[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:03.09,Default,,0000,0000,0000,,So let's say that this is an equilateral triangle.
Dialogue: 0,0:00:03.09,0:00:05.05,Default,,0000,0000,0000,,And what I wanna do is make another shape
Dialogue: 0,0:00:05.05,0:00:06.54,Default,,0000,0000,0000,,out of this equilateral triangle.
Dialogue: 0,0:00:06.54,0:00:08.98,Default,,0000,0000,0000,,And I'm gonna do that by taking each of the sides of this triangle,
Dialogue: 0,0:00:09.00,0:00:14.54,Default,,0000,0000,0000,,and divide them into three equal sections, into three equal sections.
Dialogue: 0,0:00:14.54,0:00:18.79,Default,,0000,0000,0000,,So my equilateral triangle wasn't drawn super ideally,
Dialogue: 0,0:00:18.79,0:00:20.11,Default,,0000,0000,0000,,but I think you'll get the point.
Dialogue: 0,0:00:20.11,0:00:21.43,Default,,0000,0000,0000,,And in the middle section,
Dialogue: 0,0:00:21.45,0:00:23.29,Default,,0000,0000,0000,,I wanna construct another equilateral triangle.
Dialogue: 0,0:00:23.29,0:00:25.51,Default,,0000,0000,0000,,So the middle section right over here,
Dialogue: 0,0:00:25.54,0:00:28.64,Default,,0000,0000,0000,,I am going to construct another equilateral triangle.
Dialogue: 0,0:00:28.64,0:00:31.55,Default,,0000,0000,0000,,So it's going to look something like this.
Dialogue: 0,0:00:31.55,0:00:33.86,Default,,0000,0000,0000,,And then, right over here,
Dialogue: 0,0:00:33.86,0:00:37.13,Default,,0000,0000,0000,,I'm gonna put another equilateral triangle.
Dialogue: 0,0:00:37.13,0:00:40.32,Default,,0000,0000,0000,,And so now I went from that equilateral triangle
Dialogue: 0,0:00:40.34,0:00:43.32,Default,,0000,0000,0000,,to something that's looking like a star or star of David.
Dialogue: 0,0:00:43.37,0:00:45.42,Default,,0000,0000,0000,,And then I'm gonna do it again.
Dialogue: 0,0:00:45.42,0:00:48.39,Default,,0000,0000,0000,,So, each of the sides now, I'm gonna divide into three equal sides.
Dialogue: 0,0:00:48.39,0:00:51.49,Default,,0000,0000,0000,,In that middle segment, I'm gonna put an equilateral triangle.
Dialogue: 0,0:00:51.49,0:00:54.15,Default,,0000,0000,0000,,I am going to put an equilateral triangle.
Dialogue: 0,0:00:54.15,0:00:59.28,Default,,0000,0000,0000,,So in the middle segment, I am going to put an equilateral triangle.
Dialogue: 0,0:00:59.28,0:01:01.66,Default,,0000,0000,0000,,So I'm gonna do it for every one of the sides.
Dialogue: 0,0:01:01.66,0:01:04.56,Default,,0000,0000,0000,,So let me do it right there, and right there.
Dialogue: 0,0:01:04.56,0:01:10.86,Default,,0000,0000,0000,,I think you get the idea, but I wanna make it clear, so let me just...
Dialogue: 0,0:01:10.86,0:01:16.27,Default,,0000,0000,0000,,So then, like that, and then, look like that, like that.
Dialogue: 0,0:01:16.27,0:01:20.85,Default,,0000,0000,0000,,And then, almost done for this iteration, this pass.
Dialogue: 0,0:01:20.85,0:01:22.95,Default,,0000,0000,0000,,And then it'll look like that.
Dialogue: 0,0:01:22.99,0:01:24.21,Default,,0000,0000,0000,,Then I can do it again.
Dialogue: 0,0:01:24.21,0:01:27.02,Default,,0000,0000,0000,,Each of the segments I can divide into three equal sides
Dialogue: 0,0:01:27.02,0:01:28.34,Default,,0000,0000,0000,,and draw another equilateral triangles,
Dialogue: 0,0:01:28.34,0:01:32.21,Default,,0000,0000,0000,,like there, there, there, there, there, there.
