Here, we look at the math behind an animation

like this, what’s known as a “complex Fourier series”. Each little vector is rotating

at some constant integer frequency, and when you add them all together, tip to tail, they

draw out some shape over time. By tweaking the initial size and angle of each vector,

we can make it draw anything we want, and here you’ll see how. Before diving in, take a moment to linger

on just how striking this is. This particular animation has 300 rotating arrows in total.

Go full screen for this is you can, the intricacy is worth it. Think about this, the action

of each individual arrow is perhaps the simplest thing you could imagine: Rotation at a steady

rate. Yet the collection of all added together is anything but simple. The mind-boggling

complexity is put into even sharper focus the farther we zoom in, revealing the contributions

of the littlest, quickest arrows. Considering the chaotic frenzy you’re looking

at, and the clockwork rigidity of the underlying motions, it’s bizarre how the swarm acts

with a kind of coordination to trace out some very specific shape. Unlike much of the emergent

complexity you find elsewhere in nature, though, this is something we have the math to describe

and to control completely. Just by tuning the starting conditions, nothing more, you

can make this swarm conspire in all the right ways to draw anything you want, provided you

have enough little arrows. What’s even crazier, as you’ll see, is the ultimate formula for

all this is incredibly short. Often, Fourier series are described in terms

of functions of real numbers being broken down as a sum of sine waves. That turns out

to be a special case of this more general rotating vector phenomenon that we’ll build

up to, but it’s where Fourier himself started, and there’s good reason for us to start

the story there as well. Technically, this is the third video in a

sequence about the heat equation, what Fourier was working on when he developed his big idea.

I’d like to teach you about Fourier series in a way that doesn’t depend on you coming

from those chapters, but if you have at least a high-level idea of the problem form physics

which originally motivated this piece of math, it gives some indication for how unexpectedly

far-reaching Fourier series are. All you need to know is that we had this equation,

describing how the temperature on a rod will evolve over time (which incidentally also

describes many other phenomena unrelated to heat), and while it’s hard to directly use

it to figure out what will happen to an arbitrary heat distribution, there’s a simple solution

if that initial function looks like a cosine wave with a frequency tuned to make it flat

at each endpoint. Specifically, as you graph what happens over time, these waves simply

get scaled down exponentially, with higher frequency waves decaying faster. The heat equation happens to be what’s known

in the business as a “linear” equation, meaning if you know two solutions and you

add them up, that sum is also a new solution. You can even scale them each by some constant,

which gives you some dials to turn to construct a custom function solving the equation. This is a fairly straightforward property

that you can verify for yourself, but it’s incredibly important. It means we can take

our infinite family of solutions, these exponentially decaying cosine waves, scale a few of them

by some custom constants of our choosing, and combine them to get a solution for a new

tailor-made initial condition which is some combination of cosine waves. Something important I want you to notice about

combining the waves like this is that because higher frequency ones decay faster, this sum

which you construct will smooth out over time as the high-frequency terms quickly go to

zero, leaving only the low-frequency terms dominating. So in some sense, all the complexity

in the evolution that the heat equation implies is captured by this difference in decay rates

for the different frequency components. It’s at this point that Fourier gains immortality.

I think most normal people at this stage would say “well, I can solve the heat equation

when the initial temperature distribution happens to look like a wave, or a sum of waves,

but what a shame that most real-world distributions don’t at all look like this!” For example, let’s say you brought together

two rods, each at some uniform temperature, and you wanted to know what happens immediately

after they come into contact. To make the numbers simple, let’s say the temperature

of the left rod is 1 degree, and the right rod is -1 degree, and that the total length

L of the combined rod is 1. Our initial temperature distribution is a step function, which is

so obviously different from sine waves and sums of sine waves, don’t you think? I mean,

it’s almost entirely flat, not wavy, and for god’s sake, it’s even discontinuous! And yet, Fourier thought to ask a question

which seems absurd: How do you express this as a sum of sine waves? Even more boldly,

how do you express any initial temperature distribution as a sum of sine waves? And it’s more constrained than just that!

You have to restrict yourself to adding waves which satisfy a certain boundary condition,

which as we saw last video means working only with these cosine functions whose frequencies

are all some whole number multiple of a given base frequency. (And by the way, if you were working with

a different boundary condition, say that the endpoints must stay fixed, you’d have a

different set of waves at your disposal to piece together, in this case simply replacing

the cosine functions with sines) It’s strange how often progress in math

looks like asking a new question, rather than simply answering an old one. Fourier really does have a kind of immortality,

with his name essentially synonymous with the idea of breaking down functions and patterns

as combinations of simple oscillations. It’s really hard to overstate just how important

and far-reaching that idea turned out to be, well beyond anything Fourier could have imagined.

