This is the beginning of the revolution. I hope they are comfortable with quaternions

His understanding is getting advanced enough to understand that there will be more “wtf” moments the higher up you go.

Truly the most pertinent question.

"What the %$!* is going on?" [music stops]

…someone needs to turn that into a 3 second reaction video.

I was about to comment the same thing lmao

he used a microscope ig

lol

@@Benweiner0 and i wanted to say what you said...smh

@@Mughal92_ and i wanted to say "and i wanted to say what you said...smh"... smh

Um... this is not the short though. This is a 2-year-old, full-length video that Grant featured parts of in a short. I think your comment made it to the wrong video lol.

@@joshyoung1440yeah, that’s what OP was referring to. The way they used shorts to highlight and lead people to the full length videos

bro thats the whole point hes making??? the shorts leading the older videos by just using snip of it@@joshyoung1440

Sike! My computer still takes several minutes to render it.

I first did Conway's Game of Life on my Commodore 64. I think my maximum grid size was something like 30x40 cells. I know that sounds ridiculously small and primitive, but compared to doing them on graph paper? Oh, my!

I'm still waiting for it. it looks like it will take infinite amount of time.

a lot of it is just GPUs, its still relatively slow to do on a CPU

@@Astrobrant2 wait, Conway's game of life on GRAPH PAPER was a thing? Oh my.... That's impressive!

RUN! We cannot hold off such power!! --the concepts of us not knowing the things he's teaching us

How on earth do you make python do that?

@@polterp manim library

@@polterp In short it's a python library that he developed called manim. Manim being short for math animation.

Damn I gotta get a hold of that

Haha, totally!

That part blew my mind!

I feel like there's some mandelbrot lore I haven't come across. I should check up on that.

Imagine showing this to a highschool student

@@mohamedbelafdal6362 i mean i'm a high school student

Same here

Is this maybe related to the fact that polynomials are dense in the set of all continuous functions? So any function can be approximated arbitrarily closely by a polynomial?

@CodeParade Thanks for that insight. I was wondering the exact same thing. What would the fractal patterns look like with other methods such as Laguerre's method, Bisection Method or Regula Fali methods.

This is the clue. For a real-valued function, applying Newton's method when the initial guess is near a flat part of the graph (derivative near zero) results in a very big change for the next iteration. Points fairly close to each other may have opposite signed derivatives, which would cause some to shoot off in one direction, and others in the opposite direction. The same situation applies for complex-valued functions, but the derivative is a vector whose direction varies widely and magnitude is small when near these "flat" areas.

At this point it doesn't really surprise me that fractals emerging from iteration don't really come from the function itself, like the bifurcation diagram. Somehow it's just in the nature of iteration (with respect to a particular property).

well, Id assume it can be extended to analytic functions at least

Are you sure it's not saying - What the math is going on here!? Just kidding.

Finally, a fine quote you can use in math class.

Or from his first differential equations video: “They’re really freakin’ hard to solve!” Especially with the setup to that.

His amazing visualizations really do help

Its beauty of understanding.

Same, its like im looking at an artwork representing the question to the answer of 42

This is what maths should be taught

Bla bla bla, mostly bullshit.

this video is stunningly ugly. I've never seen a more poor approach at math.

But also that is such an evil challenge to give an artist without explaining the fractal nature of it.

This guy is a gem of teaching. Society if all math teachers were like him: *picture of futuristic landscape with flying cars*

Does anyone have an idea if there is a way to compute where the boundaries of this fractal actually are? The blops are all aligned on curves and it looks like mostly the same curves are recursively stacked on top of each other the deeper you go. Is there a way to find the "zero" points, where all the five colors converge into a single point when you let the zoom approach infinity? Because those points *are* the boundary. We know it has to exist, so where is it?

I got some strong Vihart vibes from the paper craft

@@jasonreed7522 Actually, for three colors there's an easier solution if you allow regions of zero width. You look for boundaries at which exactly two colors meet and color those boundaries with the third color. For four colors and beyond you would have to do some Dirichlet function stuff, like mapping the boundary segment in question to the real axis, color all rational numbers one color and all irrational numbers another, and map it back.