Dialogue: 0,0:01:32.21,0:01:33.27,Default,,0000,0000,0000,,I think you see where this is going.
Dialogue: 0,0:01:33.27,0:01:37.02,Default,,0000,0000,0000,,And I could keep going on forever and forever.
Dialogue: 0,0:01:37.02,0:01:39.71,Default,,0000,0000,0000,,So what I wanna do in this video is think about
Dialogue: 0,0:01:39.71,0:01:40.86,Default,,0000,0000,0000,,what's going on here.
Dialogue: 0,0:01:40.86,0:01:42.49,Default,,0000,0000,0000,,And what I'm actually drawing,
Dialogue: 0,0:01:42.49,0:01:45.09,Default,,0000,0000,0000,,if we just keep on doing this forever and forever,
Dialogue: 0,0:01:45.09,0:01:48.10,Default,,0000,0000,0000,,every iteration, we look at each side,
Dialogue: 0,0:01:48.13,0:01:49.52,Default,,0000,0000,0000,,we divide them in three equal side,
Dialogue: 0,0:01:49.52,0:01:52.46,Default,,0000,0000,0000,,and then the next iteration were three equal segments,
Dialogue: 0,0:01:52.46,0:01:53.32,Default,,0000,0000,0000,,and the next iteration,
Dialogue: 0,0:01:53.32,0:01:55.48,Default,,0000,0000,0000,,the middle segment we turn to another equilateral triangle.
Dialogue: 0,0:01:55.48,0:01:58.24,Default,,0000,0000,0000,,The shape that we're describing right here
Dialogue: 0,0:01:58.24,0:02:00.20,Default,,0000,0000,0000,,is called the Koch Snowflake.
Dialogue: 0,0:02:00.20,0:02:02.89,Default,,0000,0000,0000,,And I'm sure I'm mispronouncing the Koch part.
Dialogue: 0,0:02:02.89,0:02:05.18,Default,,0000,0000,0000,,The Koch Snowflake,
Dialogue: 0,0:02:05.23,0:02:07.81,Default,,0000,0000,0000,,and was first described by this gentleman right over here,
Dialogue: 0,0:02:07.81,0:02:12.49,Default,,0000,0000,0000,,who was a Swedish mathematician Niels Fabian Helge von Koch.
Dialogue: 0,0:02:12.49,0:02:14.64,Default,,0000,0000,0000,,I'm sure I'm mispronouncing it.
Dialogue: 0,0:02:14.67,0:02:17.25,Default,,0000,0000,0000,,And this is one of the earliest described fractals.
Dialogue: 0,0:02:17.27,0:02:19.85,Default,,0000,0000,0000,,So this is a fractal.
Dialogue: 0,0:02:19.85,0:02:22.00,Default,,0000,0000,0000,,And the reason why it is considered a fractal,
Dialogue: 0,0:02:22.00,0:02:23.79,Default,,0000,0000,0000,,is that it looks the same,
Dialogue: 0,0:02:23.81,0:02:26.34,Default,,0000,0000,0000,,or it looks very similar on any scale you look at it.
Dialogue: 0,0:02:26.34,0:02:29.89,Default,,0000,0000,0000,,So when you look at it at this scale, so if you look at this,
Dialogue: 0,0:02:29.91,0:02:32.41,Default,,0000,0000,0000,,it looks like you see a bunch of triangles with some bumps on it.
Dialogue: 0,0:02:32.41,0:02:34.89,Default,,0000,0000,0000,,But then if you were to zoom in right over there,
Dialogue: 0,0:02:34.91,0:02:37.86,Default,,0000,0000,0000,,then you would still see that same type of pattern.
Dialogue: 0,0:02:37.86,0:02:39.84,Default,,0000,0000,0000,,And then if you were to zoom in again,
Dialogue: 0,0:02:39.86,0:02:41.52,Default,,0000,0000,0000,,you would see it again and again.
