0:00:00.000,0:00:03.090 So let's say that this is an equilateral triangle. 0:00:03.090,0:00:05.050 And what I wanna do is make another shape 0:00:05.050,0:00:06.540 out of this equilateral triangle. 0:00:06.540,0:00:08.980 And I'm gonna do that by taking each of the sides of this triangle, 0:00:09.000,0:00:14.540 and divide them into three equal sections, into three equal sections. 0:00:14.540,0:00:18.790 So my equilateral triangle wasn't drawn super ideally, 0:00:18.790,0:00:20.110 but I think you'll get the point. 0:00:20.110,0:00:21.430 And in the middle section, 0:00:21.450,0:00:23.290 I wanna construct another equilateral triangle. 0:00:23.290,0:00:25.510 So the middle section right over here, 0:00:25.540,0:00:28.640 I am going to construct another equilateral triangle. 0:00:28.640,0:00:31.550 So it's going to look something like this. 0:00:31.550,0:00:33.860 And then, right over here, 0:00:33.860,0:00:37.130 I'm gonna put another equilateral triangle. 0:00:37.130,0:00:40.320 And so now I went from that equilateral triangle 0:00:40.340,0:00:43.320 to something that's looking like a star or star of David. 0:00:43.370,0:00:45.420 And then I'm gonna do it again. 0:00:45.420,0:00:48.390 So, each of the sides now, I'm gonna divide into three equal sides. 0:00:48.390,0:00:51.490 In that middle segment, I'm gonna put an equilateral triangle. 0:00:51.490,0:00:54.150 I am going to put an equilateral triangle. 0:00:54.150,0:00:59.280 So in the middle segment, I am going to put an equilateral triangle. 0:00:59.280,0:01:01.660 So I'm gonna do it for every one of the sides. 0:01:01.660,0:01:04.560 So let me do it right there, and right there. 0:01:04.560,0:01:10.860 I think you get the idea, but I wanna make it clear, so let me just... 0:01:10.860,0:01:16.270 So then, like that, and then, look like that, like that. 0:01:16.270,0:01:20.850 And then, almost done for this iteration, this pass. 0:01:20.850,0:01:22.950 And then it'll look like that. 0:01:22.990,0:01:24.210 Then I can do it again. 0:01:24.210,0:01:27.020 Each of the segments I can divide into three equal sides 0:01:27.020,0:01:28.340 and draw another equilateral triangles, 0:01:28.340,0:01:32.210 like there, there, there, there, there, there. 0:01:32.210,0:01:33.270 I think you see where this is going. 0:01:33.270,0:01:37.020 And I could keep going on forever and forever. 0:01:37.020,0:01:39.710 So what I wanna do in this video is think about 0:01:39.710,0:01:40.860 what's going on here. 0:01:40.860,0:01:42.490 And what I'm actually drawing, 0:01:42.490,0:01:45.090 if we just keep on doing this forever and forever, 0:01:45.090,0:01:48.100 every iteration, we look at each side, 0:01:48.130,0:01:49.520 we divide them in three equal side, 0:01:49.520,0:01:52.460 and then the next iteration were three equal segments, 0:01:52.460,0:01:53.320 and the next iteration, 0:01:53.320,0:01:55.480 the middle segment we turn to another equilateral triangle. 0:01:55.480,0:01:58.240 The shape that we're describing right here 0:01:58.240,0:02:00.200 is called the Koch Snowflake. 0:02:00.200,0:02:02.890 And I'm sure I'm mispronouncing the Koch part. 0:02:02.890,0:02:05.180 The Koch Snowflake, 0:02:05.230,0:02:07.810 and was first described by this gentleman right over here, 0:02:07.810,0:02:12.490 who was a Swedish mathematician Niels Fabian Helge von Koch. 0:02:12.490,0:02:14.640 I'm sure I'm mispronouncing it. 0:02:14.670,0:02:17.250 And this is one of the earliest described fractals. 0:02:17.270,0:02:19.850 So this is a fractal. 