I'm a non-mathematician that stumbled on this video, and it was really interesting. I was never that fascinated by fractal images, but this demonstration made them really click.

Jailbreak

@@skiney lol jailbreak

I absolutely agree 👍

I don't know if this I done by only one person, but that's some great talent to program rendering pipelines! I'd be curious to know if he/they use Vulkan/DirectX...?

knowing math can solve half of the world's problems; the other half are not solvable anyway

As somebody who doesn't know jack shit, math is literally everything. Doesn't everything in the universe break down to some sort of numbers/ math? It's crazy and we definitely live in a simulation. Source: bro trust me

mathematics is the understanding of quantitive logic. This particular example in this video is not logical, as it's a random method of solving a problem, and the complexity in its randomness has no purpose, making it actually poor math.

@@jaakezzz_G are you seriously trying to quantify mathematics to such a simple subset of reality? Mathematics is, quite frankly, the study of patterns...ask anyone

So you're saying we will compute 1/P(x) at some x and if it's very large (close to 1/0) then we use the Newton-Raphson method to get to the precise root from there. Or is there something else I'm missing about this Residue Theorem?

@@shannu_boi It's a little more involved than that. The Residue Theorem is used by itself as a method of searching for roots. Instead of iterating over points, you start with a region that contains one or more roots you are looking for. Often, you'd have rectangular regions for simplicity. You evaluate the value of 1/P(x) at a number of points along the contour to compute the integral. (You can actually get away with quite a few points for polynomials, but it's definitely more computationally expensive than taking one value and one derivative.) Residue Thm relates the integral to the number of roots contained within that region. So you can now subdivide the region repeatedly, discarding any that yield a zero integral, until you have suitably well separated regions, each containing a single root. Note that if regions are rectangular, this basically comes down to taking an integral of Im(1/P(x)) or Re(1/P(x)) over a line segment and seeing if it's positive, negative, or zero. In principle, nothing stops you from shrinking the contours until you are within desired error of the solution, but in practice, you start getting numerically close to dividing by zero, which introduces errors until at some point you can't distinguish between positive and negative result of the integral. Fortunately, you can safely switch to Newton-Raphson long before it gets that bad in most practical cases driving result to whatever precision you need. Of course, you aren't protected from some particularly "bad" polynomial, where roots are nearly-degenderate, meaning, they are so close together that they almost appear as the same root with multiplicity. In that case, you won't be able to separate them by residue and Newton-Raphson might not settle on either of these roots either. But if that's the case, there is no magic bullet that will solve the problem. Simplifying the polynomial by dividing out other roots might help you in some cases, but can also introduce even more numerical noise. Realistically, you might end up having to treat such near-degenerate roots as truly degenerate. All of this is getting pretty involved. Most of the time, for practical problems, you have some additional information about the problem that either lets you skip a lot of this, have a decent guess, and just do pure Newton-Raphson, or you just need a few quick passes of residue tests to exclude some regions and then you can do Newton-Raphson. When you are doing optimizations, sometimes you go even dumber, establish a feasible region, drop a grid of starting points, do a fixed number of Newton-Raphson iterations on these, and then see which ones settled down. You might not get absolute optimal solutions, but you have to ask yourself if you actually need to for a particular task. But it's always good to know what methods are available to you so that at least you know what the overkill looks like.

@@konstantinkh wow what a great explanation. THANK YOU SO MUCH. About using the rectangular region, if I remember correctly, didn't 3b1b make I video on this topic. How to find the zeros of a complex function. It's called winding numbers and domain colouring. Is what you described the same as what was explained in that video?

@@shannu_boi Looked it up. Yes! And the notes near the end closely relate to where the Residue Thm comes from. There really is a 3B1B video for everything.

@@konstantinkh Once again, thankyou so much. ❤❤❤

Dang, came here just to say that!

Yea, should have done a guest hand appearance. Vi would have found a way to make it into a burrito.

totally forgot about her. ty for the nostalgia trip

@@1088lol She still makes videos, uploads one every couple of months

It's started. All our favourite math channels are converging. Soon there will be just one channel called Standupflammableviologeyam3red1bluepen.