And yet, the origin of all this is in a piece of physics which upon first glance has nothing

to do with frequencies and oscillations. If nothing else this should give a hint and how

generally applicable Fourier series are. “Now hang on,” I hear some of you saying,

“none of these sums of sine waves being shown are actually the step function.” It’s

true, any finite sum of sine waves will never be perfectly flat (except for a constant function),

nor discontinuous. But Fourier thought more broadly, considering infinite sums. In the

case of our step function, it turns out to be equal to this infinite sum, where the coefficients

are 1, -⅓, +⅕, -1/7 and so on for all the odd frequencies, all rescaled by 4/pi.

I’ll explain where these numbers come from in a moment. Before that, I want to be clear about what

we mean with a phrase like “infinite sum”, which runs the risk of being a little vague.

Consider the simpler context of numbers, where you could say, for example, this infinite

sum of fractions equals pi / 4. As you keep adding terms one-by-one, at all times what

you have is rational; it never actually equals the irrational pi / 4. But this sequence of

partial sums approaches pi / 4. That is to say, the numbers you see, while never equal

to pi / 4, get arbitrarily close to that value, and stay arbitrarily close to that value.

That’s a mouthful, so instead we abbreviate and say the infinite sum “equals” pi / 4. With functions, you’re doing the same thing

but with many different values in parallel. Consider a specific input, and the value of

all these scaled cosine functions for that input. If that input is less than 0.5, as

you add more and more terms, the sum will approach 1. If that input is greater than

0.5, as you add more and more terms it would approach -1. At the input 0.5 itself, all

the cosines are 0, so the limit of the partial sums is 0. Somewhat awkwardly, then, for this

infinite sum to be strictly true, we do have to prescribe the value of the step function

at the point of discontinuity to be 0. Analogous to an infinite sum of rational number

being irrational, the infinite sum of wavy continuous functions can equal a discontinuous

flat function. Limits allow for qualitative changes which finite sums alone never could. There are multiple technical nuances I’m

sweeping under the rug here. Does the fact that we’re forced into a certain value for

the step function at its point of discontinuity make any difference for the heat flow problem?

For that matter what does it really mean to solve a PDE with a discontinuous initial condition?

Can we be sure the limit of solutions to the heat equation is also a solution? Do all functions

have a Fourier series like this? These are exactly the kind of question real analysis

is built to answer, but it falls a bit deeper in the weeds than I think we should go here,

so I’ll relegate that links in the video’s description. The upshot is that when you take the heat

equation solutions associated with these cosine waves and add them all up, all infinitely

many of them, you do get an exact solution describing how the step function will evolve

over time. The key challenge, of course, is to find these

coefficients? So far, we’ve been thinking about functions with real number outputs,

but for the computations I’d like to show you something more general than what Fourier

originally did, applying to functions whose output can be any complex number, which is

where those rotating vectors from the opening come back into play. Why the added complexity? Aside from being

more general, in my view the computations become cleaner and it’s easier to see why

they work. More importantly, it sets a good foundation for ideas that will come up again

later in the series, like the Laplace transform and the importance of exponential functions.

The relation between cosine decomposition and rotating vector decomposition

We’ll still think of functions whose input is some real number on a finite interval,

say the one from 0 to 1 for simplicity. But whereas something like a temperature function

will have an output confined to the real number line, we’ll broaden our view to outputs

anywhere in the two-dimensional complex plane. You might think of such a function as a drawing,

with a pencil tip tracing along different points in the complex plane as the input ranges

from 0 to 1. Instead of sine waves being the fundamental building block, as you saw at

the start, we’ll focus on breaking these functions down as a sum of little vectors,

all rotating at some constant integer frequency. Functions with real number outputs are essentially

really boring drawings; a 1-dimensional pencil sketch. You might not be used to thinking

of them like this, since usually we visualize such a function with a graph, but right now

the path being drawn is only in the output space. When we do the decomposition into rotating

vectors for these boring 1d drawings, what will happen is that all the vectors with frequency

