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{
"caption": "In my video on the circle division problem, I referenced Euler's characteristic formula, and here I would like to share a particularly nice proof of this fact. It's very different from the inductive proof, typically given, but I'm not trying to argue that this is somehow better or easier to understand t... | -9OUyo8NFZg_1-8 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Moreover, since the remaining edges make up a spanning tree for Mortimer's dual graph, the number of edges he gets is one more than the number of vertices in the dual graph, which are faces of the original graph. Putting this together, it means the total number of edges is two more than the number of ve... | -9OUyo8NFZg_103-107 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "which was only later reframed in terms of planar graphs, instead of saying dots, we say vertices, instead of saying lines, we say edges, and instead of saying regions, we say faces. Hence, we write Euler's discovery as V minus E plus F equals 2. Before describing the proof, I need to go through three pi... | -9OUyo8NFZg_15-22 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "I describe them, feel free to click the appropriate annotation and skip ahead. Imagine a tiny creature sitting on one of the vertices. Let's name him Randolph. If we think of edges as something Randolph might travel along from one vertex to the next, we can sensibly talk about a path as being a sequence... | -9OUyo8NFZg_23-29 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "You might be able to guess how cycles will be important for our purposes, since they will always enclose a set of faces. Now imagine that Randolph wants access to all other vertices, but edges are expensive, so he'll only buy access to an edge if it gives him a path to an untouched vertex. This frugalit... | -9OUyo8NFZg_30-35 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "In general, a connected graph without cycles is called a tree, so named because we can move things around and make it look like a system of branches. And any tree inside a graph which touches all the vertices is called a spanning tree. Before defining the dual graph, which runs the risk of being confusi... | -9OUyo8NFZg_36-41 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Really, it's a set of anything with any notion of connection, but we typically represent those things with dots and those connections with lines. For instance, Facebook stores an enormous graph where vertices are accounts and edges are friendships. Although we could use drawings to represent this graph,... | -9OUyo8NFZg_42-49 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "connected when they differ by one letter, mathematicians considered connected if they've written a paper together, neurons connected by synapses. Or, maybe, for those of us reasoning about the actual drawing of a graph on the plane, we can take the set of faces this graph cuts the plane into and conside... | -9OUyo8NFZg_50-55 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "you automatically get a second, as of yet undrawn, graph whose vertices are the faces and whose edges are, well, edges of the original graph. We call this the dual of the original graph. If you want to represent the dual graph with dots and lines, first put a dot inside each one of the faces. I personal... | -9OUyo8NFZg_56-62 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Next, connect these new dots with new lines that pass through the centers of the old lines, where lines connected to that point at infinity can go off the screen in any direction, as long as it's understood that they all meet up at the same one point. But keep in mind, this is just the drawing of the du... | -9OUyo8NFZg_63-69 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "The reason I stress this point is to emphasize that edges of the original graph and edges of the dual graph are not just related, they're the same thing. You see, what makes the dual graph all kinds of awesome is the many ways that it relates to the original graph. For example, cycles in the original gr... | -9OUyo8NFZg_70-76 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Now for the cool part. Suppose our friend Randolph has an alter ego, Mortimer, living in the dual graph, traveling from face to face instead of from vertex to vertex, passing over edges as he does so. Let's say Randolph has bought all the edges of a spanning tree and that Mortimer is forbidden from cros... | -9OUyo8NFZg_77-84 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "To see why, we only need to check the two defining properties of spanning trees. They must give Mortimer access to all faces and there can be no cycles. The reason he still has access to all faces is that it would take a cycle in Randolph's spanning tree to insulate him from a face, but trees cannot hav... | -9OUyo8NFZg_85-89 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "If you draw some dots and some lines between them, that is, a graph, and if none of these lines intersect, which is to say you have a planar graph, and if your drawing is connected, then Euler's formula tells us that the number of dots minus the number of lines plus the number of regions these lines cut... | -9OUyo8NFZg_9-14 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "If he could, he would separate one set of Randolph's vertices from the rest so the spanning tree from which he is banned could not have spanned the whole graph. So not only does the planar graph have a dual graph, any spanning tree within that graph always has a dual spanning tree in the dual graph. Her... | -9OUyo8NFZg_90-95 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "To see this, note that after you start with the root vertex, each new edge gives exactly one new vertex. Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree. Since this tr... | -9OUyo8NFZg_96-102 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "You've seen the title, so you know this is leading to a certain fractal. And actually it's an infinite family of fractals. And yeah, it'll be one of those mind-bogglingly intricate shapes that has infinite detail no matter how far you zoom in. But this is not really a video about generating some pretty ... | -RdOwhmqP5s_1-8 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Almost certainly, the output of your polynomial at x0 is not 0, so you haven't found a solution, it's some other value visible as the height of this graph at that point. So to improve the guess, the idea is to ask, when does a linear approximation to the function around that value equal 0? In other word... | -RdOwhmqP5s_102-109 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "function in the loose vicinity of some true root, the place where this approximation equals 0 should take you closer to that true root. As long as you're able to take a derivative of this function, and with polynomials you'll always be able to do that, you can concretely compute the slope of this line. ... | -RdOwhmqP5s_110-118 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "its rise over run, looks like the height of this graph divided by the length of that step. But on the other hand, of course, the slope of the tangent line is the derivative of the polynomial at that point. If we kind of rearrange this equation here, this gives you a super concrete way that you can compu... | -RdOwhmqP5s_119-126 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "You compute the value of this function, and the slope, at this new guess, which gives you a new linear approximation, and then you make the next guess, x2, wherever that tangent line crosses the x-axis. And then apply the same calculation to x2, and this gives you x3, and before too long you find yourse... | -RdOwhmqP5s_127-134 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "If p of x is large, meaning the graph is really high, you need to take a bigger step to get down to a root. But if p' of x is also large, meaning the graph is quite steep, you should maybe ease off on just how big you make that step. Now as the name suggests, this was a method that Newton used to solve ... | -RdOwhmqP5s_135-141 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "more like what you and I are looking at now, so you also often hear this algorithm called the Newton-Rafson method. These days it's a common topic in calculus classes. One nice little exercise to try to get a feel for it, by the way, is to try using this method to approximate square roots by hand. But w... | -RdOwhmqP5s_142-149 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "You see, while Newton's method works great if you start near a root, where it converges really quickly, if your initial guess is far from a root, it can have a couple foibles. For example, let's take the function we were just looking at, but shift it upward, and play the same game with the same initial ... | -RdOwhmqP5s_150-155 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "around the local minimum of this function sitting above the x-axis. This should kind of make sense, I mean, a linear approximation of the function around these values all the way to the right is pretty much entirely unrelated to the nature of the function around the one true root that it has off to the ... | -RdOwhmqP5s_156-162 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "where it crosses the x-axis, and you can kind of eyeball what those values might be. We'd call those the roots of the polynomial. But how do you actually compute them exactly? Now this is the kind of question where if you're already bought into math, maybe it's interesting enough in its own right to mov... | -RdOwhmqP5s_16-24 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "guesses does anything productive and actually approaches that true root. Where things get especially interesting is if we ask about finding roots in the complex plane. Even if a polynomial like the one shown here has only a single real number root, you'll always be able to factor this polynomial into fi... | -RdOwhmqP5s_163-169 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Now in the happy-go-lucky land of functions with real number inputs and real number outputs, where you can picture the association between inputs and outputs as a graph, Newton's method has this really nice visual meaning with tangent lines and intersecting the x-axis. But if you want to allow these inp... | -RdOwhmqP5s_170-177 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "You can still play the same game, starting with a random guess, and evaluating the polynomial at this point, as well as its derivative, then using this update rule to generate a new guess, and hopefully that new guess is closer to the true root. But I do want to be clear, even if we can't visualize thes... | -RdOwhmqP5s_178-184 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "equal zero, and then you use that zero of the linear approximation as your next guess. It's not like we're blindly applying the rule to a new context with no reason to expect it to work. And indeed, with at least the one I'm showing here after a few iterations, you can see that we land on a value whose ... | -RdOwhmqP5s_185-192 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "this particular polynomial in the complex plane. With each iteration, each one of our little dots takes some step based on Newton's method. Most of the dots will quickly converge to one of the five true roots, but there are some noticeable stragglers which seem to spend a while bouncing around. In parti... | -RdOwhmqP5s_193-200 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Now what I'm going to do is color each one of these dots based on which of those five roots it ended up closest to, and then we'll kind of roll back the clock so that every dot goes back to where it started. Now as I've done it here, this isn't quite enough resolution to get the full story, so let me sh... | -RdOwhmqP5s_201-208 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "coloring each dot based on what root it lands on, then rolling back the