That is not the purpose of math, but what is the purpose of it anyway? What is the purpose itself? If you really think about it, you define the purpose of math. It is just a tool, a very beautiful and elegant one, but you define how to use it. So yeah, that could be one of the cases :)

I often use math for exactly that purpose. The world doesn't make sense but math does and that comforts me. 3blue1brown videos are especially good at it.

Hey, same here... Hope you get better by tomorrow! Remember you're never alone :)

👏👏👏👏👏

Maybe Newton enjoyed math because it kept him distracted from the world that he was outcast from for being a total nerd 🤓

I don't think that was a "heck", but no way to prove it lol

Bless your heart

@@HAL--vf6cg There's no way of knowing for sure!

@@HAL--vf6cg proof by contradiction should work ig

@@thisisthemactan Maybe there is... If he posted it un-bleeped on Patreon...

Waiting for your next complex analysis vid too mate

That was an effective cliffhanger

Me who got here from a short: 🫣

Mandelbrot Set is always waiting. Watching. IT KNOWS.

I agree! There is something disturbingly ethereal about a boundary that does not exist, in the sense that it never separates anything -- the three colors always meet. Vertices all the way down.

Existential anxiety is my favorite part about learning mathematics.

In my grubby pragmatic way, as an engineer, I see a fundamental lesson : since an irrational number cannot be expressed in concrete form, this is analogous to the graphic property of the boundary of a fractal curve - you can get close to it but never actually find where it is. It's easy to see why the ancient Greek geometers were upset by the concept of square root of 2, which in their geometry should be visible, but in concrete terms could not be found.

the part that blew my mind the most about this is when I just thought: Imagine standing within one of the big open regions, in an infinite-iteration "complete" version of the fractal. Now start walking towards a boundary. *which other color will you walk into first?* I can only imagine the answer is "no"

quadratic convergence is real shit

melhor presente

Exato!

@@mayconbruno2676 presente tão bonito quanto um fractal.

same, but for Mandelbrot

Newton: What in the world is a 'Video'???

Just the idea of him being able to see this. Without context of what computers or youtube are. Just the sheet amount of knowledge and joy he could derive from it (possibly).

@@caniggiaful Reminds me of the Van Gogh episode of Doctor Who. I demand a Newton episode now!

Newton would probably copyright strike the video if he saw it cos that's how much of a prick he was lol

If you take one polynomial you get one fractal, then if you vary coefficients you get the fractal variation, which you can call fractal family.

Depends on your definition, I guess.

@@lonestarr1490 idk but i think that is what maththematician defined them... Gud luck with ur own definition

It sounded like he was saying "an infinite family of fractal", which confused me. I wondered if there was some abstruse fact about the etymology of the word "fractal" which meant that the plural should be the same as the singular. Then I zoomed in by a scale factor of 10^24 and found the "s".

No, the mandelbrot set is a single fractal. But there are infinitely many Julia sets though. Another simple example is the Koch curve. It's a fractal, but there is not an infinite amount of Koch curves, there is only one.

Hi

if I had math teachers like you at high school I would have 100% studied maths at university keep up your great work

Grant, you've got a really high Ab (slightly sharp, ~6.7kHz) quietly ringing on your vocal track. You might want to check that out.

I dare you to make a video about the quartic formula :) Edit: I actually always wondered why is it that with all the currently known operations we can only have formulas for the first 4 grades of polynomials but not for higher order ones... well, except for higher order ones which can be fully divided into order 1,2,3 and 4 polynomials. Could this be related to (probably) not being able to geometrically calculate the fifth root ? I say probably because technically one can’t compute the third root geometrically with just a line and a compass... but one could do it with an origami trick so I’m gonna be naughty and still count it as a “geometric construct”.