1 and -1 will have the same length, and they’ll be horizontal reflections of each other. When

you just look at the sum of these two as they rotate, that sum stays fixed on the real number

line, and oscillates like a sine wave. This might be a weird way to think about a sine

wave, since we’re used to looking at its graph rather than the output alone wandering

on the real number line. But in the broader context of functions with complex number outputs,

this is what sine waves look like. Similarly, the pair of rotating vectors with frequency

2, -2 will add another sine wave component, and so on, with the sine waves we were looking

at earlier now corresponding to pairs of vectors rotating in opposite directions. So the context Fourier originally studied,

breaking down real-valued functions into sine wave components, is a special case of the

more general idea with 2d-drawings and rotating vectors. At this point, maybe you don’t trust me

that widening our view to complex functions makes things easier to understand, but bear

with me. It really is worth the added effort to see the fuller picture, and I think you’ll

be pleased by how clean the actual computation is in this broader context. You may also wonder why, if we’re going

to bump things up to 2-dimensions, we don’t we just talk about 2d vectors; What’s the

square root of -1 got to do with anything? Well, the heart and soul of Fourier series

is the complex exponential, e^{i * t}. As the value of t ticks forward with time, this

value walks around the unit circle at a rate of 1 unit per second. In the next video, you’ll see a quick intuition

for why exponentiating imaginary numbers walks in circles like this from the perspective

of differential equations, and beyond that, as the series progresses I hope to give you

some sense for why complex exponentials are important. You see, in theory, you could describe all

of this Fourier series stuff purely in terms of vectors and never breathe a word of i.

The formulas would become more convoluted, but beyond that, leaving out the function

e^x would somehow no longer authentically reflect why this idea turns out to be so useful

for solving differential equations. For right now you can think of this e^{i t} as a notational

shorthand to describe a rotating vector, but just keep in the back of your mind that it’s

more significant than a mere shorthand. I’ll be loose with language and use the

words “vector” and “complex number” somewhat interchangeably, in large part because

thinking of complex numbers as little arrows makes the idea of adding many together clearer. Alright, armed with the function e^{i*t},

let’s write down a formula for each of these rotating vectors we’re working with. For

now, think of each of them as starting pointed one unit to right, at the number 1. The easiest vector to describe is the constant

one, which just stays at the number 1, never moving. Or, if you prefer, it’s “rotating”

at a frequency of 0. Then there will be a vector rotating 1 cycle every second which

we write as e^{2pi * i * t}. The 2pi is there because as t goes from 0 to 1, it needs to

cover a distance of 2pi along the circle. In what’s being shown, it’s actually 1

cycle every 10 seconds so that things aren’t too dizzying, but just think of it as slowed

down by a factor of 10. We also have a vector rotating at 1 cycle

per second in the other direction, e^{negative 2pi * i * t}. Similarly, the one going 2 rotations

per second is e^{2 * 2pi * i * t}, where that 2 * 2pi in the exponent describes how much

distance is covered in 1 second. And we go on like this over all integers, both positive

and negative, with a general formula of e^{n * 2pi * i * t} for each rotating vector. Notice, this makes it more consistent to write

the constant vector is written as e^{0 * 2pi * i * t}, which feels like an awfully complicated

to write the number 1, but at least then it fits the pattern. The control we have, the set of knobs and

dials we get to turn, is the initial size and direction of each of these numbers. The

way we control that is by multiplying each one by some complex number, which I’ll call

c_n. For example, if we wanted that constant vector

not to be at the number 1, but to have a length of 0.5, we’d scale it by 0.5. If we wanted

the vector rotating at one cycle per second to start off at an angle of 45o, we’d multiply

it by a complex number which has the effect of rotating it by that much, which you might

write as e^{pi/4 * i}. If it’s initial length needed to be 0.3, the coefficient would be

0.3 times that amount. Likewise, everyone in our infinite family

of rotating vectors has some complex constant being multiplied into it which determines

its initial angle and magnitude. Our goal is to express any arbitrary function f(t),

say this one drawing an eighth note, as a sum of terms like this, so we need some way

to pick out these constants one-by-one given data of the function. The easiest one is the constant term. This

term represents a sort of center of mass for the full drawing; if you were to sample a

bunch of evenly spaced values for the input t as it ranges from 0 to 1, the average of

all the outputs of the function for those samples will be the constant term c_0. Or

more accurately, as you consider finer and finer samples, their average approaches c_0

in the limit. What I’m describing, finer and finer sums of f(t) for sample of t from

the input range, is an integral of f(t) from 0 to 1. Normally, since I’m framing this

in terms of averages, you’d divide this integral by the length of the interval. But

that length is 1, so it amounts to the same thing. There’s a very nice way to think about why

this integral would pull out c0. Since we want to think of the function as a sum of

these rotating vectors, consider this integral (this continuous average) as being applied

to that sum. This average of a sum is the same as a sum over the averages of each part;

you can read this move as a subtle shift in perspective. Rather than looking at the sum

of all the vectors at each point in time, and taking the average value of the points

they trace out, look at the average value for each individual vector as t goes from