clock to see where it originally came from. But even this isn't really a high enough resolution to appreciate the pattern. If we did this process for every single pixel on the plane, here's what you would get. And at this level of d... | -RdOwhmqP5s_209-215 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "be enough, because the finer details of the shape we get go on with endless complexity. But take a moment to think about what this is actually saying. It means that there are regions in the complex plane where if you slightly adjust that seed value, you know, you just kind of bump it to the side by 1,1 ... | -RdOwhmqP5s_216-220 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "We saw some foreshadowing of this kind of chaos with the real graph and the problematic guess shown earlier, but picturing all of this in the complex plane really shines a light on just how unpredictable this kind of root finding algorithm can be, and how there are whole swaths of initial values where t... | -RdOwhmqP5s_221-228 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "And notice for example how the regions around a given root always have the same color, since those are the points that are close enough to the root where this linear approximation scheme works as a way of finding that root with no problem. All of the chaos seems to be happening at the boundaries between... | -RdOwhmqP5s_229-236 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "this seems to be a general fact for any given polynomial. Another facet we can tweak here just to better illustrate what's going on is how many steps of Newton's method we're using. For example, if I had the computer just take zero steps, meaning it just colors each point of the plane based on whatever ... | -RdOwhmqP5s_237-244 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "and then color it based on what root that single step result is closest to, here's what we would get. Similarly, if we allow for two steps, we get a slightly more intricate pattern, and so on and so on, where the more steps you allow, the more intricate an image you get, bringing us closer to the origin... | -RdOwhmqP5s_245-250 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "setting of computer graphics, where polynomials are just littered all over the place. So it's not uncommon that when you're figuring out how a given pixel should be colored, that somehow involves solving an equation that uses these polynomials. Here let me give you one fun example. When a computer rende... | -RdOwhmqP5s_25-32 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "one of these, it's the limit as we allow for an arbitrarily large number of iterations. At this point, there are so many questions we might ask. Maybe you want to try this out with some other polynomials, see how general it is, or maybe you want to dig deeper into what dynamics are exactly possible with... | -RdOwhmqP5s_251-257 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "what the **** is going on here? I mean, all we're doing here is repeatedly solving linear approximations. Why would that produce something that's so endlessly complicated? It almost feels like the underlying rule here just shouldn't carry enough information to actually produce an image like this. And be... | -RdOwhmqP5s_258-265 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "it lands on and move it back to the original position, the final image would look like one of these Voronoi diagrams with straight-line boundaries. And since I referenced earlier the unsolvability of the quintic, maybe you would wonder if the complexity here has anything to do with that. That would be c... | -RdOwhmqP5s_266-274 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Notice how, again, while most points nestle into a root, some of them are kind of flying all over the place more chaotically. In fact, those ones are the most noticeable ones in an animation like this, with the ones going towards the roots just quietly nestled in in their ending points. And again, if we... | -RdOwhmqP5s_275-282 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "However, quadratic polynomials with only two roots are different. In that case, each seed value does simply tend towards whichever root it's closest to, the way you might expect. There is a little bit of meandering behavior from all the points that are an equal distance from each root, it's kind of like... | -RdOwhmqP5s_283-290 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "So something new seems to happen when you jump from 2 to 3, and the question is what, exactly? And if you had asked me a month ago, I probably would have shrugged and just said, you know, math is what it is, sometimes the answers look simple, sometimes not, it's not always clear what it would mean to as... | -RdOwhmqP5s_291-298 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Focus your attention on just one of the colored regions, say this blue one, in other words, the set of all points that eventually tend towards just one particular root of the polynomial. Now consider the boundary of that region, which for the example shown on screen has this kind of nice threefold symme... | -RdOwhmqP5s_299-306 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "but mathematicians have a pretty clever way to formalize it, and this makes it easier to reason about in the context of more wild sets like our fractal. We say that a point is on the boundary of a set if when you draw a small circle centered at that point, no matter how small, it will always contain poi... | -RdOwhmqP5s_307-316 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "But when it's on the boundary, what it means to be on the boundary is that your tiny tiny circles will always contain both. So looking back at our property, one way to read it is to say that if you draw a circle, no matter how small that circle, it either contains all of the colors, which happens when t... | -RdOwhmqP5s_317-323 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "In particular, what this implies is you should never be able