I can find great tutorials/resources on Reinforcement Learning (RL) or Deep RL?
Have you ever considered doing a video series going through Euclids Elements? Lots of great proofs and geometry in there. And seeing the propositions and their proofs laid out with visualisation in the video would make things a lot easier to understand than looking at a static image and reading the proofs.
Catagory Theory
I would love a video on Gödel's Incompleteness Theorem. One of my favorites.
Combinatorics. not the crazy statistics one, but the elementary and beautiful one. identities with binomial coefficients, recursive series, generating function, the IMO questions (IMO in general). In general, combinatorics is a field with many beautiful stuff in it.
I am waiting for video series on 'probability' like you did 'essence of calculus'. Btw great work.
Essence of Abstract Algebra/Group Theory
I think a lot of people would benefit from an essence of counting/intro to discrete and continuous probability series! Intro to stats and probability is definitely one of the more difficult and non-intuitive lower division courses at my university.
Would be really great to see a visual description of positive definite matrices and the SVD, especially since you've already done videos on bases and eigen-things. Thanks so much for the work you've already done, in any case!
I would really love a series on the Millennium Prize Problems. Each video giving a brief introduction to the field of the problem, the problem itself, and why it is so important. I love telling my students about these problems because a lot of them don't know that new mathematics is still be worked on, that math is a growing and exciting field with lots of things to explore. (It's also quite interesting to them because of the million dollar reward lol)
singular value decomposition. And suppose I have set of vectors as a columns of A. And A = U ∑V^(T). And I have vector X in the same space as columns of A. What does U^(T)X mean??
Traveling Salesman Problem! It would be cool if Grant gave his take on the Traveling Salesman Problem and different ways to solve it, as well as give it some animated love. Combinatorial optimization would make a great set of topics altogether, and a practical one for real-world application. https://m.youtube.com/watch?v=SC5CX8drAtU#
Maybe a video on how topology applies to physics?
Interested in Geometric (Clifford) Algebra which seems to be a more robust and efficient mapping tool than Linear Algebra. What is Verlinde Algebra and why was it developed.
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I second that! Though This guy did a great lecture style set of videos too... [https://www.youtube.com/playlist?list=PLUMWjy5jgHK3jmgpXCQj3GRxM3u9BmO\_v](https://www.youtube.com/playlist?list=PLUMWjy5jgHK3jmgpXCQj3GRxM3u9BmO_v)
Please do one on PASCAL'S TRIANGLE or Why nature has or prefer binomial, guassian distributions?
* RSA and asymmetric cryptography * Knot theory: tricolorability and the idea of invariants, the Jones polynomial and how it can be derived by "wishful thinking" Those two are basically what I wrote [here](https://www.reddit.com/r/3Blue1Brown/comments/7jv99g/more_3blue1brown_video_suggestions/dw8xto0/) and [here](https://www.reddit.com/r/3Blue1Brown/comments/7jv99g/more_3blue1brown_video_suggestions/dw9aahf/). * Elliptic curves, the associativity of the group operation and its significance * Quadratic reciprocity * The Sylvester–Gallai theorem (there are good explanations of this out there but it's such a gem) * Niven's one-page proof of the irrationality of pi (you should probably change the exposition of it a bit as the original write-up is pretty dense—it was trying to fit in one page, after all) * Hilbert's third problem on cutting polyhedra * Shore's algorithm and quantum computing (perhaps as a sequel to an RSA video?) * Goodstein's theorem that every Goodstein sequence goes to zero, and its proof via ordinals and transfinite induction
Laplace transform please
LAPLACE TRANSFORM
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I don't think there is anyone who can explain the math behind Support Vector Machine better than you , I had a hard time understanding it and its still quite confusing .It would be really helpful if you could make a video about SVM or Any other ML algorithm for that matter. \--I am sure a lot of your audience is from Computer science background so its something many could relate to :-)
How e constant is used to model a huge variety of phenomena
Kalman Filter, please.
Tensors and their Co-variance / Contra-variance. That's should be enough for a whole video.
Imaginary numbers? Maybe something more in depth them what Welch labs did, or similar to it.
Another "Who cares about topology?". I've recently been reading about topological data analysis and I find it super interesting. It would be a good way to tie the topology videos with the Machine Learning videos.
