Koch snowflake fractal
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Koch snowflake fractal
So let's say that this is an equilateral triangle. And what I want to do is make another shape out of this equilateral triangle. And I'm going to do that by taking each of the sides of this triangle, and divide them into three equal sections. So my equilateral triangle wasn't drawn super ideally. But I think you'll get the point. And in the middle section I want to construct another equilateral triangle. So it's going to look something like this. And then right over here I'm going to put another equilateral triangle. And so now I went from that equilateral triangle to something that's looking like a star, or a Star of David. And then I'm going to do it again. So each of the sides now I'm going to divide into three equal sides. And in that middle segment I'm going to put an equilateral triangle. So in the middle segment I'm going to put an equilateral triangle. So I'm going to do it for every one of the sides. So let me do it right there. And then right there. I think you get the idea, but I want to make it clear. So let me just, so then like that, and then like that, like that, and then almost done for this iteration. This pass. And it'll look like that. Then I could do it again. Each of the segments I can divide into three equal sides and draw another equilateral triangle. So I could just there, there, there, there. I think you see where this is going. And I could keep going on forever and forever. So what I want to do in this video is think about what's going on here. And what I'm actually drawing, if we just keep on doing this forever and forever, every side, every iteration, we look at each side, we divide into three equal sides. And then the next iteration, or three equal segments, the next iteration, the middle segment we turn to another equilateral triangle. This shape that we're describing right here is called a Koch snowflake. And I'm sure I'm mispronouncing the Koch part. A Koch snowflake, and it was first described by this gentleman right over here, who is a Swedish mathematician, Niels Fabian Helge von Koch, who I'm sure I'm mispronouncing it. And this was one of the earliest described fractals. So this is a fractal. And the reason why it is considered a fractal is that it looks the same, or it looks very similar, on any scale you look at it. So when you look at it at this scale, so if you look at this, it like you see a bunch of triangles with some bumps on it. But then if you were to zoom in right over there, then you would still see that same type of pattern. And then if you were to zoom in again, you would see it again and again. So a fractal is anything that at on any scale, on any level of zoom, it kind of looks roughly the same. So that's why it's called a fractal. Now what's particularly interesting, and why I'm putting it at this point in the geometry playlist, is that this actually has an infinite perimeter. If you were to keep doing it, if you were actually to make the Koch snowflake, where you keep an infinite number of times on every smaller little triangle here, you keep adding another equilateral triangle on its side. And to show that it has an infinite perimeter, let's just consider one side over here. So let's say that this side, so let's say we're starting right when we started with that original triangle, that's that side. Let's say it has length s. And then we divide it into three equal segments. So those are going to be s/3, s/3-- actually, let me write it this way. s/3, s/3, and s/3. And in the middle segment, you make an equilateral triangle. So each of these sides are going to be s/3. s/3, s/3. And now the length of this new part-- I can't call it a line anymore, because it has this bump in it-- the length of this part right over here, this side, now doesn't have just a length of s, it is now s/3 times 4. Before it was s/3 times 3. Now you have 1, 2, 3, 4 segments that are s/3. So now, after one time, after one pace, after one time of doing this adding triangles, our new side, after we add that bump, is going to be four times s/3. Or it equals 4/3 s. So if our original perimeter when it was just a triangle is p sub 0. After one pass, after we add one set of bumps, then our perimeter is going to be 4/3 times the original one. Because each of the sides are going to be 4/3 bigger now. So this was made up of three sides. Now each of those sides are going to be 4/3 bigger. So the new perimeter's going to be 4/3 times that. And then when we take a second pass on it, it's going to be 4/3 times this first pass. So every pass you take, it's getting 4/3 bigger. Or it's getting, I guess, a 1/3 bigger on every, it's getting 4/3 the previous pass. And so if you do that an infinite number of times, if you multiply any number by 4/3 an infinite number of times, you're going to get an infinite number of infinite length. So P infinity. The perimeter, if you do this an infinite number of times, is infinite. Now that by itself is kind of cool, just to think about something that has an infinite perimeter. But what's even neater is that it actually has a finite area. And when I say a finite area, it actually covers a bounded amount of space. And I could actually draw a shape around this, and this thing will never expand beyond that. And to think about it, I'm not going to do a really formal proof, just think about it, what happens on any one of these sides. So on that first pass we have that this triangle gets popped out. And then, if you think about it, if you just draw what happens, the next iteration you draw these two triangles right over there. And these two characters right over there. And then you put some triangles over here, and here, and here, and here, and here. So on and so forth. But notice, you can keep adding more and more. You can add essentially an infinite number of these bumps, but you're never going to go past this original point. And the same thing is going to be true on this side right over here. It's also going to be true of this side over here. Also going to be true at this side over here. Also going to be true this side over there. And then also going to be true that side over there. So even if you do this an infinite number of times, this shape, this Koch snowflake will never have a larger area than this bounding hexagon. Or which will never have a larger area than a shape that looks something like that. I'm just kind of drawing an arbitrary, well I want to make it outside of the hexagon, I could put a circle outside of it. So this thing I drew in blue, or this hexagon I drew in magenta, those clearly have a fixed area. And this Koch snowflake will always be bounded. Even though you can add these bumps an infinite number of times. So a bunch of really cool things here. One, it's a fractal. You can keep zooming in and it'll look the same. The other thing, infinite, infinite perimeter, and finite, finite area. Now you might say, wait Sal, OK. This is a very abstract thing. Things like this don't actually exist in the real world. And there's a fun thought experiment that people talk about in the fractal world, and that's finding the perimeter of England. Or you can actually do it with any island. And so England looks something like-- and I'm not an expert on, let's say it looks something like that-- so at first you might approximate the perimeter. And you might measure this distance, you might measure this distance, plus this distance, plus this distance, plus that distance, plus that distance, plus that distance. And you're like look, it has a finite perimeter. It clearly has a finite area. But you're like, look, that has a finite perimeter. But you're like, no, wait that's not as good. You have to approximate it a little bit better than that. Instead of doing it that rough, you need to make a bunch of smaller lines. You need to make a bunch of smaller lines so you can hug the coast a little bit better. And you're like, OK, that's a much better approximation. But then, let's say you're at some piece of coast, if we zoom in enough, the actual coast line is going to look something like this. The actual coast line will have all of these little divots in it. And essentially, when you did that first, when did this pass, you were just measuring that. And you're like, that's not the perimeter of the coastline. You're going to have to do many, many more sides. You're going to do something like this, to actually get the perimeter of the coast line. And you're just like, hey, now that is a good approximation for the perimeter. But if you were to zoom in on that part of the coastline even more, it'll actually turn out that it won't look exactly like that. It'll actually come in and out like this. Maybe it'll look something like that. So instead of having these rough lines that just measure it like that, you're going to say, oh wait, no, I need to go a little bit closer and hug it even tighter. And you can really keep on doing that until you get to the actual atomic level. So the actual coastline of an island, or a continent, or anything, is actually somewhat kind of fractalish. And it is, you can kind of think of it as having an almost infinite perimeter. Obviously at some point you're getting to kind of the atomic level, so it won't quite be the same. But it's kind of the same phenomenon. It's an interesting thing to actually think about.