Dialogue: 0,0:02:41.58,0:02:43.47,Default,,0000,0000,0000,,So a fractal is anything that, on any scale,
Dialogue: 0,0:02:43.47,0:02:46.81,Default,,0000,0000,0000,,on any level of zoom, it kind of looks roughly the same.
Dialogue: 0,0:02:46.81,0:02:48.70,Default,,0000,0000,0000,,So that's why it's called a fractal.
Dialogue: 0,0:02:48.72,0:02:50.15,Default,,0000,0000,0000,,Now what's particularly interesting,
Dialogue: 0,0:02:50.20,0:02:53.53,Default,,0000,0000,0000,,and why I'm putting it at this point in the geometry playlist,
Dialogue: 0,0:02:53.53,0:02:56.79,Default,,0000,0000,0000,,is that this actually has an infinite perimeter.
Dialogue: 0,0:02:56.79,0:02:58.33,Default,,0000,0000,0000,,If you were to keep doing it,
Dialogue: 0,0:02:58.37,0:02:59.90,Default,,0000,0000,0000,,if you were actually to make the Koch Snowflake
Dialogue: 0,0:02:59.90,0:03:03.26,Default,,0000,0000,0000,,where you keep an infinite number of times
Dialogue: 0,0:03:03.28,0:03:05.24,Default,,0000,0000,0000,,on every smaller little triangle here,
Dialogue: 0,0:03:05.28,0:03:09.91,Default,,0000,0000,0000,,you keep adding another equilateral triangle on its side.
Dialogue: 0,0:03:09.93,0:03:11.68,Default,,0000,0000,0000,,And to show that it has an infinite perimeter,
Dialogue: 0,0:03:11.68,0:03:13.44,Default,,0000,0000,0000,,let's just consider one side over here.
Dialogue: 0,0:03:13.44,0:03:16.00,Default,,0000,0000,0000,,So let's say that this side,
Dialogue: 0,0:03:16.00,0:03:18.55,Default,,0000,0000,0000,,so let's say we're starting right where we started
Dialogue: 0,0:03:18.55,0:03:20.05,Default,,0000,0000,0000,,with that original triangle, that's that side.
Dialogue: 0,0:03:20.08,0:03:21.48,Default,,0000,0000,0000,,And let's say it has length S.
Dialogue: 0,0:03:21.52,0:03:23.93,Default,,0000,0000,0000,,And then we divide it into three equal segments.
Dialogue: 0,0:03:23.96,0:03:26.29,Default,,0000,0000,0000,,We divide it into three equal segments.
Dialogue: 0,0:03:26.31,0:03:30.81,Default,,0000,0000,0000,,So those are gonna be S/3, S/3, let me write it this way.
Dialogue: 0,0:03:30.81,0:03:35.94,Default,,0000,0000,0000,,S/3, S/3, and S/3.
Dialogue: 0,0:03:35.94,0:03:38.82,Default,,0000,0000,0000,,In the middle segment, you make an equilateral triangle.
Dialogue: 0,0:03:38.82,0:03:41.91,Default,,0000,0000,0000,,In the middle segment, you make an equilateral triangle.
Dialogue: 0,0:03:41.91,0:03:44.09,Default,,0000,0000,0000,,So each of these sides are going to be S/3.
Dialogue: 0,0:03:44.09,0:03:47.00,Default,,0000,0000,0000,,S/3, S/3.
Dialogue: 0,0:03:47.00,0:03:50.70,Default,,0000,0000,0000,,And now the length of this new part,
Dialogue: 0,0:03:50.70,0:03:53.27,Default,,0000,0000,0000,,I can't call it a line anymore 'because it has its bump in it.
Dialogue: 0,0:03:53.29,0:03:56.88,Default,,0000,0000,0000,,The length of this part right over here, this side,
Dialogue: 0,0:03:56.88,0:03:59.11,Default,,0000,0000,0000,,now doesn't have just the length of S.