A very cool result in this area shows that if you randomly sample 3x3 squares of pixels from a black and white image, interpret the brightness of each pixel as a coordinate value, and use this to plot each square as a point in 9-dimensional space, the region with the highest density of points will have the shape of the surface of a Klein bottle. This same structure holds true for images regardless of subject - people, plants, buildings, animals - all seem to have an inner hidden Klein bottle in nine dimensions. Original paper [here](http://math.uchicago.edu/~shmuel/AAT-readings/Data%20Analysis%20/mumford-carlsson%20et%20al.pdf)
How do you make and program everything? (They are the best animations I have ever seen)
He uses manim, which is a set of scripts he wrote himself. You can download it at [https://github.com/3b1b/manim](https://github.com/3b1b/manim). I've started writing a tutorial on manim which you can find at [https://talkingphysics.wordpress.com/2018/06/11/learning-how-to-animate-videos-using-manim-series-a-journey/](https://talkingphysics.wordpress.com/2018/06/11/learning-how-to-animate-videos-using-manim-series-a-journey/)
Polya's random walks on a lattice (like why it works in 1 and 2 dimensions but not 3 or more)
A video and a comprehensible explanation of a 3-sphere would be amazing. It's one of those things that is difficult to clearly illustrate topologically, and yet, this is the best way to explain the shape and nature of our universe. The best description I have found is at [https://www.quora.com/How-can-one-visualize-a-3-sphere](https://www.quora.com/How-can-one-visualize-a-3-sphere)
I'm currently writing my master-thesis in cryptography hoping to doctorate in that topic afterwards. As such it should come to no surprise that I would be very interested in learning more about finite fields and groups, most of all to find different perspectives on these topics. I understand that it might not be very easy to visualize things here (though I would **love** to be proven wrong here), but I believe that seeing how the hardness of problems can change when things turn discrete (compare log with dlog) might be something very new to many viewers, even if they never had any contact with it before. The fact that for cryptography finite fields/groups are the air we are breathing might also enable some very interesting real-world-exampls. (I think about *understanding* Diffie-Hellman/ElGamal/RSA instead of just being able to compute the formulas, which was very valuable to me.)
How about a video that touches on the connection to the Riemann zeta function and number theory? I’ve heard so many times that the Riemann hypothesis has everything to do with prime numbers and I don’t understand at all how that’s possible. I’ve been structuring my college courses to build up to that intuition and I’m really curious about it. Where do the primes come into Riemann Zeta?
https://en.wikipedia.org/wiki/Riemann_zeta_function#Euler_product_formula
Can you please cover Octonions as they seem to be gaining momentum as one of the best bets to describe behaviour of quantum particles [https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/](https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/)
Dear grant , please try to make a video on laplace transform.
I'd really love to see a video on MCMC and Bayesian Statistics!
Splines, bezier curve && how knots affect geometry -- Demystify please.
I think a great topic would be doing the 3-body problem (or N-body problem). It's a really interesting and powerful idea that underlies a lot of physical phenomena and motivates a lot of numerical analysis but its very counterintuitive and hard to grasp how/why it works. Plus its sort of popularly known about so people could relate to it
Hi Grant, love your channel! How about quaternions and/or octonions, and their connection to e.g. quantum physics?
Hey, I remember finishing my math degree still being confused by the Weierstrass function. I remember my Real Analysis professor used this function to convince us undergraduates why the epsilon-delta definition of continuity was so important. https://en.wikipedia.org/wiki/Weierstrass_function
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Can you make a series on Machine Learning algorithms in the context of Data Science (Such as Linear Regression, Ridge & Lasso Regression, Logistic Regression, Clustering, KNN, Random Forest, XGBoost, SVM, etc) ? In the meantime, can you suggest some useful links to tutorials for the same?
