Hilbert's Curve is specialy significant to me, because roughly said, it might be a fundamental key to understand the phenomena that preceded the creation of the Universe. As I see it, using the modern take on the Standard physics model including now the use of supersimmetry and specialy effective mathematical models like the Mandelbort series, we might grasp how "something" can emerge from "nothing".
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.
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
@@TheSanpletext well adding more infill doesn’t increase strength as much as adding more perimeters. So firstly the question shouldn’t be ‘how do we efficiently hit all the points in this area’, but instead, ‘Do we even need to?’. But if you actually do need to there are other things you want to consider. The Hilbert curve is one application specific optimal path algorithm, the ‘Snake curve’ is another and it’s the default for 3d printers because we work in the real world with physical (many direction changes isn’t speed optimal) and memory constraints. There are probably infinitely many optimal path algorithms that satisfy any and all definitions of optimal we could conceive. Maybe even infinitely many algorithms that generate optimal algorithms with varying levels of definitions of optimal. Infinity is mind boggling
🟪🟪🟪🟪🟪🟪🟪🟪🟪🟪🟪 🟪🟪🟪⬜️⬛️🟪⬛️⬜️🟪🟪🟪 🟪🟪🟪🟪🟪🟪🟪🟪🟪🟪🟪 🟫🟫🟫🟫🟫🟫🟫🟫🟫🟫🟫🟫🟫 🟫🟫 So I'm gonna do this wood cutting challenge But 8t wasn't my fault so so I made it 1.5 certain of it center wood long so it's easy for somebody Who want to do this?
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!
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.
It's also very helpful in visualizing a continuous signal. It helps keep parts that are close together in 1D space also close together in 2D space, so the patterns aren't distorted beyond recognition.
It's also used in numerical simulations in astrophysics, except we use a 3D Hilbert curve to partition space. Basically, when performing expensive computations, you want a) to divide space into small cubes, b) to spread these cubes equally over multiple CPUs (or even different computers) so that they have similar workloads, and c) that each CPU gets to work on chunks of data that are close spatially. All three properties are satisfied by a Hilbert curve! The way we do this is that we split our computational domain into cells and order them by their index on the Hilbert curve. If we have four CPUs, we then feed the first CPU the first 25% of the cells along the curve, then the next 25% to the second CPU, etc. You end up with chunks of data that are close spatially and you have easy control over how much load each CPU receives. For example, if one CPU is slower than the others for some reason, you can feed it 20% of the cells instead of 25% to release the pressure, and spread the remaining ones on the neighbouring CPUs. What's even better is that you can also use the CPUs physical location and index them by a Hilbert curve (so CPUs that are close have a similar Hilbert index). When feeding data to each CPU, you can then make sure that neighbouring CPUs receive neighbouring data thus limiting the distance data need to travel if one CPU wants to communicate with its neighbour.
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.
When people say they don't find something miraculous or magical in this world, I want to point to comments like yours. The intersection of expression and reason is so cool. 🥺
I find it amazing that a person that knows about hilburt curves and a person who could practically apply a pseudo hilburt curve met at a craft show and shared information that exists in a digital space using the 10,000 year old tradition of mouthing noises at each other. When do we realize that we ARE the aliens lol
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.
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).
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
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!
Every field has history that defines the language it uses. The danger of updating the language is the loss of coherence of all previous research that used the previous terminology (or the massive effort and expense of translating old research to the new terminology). Mathematics and history are both universally inescapable.
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.
Thank you for your answer @columbus8myhw; I did more research of my own, and yes, it is to do with efficiency of neurons with surface area. The wrinkles, called quite fantastically, gyri and sulci, allow the brain to fold in to a space that can contain more neural connections while still being close enough together to allow the brain to map more links across wider areas. It is actually very complicated topology, which is why the brain has it's very abstract shape, but it is incredibly optimized.
Jeremiah Pratt The same phenomenon can be found in your lungs as well, since they have to maximise inner surface area to extract enough oxygen out of the atmosphere
If you want to see something really cool, look up The 8-Bit Guy’s video on cassette tapes. It shows how you can convert a video game to sound. But don’t expect a symphony unless you are a real geek, as most people outside of the nostalgia crowd don’t care for the screeching sounds.