0 to 1, and add up all these averages. But each of these vectors makes a whole number

of rotations around 0, so its average value as t goes from 0 to 1 will be 0. The only

exception is that constant term; since it stays static and doesn’t rotate, it’s

average value is just whatever number it started on, which is c0. So doing this average over

the whole function is sort of a way to kill all terms that aren’t c0. But now let’s say you wanted to compute

a different term, like c_2 in front of the vector rotating 2 cycles per second. The trick

is to first multiply f(t) by something which makes that vector hold still (sort of the

mathematical equivalent of giving a smartphone to an overactive child). Specifically, if

you multiply the whole function by e^{negative 2 * 2pi*i * t}, think about what happens to

each term. Since multiplying exponentials results in adding what’s in the exponent,

the frequency term in each of the exponents gets shifted down by 2. So now, that c_{-1} vector spins around -3

times, with an average of 0. The c_0 vector, previously constant, now rotates twice as

t ranges from 0 to 1, so its average is 0. And likewise, all vectors other than the c_2

term make some whole number of rotations, meaning they average out to 0. So taking the

average of this modified function, all terms other than the second one get killed, and

we’re left with c_2. Of course, there’s nothing special about

2 here. If we replace it with any other n, you have a formula for any other term c_n.

Again, you can read this expression as modifying our function, our 2d drawing, so as to make

the n-th little vector hold still, and then performing an average so that all other vectors

get canceled out. Isn’t that crazy? All the complexity of this decomposition as a

sum of many rotations is entirely captured in this expression. So when I’m rendering these animations,

that’s exactly what I’m having the computer do. It treats this path like a complex function,

and for a certain range of values for n, it computes this integral to find each coefficient

c_n. For those of you curious about where the data for the path itself comes from, I’m

going the easy route having the program read in an svg, which is a file format that defines

the image in terms of mathematical curves rather than with pixel values, so the mapping

f(t) from a time parameter to points in space basically comes predefined. In what’s shown right now, I’m using 101

rotating vectors, computing values of n from -50 up to 50. In practice, the integral is

computed numerically, basically meaning it chops up the unit interval into many small

pieces of size delta-t and adds up this value f(t)e^{-n * 2pi * i * t} * delta-t for each

one of them. There are fancier methods for more efficient numerical integration, but

that gives the basic idea. After computing these 101 values, each one

determines an initial position for the little vectors, and then you set them all rotating,

adding them all tip to tail, and the path drawn out by the final tip is some approximation

of the original path. As the number of vectors used approaches infinity, it gets more and

more accurate. Relation to step function

To bring this all back down to earth, consider the example we were looking at earlier of

a step function, which was useful for modeling the heat dissipation between two rods of different

temperatures after coming into contact. Like any real-valued function, and step function

is like a boring drawing confined to one-dimension. But this one is and especially dull drawing,

since for inputs between 0 and 0.5, the output just stays static at the number 1, and then

it discontinuously jumps to -1 for inputs between 0.5 and 1. So in the Fourier series

approximation, the vector sum stays really close to 1 for the first half of the cycle,

then really quickly jumps to -1 for the second half. Remember, each pair of vectors rotating

in opposite directions correspond to one of the cosine waves we were looking at earlier. To find the coefficients, you’d need to

compute this integral. For the ambitious viewers among you itching to work out some integrals

by hand, this is one where you can do the calculus to get an exact answer, rather than

just having a computer do it numerically for you. I’ll leave it as an exercise to work

this out, and to relate it back to the idea of cosine waves by pairing off the vectors

rotating in opposite directions. For the even more ambitious, I’ll also leave

another exercises up on screen on how to relate this more general computation with what you

might see in a textbook describing Fourier series only in terms of real-valued functions

with sines and cosines. By the way, if you’re looking for more Fourier

series content, I highly recommend the videos by Mathologer and The Coding Train on the

topic, and the blog post by Jezzamoon. So on the one hand, this concludes our discussion

of the heat equation, which was a little window into the study of partial differential equations. But on the other hand, this foray into Fourier

series is a first glimpse at a deeper idea. Exponential functions, including their generalization

into complex numbers and even matrices, play a very important role for differential equations,

especially when it comes to linear equations. What you just saw, breaking down a function

as a combination of these exponentials, comes up again in different shapes and forms.