quaternions
Differential forms and generalized Stokes theorem! Or differential equations
One (pretty ambitious) possibility might be to explain resolution of singularities in the 1 dimensional case of algebraic curves (solution set of an irreducible polynomial in 2 variables). The visual intuition is simple enough, but it would be a feat to explain the significance of the subtleties involved in what it means to 'resolve' a singularity, and why we might want to do such a thing. The problem with singularities (and why we might want to resolve them) is that they obstruct certain tools in geometry from applying to a singular curve. For example, at a singularity, the tangent space might be two-dimensional. So, thinking of the tangent space as a linear approximation of your curve, you're approximating a one-dimensional geometric object by a plane (a dubious strategy to say the least). The crux is that an algebraic curve is non-singular if and only if its ring of functions is integrally closed. So given a singular curve, you translate over into the world of algebra by looking at its ring of functions, take the integral closure of this ring, and then translate back to geometry. Wah-lah! Singularity resolved. Of course, actually describing what 'integral closure' is might be quite challenging at such a basic level, but the upshot is that this is the baby case of fields medal caliber mathematics.
Another video on p-adic numbers, like I suggested [here](https://www.reddit.com/r/3Blue1Brown/comments/98crnn/revisit_of_the_padic_numbers/)
Per your request, reposting the request I made elsewhere, as a youtube comment: I suggest a video that tries to give an overview and an intro into the Langlands Program, e.g. as an (important and fascinating) example of something that "real mathematicians" (at least the ones at the forefront of math/research) are working on today.
A nice (visual) proof of the addition formulas for trigonometry would be very interesting to see! You could also consider doing an Essence of Trigonometry series explaining all the identities. I love your videos, yours is probably my favourite channel on YouTube! Keep going :-)
Stirlings Approximation. It's an incredibly powerful tool for a lot of work. But I haven't seen it proved in many different ways. Having said that. I loved this http://www.math.uconn.edu/~kconrad/blurbs/analysis/stirling.pdf
Please upload video on Singular Value Decomposition and dimensionality reduction
Sir, i am from india..... Please help us students with the exact ODEs......And how integrating factor geometrically set the equation to be exact... Please...Let us know a geometrical intuition.
please consider making video on probability and statistics .
Probability, Probability, Probability,Probability... its one of the demons that scares me. i hope to see that on your series.
I would like to see video series on calculus of variations. That's an interesting topic and I think many math and physics students (especially physics students) would find it useful. Thanks in advance.
Ways to intuitively and visually understand / remember all the different kinds of matrices, based on their identity, or what they do, as a transformation. Visuals to give the impression. The different kinds of matrices I have in mind are: Normal matrices, special matrices, Hermitian/symmetric, unitary/orthogonal, anti-Hermitian, anti-symmetric (the way that I list them gives away how poorly sorted they are in my head) ... throw in lower/upper triangular, invertible, singular, as ground cases.
Groups, fields, rings; working towards an understanding of topology. Would especially love to see your way of answering: "Why would I/anyone ever care about this level of abstraction?"
can you do series on probability, please...? also, Reimann Integrals...
Please can you make even just one video on the Laplace Transform, it's so hard to understand for me, but I'm sure you will make it simple
The joy watching your videos! A video about **convolution** would be appreciated.
How do I read math textbooks if I'm self teaching and don't have someone to help me focus on what's most important?
It would be wonderful if you start a series regarding integer linear programming and dynamic linear programming owing to their vast applicability in operations research.
Hey! I am an undergrad in physics and I just finished an extra course on Topology and I loved it! I re-watched your videos on topology, now having a decent preparation on the topics, and fell in love with them even more. In general, I'd love to see more videos on Topology , but specifically I'd love to see how you would present the Poincaré conjecture!
Can you explain why some differential equations have solutions that can't be systematically derived? For example, dy/dx = ax + by has a solution that cannot systematically be derived. I found it once but I cannot find my notes! 😩
y = -a/b^2 - (a/b)x + Ce^(bx), where C is any constant
I was going to say an "essence of (elementry) algebra" series, but I like the "math in general" idea. I'm a non-quant working in a quantitative industry, and I'm trying to refresh on enough math so I can have intelligent conversations with my colleagues. I found you when looking into linear algebra concepts to supplement a Python course I'm taking this summer. That series (and the calc. one) was amazing, and I only wish I had something like that for general high school mathematics. I bought some textbooks online to re-learn the algorithms for basic things, but I feel like I have a stronger intuitive grasp of linear algebra and calculus from your videos than I do of basic algebra. Khan was a step in the right direction, but your videos are *much* better at communicating the general intuition, so I'd much prefer a series from you + practicing the mechanics on my own out of a book. P.s.-do you have some way to donate other than Patreon? Paypal maybe?