Instead of sound, electrical signal surge are use to represent the color of the exact point on the curve. This I assume is how Analog TVs and screens converts image to signal and vice versa
@lol fert What if someone made the program? would it be possible to learn to see sound with enough focus and knowledge about frequencies and sound? It looks like a lot of work for a new language, doesn't it? Edit: or a form of communication rather
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.
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.
I'm in a very similar boat to you. I grew up thinking I hated math, but as an adult I discovered this channel along with numberphile, stand-up maths, and a few other educational channels that completely turned me around. I still don't exactly enjoy solving math problems and struggle at times to even keep count, but I definitely like learning about it and understanding it in a way I never did in school
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
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.
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
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.
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.
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!
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.
3Blue1Brown Also could you make a video on the 1+1=2 proof which took 200 to 300 pages to prove. I really would like to know how they made the axioms and applied them. Plus some applications of maths in subjects like economics,sociology would also be appreciated.
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 have a mathematical physics degree and took multiple calculus courses. Your explanation of continuity is better than any I've seen. So much easier to understand. In my degree they just told us to memorise these proofs.. For epsilon > 0 blah blah. I can see that it's exactly what you described
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.
That's actually a really cool ideia. Using 2D vision + sound with space filling curves to get to know the 4th dimension is a brilliant ideia! Has anyone made this before?
Like a manifold filling a volume? It would be like balling up a piece of notebook paper. That would be a really interesting idea for visualizing high dimensional data.
tj12711 you wouldn't have to mod it, since it can learn from what it sees. Do d a way to map a Hilbert curve to 4 dimensional space, and find someone blind (why does autocorrect wants me to write blond instead?) who is willing to take part in your experiment. Also program a virtual 4d environment that you can map the Hilbert curve to And think about how you could controller a character in 4D and you are done. Only problem is that even for 2D images your test subject is not going to think about it as a image but rather as a sound.
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
When I looked up Hilbert's Curves because they looked cool I was not expecting to find a solution for a computer science project I'm currently stuck on. Thank you.
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.
Yes, you can take the order 1 Hilbert curve, changing the 2x2 squares into four cubes. The put another 2x2 cubes behind those, giving a 2x2x2 block. Put another order 1 curve in the empty cubes and a short link to join them to the first curve. You now have a 2x2x2 block with a 3d space filling curve inside. Just keep stacking them together to make higher order curves.
There actually ARE good channels on UA-cam; 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.
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!
@@antanis I'm not exactly sure, really. It varies a lot. Currently I like Suo Gân from Empire of The Sun, and Outro by M83. Suo Gân is a very warm song with lots of orange colors, which I made some artwork on. Outro is much more green and reminds me of forests and rolling landscapes, but also of things like green stained glass that's bubbly almost like an ocean that was made soft.
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.
Cole Turner Actually, the curve never touches itself nor the same point twice, because the reverse mapping from a point to a value along the curve is a function as well, as in that it has exactly one output for each input.
Cole Turner, you are correct. A function which has a unique input for any output is called "injective"-what you noticed is that, even though the (finite) Pseudo-Hilbert curves _are_ injective, the true (infinite) Hilbert curve is not. There do exist objective functions from the line to the square that hit every point; however, none of these are continuous. A function that hits every point is called "surjective"-there do not exist any continuous, injective, surjective functions from the line to the square.
You can see an example at 13:05; the bottom two points approach each other, and in the limit (that is, in the true Hilbert curve) they equal each other.
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!
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.
Hmm... one wonders why are there stil only interlaced and progressive scanning of video (1080p, 1080i, etc.).. When we will see a Hilbert curve scanning and tv screens with 1080hc? As I undurstand from this video, detailed HC of an image will be compression-friendly...
VibratorDefibrilator Back in the early days of TV, left to right interlaced scanning was very simple to implement - and it worked just fine. And so we've stuck with it, just as cars still use petrol/gasoline, and the carburettor has only relatively recently been replaced by fuel injection. With modern technology, it is no doubt possible to create a Hilbert scanning system, and I expect someone has done it somewhere, but that same modern technology also allows us to use software to overcome the limitations of raster scanning. We can rapidly and reliably reformat, compress, expand, etc raster images in software so there's no pressure to move to a different scanning system.
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!
I can often see images associated with the sound I am listening to, Like subtitles from voices. This idea fascinated me when I saw the short and I immediately came here!
So quick. I am a student in 11th standard and have no background of many topics you teach but your videos are so intuitive that (I think) I understand the whole thing without any pains..
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.
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.
