Hilbert's Curve: Is infinite math useful? 

That background music sounds like a parallelogram

Everyone: can you give us a practical example of why is math useful? 3blue1brown: Sure. Okay, imagine you want to see with your ears...

Fun fact (which no-one will see): I watched this video about a year ago and found it pretty interesting, and I remembered how each order of pseudo Hilbert curves was made. Recently, I saw it in action, in the form of a 3d render. Specifically the area is divided into equal sized 'panels' and these panels trace a path of order 2 curves spiralling from the center, watching that reminded me of this video, so I thought I'd share it with the one random person to stumble across this comment.

The goal of mapping 2D space into 1D space such that points close in 2D are also close in 1D is exactly how GPU's accelerate texture fetches. Both CPU's and GPU's use caches to speed up calculations, because it is expected that if I request data at some location, I will most likely request data close to that location in the near future. Basically, when you ask for a few bytes, they store an entire section of the memory in anticipation that you will use it. Memory is of course just a long line of bytes. Textures, whether they are 2D or 3D, are slightly different. If I request a color from a texture at some 2D or 3D point, the same logic would mean that I would want to store an entire 2D or 3D chunk around that point in anticipation of the future. So in order to reuse the same caching mechanism that already works for 1D arrays, they use a curve, in this case, the Morton's Curve or Z-Curve. It is not as mathematically optimal as a Hilbert curve in terms of keeping nearby 2D points nearby in 1D, but it is extremely simple to compute from the 2D coordinate by just turning X and Y into binary, interleaving their bits, and then converting back into a single decimal location. And that is how basically every GPU provide a "2D" cache or "3D" cache optimized for textures.

This is actually useful for 3D printers, the first layer in some slicers has the option to use them. As they change directions constantly the warping due to thermal contraction is evened out

Re-watching this again after two years of math, physics, computational physics, and computer science. I'm understanding this video on levels I've never before experienced. Talk about "pause and ponder", even over two years!

Believe it or not, I was directed to this video by a gentleman in my craft show booth last month. I am a lacemaker and had made a doily which is essentially a fractal design. He suggested a Hilbert Curve might work also and I have to agree. Art & math combine so beautifully.

Two more interesting points: (1) the limiting curve is continuous but everywhere non-differentiable; (2) it "preserves measure" in the sense that it takes 1-dimensional Lebesgue measure (length) to 2-dimensional (area).

"And to make my own animation efforts easier..." *does fancy swoosh animation*

i love the fact that vsause is promoting you, you are youtube's gem

“But ¡hey!, it’s math, we live with bad terminology...” I couldn’t agree more xD

14:44 Mathematicians: Ok I will try it Physicists: Nah, I believe you

I played your Essence of Calculus playlist at my graduation party and just wanted to say I love your work! And it was great!

I have little to no knowledge of mathematics. I barely made it to functions in high school. Yet, this was incredibly easy to understand (on a surface level, of course). Moreover it was fascinating, poetic even. It made me think about the underlying structure of an ifinite universe, about the big bang, about the fractal structures of life forms, about music, about the possibily of a soul, about fate, even about the interconnectedness of love..This video has siglehandedly changed the way I think about mathematics. There is beauty and wisdom encoded within the number's hermetic and dry appearence. Lovely work.

Oh my god, you made the (modified) epsilon-delta definition look motivated and elegant. I honestly viewed you as a useful learning tool before because I already had an intuitive sense of what you taught, but despite learning epsilon-delta for hours, I never, ever thought of this. You’re game changing.

I just wanna say thank you. I really enjoy your videos because they teach me a lot of english (I'm from Germany) because of your well pronounciation, while telling me interesting stuff about my hobby math. In my view your animations are just briliant and a beatiful way of connecting design and Art with Math. I'm not in a financial position to support you on patreon but I can like your videos and tell you in the comments what a wonderful job you make, which I now have done.

But... we never get to hear a video converted to sound by this method? I'm a little bit dissapointed... :-(

I have seen ip addresses(both version 4 and version 6) being mapped along hilbert curves. It gives an order in how the various registries and oganisations owns subnets of /24 blocks. It looks good, kind of like looking at an internet map of the world. This video helped a lot in understanding that image thanks a lot!