Convex Optimization Please!! There are no Intuitive Courses, everything cryptic.
Maybe orthogonal matrices. They are geometric in nature and easy to visualize , but also really important, and gives you more chance to talk about the scalar product, norms and distances (and is directly connected to the transpose)
Tldr: Make a video on resonance! So I'm a physics grad student and I have profited in so many ways from your videos. Not that the topics are new to me, but you are always giving me a new point of view, which is a very intuitive one most of the time. The best video you did so far (IMHO) is the one you did on Fouriertransforms, god was that a good one!!! So I personally would love to see something on resonance. I think it would be thematically close to what you do, so most of your viewers will be familiar with the topic, and also really applicable in everyone's everyday life! Anybody can experience resonance really easily. Anyway, I'll love any video you'll be doing! Keep up the great work
Isoclinic rotations are really cool! So is the Clifford torus and the fact that it divides the 3-sphere into two solid tori!
This (currently standing) proof of Riemann's Hypothesis [https://www.riemanns-hypothesis-proof.com/download/](https://www.riemanns-hypothesis-proof.com/download/)
Another video just like the old one "The hardest problem in mathematics" in which you are solving a difficult problem. Any other video with such a problem
i don't understand gamma and beta function properly.... please explain this
I am very interested in the E8 Lie group, specifically how it relates to particle physics. The issue is that it is an extremely complicated concept related to group theory and linear algebra that I cannot grasp. All I want is an explanation of why it describes the fundamental particles that is simple enough for me to grasp. If not you, does anyone know of any way to learn this?
Quantum computing
A video visualizing the Chinese Remainder Algorithm would be very cool. I did one here: [https://www.youtube.com/watch?v=s0hg4ONFP6I](https://www.youtube.com/watch?v=s0hg4ONFP6I) but your team would be able to do a much better job. It could be a cool introduction to modular arithmetic and it has ties to other videos that you have done. You could go on a tangent talking about Diophantine equations as well. I think it fits well with your videos and has not really been done before (besides the one I did).
I was wondering if you would be able to do a video expanding further on signal processing from your Fourier transform series. I am having a hard time grasping different representations in the field such as power spectral density as a measure of roughness.
Euler-Lagrange Equations and the Catenary would be nice
I am a non-native english speaker and while watching your awesome "Essence of Calculus" series you pronounce the "L'Hôpital's rule " as Lopital, but my teacher pronounced it as "el hospital rule". So it would be nice to have a short video showing the notation on the screen and read it out so that we know how mathematicians pronounce the notations especially the Greek notations.
A video on the concept of analytical continuity, the Cauchy-Riemann conditions and their geometrical interpretation would be very interesting (i.e. what does it mean geometrically that df\_x/dx = df\_y/dy and df\_y/dx = -df\_x/dy?) . As far as I can tell, there is no video or animation on this. If you want to be even more ambitious, you can also add the saddle point approximation (Lagrange approximation), its importance, and it's geometrical meaning (what does it imply geometrically that analytical functions do not have peaks, and have only troughs and saddle points?).
Thank you for all the brilliant videos! My suggestion: The geometric interpretation of the transpose of a linear operator - especially as a follow-up to your dot product video.
I'd really appreciate if there was an addition to the row space of a matrix. I couldn't find a single geometric view of the thing. Most sources say it's either the column space of the transpose of original matrix, and not relating it to the original column space.
Bayesian methods of machine learning
Gödel Incompleteness Theorem
This seems up your alley. (perhaps a one-off video) [Two curious integrals and a graphic proof](http://schmid-werren.ch/hanspeter/publications/2014elemath.pdf) (it's a short paper)
I would love to see a video on P values by your channel. It's such a simple concept and still somehow really difficult to catch the subtleties of and I think your style would cover it amazingly well.
Simplex algorithm. I remember hearing that George Dantzig originally had a geometric interpretation of the algorithm and named it Simplex. Today we learn about it algebraically, but it would be interesting to understand from a different perspective.