0:02:19.850,0:02:22.000 And the reason why it is considered a fractal, 0:02:22.000,0:02:23.790 is that it looks the same, 0:02:23.810,0:02:26.340 or it looks very similar on any scale you look at it. 0:02:26.340,0:02:29.890 So when you look at it at this scale, so if you look at this, 0:02:29.910,0:02:32.410 it looks like you see a bunch of triangles with some bumps on it. 0:02:32.410,0:02:34.890 But then if you were to zoom in right over there, 0:02:34.910,0:02:37.860 then you would still see that same type of pattern. 0:02:37.860,0:02:39.840 And then if you were to zoom in again, 0:02:39.860,0:02:41.520 you would see it again and again. 0:02:41.580,0:02:43.470 So a fractal is anything that, on any scale, 0:02:43.470,0:02:46.810 on any level of zoom, it kind of looks roughly the same. 0:02:46.810,0:02:48.700 So that's why it's called a fractal. 0:02:48.720,0:02:50.150 Now what's particularly interesting, 0:02:50.200,0:02:53.530 and why I'm putting it at this point in the geometry playlist, 0:02:53.530,0:02:56.790 is that this actually has an infinite perimeter. 0:02:56.790,0:02:58.330 If you were to keep doing it, 0:02:58.370,0:02:59.900 if you were actually to make the Koch Snowflake 0:02:59.900,0:03:03.260 where you keep an infinite number of times 0:03:03.280,0:03:05.240 on every smaller little triangle here, 0:03:05.280,0:03:09.910 you keep adding another equilateral triangle on its side. 0:03:09.930,0:03:11.680 And to show that it has an infinite perimeter, 0:03:11.680,0:03:13.440 let's just consider one side over here. 0:03:13.440,0:03:16.000 So let's say that this side, 0:03:16.000,0:03:18.550 so let's say we're starting right where we started 0:03:18.550,0:03:20.050 with that original triangle, that's that side. 0:03:20.080,0:03:21.480 And let's say it has length S. 0:03:21.520,0:03:23.930 And then we divide it into three equal segments. 0:03:23.960,0:03:26.290 We divide it into three equal segments. 0:03:26.310,0:03:30.810 So those are gonna be S/3, S/3, let me write it this way. 0:03:30.810,0:03:35.940 S/3, S/3, and S/3. 0:03:35.940,0:03:38.820 In the middle segment, you make an equilateral triangle. 0:03:38.820,0:03:41.910 In the middle segment, you make an equilateral triangle. 0:03:41.910,0:03:44.090 So each of these sides are going to be S/3. 0:03:44.090,0:03:47.000 S/3, S/3. 0:03:47.000,0:03:50.700 And now the length of this new part, 0:03:50.700,0:03:53.270 I can't call it a line anymore 'because it has its bump in it. 0:03:53.290,0:03:56.880 The length of this part right over here, this side, 0:03:56.880,0:03:59.110 now doesn't have just the length of S. 0:03:59.150,0:04:01.620 It is now S/3 * 4. 0:04:01.620,0:04:03.360 Before it was S/3 * 3, 0:04:03.360,0:04:07.550 now you have one, two, three, four segments that are S/3. 0:04:07.550,0:04:10.500 So now after one time, after one pace, 0:04:10.500,0:04:14.930 after one time of doing this adding triangles, 0:04:14.930,0:04:16.300 our new side, 0:04:16.340,0:04:23.560 after we had that bump is going to be 4 * S/3, or equals 4/3 s. 0:04:23.560,0:04:30.950 So if our original perimeter when it was just a triangle is P sub 0, 0:04:30.950,0:04:34.230 after one pass, after we had one set of bumps, 0:04:34.230,0:04:35.670 then our perimeter is going to be, 0:04:35.710,0:04:39.880 so it's going to be 4/3 * the original one. 0:04:39.880,0:04:42.660 Because each of the sides are gonna be 4/3 bigger now. 0:04:42.660,0:04:44.270 So if this was made up of three sides, 0:04:44.290,0:04:46.690 now each of those sides are going to be 4/3 bigger. 