to find a circle that contains just two of the colors, since that would require that you have points on the boundary between two regions, but not all of them. And before explaining where this fact actually comes from, it's fun to try just wrap... | -RdOwhmqP5s_324-331 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "And any of you who've messed around with vector graphics, maybe in some design software, would be well familiar with these kinds of curves. But to actually display one of them on the screen, you need a way to tell each one of the pixels of your screen whether it should be colored in or not. These curves... | -RdOwhmqP5s_33-41 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "maybe we say red, green, and blue, so that the boundary of one color is the boundary of all of them. So if you started with something simple like this, that clearly doesn't work because we have this whole line of points that are on the boundary of green and red, but not touching any blue, and likewise y... | -RdOwhmqP5s_332-340 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "now the boundary of those blobs are a problem, for example, touching just blue and red, but no green. So maybe you go and try to add even smaller blobs, with the relevant third color around those smaller boundaries to help try to correct. And likewise you have to do this for every one of the blobs that ... | -RdOwhmqP5s_341-347 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "And if you look at Newton's fractal itself, this sort of blobs on blobs on blobs pattern seems to be exactly what it's doing. The main thing I want you to notice is how this property implies you could never have a boundary which is smooth, or even partially smooth on some small segment, since any smooth... | -RdOwhmqP5s_348-352 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Instead, the boundary has to consist entirely of sharp corners, so to speak. So if you believe the property, it explains why the boundary remains rough no matter how far you zoom in. And for those of you who are familiar with the concept of fractal dimension, you can measure the dimension of the particu... | -RdOwhmqP5s_353-361 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "It says that if you're near a sensitive point where some of the seed values go to one root but other seed values nearby would go to another root, then in fact every possible root has to be accessible from within that small neighborhood. For any tiny little circle that you draw, either all of the points ... | -RdOwhmqP5s_362-368 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "a cluster like the one I'm showing on screen undergo this process. It starts off mostly sticking together, but at one iteration they all kind of explode outward, and after that it feels a lot more reasonable that any root is up for grabs. And keep in mind I'm just showing you finitely many points, but i... | -RdOwhmqP5s_369-374 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "This property also kind of explains why it's okay for things to look normal in the case of quadratic polynomials, with just two roots, because there a smooth boundary is fine, there's only two colors to touch anyway. To be clear, it doesn't guarantee that the quadratic case would have a smooth boundary,... | -RdOwhmqP5s_375-381 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "But of course all of this simply raises the question of why this bizarre boundary property would have to be true in the first place, where does it even come from? For that I'd like to tell you about a field of math which studies this kind of question, it's called holomorphic dynamics. And I think we've ... | -RdOwhmqP5s_382-389 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Newton had no clue about any of this, and could never have possibly played with these images the way you and I can with modern technology. And it happens a lot through math that people's names get attached to things well beyond what they could have dreamed of. Hamiltonians are central to quantum mechani... | -RdOwhmqP5s_390-397 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "It reflects how even the simple ideas, ones that could be discovered centuries ago, often hold within them some new angle or a new domain of relevance that can sit waiting to be discovered hundreds of years later. It's not just that Newton had no idea about Newton's fractal. There are probably many othe... | -RdOwhmqP5s_398-405 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "For example, if you were to ask about whether this process we've been talking about today ever gets trapped in a cycle, it leads you to a surprising connection with the Mandelbrot set, and we'll talk a bit about that in the next part. At the time that I'm posting this, that second part by the way is ava... | -RdOwhmqP5s_406-415 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "I know that in recent history new videos have been a little slow coming. Part of this has to do with other projects that have been in the works. Things I'm proud of, by the way, things like the Summer of Math Exposition, which was a surprising amount of work, to be honest, but so worth it given the outc... | -RdOwhmqP5s_416-423 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "knows whether it should be colored in or not, just based on the pure mathematical curve. I mean, take the case of stroke width. This comes down to understanding how far away a given pixel is from this pure mathematical curve, which itself is some platonic ideal, it has zero width. You would think of it ... | -RdOwhmqP5s_42-49 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "even during times of trying a few new things. It means a lot to me, it's what keeps the channel going, and I'll do my best to make the new lessons in the pipeline live up to your vote of confidence there.",
"caption_code": "",
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{