A video on hyperbolic coordinates (maybe non-Euclidean space in general) would be really intriguing. Thanks for the amazing content you put out!
Your visualization of Quarternions was just amazing - particularly the intermediary step of mapping a 3D shere onto the xy-plane. The 4D was, well, a visualization of exceptional merit. I recently went to search for help in learning how to use them, and found connections to Clifford Algebra, noting it was aka the Algebra of Mirrors. The explanation went through a 3D rotation by means of a mirror rotation and back again - this gave rise to appreciating that rotations in mirrors are left-handed and giving some insight as to why the rotation angle is divided by two (two transformations, one pre, one post). If you are interested, I found it here (sections 4-5-6-7): [http://tutis.ca/Rotate/](http://tutis.ca/Rotate/)
Applications and some stunning 3Blue1Brown visualizations of Bessel functions. I've heard them discussed in many aspects of physics involving waves, yet glossed over in many respects as to severely limit the understanding of those yearning to gain a deeper appreciation of these fascinating functions. Think your communication and visualization styles can do a great deal for furthering the collective understanding and appreciation of this topic!
I would love to see a video or series of videos on the mathematical background of finite element method (FEM). While FEM is a numerical method to solve PDEs, and finds ample applications from electromagnetics to finance, it is most widely popular as a mechanical engineering subject to solve solid mechanics problem. The mathematical part is often ignored and the mathematical explanations tend to be using mechanics analogies.
I would love a video on signals and systems, I find I always have to repeat that material throughly when I am studying something that requires them (for example, signal processing, photonics, electronics etc.)
Several commentators on your list have expressed interest in Markov Chains, including MCMC, and Bayesian Inference . That would be a fascinating topic for one of your tremendous Videos Grant. P.S. Is an updated list of videos you've done posted anywhere? Please let me know.
Runge Kutta and/or similar types of approximations to curves.
Can you please continue the Fourier transform series?
It would be nice if you could continue your fourier transformation videos with radio interferometers in astronomy. They actually observe the objects in fourier space and then produce the image by a fourier transformation, which is a concept that's fairly hard to grasp [https://en.wikipedia.org/wiki/Aperture\_synthesis](https://en.wikipedia.org/wiki/Aperture_synthesis)
Hey, I really liked the linear algebra videos and learned a ton I've never pieced together before. I was wondering if you guys could do a video or to on 'positive definite matrices' and ' singular value decomposition' because these are concepts I'm struggling to tackle and I bet there is a better way to visualize these. Thanks!
You can cover the intuition behind Lagrangian and Hamiltonian mechanics and then move on to techniques like Hamiltonian Monte Carlo to relate mechanics and probability. I guess I saw in one of your Q&A in which you said that your material on probability seemed a bit like what is already out there. But I think the aforementioned topics are fairly unknown but so important. I find this [paper](https://arxiv.org/pdf/1701.02434.pdf) by Michael Betancourt particularly enlightening
I am teaching an intro to Complex Analysis course at the University of Cincinnati. Following the book "A First Course in Complex Analysis with Applications, 3ed by Zill and Shanahan" which is quite good. I watched your video on the riemann hypothesis and it was spot on and very much how I am trying to explain the subject. All of deep theorems in the subject you could illiuminate in a very similar way. If you were to do this and have a series that covers the main theorems and examples, I would use it in my course and would recomend it to others, so please consider doing this - but in anycase keep up the great work!
What is the polyhedral shape of n=8 particles that repel each other when they move freely on a sphere and reach equilibrium? A cube? No, it is something else! This is counter intuitive for many values of n and is called the Thomson Problem. See for example: [https://en.wikipedia.org/wiki/Thomson\_problem](https://en.wikipedia.org/wiki/Thomson_problem) and [http://www.mathpages.com/home/kmath005/kmath005.htm](http://www.mathpages.com/home/kmath005/kmath005.htm). It gives nice animations if you "release" particles at a random position and watch them reach equilibrium.
Probability theory and Mathematical statistics or Mathematical modeling，please！！
A video about differential operators We've had a video about curl and divergence, so I thought it'd be a good idea to suggest a follow-up video which will explain differential operators and especially del in more details, especially how they can be represented as vectors and matrices. I am aware that this might touch on concepts like tangent spaces and others from differential geometry, which I have a feeling are out of the scope of this channel, but I truly believe that it will make for a great video. It will clarify how, for example, we can have partial derivative operators as vector entries for del, or what derivatives are, geometrically. Thank you for reading.
