0:00:00.000,0:00:00.620


0:00:00.620,0:00:03.122
So let's say that this is[br]an equilateral triangle.

0:00:03.122,0:00:05.080
And what I want to do is[br]make another shape out

0:00:05.080,0:00:06.205
of this equilateral triangle.

0:00:06.205,0:00:07.621
And I'm going to[br]do that by taking

0:00:07.621,0:00:09.370
each of the sides[br]of this triangle,

0:00:09.370,0:00:11.854
and divide them into[br]three equal sections.

0:00:11.855,0:00:15.010


0:00:15.010,0:00:18.736
So my equilateral triangle[br]wasn't drawn super ideally.

0:00:18.736,0:00:20.110
But I think you'll[br]get the point.

0:00:20.110,0:00:21.810
And in the middle section[br]I want to construct

0:00:21.810,0:00:23.104
another equilateral triangle.

0:00:23.105,0:00:29.510


0:00:29.510,0:00:32.680
So it's going to look[br]something like this.

0:00:32.680,0:00:35.090
And then right[br]over here I'm going

0:00:35.090,0:00:37.740
to put another[br]equilateral triangle.

0:00:37.740,0:00:40.270
And so now I went from[br]that equilateral triangle

0:00:40.270,0:00:43.250
to something that's looking[br]like a star, or a Star of David.

0:00:43.250,0:00:44.670
And then I'm going[br]to do it again.

0:00:44.670,0:00:46.700
So each of the sides[br]now I'm going to divide

0:00:46.700,0:00:48.410
into three equal sides.

0:00:48.410,0:00:50.000
And in that middle[br]segment I'm going

0:00:50.000,0:00:51.295
to put an equilateral triangle.

0:00:51.295,0:00:54.130


0:00:54.130,0:00:56.370
So in the middle[br]segment I'm going

0:00:56.370,0:00:59.349
to put an equilateral triangle.

0:00:59.350,0:01:01.950
So I'm going to do it for[br]every one of the sides.

0:01:01.950,0:01:05.600
So let me do it right there.

0:01:05.600,0:01:06.759
And then right there.

0:01:06.760,0:01:10.030
I think you get the idea,[br]but I want to make it clear.

0:01:10.030,0:01:15.260
So let me just, so then like[br]that, and then like that,

0:01:15.260,0:01:20.060
like that, and then almost[br]done for this iteration.

0:01:20.060,0:01:21.840
This pass.

0:01:21.840,0:01:23.260
And it'll look like that.

0:01:23.260,0:01:24.330
Then I could do it again.

0:01:24.330,0:01:26.663
Each of the segments I can[br]divide into three equal sides

0:01:26.663,0:01:28.259
and draw another[br]equilateral triangle.

0:01:28.260,0:01:31.440
So I could just there,[br]there, there, there.

0:01:31.440,0:01:33.259
I think you see[br]where this is going.

0:01:33.260,0:01:37.534
And I could keep going[br]on forever and forever.

0:01:37.534,0:01:38.950
So what I want to[br]do in this video

0:01:38.950,0:01:40.930
is think about[br]what's going on here.

0:01:40.930,0:01:43.000
And what I'm actually[br]drawing, if we just

0:01:43.000,0:01:45.000
keep on doing this[br]forever and forever,

0:01:45.000,0:01:47.990
every side, every iteration,[br]we look at each side,

0:01:47.990,0:01:49.520
we divide into[br]three equal sides.

0:01:49.520,0:01:52.176
And then the next iteration,[br]or three equal segments,

0:01:52.177,0:01:53.760
the next iteration,[br]the middle segment

0:01:53.760,0:01:55.799
we turn to another[br]equilateral triangle.

0:01:55.799,0:01:57.590
This shape that we're[br]describing right here

0:01:57.590,0:01:59.950
is called a Koch snowflake.

0:01:59.950,0:02:03.300
And I'm sure I'm[br]mispronouncing the Koch part.

0:02:03.300,0:02:05.480
A Koch snowflake,[br]and it was first

0:02:05.480,0:02:08.229
described by this gentleman[br]right over here, who

0:02:08.229,0:02:11.270
is a Swedish mathematician,[br]Niels Fabian Helge von

0:02:11.270,0:02:14.610
Koch, who I'm sure[br]I'm mispronouncing it.

0:02:14.610,0:02:17.740
And this was one of the[br]earliest described fractals.

0:02:17.740,0:02:19.690
So this is a fractal.

0:02:19.690,0:02:22.120
And the reason why it[br]is considered a fractal

0:02:22.120,0:02:24.890
is that it looks the same,[br]or it looks very similar,

0:02:24.890,0:02:26.630
on any scale you look at it.