No. At least not one in the way you think of it. You can give a formula for the Pseudo-Hilbert-Curve, however even those are not really beautiful. But giving a concrete formula for the actual Hilbert Curve is pretty much impossible, just like you can't really give a formula for the Cantor function.
In wikipedia you can find the algorithm that used for generation a pseudo hilbert curve of order n. But an even better way to represent the general Hilbert Curve is using a rewriting system, like the L-Systems. They are very important in many areas, like the theory of computation, and can be used to represent pretty much every fractal you can imagine in a very simple way
Just like the function simply needs to take the point (the simple number) on the 1d line and turn it into the two coordinate numbers of the 2d space, the same would be true by turning every point in 1d space into 3 number coordinates. This is true for all N-dimensional spaces, by the way - same thing goes for all of them, and they all have the same Ordinal number as an infinite set, which is why it works.
Thank you for introducing me to the space filling curve. Unlike studying only with text books, it was a good experience because I could understand intuitively through visual data.
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.
Doesn't the spiral coverage of the plane break continuity? I mean, approaching HC(1) from below (in the domain of the first tile) and from above (in the domain of the second tile) would generally result in a different point on the plane, no?
Panagiotis Rentzepopoulos I was going to ask a similar question: wouldn't it make more sense to fill the tiles with Hilbert Curves via Hilbert Curves? Or alternatively, just scale the plane down to ((0,0),(1,1))?
So it has a small bug (see main.c) and it's not python but i saw your comment and decided to write it! Well you know youtube is very weird with links so it's on github, unsafecast/soundplot
I don't even like math, I just really like your animations and your voice, helps me fall asleep. Also I like your videos because of the black background. It's easier on the eyes that way.
There's a TED talk of a guy who used a vest that produced an array of tactile points that one could feel with the skin of the back, and used it to gradually train brains to figure out very complex patterns that would be impossible to parse visually (I might be misremembering the details)
Everyone: can you give us a practical example of why is math useful?
3blue1brown: Sure. Okay, imagine you want to see with your ears...
I lol'd :). (love these videos though -- sooo good)
endogeneticgenetics Maybe good for blind people, but regular sound works quite well.
Hilbert's Curve is specialy significant to me, because roughly said, it might be a fundamental key to understand the phenomena that preceded the creation of the Universe. As I see it, using the modern take on the Standard physics model including now the use of supersimmetry and specialy effective mathematical models like the Mandelbort series, we might grasp how "something" can emerge from "nothing".
"Okay, imagine you want to establish an Advanced Capitalist super power."
^^ "... with sound-vision."
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.
Atleast 36 people stumbled upon your comment.
I write this comment so could be reminded thus video
@@salvador1683 after 2 days your wish has finally came true
*71 people liked that*
2^7 likes yay
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
Infill can also be done with Hilbert. IDK why, since it's not fast, strong nor saving filament, but you can.
@@TheSanpletext well adding more infill doesn’t increase strength as much as adding more perimeters. So firstly the question shouldn’t be ‘how do we efficiently hit all the points in this area’, but instead, ‘Do we even need to?’. But if you actually do need to there are other things you want to consider.
The Hilbert curve is one application specific optimal path algorithm, the ‘Snake curve’ is another and it’s the default for 3d printers because we work in the real world with physical (many direction changes isn’t speed optimal) and memory constraints.
There are probably infinitely many optimal path algorithms that satisfy any and all definitions of optimal we could conceive. Maybe even infinitely many algorithms that generate optimal algorithms with varying levels of definitions of optimal. Infinity is mind boggling
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So I'm gonna do this wood cutting challenge
But 8t wasn't my fault so so I made it 1.5 certain of it center wood long so it's easy for somebody
Who want to do this?
@@gcod3d161i love how everyone is thinking of how best to apply this, but the authors in pure maths just think different they just do it just because
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!
lmao
How do you feel about this video 3 years later?
@@yourfacelookslikebut like a God of math.)
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.
It's also very helpful in visualizing a continuous signal. It helps keep parts that are close together in 1D space also close together in 2D space, so the patterns aren't distorted beyond recognition.
Thanks for this comment! Very interesting!
It's also used in numerical simulations in astrophysics, except we use a 3D Hilbert curve to partition space. Basically, when performing expensive computations, you want a) to divide space into small cubes, b) to spread these cubes equally over multiple CPUs (or even different computers) so that they have similar workloads, and c) that each CPU gets to work on chunks of data that are close spatially. All three properties are satisfied by a Hilbert curve!