"caption": "But that's both inefficient and imprecise. Better is to get a little mathematical and acknowledge that this distance to the curve at all the possible points is itself some smooth function of the parameter. And as it happens, the square of that distance will itself be a polynomial, which makes it pretty ... | -RdOwhmqP5s_50-57 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "this function whose minimum you want to know is some polynomial. Finding this minimum, and hence determining how close the pixel is to the curve and whether it should get filled in, is now just a classic calculus problem. What you do is figure out the slope of this function graph, which is to say its de... | -RdOwhmqP5s_58-64 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "wouldn't it be nice if you had a systematic and general way to figure out when a given polynomial equals zero? Of course, we could draw 100 other examples from 100 other disciplines, I just want you to keep in mind that as we seek the roots of polynomials, even though we always display it in a way that'... | -RdOwhmqP5s_65-71 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "If whatever problem you're working on leads you to a quadratic function, then happy days, you can use the quadratic formula that we all know and love. And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considerin... | -RdOwhmqP5s_72-78 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "trillions of times in the production of that film. Now, when your problem leads you to a higher order polynomial, things start to get trickier. For cubic polynomials, there is also a formula, which Mathologer has done a wonderful video on, and there's even a quartic formula, something that solves degree... | -RdOwhmqP5s_79-86 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "But after that, and I find this one of the most fascinating results in all of math, you cannot have an analogous formula to solve polynomials that have a degree 5 or more. More specifically, for a pretty extensive set of standard functions, you can prove that there is no possible way that you can combin... | -RdOwhmqP5s_87-93 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "meaningful if we make an effort to understand why, given what they represent, they kind of have to look as complicated as they do, and what this complexity reflects about an algorithm that is used all over the place in engineering. The starting point here will be to assume that you have some kind of pol... | -RdOwhmqP5s_9-15 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "we can hopefully get into it some other time, but in practice it kind of doesn't matter, because we have algorithms to approximate solutions to these kinds of equations with whatever level of precision you want. A common one, and the main topic for you and me today, is Newton's method. And yes, this is ... | -RdOwhmqP5s_94-101 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
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"caption_code": "",
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{
"caption": "[There is just piano here.]",
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{
"caption": "Today, I want to share with you a neat way to solve the Towers of Hanoi puzzle just by counting in a different number system. And surprisingly, this stuff relates to finding a curve that fills Sierpinski's triangle. I learned about this from a former CS lecturer of mine, his name's Keith Schwartz, and I... | 2SUvWfNJSsM_1-7 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "I remember putting together an animation for this when I was teaching this, and just like, I know how this works. I know all the things in it. It's still fun to just sit and just watch it play out. Oh yeah. I mean, it's not even clear at first that this is always going to give legal moves. For example, ... | 2SUvWfNJSsM_104-111 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than having to do 2 to the n minus 1 steps? It turns out, not only does this solve Towers of Hanoi, but it does it in the most efficient way... | 2SUvWfNJSsM_112-118 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Disk 3 is thinking, okay, 2, 1, and 0, you have to get off of me. I can't really function under this much weight and pressure. And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That... | 2SUvWfNJSsM_119-126 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. You never need to move me again. Everyone else just figure out how to get here. And in a sense, you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get... | 2SUvWfNJSsM_127-136 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "Well, it's exactly the same thing. Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space. Get off. They need to move somewhere. Then disk 2 can move to where it needs to go. Then disk 1 and 0 can do this. But the interesting point is that every single disk pretty much has... | 2SUvWfNJSsM_137-147 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear that this has to be the most efficient solut... | 2SUvWfNJSsM_148-154 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "So they all start up stacked up from biggest to smallest on one spindle, and the goal is to move the entire tower from one spindle to another. The rule is you can only move one disk at a time, and you can't move a bigger disk on top of a smaller disk. For example, your first move must involve moving dis... | 2SUvWfNJSsM_15-21 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar | |
{
"caption": "And you have to move disk 3. And then you have to move disk 0 through 2 back onto it. There's just not any room for inefficiency from this perspective. So why does counting in binary capture this algorithm? Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then ... | 2SUvWfNJSsM_155-161 | hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar |
End of preview. Expand in Data Studio
English Audio Dataset from YouTube
This dataset contains English audio segments and creator uploaded transcripts (likely higher quality) extracted from various YouTube channels, along with corresponding transcript metadata. The data is intended for training automatic speech recognition (ASR) models.