Keep it up ❤️

*aesthetic

I guess if we consider an extended complex plane (a Riemann sphere) and consider the "infinity" as a single point where all these tangents meet, then each of these fractal "meeting" points on boundaries (as at 12:45) will be just local reflections (images) of the "infinity" point. And it looks like all these meeting point retain properties of the infinity point, e.g. all have the same ratio and order (up to the sign) of each color in the neighborhood, equal to the ratio and order of the solid-colored areas as they approach infinity.

That is a wicked helpful insight on it, thank you for sharing : )

I have good news for you, my friend: www.3blue1brown.com/lessons/newtons-fractal

yeees :D

@@3blue1brown "an unexpected error has occurred" - I'm on a phone so I'm not shocked, but it should be capable?

@@3blue1brown i'm having an issue where my scroll wheel up is causing it to super zoom and scroll wheel down is normal. Is this a problem on my end? on youtube scroll wheel up and scroll wheel down move the page the same amount so it seems not, but I'm curious if you have the same issue. Tried it on firefox and chrome both, same issue.

@@colecarter2829 I have the same problem

Once I tried to implement the newton's method in Python using numerical derivatives. For most of the starting points, the solution always oscillated all around the place and I thought my numerical derivative implementation was wrong. But looks like this could happen for high-dimensional functions. My numerical derivative implementation could have been correct after all :)

Didn't something like that happen to Mandelbrot when he was trying to print out what the set looked like? Everyone thought something was wrong with the printer

that is why programmers should learn math.

@@OrangeC7 No, the first visualization is not that high of a resolution. Think of ASCII Art. And, the idea came from an index of Julia sets, which also look like that, so he knew what to expect.

@@OrangeC7 As far as I know, the operators of the printer cleaned up printing artifacts that were actually what Mandelbrot tried to see (They were doing that automatically). And so, he always recieved something that was different from what he was expecting.

Yeah, that sounds like a good addition to this.

You can try generate newton's fractal for f(x)=sin(x), which in some sense is a polynomial with infinite roots. The boundary is pretty rough, but the interior is still well-defined

@@wangweiyi8478 I'd love to see that animated

For your second question, although I see no simple way to prove it here, I think it's likely the boundary is negligible in the plane (i.e. it has measure zero) which would be the reason we still use Newton's method anyway: For a randomly selected point(*), it would then have zero probability of being on the boundary. (*) It is obviously not true for some specific distributions, but maybe for a uniform probability distribution over a square bounded region. A square cannot be the boundary of a polynomial.

Interesting. I think the fractal dimension would increase (i.e. would be rougher) with an increasing number of roots so that the limit of the fractal dimension equals 2 as the order of the polynomial approachs infinity (assuming no repeated roots)

@@AwkwardDemon that leads to another rabit hole question: "Are/which polynomials of degree n≈infinity with no repeated roots?" Not sure if the answer has a meaning beyond helping another math proof along. (Pure math doesn't always reveal its value anytime soon after discovery like prime finding algorithms being critical foe computer encrytion but also being 200+ years old)

I love the turns of phrase he uses. "Embrace the mess" and "godawful formula" are great ways of indicating that even someone who really likes math, doesn't think everything about it is sunshine and rainbows.

Grant's personality and storytelling are a big reason why his videos are so fantastic I think 😆

Yeah, it felt like they're running around looking for their friends, like, "Guys, where did you go? Guys?"

Well would you think the same if they were *black* ?

@@Yazan_Majdalawi Yes

I can agree, just look at Conway's Game of Life! Without the right initial conditions, you won't end up with interesting continuous stable patterns.

I agree with you. I think the initial condition is most important thing in mathematics. If it were wrong, we would be wrong forever.

Garbage in, garbage out.

If you can't explain a topic to someone who knows nothing about it, then you yourself don't know it.

@@thecodeking91 good point, what I meant was, with carefully thought, if you could not write down a concept in simple terms then you don’t know it.

always

so true

In general, just about any iterated map like this with even the slightest bit of nonlinearity shows chaotic behavior.

Also an important definition chaos ≠ random. Random has no rhyme or reason. Chaos is 100% deterministic, but so complex we can't predict it.* *Depending on degree and accuracy requirements, weather is an example of a chaotic system we have bad long term but decent short term predictions of