The way we do this is that we split our computational domain into cells and order them by their index on the Hilbert curve. If we have four CPUs, we then feed the first CPU the first 25% of the cells along the curve, then the next 25% to the second CPU, etc. You end up with chunks of data that are close spatially and you have easy control over how much load each CPU receives. For example, if one CPU is slower than the others for some reason, you can feed it 20% of the cells instead of 25% to release the pressure, and spread the remaining ones on the neighbouring CPUs.
What's even better is that you can also use the CPUs physical location and index them by a Hilbert curve (so CPUs that are close have a similar Hilbert index). When feeding data to each CPU, you can then make sure that neighbouring CPUs receive neighbouring data thus limiting the distance data need to travel if one CPU wants to communicate with its neighbour.
Do that with temporal locality to achieve 1d cache that never refills.
A ROM cache
i love the fact that vsause is promoting you, you are youtube's gem
He's not very subtle about it
Vsauce*
I love that other Greeks actually watch this. Faith in humanity re- _well,_ partially restored.
When Greek meets Greek.......
I've heard about this channel, before vsauce promoted it.
That background music sounds like a parallelogram
I really want to see it
I think it sounds like a dodecahedron
That background music IS a parallelogram
what?
aren’t those shapes?
I played your Essence of Calculus playlist at my graduation party and just wanted to say I love your work! And it was great!
BweDaTub z you must be fun at parties
I bet he is way more fun than assuming that fun has a universal meaning :)
Zartymil probably
We are all fun in our own ways. There is no need to shame people about something they like.
Zartymil chill. It was just a joke
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.
When people say they don't find something miraculous or magical in this world, I want to point to comments like yours. The intersection of expression and reason is so cool. 🥺
I find it amazing that a person that knows about hilburt curves and a person who could practically apply a pseudo hilburt curve met at a craft show and shared information that exists in a digital space using the 10,000 year old tradition of mouthing noises at each other. When do we realize that we ARE the aliens lol
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.
As someone who does not have an intuitive sense of most of what he teaches:
He somehow does this to almost every topic he touches.
Yeah, really.
I have my algebra+calculus exam next week and I was just dazzled by his explanation
"And to make my own animation efforts easier..."
*does fancy swoosh animation*
*r/*_woooosh_
S/wooooosh
Timestamp?
@@vari1535 @1:16
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.
"well" ist ein Adverb. Das dazugehörige Adjektiv heißt "good".
I just like this comment
Your English is quite good!
@@NoriMori1992
It is. And watching English media is a great way to learn the language,
I am speaking from experience.
Yes! It’s so easy to understand him and his voice is so soothing! (I’m a spanish speaker)
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).
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
I was looking for this comment.
That's because this is a Fibonacci sequence
At this point, just assume anything that is recursive or self-similar is Fibonacci and/or Golden Ratio.
Just observed it and saw this comment the next second .
@@benjiusofficial Exactly :)
You make some of the most interesting math videos on UA-cam. You definitely deserve more subscribers :)
But then again, not that many people like math.
From the date you wrote this comment, the number of his suscribers did more than triple itself.
And now its up to 4.23 million lol
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!
14:44
Mathematicians: Ok I will try it
Physicists: Nah, I believe you
lol so true
This comment is gold. XDDD
I literally ended the video there 😂😂
I feel personally attacked but is so true xD
Engineers: You guys have proofs?
“But ¡hey!, it’s math, we live with bad terminology...” I couldn’t agree more xD
Every field has history that defines the language it uses.
The danger of updating the language is the loss of coherence of all previous research that used the previous terminology (or the massive effort and expense of translating old research to the new terminology).
Mathematics and history are both universally inescapable.
This is amazing! Imagine a art museum, that plays a corresponding tone for each painting. 😀
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.
Brains have to be wrinkly - I think it has to do with maximizing surface area, but I forget the details or why
Thank you for your answer @columbus8myhw; I did more research of my own, and yes, it is to do with efficiency of neurons with surface area. The wrinkles, called quite fantastically, gyri and sulci, allow the brain to fold in to a space that can contain more neural connections while still being close enough together to allow the brain to map more links across wider areas. It is actually very complicated topology, which is why the brain has it's very abstract shape, but it is incredibly optimized.