0:04:46.690,0:04:48.950 So the new perimeter's gonna be 4/3 times that. 0:04:48.950,0:04:51.980 And then we take a second pass on it. 0:04:51.980,0:04:54.470 That's gonna be 4/3 times this first pass. 0:04:54.470,0:04:57.740 So every pass you take it's getting 4/3 bigger, 0:04:57.790,0:05:00.190 it's getting I guess a third bigger on every, 0:05:00.190,0:05:03.550 it's getting 4/3 the previous pass. 0:05:03.610,0:05:05.590 And so if you do that in infinite number of times, 0:05:05.590,0:05:10.740 if you multiply any number by 4/3 an infinite number of times, 0:05:10.740,0:05:13.760 you are gonna get an infinite number of！an infinite length. 0:05:13.760,0:05:16.340 So, P infinity, P infinity, 0:05:16.360,0:05:19.910 the perimeter if you do it an infinite number of times, is infinite. 0:05:19.940,0:05:22.140 Now that by itself, is kind of cool, 0:05:22.190,0:05:24.300 just to think about something that has an infinite perimeter. 0:05:24.300,0:05:28.260 But what's even neater is that it actually has a finite area. 0:05:28.260,0:05:30.120 And when I say a finite area, 0:05:30.120,0:05:32.480 it actually covers a bounded amount of space. 0:05:32.480,0:05:34.490 That I could actually draw a shape around this 0:05:34.490,0:05:36.340 and this thing will never expand beyond that. 0:05:36.340,0:05:38.960 And to think about, I'm not gonna do a really formal proof, 0:05:38.960,0:05:41.600 just think about what happens on any one of these sides. 0:05:41.600,0:05:45.550 So in that first pass, we have this triangle gets popped out. 0:05:45.550,0:05:49.540 And then, if you think about it, if you just draw what happens, 0:05:49.540,0:05:52.280 the next iteration you draw these two triangles right over there, 0:05:52.310,0:05:53.940 and these two characters right over there. 0:05:53.940,0:05:56.230 And then you put some triangles over here, 0:05:56.260,0:05:59.600 and here, and here, and here, and here, so on and so forth. 0:05:59.630,0:06:02.520 But notice, you could keep adding more and more, 0:06:02.520,0:06:04.980 you can add an infinite number of these bumps, 0:06:05.020,0:06:07.070 but you're never gonna go past this original point. 0:06:07.070,0:06:11.220 And the same thing is gonna be true on this side right over here. 0:06:11.220,0:06:13.840 It's also gonna be true on this side right over here. 0:06:13.870,0:06:17.540 Also going to be true at this side over here. 0:06:17.540,0:06:19.550 Also going to be true at this side over there. 0:06:19.550,0:06:22.330 And then also going to be true at that side over there. 0:06:22.350,0:06:24.590 So even if you do this an infinite number of times, 0:06:24.590,0:06:27.120 this shape, this Koch Snowflake 0:06:27.160,0:06:30.130 will never have a larger area than this bounding hexagon. 0:06:30.130,0:06:32.070 Or it will not have a larger area 0:06:32.070,0:06:34.530 than a shape that looks something like that. 0:06:34.530,0:06:36.450 And I'm just kind of drawing an arbitrary, 0:06:36.450,0:06:38.200 well I wanna make it outside of the hexagon, 0:06:38.200,0:06:39.780 I could put a circle outside of it. 0:06:39.780,0:06:44.630 So this thing I drew in blue, or this hexagon I drew in magenta, 0:06:44.630,0:06:46.820 those clearly have a fixed area. 0:06:46.820,0:06:49.480 And this Koch Snowflake will always be bounded, 0:06:49.480,0:06:52.450 eventhough you can add these bumps an infinite number of times. 0:06:52.450,0:06:55.380 So a bunch of really cool things here. 0:06:55.420,0:06:56.330 One, it's a fractal. 0:06:56.330,0:06:58.760 You can keep zooming in and, it'll look the same. 0:06:58.780,0:07:04.950 Another thing, infinite perimeter, and finite area. 0:07:04.950,0:07:07.830 Now you might say, "Wait, uh, okay, this is a very abstract thing. 0:07:07.830,0:07:10.120 Things like this don't actually exist in the real world." 