Dialogue: 0,0:03:59.15,0:04:01.62,Default,,0000,0000,0000,,It is now S/3 * 4.
Dialogue: 0,0:04:01.62,0:04:03.36,Default,,0000,0000,0000,,Before it was S/3 * 3,
Dialogue: 0,0:04:03.36,0:04:07.55,Default,,0000,0000,0000,,now you have one, two, three, four segments that are S/3.
Dialogue: 0,0:04:07.55,0:04:10.50,Default,,0000,0000,0000,,So now after one time, after one pace,
Dialogue: 0,0:04:10.50,0:04:14.93,Default,,0000,0000,0000,,after one time of doing this adding triangles,
Dialogue: 0,0:04:14.93,0:04:16.30,Default,,0000,0000,0000,,our new side,
Dialogue: 0,0:04:16.34,0:04:23.56,Default,,0000,0000,0000,,after we had that bump is going to be 4 * S/3, or equals 4/3 s.
Dialogue: 0,0:04:23.56,0:04:30.95,Default,,0000,0000,0000,,So if our original perimeter when it was just a triangle is P sub 0,
Dialogue: 0,0:04:30.95,0:04:34.23,Default,,0000,0000,0000,,after one pass, after we had one set of bumps,
Dialogue: 0,0:04:34.23,0:04:35.67,Default,,0000,0000,0000,,then our perimeter is going to be,
Dialogue: 0,0:04:35.71,0:04:39.88,Default,,0000,0000,0000,,so it's going to be 4/3 * the original one.
Dialogue: 0,0:04:39.88,0:04:42.66,Default,,0000,0000,0000,,Because each of the sides are gonna be 4/3 bigger now.
Dialogue: 0,0:04:42.66,0:04:44.27,Default,,0000,0000,0000,,So if this was made up of three sides,
Dialogue: 0,0:04:44.29,0:04:46.69,Default,,0000,0000,0000,,now each of those sides are going to be 4/3 bigger.
Dialogue: 0,0:04:46.69,0:04:48.95,Default,,0000,0000,0000,,So the new perimeter's gonna be 4/3 times that.
Dialogue: 0,0:04:48.95,0:04:51.98,Default,,0000,0000,0000,,And then we take a second pass on it.
Dialogue: 0,0:04:51.98,0:04:54.47,Default,,0000,0000,0000,,That's gonna be 4/3 times this first pass.
Dialogue: 0,0:04:54.47,0:04:57.74,Default,,0000,0000,0000,,So every pass you take it's getting 4/3 bigger,
Dialogue: 0,0:04:57.79,0:05:00.19,Default,,0000,0000,0000,,it's getting I guess a third bigger on every,
Dialogue: 0,0:05:00.19,0:05:03.55,Default,,0000,0000,0000,,it's getting 4/3 the previous pass.
Dialogue: 0,0:05:03.61,0:05:05.59,Default,,0000,0000,0000,,And so if you do that in infinite number of times,
Dialogue: 0,0:05:05.59,0:05:10.74,Default,,0000,0000,0000,,if you multiply any number by 4/3 an infinite number of times,
Dialogue: 0,0:05:10.74,0:05:13.76,Default,,0000,0000,0000,,you are gonna get an infinite number of！an infinite length.
Dialogue: 0,0:05:13.76,0:05:16.34,Default,,0000,0000,0000,,So, P infinity, P infinity,
Dialogue: 0,0:05:16.36,0:05:19.91,Default,,0000,0000,0000,,the perimeter if you do it an infinite number of times, is infinite.
Dialogue: 0,0:05:19.94,0:05:22.14,Default,,0000,0000,0000,,Now that by itself, is kind of cool,
Dialogue: 0,0:05:22.19,0:05:24.30,Default,,0000,0000,0000,,just to think about something that has an infinite perimeter.
Dialogue: 0,0:05:24.30,0:05:28.26,Default,,0000,0000,0000,,But what's even neater is that it actually has a finite area.