👍👍👍👍

양자 컴퓨터가 개발 되고 상용화 되면 이러한 방식으로 이미지을 표현하게 되겠네요. 기존의 컴퓨터의 처리속도와는 차원이 다른 처리속도로 결국, 경험하지 못해본 초 고화질 이미지를 표현하겠네요. 과학의 발전은 언제나 신기하면서도 무섭습니다.

Is there a generalization of this to 3D or higher dimensions? I mean, for example, to make a sculpture only using rotating vectors inside spheres?

TUSK ACT IV

I feel like I have PTSD from all those channels who sneak ads into their videos. Every time you say: "before that, let me explain…" it sounds like you're going to say: "before that, this video is brought to you by…" Ugh.

I'm nodding my head and pretending I know exactly what he's talking about

1. Why I am watching this shit

2. How the fuck did he come up with these ideas

3. Stop taking my time away!

Mind-blowing !!!!

"Taylor made polynomial" and then not acknowledging the pun, I love it

The video is amazing as always, but I'd like to point out a confusing detail.

I do not think that talking about i as a "square root of -1" is right. i has not been defined that way because it violates the definition of a square root of real numbers and ruins real analysis. It has been defined as i^2=-1, for exactly that reason, so it doesn't contradict anything.

Hey guys I have a library (for sale) full of mathematics contains 3000 books in English and arranged levels added to it covers all fields of mathematics with this library contains more than 10000 research paper

in various mathematical disciplines of the subject matter Valsalttalat with me on the following WhatsApp 212699501527💪❤❤❤

Türkçe altyazılı videolar yayınlayın lütfen

Videolarınıza türkçe altyazı ekleyin lütfen

Thanks

seeing the patterns evolve in a two dimensional plane goes beyond what I could imagine

Mind blown

what classes do i take in college to learn how to do this visualization and math? I don’t know what’s going on but i’m in love

3b1b Has allowed me to find beauty (and even a little love for maths) a subject I previously (and kind of still hate/dislike). I've went from barely passing to acing my college math exams, much thanks to the conceptually and visually (yet incredibly accurate) descriptions Grant has given. The overview stipped of the detailed theoretical information is priceless. So much respect to you! Please join his patreon!

6:13 x=φ

6:19 x=τ

I've watched this video about a hundred times and I understand just about every word of it. One thing that's got me scratching my head is the final product. The way I'm interpreting it, it seems like f(t) is dependent upon the c-sub-n values/coefficients of each term but at the same time, it looks each individual c-sub-n is dependent upon f(t) itself. I'm not sure I see how I'm supposed to determine one because the evaluation of each one seems to be dependent upon the other. I would greatly appreciate it if anyone were able to clarify.

Damn your good!

Please which software do you use to create these graphs?? Thanks

Which software did you use for the letter and image animation?? Thanks

Hey Grant, could you do something about Laplace Transform and its applications in solving differential equations and in topics such as Control Engineering?

Grant, did you ever struggle with Math? This is amazing

At the end a Software and hardware combination

Very useful for me to understand!thank you!

Absolutely fascinating visualisation of an otherwise quite complex concept.

Wait what.. no… what about the Gibbs phenomenon ?

Dude you're awesome!

let's watch this for the 6th time, this is like when you go back and watch rock lee vs gaara

I've never more related to a little blue pi before

Is there a software that could show the data of the circles?

You mentioned that you would eventually do a video on the Laplace transforms. When would that be? I would love to see a 3b1b explanation of it 🙂

Amazing; truly.

I didn't understand a single word he said. (Normally I can understand though)

does the order of the circles matter? is aka is it commutative?

What the heck. So baller. This guy makes studying a lot fun.

Hi, how do u do ur animations?

I see you are a Tim as well

My brain is dripping out of my ears.

This is UHmazing.

Tim!

Blows all the university away

10:10 damn, now I need an animation of 2sin(x)=(e^jx-e^-jx)/j

Turkce altyazi lutfenn

I’d say everything is frequency

How would you generalize this to higher dimensions?

Wow 👍👍❤️❤️

After seeing this maths became my favorite subject because in high school no one is there for me to teach like this . Maths is most interesting subject to learn and apply in our day to day life.

Drawings are nice and all but what about quantum gravity?