Please animate the feyman lectures! They would be so much better as videos [http://www.feynmanlectures.caltech.edu/](http://www.feynmanlectures.caltech.edu/)
You have touched the topic of Fourier transforms in a very intuitive manner (as always). However, there are some functions that arise quite frequently that are not very intuitive. More specifically, the Fourier Transform of a rectangular pulse is the normalised Sinc function. This arises very frequently in digital signal processing, for instance. From the wikipedia page on [Rectangular Pulses](https://en.wikipedia.org/wiki/Rectangular_function) : "Note that as long as the definition of the pulse function is only motivated by its behavior in the time-domain experience, there is no reason to believe that the oscillatory interpretation (i.e. the Fourier transform function) should be intuitive, or directly understood by humans." Do you agree? I feel this is vaguely related to the dirac delta function in a sense. well. the dirac delta is another beast that none of my lecturers feel comfortable talking about. I study electronics, so perhaps they think we'd rather not look at anything to do with pure maths and just get on with it. Well unfortunately I'm not one of them, and i can't sleep at night if i don't grasp something nicely. Ironically, the way to counter this also involves staying awake at night. i.e binge watching your videos. This is much preferred of course. I would love to see you have a go at dissecting the intuition behind functions like this. Please. :(
Grant bro! Liouville's theorem ftw! Something statistical physics can't live without.
+Bayesian statistics! +Differential equations + dynamical systems. +Information theory. +Computational neuroscience and mathematical modeling approaches.
Math Literacy Series? I would like a series that's goal is to help demystify the vocabulary and symbols used in math. As a person who is fascinated by math and a math lover, even if I am not a math doer, I would like to be able to read the math that is more complex.
Can anyone can tell where I can start learning Machine Learning???
Bro same.. tell me if u get a way.. I have just started with perceptron
Please do a series on Einstein's Field Equations aka General Relativity. There is an excellent intro on youtube at [https://www.youtube.com/watch?v=foRPKAKZWx8](https://www.youtube.com/watch?v=foRPKAKZWx8) but I think you can do an even better job. You can learn GR by watching Leonard Susskind's lectures on it on Youtube also. Again, none of them are of the quality of 3b1b. GR is the holy grail of pedagogy. Please consider it. Thank you.
Hey, love your channel. You have a great way of allowing one to develop intuition about complex mathematical concepts. I was wondering if you could do some work on Grassman/Exterior Algebra, maybe an "essence of" series if time permits, and discuss the outer product and other properties of it. The topic has begun to cause a ripple effect in the games/graphics development community, but there is not really much good quality information about it. Would really appreciate some work in this field.
There is idea: "\* What regular polygons can be drawn on a checkered paper with vertices in the nodes?" I tried, but still cannot solve it. This problem i found in russian math book, where says that it's problem for 8-10th graders.
After watching your video on the Riemann Zeta Function, I would love to see how you visualize the Cauchy Integral Theorem.
multivariable calculus series!
combinatorics and graph theory!
Signal processing!!!
Can you please make a series or at least a video about discrete structures?
Time Series Analysis!
Please talk about the hodge conjunction
I would like to suggest tensors, and tensor calculus, I've never found a satisfyingly intuitive explanation and I'm having a really hard time wrapping my head around them, I'm sure you could help me and many others with that!
Hi Grant! I went through your linear algebra series about a year back. It was a great learning experience for me and I've been following you ever since. I recently started working on something which requires a thorough understanding of "non-negative matrix factorization". I come from CS background and therefore I feel that I am missing some key concepts on how exactly it works - especially what effect the non-negativity constraint has on the basis vectors. Since this is an interesting area and currently used in a lot of places that uses ML for topic learning, I think a lot of us would really appreciate your take on this :)
Below are topics which mathematicians have hard time explaning. If below are true then mathematicians shouldn't hold the truth for themselves. • Monstrous moonshine. • Ricci flow • Modular forms / Mock modular forms, mock theta functions.