Jeremiah Pratt The same phenomenon can be found in your lungs as well, since they have to maximise inner surface area to extract enough oxygen out of the atmosphere
I was gonna comments that it goes indeed.
I am so desperate to jump from front end Dev to AI but the entry level is so high I need a lot to catch up.
Koray Acar so how are the lungs structured?
"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)
But... we never get to hear a video converted to sound by this method?
I'm a little bit dissapointed... :-(
If you think that you should check the resources in the description
If you want to see something really cool, look up The 8-Bit Guy’s video on cassette tapes. It shows how you can convert a video game to sound. But don’t expect a symphony unless you are a real geek, as most people outside of the nostalgia crowd don’t care for the screeching sounds.
Instead of sound, electrical signal surge are use to represent the color of the exact point on the curve. This I assume is how Analog TVs and screens converts image to signal and vice versa
Lol
@lol fert
What if someone made the program?
would it be possible to learn to see sound with enough focus and knowledge about frequencies and sound?
It looks like a lot of work for a new language, doesn't it?
Edit: or a form of communication rather
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.
That explanation of continuity using circles was one of the best I've ever seen.
Think that is a standard explanation from Real Analysis - i.e the Epsilon.
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.
I recommend you watch more of this guy's videos. You may learn a lot and maybe you can become the next Albert Einstein
I'm in a very similar boat to you. I grew up thinking I hated math, but as an adult I discovered this channel along with numberphile, stand-up maths, and a few other educational channels that completely turned me around. I still don't exactly enjoy solving math problems and struggle at times to even keep count, but I definitely like learning about it and understanding it in a way I never did in school
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
I was hoping I'd get to hear a picture of a lion before the end of this Vid.
The way you talk is really pleasant to listen to, and I love how you explain complex thoughts in simple ways!
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.
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
Seeing an old, educational video of a true intelligent man, fills you with DETERMINATION.
Hello, fellow Undertale enjoyer.
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.
a Lecture Saved My Night - Sigrid
10:44-12:27 : This is the best precise explanation of continuity that I have ever seen.
This channel is so damn cool... the animations are amazing intuitive and smooth, what an essential bonus
like if you want to study in a 3blue1brown school
um yes pls!
@@nadine7142 what a madlad
Vishwas Dubey do not ask 4 likes
@@maulwurf9414 why ?
maybe
I'm a simple man. I see 3blue1brown, I click.
I am a simple man. But 3blue1brown's ability to make simple man like me understand, makes me really happy.
Mikko Finell memes are generally inevitable, I believe. :P Maybe that's a self fulfilling prophecy tho
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.
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!
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.
"...mathematicians, interested in filling continuous space..."
In your local area
Click now
@@sirhasslich536 its not working...
that's what she said hahaha
Please make a video about partial differentiation
Have you seen the work I did for Khan Academy?
3Blue1Brown wait you worked for khan academy where can i see them
Check out the multivariable calculus playlist.
3Blue1Brown Also could you make a video on the 1+1=2 proof which took 200 to 300 pages to prove. I really would like to know how they made the axioms and applied them. Plus some applications of maths in subjects like economics,sociology would also be appreciated.
Thank you sir, love your work
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 have a mathematical physics degree and took multiple calculus courses. Your explanation of continuity is better than any I've seen. So much easier to understand. In my degree they just told us to memorise these proofs.. For epsilon > 0 blah blah. I can see that it's exactly what you described
Agree. Students are not stupid, the education is bad.
I’m a visual learner, the 3B1B videos are such a pleasure.
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.
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
now we need a 3d or 4d filling curve to realise 3d vision + sound as sound
oh man I forgot about colour
That's actually a really cool ideia. Using 2D vision + sound with space filling curves to get to know the 4th dimension is a brilliant ideia! Has anyone made this before?
Like a manifold filling a volume? It would be like balling up a piece of notebook paper. That would be a really interesting idea for visualizing high dimensional data.
Crimson Vale Unless we mod the human brain, which is ideally the end result of the computing revolution
tj12711 you wouldn't have to mod it, since it can learn from what it sees. Do d a way to map a Hilbert curve to 4 dimensional space, and find someone blind (why does autocorrect wants me to write blond instead?) who is willing to take part in your experiment.
Also program a virtual 4d environment that you can map the Hilbert curve to
And think about how you could controller a character in 4D and you are done.
Only problem is that even for 2D images your test subject is not going to think about it as a image but rather as a sound.