Dialogue: 0,0:05:28.26,0:05:30.12,Default,,0000,0000,0000,,And when I say a finite area,
Dialogue: 0,0:05:30.12,0:05:32.48,Default,,0000,0000,0000,,it actually covers a bounded amount of space.
Dialogue: 0,0:05:32.48,0:05:34.49,Default,,0000,0000,0000,,That I could actually draw a shape around this
Dialogue: 0,0:05:34.49,0:05:36.34,Default,,0000,0000,0000,,and this thing will never expand beyond that.
Dialogue: 0,0:05:36.34,0:05:38.96,Default,,0000,0000,0000,,And to think about, I'm not gonna do a really formal proof,
Dialogue: 0,0:05:38.96,0:05:41.60,Default,,0000,0000,0000,,just think about what happens on any one of these sides.
Dialogue: 0,0:05:41.60,0:05:45.55,Default,,0000,0000,0000,,So in that first pass, we have this triangle gets popped out.
Dialogue: 0,0:05:45.55,0:05:49.54,Default,,0000,0000,0000,,And then, if you think about it, if you just draw what happens,
Dialogue: 0,0:05:49.54,0:05:52.28,Default,,0000,0000,0000,,the next iteration you draw these two triangles right over there,
Dialogue: 0,0:05:52.31,0:05:53.94,Default,,0000,0000,0000,,and these two characters right over there.
Dialogue: 0,0:05:53.94,0:05:56.23,Default,,0000,0000,0000,,And then you put some triangles over here,
Dialogue: 0,0:05:56.26,0:05:59.60,Default,,0000,0000,0000,,and here, and here, and here, and here, so on and so forth.
Dialogue: 0,0:05:59.63,0:06:02.52,Default,,0000,0000,0000,,But notice, you could keep adding more and more,
Dialogue: 0,0:06:02.52,0:06:04.98,Default,,0000,0000,0000,,you can add an infinite number of these bumps,
Dialogue: 0,0:06:05.02,0:06:07.07,Default,,0000,0000,0000,,but you're never gonna go past this original point.
Dialogue: 0,0:06:07.07,0:06:11.22,Default,,0000,0000,0000,,And the same thing is gonna be true on this side right over here.
Dialogue: 0,0:06:11.22,0:06:13.84,Default,,0000,0000,0000,,It's also gonna be true on this side right over here.
Dialogue: 0,0:06:13.87,0:06:17.54,Default,,0000,0000,0000,,Also going to be true at this side over here.
Dialogue: 0,0:06:17.54,0:06:19.55,Default,,0000,0000,0000,,Also going to be true at this side over there.
Dialogue: 0,0:06:19.55,0:06:22.33,Default,,0000,0000,0000,,And then also going to be true at that side over there.
Dialogue: 0,0:06:22.35,0:06:24.59,Default,,0000,0000,0000,,So even if you do this an infinite number of times,
Dialogue: 0,0:06:24.59,0:06:27.12,Default,,0000,0000,0000,,this shape, this Koch Snowflake
Dialogue: 0,0:06:27.16,0:06:30.13,Default,,0000,0000,0000,,will never have a larger area than this bounding hexagon.
Dialogue: 0,0:06:30.13,0:06:32.07,Default,,0000,0000,0000,,Or it will not have a larger area
Dialogue: 0,0:06:32.07,0:06:34.53,Default,,0000,0000,0000,,than a shape that looks something like that.
Dialogue: 0,0:06:34.53,0:06:36.45,Default,,0000,0000,0000,,And I'm just kind of drawing an arbitrary,
Dialogue: 0,0:06:36.45,0:06:38.20,Default,,0000,0000,0000,,well I wanna make it outside of the hexagon,
Dialogue: 0,0:06:38.20,0:06:39.78,Default,,0000,0000,0000,,I could put a circle outside of it.
Dialogue: 0,0:06:39.78,0:06:44.63,Default,,0000,0000,0000,,So this thing I drew in blue, or this hexagon I drew in magenta,
Dialogue: 0,0:06:44.63,0:06:46.82,Default,,0000,0000,0000,,those clearly have a fixed area.