"How can these results be useful in the finite context?" Euler's formula proof is my favorite by far (that I've learned so far)

This may just be my biased association, but the pattern created by higher order psuedo-hilbert curves, such as the order 7 or 8, reminds of the patter seen in brains. This gave me an interesting thought wondering about the exact reasoning the brain has the pattern it does. I suspect it has something to do with the efficiency of neural connections, but the pattern is very abstract looking, and seems like a great topic to do a video on or include in a video. I'll probably do further research anyway, but thought I'd just share this.

14:00 Did anybody else take notice to the fact that every time the points moved into a higher resolution, it made a sort of Fibonacci curve? Say if you were to map the points out, it looks like it would make something similar

That explanation of continuity using circles was one of the best I've ever seen.

I was hoping I'd get to hear a picture of a lion before the end of this Vid.

You make some of the most interesting math videos on RU-vid. You definitely deserve more subscribers :)

This is amazing! Imagine a art museum, that plays a corresponding tone for each painting. 😀

This is one of my favorites from you. Neat and simple. Also gave me insight into why the "Hilbert Spiral" in Blender's cycles tile-based renderer moves the way it does. Hadn't thought about it before.

Came here from Steve Mould's channel. I was still having trouble getting my head round it. This video helped get me over the line. Cheers, now I can sleep without my brain trying to work it out.

I actually used the hilbert curve once in some software i wrote. i needed an algorithm that could traverse every point in a grid of unknown size. i considered the zamboni method, but realized that doesn't work if you don't know any side lengths of your grid. there are probably other solutions, but i realized the hilbert curve doesn't need a side length to fill a square grid. you can just start drawing it until you hit a wall. once you hit a wall, you've filled the grid entirely. i had my algorithm walk the hilbert curve as a path until it hit a wall, and in doing so it traversed every cell in the grid. i never though i'd actually use the hilbert curve for something other than doodling, but there you go lol

Quick comment from someone who hadn't seen the original version. This is a great introduction to space-filling curves, and one of the most intuitive introductions to continuity that I've seen. Just one small change that I'd make: when you show how to tile all of space with copies of the unit square, instead of wrapping copies around in a spiral, you could lay them out in pseudo-Hilbert curves of increasing order. This way, the curves in the individual squares join end-to-end without too much messing around, and it also ties in nicely with the message about changing resolutions.

The way you talk is really pleasant to listen to, and I love how you explain complex thoughts in simple ways!

These shorts are too good, I constantly find myself almost forced to go to the comments for the full video because i just need to know.

You're an amazing teacher. I'm not a math person, but you make me interested somehow. I watch you videos both to learn math and to learn how to teach others the way you do

Please make a video about partial differentiation

Love this video! You always find a way to put a really unique and interesting spin on math. I recommend that anyone who watched this video watch the other 3 blue 1 brown video in the description, there are some really cool animation supplements there. Thanks again for this great video.

This channel is so damn cool... the animations are amazing intuitive and smooth, what an essential bonus

This video in particular made me really delve deep into maths, although not this topic. I have posters up on my bedroom walls of this in case I ever forget how beautiful mathematics can be, given the right teacher. You are an inspiration and will continue to be. Thankyou so much for your videos.

Seeing an old, educational video of a true intelligent man, fills you with DETERMINATION.

I really think that everything is equivalent to information (orderly and chaotic). Everything is fluctuating, oscillating to some clean point. But never reach it, just keeps going and going. Yet it's so beautiful and captivating that I feel fulfilled. Amazing video!

One of the best videos you've made, in my opinion. Thank you Grant :)

"...mathematicians, interested in filling continuous space..."

This, like all your videos, is just beautiful! One extra point for why a Pseudo-Hilbert Curve would be better than a Snake Curve is that it preserves the locality better. While two points that are close horizontally in an image will always be close when converted to 1D with this type of snake curve, points that are close vertically will generally not be. Much love!

I simply love these videos! It just tells me all about fractals and infinity! Keep up the good work!

And now I have language for why I like exploring infinite sequences. Thank you for all you do!

like if you want to study in a 3blue1brown school

I can't wait for that Essence of Abstract Algebra series. ;-)

This almost feels related to the idea that all of the information contained in the volume of a black hole can be discreetly expressed on the surface of that black hole. (I can’t remember the name of the theorem, but PBS SpaceTime does a great job of explaining it). I never actually understood how all of the information contained in a lower dimension could be expressed in a higher dimension until I watched this video. Great work!