And now I have language for why I like exploring infinite sequences. Thank you for all you do!
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.
I can't wait for that Essence of Abstract Algebra series. ;-)
Only scratching the surface of the tip of the iceberg.
yes!!!!!!!
There's an essence of abstract algebra series coming out?! Have my prayers been fianlly answered?
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
9:06 yeah, I completely agree that we have to cope with bad terminology!
This was beautiful. Appreciate it!
When I looked up Hilbert's Curves because they looked cool I was not expecting to find a solution for a computer science project I'm currently stuck on. Thank you.
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.
More than one generation.
Is there such thing as 3-dimentional space filling curves? Or, even better, n-dimentional space filling curves?
Yes, you can take the order 1 Hilbert curve, changing the 2x2 squares into four cubes. The put another 2x2 cubes behind those, giving a 2x2x2 block. Put another order 1 curve in the empty cubes and a short link to join them to the first curve. You now have a 2x2x2 block with a 3d space filling curve inside. Just keep stacking them together to make higher order curves.
Mandolinic Thank you, that was really helpful!
There actually ARE good channels on UA-cam; 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.
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!
"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"
What's your favorite kind of music?
@@antanis I'm not exactly sure, really. It varies a lot. Currently I like Suo Gân from Empire of The Sun, and Outro by M83.
Suo Gân is a very warm song with lots of orange colors, which I made some artwork on. Outro is much more green and reminds me of forests and rolling landscapes, but also of things like green stained glass that's bubbly almost like an ocean that was made soft.
@@MandMs05 Ever tried Indian classical, buddy?
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.
Cole Turner Actually, the curve never touches itself nor the same point twice, because the reverse mapping from a point to a value along the curve is a function as well, as in that it has exactly one output for each input.
Cole Turner, you are correct. A function which has a unique input for any output is called "injective"-what you noticed is that, even though the (finite) Pseudo-Hilbert curves _are_ injective, the true (infinite) Hilbert curve is not.
There do exist objective functions from the line to the square that hit every point; however, none of these are continuous. A function that hits every point is called "surjective"-there do not exist any continuous, injective, surjective functions from the line to the square.
You can see an example at 13:05; the bottom two points approach each other, and in the limit (that is, in the true Hilbert curve) they equal each other.
Cole Turner, thank you! The same thing was puzzling me when I saw this video. Now I can have calm sleep again.
Wow oO Good intuition. Yeah the hilbert curve is not injective only surjective... I am not used to seeing nondumb comments in the comment section :///
I watched this a second time the other day and now I'm watching this a third time... oh well!
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!
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.
Beautiful.
Can one fill D-dimensional space with a line or it only works in 2d?
Yep. For every D.
@@benjiusofficial also not whole number dimensions?
@@hierkonnteihrewerbungstehe5636 Real number dimensions are fractals and Hilbert curves are fractals, so why not? :)
Hmm... one wonders why are there stil only interlaced and progressive scanning of video (1080p, 1080i, etc.).. When we will see a Hilbert curve scanning and tv screens with 1080hc? As I undurstand from this video, detailed HC of an image will be compression-friendly...
VibratorDefibrilator
Back in the early days of TV, left to right interlaced scanning was very simple to implement - and it worked just fine. And so we've stuck with it, just as cars still use petrol/gasoline, and the carburettor has only relatively recently been replaced by fuel injection. With modern technology, it is no doubt possible to create a Hilbert scanning system, and I expect someone has done it somewhere, but that same modern technology also allows us to use software to overcome the limitations of raster scanning. We can rapidly and reliably reformat, compress, expand, etc raster images in software so there's no pressure to move to a different scanning system.
Please, next time you use a non-converging sum, put a trigger warning before. (0:17)
@@violet_flower this is "proved" with non-convergent sums and thus incorrect
@@violet_flower hey you never know what people on the internet believe, my bad
It hurts to see it. There isn't even a reference to non-convering Laurent series that do this but with a mathematical explanation.
@@violet_flower a meme is a humorous transformation of a widespread image, called a template, since it does not fit humorous, it is not a meme
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!
I can often see images associated with the sound I am listening to, Like subtitles from voices.
This idea fascinated me when I saw the short and I immediately came here!
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
Shivam Kimothi i hv also wondered about it since long time..
3b1b.co/about
3Blue1Brown a man pays his regards🙏
So quick. I am a student in 11th standard and have no background of many topics you teach but your videos are so intuitive that (I think) I understand the whole thing without any pains..