5:00 negative one degree….

kelvin

I translated subtitles to Slovak language long time ago. Still not avaliable. Watafuk?

I CAN DO IT ALL DAY

Please make a video about how Fourier and Laplace transform can solve differential equations.

How we can pass from frequency to picture with fourier transform? Thanks

did someone know how to extract those complex data from a given svg file?

https://www.youtube.com/watch?v=WKoYzxyu2S8

Honestly with maths a-level being the bane of my existence, these videos make me remember why i love maths.

Like the ideas in maths are really interesting, but the actual numbers and calculations are killing my grade just because of dumb mistakes

21:54 this must have been very hard to render, I mean you added the amount of vectors while drowing the image and changing the image

the easiest way I can think of is rendering it for all the different amounts of vectors and then present them for a small duration each. This is realy time consuming and hard to make it look this good and smooth

this has to be the best thing I've learned so far in life

Edit: I did get to the text book expression of Fourier series with help of the hint given, but I didn't understand why we should write Cn like that and how we can get to the text book expression for in the input range (0,2π)

This guy deserve an oscar for Editing video.

I wish this video was out three years ago so I won't fumble so hard while studying intro to PDE…

Please do more topics such as analytic functions,complex integration,laplace and z transform.

I am confused about output

Which is taken as output

Real part(output ) with time(time)

Imaginary part(output ) with time(time)

It makes me happy noticing that many of the commentators are anime fans.

Can this be utilized in 3d printing?

engineering-and-science.com/mathematics-basic-formulas.html

Super etchasketch

Me: Woah! that looks cool drawing pictures with rotating vectors! I want to learn!

Also me(after trying to learn how to do it myself): OHHH i finally get it! this was so well explained and i understand how to find the constants… now how do i exactly… do it

Ahh yes a fellow Tim

But how do you find the equation that draws Mr Fourier so that you can do his transform on him and get the vectors ?

Fun fact: Fourier series were used to produce this video and the audio too. The pictures and audio waveforms were analyzed by a computer and turned into numbers that describe the vectors. Playing back the video is a matter of replaying those vectors. Compressing the audio and video to make it take up less space and less bandwidth is a matter of stopping the analysis when more vectors don't add more visible or audible details anymore.

please share the code of that video created in manim

Did understand nothing, but very interesting.

Why i came to this video:

An answer to the question

I get the answer

He says he sweeps a bunch of things under the rug:

Me: huh?

Me: i came for an answer not a bunch of questions i don't even understand.

But all joking aside, i love your videos, explanations and animations so thanks a lot

nice explanation sir. Sir can u make a video or series on "WAVELET TRANSFORMS".

Waiting for the Laplace video

wut

What is this?

Bro , Really really really really really …… Infinity awesome video

You should name your chanel useless math with brilliant animations

"The power of rotation is limitless, trust in that!"

AKA

Tusk Act 4, for art

I ate three blue i made one brown 😀

In my btech computer science we had chapters about Fourier series and Fourier transform.. literally i mean literally teacher never told us where we gonna use it and.. he dont even teach us what is Fourier transform… Because in exam pattern we get a choice between series and transform questions… It was all about getting passing marks.. indian education system sucks a little but its the teachers that sucks more ..

I became confused when it said that equation…

Cn = /o1e-2pi,intf(t)dtArtists use a paintbrush to sketch.

Mathematicians use vectors rotating on a circle in a complex plane to sketch.

When you saw too much memes. Youtube show you this

The more and more we screw with the quantum realm the more feesable it is we are an ancestor simulation.

This is fascinating for the potential applied sciences that can utilize this.

You do have 3D linear equations that’s easy enough to understand; however, all drawings you have deriving from it are on 2D Cartesian graphs.

I’m interested if you will elaborate more on the possibilities of how one might go about using this concept drawing out anything in dimensions > 2.

looks like disney magic

This video make me feel like i never study mathematics

Nail and gear!

I watched your code for the brownian movement and you used a formula to find the velocity vector of a particule after it collides with another one. How did you find it?

Thank you! You made me realize 2 things. First, what does all that math means in my Signal & Systems undergraduate class. Second, that I am a drooling idiot.

What if the rotation is infinite?

…

Ответ на парадокс Ахиллеса и черепахи

watching this because my Fourier lecture notes are wank

If only someone had explained to me in this manner, learning would have been so fun and practical in the 11th standard and also in college.

could tell me the example of non linear functions?

This math is so complicated it just looks like your swearing