Dialogue: 0,0:06:46.82,0:06:49.48,Default,,0000,0000,0000,,And this Koch Snowflake will always be bounded,
Dialogue: 0,0:06:49.48,0:06:52.45,Default,,0000,0000,0000,,eventhough you can add these bumps an infinite number of times.
Dialogue: 0,0:06:52.45,0:06:55.38,Default,,0000,0000,0000,,So a bunch of really cool things here.
Dialogue: 0,0:06:55.42,0:06:56.33,Default,,0000,0000,0000,,One, it's a fractal.
Dialogue: 0,0:06:56.33,0:06:58.76,Default,,0000,0000,0000,,You can keep zooming in and, it'll look the same.
Dialogue: 0,0:06:58.78,0:07:04.95,Default,,0000,0000,0000,,Another thing, infinite perimeter, and finite area.
Dialogue: 0,0:07:04.95,0:07:07.83,Default,,0000,0000,0000,,Now you might say, "Wait, uh, okay, this is a very abstract thing.
Dialogue: 0,0:07:07.83,0:07:10.12,Default,,0000,0000,0000,,Things like this don't actually exist in the real world."
Dialogue: 0,0:07:10.12,0:07:13.24,Default,,0000,0000,0000,,And there's an experiment
Dialogue: 0,0:07:13.24,0:07:14.82,Default,,0000,0000,0000,,that people talk about in the fractal world.
Dialogue: 0,0:07:14.87,0:07:17.77,Default,,0000,0000,0000,,And that's finding the perimeter of England,
Dialogue: 0,0:07:17.82,0:07:19.20,Default,,0000,0000,0000,,or you can actually do it with any island.
Dialogue: 0,0:07:19.20,0:07:21.17,Default,,0000,0000,0000,,And so England looks something like,
Dialogue: 0,0:07:21.17,0:07:22.73,Default,,0000,0000,0000,,you know, I'm not an expert on the, you know,
Dialogue: 0,0:07:22.73,0:07:24.23,Default,,0000,0000,0000,,let's say it looks something like that.
Dialogue: 0,0:07:24.23,0:07:26.23,Default,,0000,0000,0000,,So at first, you might approximate the perimeter,
Dialogue: 0,0:07:26.23,0:07:27.48,Default,,0000,0000,0000,,and you might measure this distance.
Dialogue: 0,0:07:27.55,0:07:32.35,Default,,0000,0000,0000,,You might measure this distance + this distance
Dialogue: 0,0:07:32.35,0:07:36.07,Default,,0000,0000,0000,,+this distance + that distance + that distance + that distance.
Dialogue: 0,0:07:36.07,0:07:37.66,Default,,0000,0000,0000,,You know, look.
Dialogue: 0,0:07:37.66,0:07:38.59,Default,,0000,0000,0000,,It has a finite perimeter.
Dialogue: 0,0:07:38.62,0:07:40.30,Default,,0000,0000,0000,,Clearly, it has a finite area, but you know,
Dialogue: 0,0:07:40.30,0:07:42.30,Default,,0000,0000,0000,,look that has a finite perimeter.
Dialogue: 0,0:07:42.34,0:07:43.72,Default,,0000,0000,0000,,But you're like, "No, no, that's not as good.
Dialogue: 0,0:07:43.75,0:07:45.38,Default,,0000,0000,0000,,You have to approximate it a little better than that."
Dialogue: 0,0:07:45.40,0:07:46.96,Default,,0000,0000,0000,,Instead of doing it that rough,
Dialogue: 0,0:07:46.98,0:07:48.68,Default,,0000,0000,0000,,you need to make a bunch of smaller lines.
Dialogue: 0,0:07:48.68,0:07:50.74,Default,,0000,0000,0000,,You got to make a bunch of smaller lines
Dialogue: 0,0:07:50.77,0:07:52.57,Default,,0000,0000,0000,,so you can hug the coast a little bit better.
Dialogue: 0,0:07:52.62,0:07:55.01,Default,,0000,0000,0000,,And you're like, "Okay, that's a much better approximation."
