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