Roop chand Jain
Then don't limit yourself to your school textbooks explore the abstract world of maths around you trust me you will love it
Anyone else notice the interrobang used at 3:39 ‽‽
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.
He uses manim, sorry if it's a bit late
One of the best and advanced channel on UA-cam
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.
Is there a written function for the Hilbert curve
No. At least not one in the way you think of it. You can give a formula for the Pseudo-Hilbert-Curve, however even those are not really beautiful. But giving a concrete formula for the actual Hilbert Curve is pretty much impossible, just like you can't really give a formula for the Cantor function.
i guess there is not a formula, but rather an algorithm for such curves
It's possible, but you need to use esoteric notations for it to deal with it.
L systems?
In wikipedia you can find the algorithm that used for generation a pseudo hilbert curve of order n. But an even better way to represent the general Hilbert Curve is using a rewriting system, like the L-Systems. They are very important in many areas, like the theory of computation, and can be used to represent pretty much every fractal you can imagine in a very simple way
So how does the lion picture actually sound like?!
RRRRrooooooAAAAARRrrr
kshhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
Surprisingly, it sounds quite similar to one of my recent beer farts.
aaaAAAAAAAAAAAAA *AAAAAAAAAAAA **_AAAAAAAAAAAAAAAAAA_*
rooarr
I want to learn how to see with sound so I can just randomly say to people “sounds like a tv remote”
This video contained the cleverest and most accessible explanation of continuity that I have ever seen. I'm amazed, as always.
I agree!
The ending really shows how knowledge can be so useful if there's a mind creative enough to use that knowledge
"def synesthesia"
That was a nice touch
Can this thing be done in 3 d also?
Just like the function simply needs to take the point (the simple number) on the 1d line and turn it into the two coordinate numbers of the 2d space, the same would be true by turning every point in 1d space into 3 number coordinates. This is true for all N-dimensional spaces, by the way - same thing goes for all of them, and they all have the same Ordinal number as an infinite set, which is why it works.
Is infinite math useful?
Calculus: *Am I a Joke to you?*
Thank you for introducing me to the space filling curve. Unlike studying only with text books, it was a good experience because I could understand intuitively through visual data.
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.
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
Doesn't the spiral coverage of the plane break continuity? I mean, approaching HC(1) from below (in the domain of the first tile) and from above (in the domain of the second tile) would generally result in a different point on the plane, no?
Panagiotis Rentzepopoulos I was going to ask a similar question: wouldn't it make more sense to fill the tiles with Hilbert Curves via Hilbert Curves? Or alternatively, just scale the plane down to ((0,0),(1,1))?
I want to hear what the picture of that lion sounded like and see with my ears. Call me GooseBat
this is the first real useful application i've seen for a Hilbert curve. i didn't know about this property
It is one thing for hilbert to figure out hilbert curves... that is smart... explaining it the way you have done is genius!
Oh my God! I just found a channel that's as good as vsauce but free!
PLEASE, someone put this algorithm in a python github, I need to "see" that sound looks like 👁👂
Bumping thread
bump
So it has a small bug (see main.c) and it's not python but i saw your comment and decided to write it! Well you know youtube is very weird with links so it's on github, unsafecast/soundplot
I hate those people who say "What Math is usefull for?".
What music is for? And art? Drawing? Litterature?
My favorite video since you connected it to real world application to more understand and relate to it
I don't even like math, I just really like your animations and your voice, helps me fall asleep.
Also I like your videos because of the black background. It's easier on the eyes that way.
3BLUE1BROWN NOTIFICATION
CLICK INSTANTLY
When he uses female pronouns for the mathematician friend T_T
Notification Squad
I simply love these videos! It just tells me all about fractals and infinity! Keep up the good work!
One of the best videos you've made, in my opinion.
Thank you Grant :)
First
you're exactly 3.14159265358979323846264338327950288419716939937510582097494459230781640628.....th
Vishwas Dubey Lies, you didn't type out the whole thing!
Also pird, not pith.
@@supersonictumbleweed Hahaha, is it so hard to spend eternity to type out π, lazy bone?
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.
There's a TED talk of a guy who used a vest that produced an array of tactile points that one could feel with the skin of the back, and used it to gradually train brains to figure out very complex patterns that would be impossible to parse visually (I might be misremembering the details)