Dialogue: 0,0:07:55.01,0:07:58.73,Default,,0000,0000,0000,,But then, let's say at some piece of coast, if we zoom in,
Dialogue: 0,0:07:58.76,0:08:01.78,Default,,0000,0000,0000,,if we zoom in enough, if we zoom in enough,
Dialogue: 0,0:08:01.78,0:08:03.98,Default,,0000,0000,0000,,the actual coastline's gonna look something like this.
Dialogue: 0,0:08:04.02,0:08:08.19,Default,,0000,0000,0000,,The actual coastline will have all these little tidbits in it.
Dialogue: 0,0:08:08.26,0:08:11.15,Default,,0000,0000,0000,,And essentially, when you did that first, when you did this pass,
Dialogue: 0,0:08:11.15,0:08:13.58,Default,,0000,0000,0000,,you were just measuring, you were just measuring that.
Dialogue: 0,0:08:13.58,0:08:15.74,Default,,0000,0000,0000,,And you're like,"That's not the perimeter of the coastline."
Dialogue: 0,0:08:15.74,0:08:17.62,Default,,0000,0000,0000,,You're gonna have to do many many more sides.
Dialogue: 0,0:08:17.65,0:08:18.85,Default,,0000,0000,0000,,You're gonna have to do something like this
Dialogue: 0,0:08:18.90,0:08:25.66,Default,,0000,0000,0000,,to actually get the perimeter of the coastline.
Dialogue: 0,0:08:25.66,0:08:29.15,Default,,0000,0000,0000,,And you say, "Hey, that is a good approximation of the perimeter."
Dialogue: 0,0:08:29.15,0:08:32.19,Default,,0000,0000,0000,,But, if you were to zoom in on that part of the coastline even more,
Dialogue: 0,0:08:32.19,0:08:35.05,Default,,0000,0000,0000,,it will actually turn out that it won't look exactly like that.
Dialogue: 0,0:08:35.05,0:08:37.33,Default,,0000,0000,0000,,It will actually come in and out, like this.
Dialogue: 0,0:08:37.36,0:08:39.45,Default,,0000,0000,0000,,Maybe look something like that.
Dialogue: 0,0:08:39.45,0:08:42.81,Default,,0000,0000,0000,,So instead of having these rough lines, that just measure it like that.
Dialogue: 0,0:08:42.89,0:08:43.85,Default,,0000,0000,0000,,You're gonna say, "Oh wait,
Dialogue: 0,0:08:43.90,0:08:46.17,Default,,0000,0000,0000,,now I need to go a little bit closer and hug it even tighter."
Dialogue: 0,0:08:46.22,0:08:48.27,Default,,0000,0000,0000,,And you can really keep on doing that
Dialogue: 0,0:08:48.31,0:08:50.15,Default,,0000,0000,0000,,until you get to the actual atomic level.
Dialogue: 0,0:08:50.15,0:08:54.73,Default,,0000,0000,0000,,So the actual coastline of an island,
Dialogue: 0,0:08:54.77,0:08:58.79,Default,,0000,0000,0000,,or a continent, or anything, is actually, somewhat kind of fractalish.
Dialogue: 0,0:08:58.84,0:09:01.21,Default,,0000,0000,0000,,And it is, you can kind of think of it
Dialogue: 0,0:09:01.21,0:09:03.13,Default,,0000,0000,0000,,as having an almost infinite perimeter.
Dialogue: 0,0:09:03.18,0:09:04.15,Default,,0000,0000,0000,,Obviously at some point,
Dialogue: 0,0:09:04.22,0:09:05.48,Default,,0000,0000,0000,,you'll get into the kind of atomic level,
Dialogue: 0,0:09:05.52,0:09:06.61,Default,,0000,0000,0000,,so it won't quite be the same,
Dialogue: 0,0:09:06.66,0:09:08.51,Default,,0000,0000,0000,,but it's kind of the same phenomenon.
Dialogue: 0,0:09:08.54,0:09:10.39,Default,,0000,0000,0000,,It's an interesting thing to actually think about.