0:07:10.120,0:07:13.240 And there's an experiment 0:07:13.240,0:07:14.820 that people talk about in the fractal world. 0:07:14.870,0:07:17.770 And that's finding the perimeter of England, 0:07:17.820,0:07:19.200 or you can actually do it with any island. 0:07:19.200,0:07:21.170 And so England looks something like, 0:07:21.170,0:07:22.730 you know, I'm not an expert on the, you know, 0:07:22.730,0:07:24.230 let's say it looks something like that. 0:07:24.230,0:07:26.230 So at first, you might approximate the perimeter, 0:07:26.230,0:07:27.480 and you might measure this distance. 0:07:27.550,0:07:32.350 You might measure this distance + this distance 0:07:32.350,0:07:36.070 +this distance + that distance + that distance + that distance. 0:07:36.070,0:07:37.660 You know, look. 0:07:37.660,0:07:38.590 It has a finite perimeter. 0:07:38.620,0:07:40.300 Clearly, it has a finite area, but you know, 0:07:40.300,0:07:42.300 look that has a finite perimeter. 0:07:42.340,0:07:43.720 But you're like, "No, no, that's not as good. 0:07:43.750,0:07:45.380 You have to approximate it a little better than that." 0:07:45.400,0:07:46.960 Instead of doing it that rough, 0:07:46.980,0:07:48.680 you need to make a bunch of smaller lines. 0:07:48.680,0:07:50.740 You got to make a bunch of smaller lines 0:07:50.770,0:07:52.570 so you can hug the coast a little bit better. 0:07:52.620,0:07:55.010 And you're like, "Okay, that's a much better approximation." 0:07:55.010,0:07:58.730 But then, let's say at some piece of coast, if we zoom in, 0:07:58.760,0:08:01.780 if we zoom in enough, if we zoom in enough, 0:08:01.780,0:08:03.980 the actual coastline's gonna look something like this. 0:08:04.020,0:08:08.190 The actual coastline will have all these little tidbits in it. 0:08:08.260,0:08:11.150 And essentially, when you did that first, when you did this pass, 0:08:11.150,0:08:13.580 you were just measuring, you were just measuring that. 0:08:13.580,0:08:15.740 And you're like,"That's not the perimeter of the coastline." 0:08:15.740,0:08:17.620 You're gonna have to do many many more sides. 0:08:17.650,0:08:18.850 You're gonna have to do something like this 0:08:18.900,0:08:25.660 to actually get the perimeter of the coastline. 0:08:25.660,0:08:29.150 And you say, "Hey, that is a good approximation of the perimeter." 0:08:29.150,0:08:32.190 But, if you were to zoom in on that part of the coastline even more, 0:08:32.190,0:08:35.050 it will actually turn out that it won't look exactly like that. 0:08:35.050,0:08:37.330 It will actually come in and out, like this. 0:08:37.360,0:08:39.450 Maybe look something like that. 0:08:39.450,0:08:42.810 So instead of having these rough lines, that just measure it like that. 0:08:42.890,0:08:43.850 You're gonna say, "Oh wait, 0:08:43.900,0:08:46.170 now I need to go a little bit closer and hug it even tighter." 0:08:46.220,0:08:48.270 And you can really keep on doing that 0:08:48.310,0:08:50.150 until you get to the actual atomic level. 0:08:50.150,0:08:54.730 So the actual coastline of an island, 0:08:54.770,0:08:58.790 or a continent, or anything, is actually, somewhat kind of fractalish. 0:08:58.840,0:09:01.210 And it is, you can kind of think of it 0:09:01.210,0:09:03.130 as having an almost infinite perimeter. 0:09:03.180,0:09:04.150 Obviously at some point, 0:09:04.220,0:09:05.480 you'll get into the kind of atomic level, 0:09:05.520,0:09:06.610 so it won't quite be the same, 0:09:06.660,0:09:08.510 but it's kind of the same phenomenon. 0:09:08.540,0:09:10.390 It's an interesting thing to actually think about.