The image to sound actually sounds like a good idea, I may actually put that in my list of ideas to program in python when I'm bored.

Infinite math, and the connection between the infinite and the finite is super interesting to me. I'd love to see you cover the Fast Growing Hierarchy and limit ordinals some time.

now we need a 3d or 4d filling curve to realise 3d vision + sound as sound

I knew about Hilbert curves, i read and watched about them many times. But i totally discovered something new about them in this video! Thank you and well done!

Hey Grant! I would love to see a video on the fractional dimensions of the chaotic strange attractors that can be found in phase space. Your animations for that would be epic. Also, all of your work is incredible and I acquire a great deal of inspiration from it so I'd just like to say thanks man.

This is just a thing of beauty. I was laughing at the brilliance of this idea, your description and the beauty of this all. You are amazing beyond my imagination! I wish I could brainstorm such amazing beautiful ideas with you.

how does these guy makes such stuffs so amazing to watch?

Thank you for the excellent video. Your teaching is an inspiration to myself and many others around the world. You are another teacher demonstrating that any field of knowledge is inherently beautiful, interesting, and can be related back to concrete reality; it just takes brilliant educators to demonstrate that.

There actually ARE good channels on RU-vid; I know it because this IS one. There are only a few really good ones about math. Yours & the Mathologer's are absolutely at the top of that list. Your channel & his are somewhat different in feel - I suppose each has a somewhat different personality - but both are A++ when it comes to explaining math. [Numberphile is also quite good, & I do watch it, but I find these 2 to be my favorites - by far.] Thank you!! To watch an explanation unfold on one of your videos is to experience revelation - it's that beautiful. I don't know how you do it, but please - keep up the good work! Rikki Tikki.

17:26 this correlation between the existence of something infinite and the existence of something similar for all finite cases reminds me a lot of the compactness-theorem from propositional logic and first-order-predicate logic. Infact, I'm thinking about using the compactness-theorem for proving the 3 exercises... :-D

I'm a simple man. I see 3blue1brown, I click.

The intuition to continuity concept is awesome, thanks a lot 3blue1brown

You are a legend man!! If learning is an art, teaching also is.. and you prove to be the best capable teacher out there, who can sink in his thoughts to any layman's mind.. and trust me, it's not that easy.. keep it up!

Huh, i did NOT know those things i drew when bored in math class were called “pseudo-hilbert curves”, because i drew EXACTLY that one time at school

Curious notes : 1. There is a variant of the Hilbert Curve called a “Moore Curve” that joins up 4 rotates Hilbert Curves such that the ends connect to form a loop. Personally I think this is a more accurate way of connecting Cartesian space with frequency space. 2. The Morton Order Curve (aka z-order) can be made simply by taking the 2 coordinate numbers for x and y and interlacing then into a single number by combining their bits in the pattern xyxyxyxy (first number is xxxx bits, second number is yyyy, and the curve position number I s the combined pair.) To make a Hilbert Curve, you can do the same process if you treat the binary numbers as “Gray Code” numbers - kinda... It only works in some dimensions (4,8,24...) in other dimensions you need to do the flipping step on the bits... This is a reflection of deep properties relating to spatial packing.

10:44-12:27 : This is the best precise explanation of continuity that I have ever seen.

again and again I am amazed by the quality of your Videos! I love how you find applications for things, that people often find to be useless mathematic masturbation ;) You are part of the reason I started to study math besides physics. Just to see how neatly things are fitting togeather, wich you think have no connection whatsoever is just fascinating. I truly think, these are the best made videos on RU-vid. I have never seen a Video, with such nice Graphics and with such a great "alternative" approach on things. I often watch videos on topics, that i think i have fully understood and i allways learn something new, or an alternative way of looking at it.

Hi, I don't know if I admitted to make this question but. Could we make a video about how you animate your videos? I mean, all these animations you make up and put in your video, simply it's awesome.

Sacrificing old view count in order make a better quality video!? Now I've seen everything.

The animations are really Breathtaking and does a great job in explaining

wonderful video; the subject is theoretically important and facinating, practically very useful [which you show with a good example] and the presentation is perfect

Amazing video!! 3Blue1Brown, thank you for making a whole generation of youtube viewers find delight in watching Math. It is because of channels like yours that, despite all the terrible content published every second, I have faith in this platform.