Data Source and Processing
The data was obtained through the following process:
- Download: Audio (
.m4a) and available English subtitles (.srtforen,en.j3PyPqV-e1s) were downloaded from selected YouTube channels. This raw data, along with video metadata (metadata.csv), is stored initially in adata/{channel_id}/directory structure. - Segmentation: The raw audio files were segmented based on the timing information in the
.srtfiles.- Audio files are splitted by SRT segments and then combined to a maximum duration less than but close to 30 seconds per group for Whisper.
- The corresponding audio portions for each group are extracted using
ffmpegand saved as.mp3files at a 16000 Hz sample rate. - Metadata for each segment, including channel/video info and the text/timing of subtitles within the segment, is saved in a corresponding
.jsonfile.
Intermediate Dataset Structure (dataset directory)
Before being packaged into TAR archives for Hugging Face, the segmented data resides in the dataset directory with the following structure:
dataset/
└── {channel_id}/ # Directory named after the YouTube channel ID
└── {video_id}/ # Directory named after the YouTube video ID
├── {video_id}_{group_name}.mp3 # Segmented audio file
├── {video_id}_{group_name}.json # Corresponding metadata file
└── ...
{channel_id}: The ID of the YouTube channel (e.g.,greenbeanmediaofficial).{video_id}: The unique identifier for the YouTube video.{group_name}: Represents the subtitles included in the segment. It's either the index of the first subtitle (e.g.,1) if the group contains only one, or a range indicating the first and last subtitle indices (e.g.,1-5) if the group contains multiple subtitles.
Dataset Summary
The dataset comprises audio from the following channels:
Channel | Videos | Duration | Percent
--------------------- | ------------ | ------------- | -------
3blue1brown | 136 videos | 37.82 hours | 1.08%
boxofficemoviesscenes | 1626 videos | 153.06 hours | 4.38%
business | 887 videos | 187.80 hours | 5.38%
markrober | 97 videos | 21.77 hours | 0.62%
marvel | 763 videos | 35.17 hours | 1.01%
mitocw | 2844 videos | 1738.07 hours | 49.79%
mkbhd | 114 videos | 27.61 hours | 0.79%
msftmechanics | 732 videos | 131.52 hours | 3.77%
neoexplains | 35 videos | 8.06 hours | 0.23%
nvidia | 134 videos | 19.42 hours | 0.56%
quantasciencechannel | 93 videos | 13.60 hours | 0.39%
teded | 1768 videos | 145.53 hours | 4.17%
theinfographicsshow | 3402 videos | 827.06 hours | 23.69%
twominutepapers | 871 videos | 79.34 hours | 2.27%
veritasium | 291 videos | 64.96 hours | 1.86%
--------------------- | ------------ | ------------- | -------
Total | 13793 videos | 3490.79 hours | 100.00%
Loading the Data
You can load the data using the Hugging Face datasets library:
import os
from datasets import load_dataset
ds = load_dataset(
"OrcinusOrca/YouTube-English",
"all", # or channel_id as config
split="train",
streaming=False, # or True
num_proc=os.cpu_count(),
)
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