Anyone else notice the interrobang used at 3:39 ‽‽

This is helping me conceptualize a new theory on reality itself, in which I postulate all existence is a single point being iterated infinite times relative to itself; i.e. the only real dimension is a "line". Specifically, the "jumps" between input space and output space fit into my theory in a very unexpected way.

Really awesome easy to understand how we went from written fractals to measuring quantum states and proving them.

For a moment I thought I saw a flaw in the space "filling" curve argument because of the asymptotic nature of the function at the divides of the quadrants. I thought that points surrounding the asymptotes get infinitely close to touching the quadrant boundaries, but never actually collide; while that is true for finite curves, infinite curves probably do join at the asymptotes eventually, even if it's only in a limit sense. Then, I thought that would break the function property of the curve. One (two dimensional) point has two outputs! Here I am breaking a century's worth of genius mathematicians' hard work. It's too bad that two-D space is the output, and points on the number-line are the inputs. It is the listener's job to decipher the line. It's perfectly fine for two points on the number-line to both output to the same two-D point. It's just like x^2 hitting 4 both at -2 and 2. It's just like the trig functions, etc... This was fun for me to work out that the function is indeed a function because it is a little bit more abstracted than what the normal way of quickly assessing a curve's functionality. You can't just do a vertical line test, or some form of a planar test. It was just a very pure form of "an output can have multiple inputs, but an input can only have one output." I didn't have this fun logic moment the first time you uploaded this, so I appreciate the re-upload.

I watched this a second time the other day and now I'm watching this a third time... oh well!

Wow, Grant. You are a genius! You make us understand hard and interesting math stuff, it's just amazing!

There is so much rich insight in this video and I absolutely love it

"def synesthesia" That was a nice touch

Are there higher dimensional equivalents of a Hilbert curve?

Your animations are amazing. What a nice work.

the sight to sound model you described reminds me a lot of an inverse fourier transform one uses in sampling k-space when producing MRI images!

Is there such thing as 3-dimentional space filling curves? Or, even better, n-dimentional space filling curves?

"Let's say you wanted to write software to let you see with your years" Me, a chromesthete: "I'm four parallel universes ahead of you"

i always remember epsilon delta definition with functions, but always have some problem remember and explaining it efficiently and this video, is just suddenly wonderful. i hope but in my days during university i hope there were videos like these. there was just youtube and funny videos.

this is the first real useful application i've seen for a Hilbert curve. i didn't know about this property

9:06 yeah, I completely agree that we have to cope with bad terminology!

How do you make this kind of animation? It's so beautiful. which language do you use to write your code? please answer both the questions

I've used Hilbert sorting when I implemented a Delaunay triangulation, which is created by incrementally adding points. If the points are sorted first along a Hilbert curve, each successive point added is very near to the existing triangulation, reducing the amount of work compared to random insertions.

The ending really shows how knowledge can be so useful if there's a mind creative enough to use that knowledge

So how does the lion picture actually sound like?!

0:43 : That's basically Active Echolocation... :D

I just found this video: a great introduction to space filling curves. Using the snake curve for comparison was a good idea. Only missing point was what the LENGTH of it is.

Been your fan since start of the channel, Wish you all the success.

Beautiful. Can one fill D-dimensional space with a line or it only works in 2d?

3Blue1Brown, Could you use space filling curves (more specifically, Hilbert's Curve) for higher dimensional functions? Example being lim(n->∞) PHCₙ(x₀,y₀) = (x,y,z). I would assume so, but who knows maybe there is some weird rule. Thanks!

I love watching this channel just out of curiosity and fun.

One of the best and advanced channel on RU-vid

Please, next time you use a non-converging sum, put a trigger warning before. (0:17)

You are such a good teacher. I am 11 and I understand this.

One practical use for pseudo hilbert curves can be found in video game physics simulations, where one of the biggest problems is to find a fast method to bundle up objects of the game that are close to each other. Linearizing space by the use of pseudo hilbert curves allows 3D positions to be mapped to a 1D position on that curve, with the nice property that objects that are close to each other in 3D space also are close in this 1D mapping. When sorting a list of game object positions by the curve parameter one can simply bundle close game objects by selecting specific segments in that list.

This has helped me in understanding about Finite Element Method and the application of Hilbert spaces in it, more than anything..... Thankyou.. :-)

I wanna see the path every 1D point makes in 2D space, all overlayed. I have a feeling it would be like a flower bouquet