0:02:26.630,0:02:29.591
So when you look at it at this[br]scale, so if you look at this,

0:02:29.591,0:02:31.090
it like you see a[br]bunch of triangles

0:02:31.090,0:02:32.320
with some bumps on it.

0:02:32.320,0:02:35.130
But then if you were to[br]zoom in right over there,

0:02:35.130,0:02:38.190
then you would still see[br]that same type of pattern.

0:02:38.190,0:02:39.926
And then if you were[br]to zoom in again,

0:02:39.926,0:02:41.299
you would see it[br]again and again.

0:02:41.300,0:02:43.320
So a fractal is anything[br]that at on any scale,

0:02:43.320,0:02:46.769
on any level of zoom, it kind[br]of looks roughly the same.

0:02:46.770,0:02:48.270
So that's why it's[br]called a fractal.

0:02:48.270,0:02:49.790
Now what's particularly[br]interesting,

0:02:49.790,0:02:53.700
and why I'm putting it at this[br]point in the geometry playlist,

0:02:53.700,0:02:56.417
is that this actually has[br]an infinite perimeter.

0:02:56.417,0:02:58.500
If you were to keep doing[br]it, if you were actually

0:02:58.500,0:03:00.690
to make the Koch[br]snowflake, where

0:03:00.690,0:03:04.130
you keep an infinite number[br]of times on every smaller

0:03:04.130,0:03:05.840
little triangle[br]here, you keep adding

0:03:05.840,0:03:09.754
another equilateral[br]triangle on its side.

0:03:09.754,0:03:11.670
And to show that it has[br]an infinite perimeter,

0:03:11.670,0:03:13.890
let's just consider[br]one side over here.

0:03:13.890,0:03:15.959
So let's say that[br]this side, so let's

0:03:15.960,0:03:18.024
say we're starting[br]right when we started

0:03:18.024,0:03:19.940
with that original[br]triangle, that's that side.

0:03:19.940,0:03:21.565
Let's say it has length s.

0:03:21.565,0:03:23.565
And then we divide it[br]into three equal segments.

0:03:23.565,0:03:26.170


0:03:26.170,0:03:29.910
So those are going to[br]be s/3, s/3-- actually,

0:03:29.910,0:03:31.670
let me write it this way.

0:03:31.670,0:03:36.170
s/3, s/3, and s/3.

0:03:36.170,0:03:38.730
And in the middle segment, you[br]make an equilateral triangle.

0:03:38.730,0:03:41.840


0:03:41.840,0:03:44.462
So each of these sides[br]are going to be s/3.

0:03:44.462,0:03:47.290
s/3, s/3.

0:03:47.290,0:03:51.471
And now the length of this new[br]part-- I can't call it a line

0:03:51.472,0:03:53.180
anymore, because it[br]has this bump in it--

0:03:53.180,0:03:57.580
the length of this part right[br]over here, this side, now

0:03:57.580,0:04:01.510
doesn't have just a length[br]of s, it is now s/3 times 4.

0:04:01.510,0:04:03.160
Before it was s/3 times 3.

0:04:03.160,0:04:07.730
Now you have 1, 2, 3, 4[br]segments that are s/3.

0:04:07.730,0:04:10.399
So now, after one[br]time, after one pace,

0:04:10.400,0:04:15.160
after one time of doing[br]this adding triangles,

0:04:15.160,0:04:17.480
our new side, after[br]we add that bump,

0:04:17.480,0:04:20.760
is going to be four times s/3.

0:04:20.760,0:04:23.730
Or it equals 4/3 s.

0:04:23.730,0:04:28.670
So if our original[br]perimeter when it was just

0:04:28.670,0:04:31.100
a triangle is p sub 0.

0:04:31.100,0:04:33.930
After one pass, after[br]we add one set of bumps,

0:04:33.930,0:04:39.840
then our perimeter is going to[br]be 4/3 times the original one.

0:04:39.840,0:04:42.630
Because each of the sides are[br]going to be 4/3 bigger now.

0:04:42.630,0:04:44.610
So this was made[br]up of three sides.

0:04:44.610,0:04:46.870
Now each of those sides[br]are going to be 4/3 bigger.

0:04:46.870,0:04:49.400
So the new perimeter's[br]going to be 4/3 times that.

0:04:49.400,0:04:51.679
And then when we take[br]a second pass on it,

0:04:51.680,0:04:54.740
it's going to be 4/3[br]times this first pass.

0:04:54.740,0:04:57.650
So every pass you take,[br]it's getting 4/3 bigger.

0:04:57.650,0:05:00.380
Or it's getting, I guess,[br]a 1/3 bigger on every,

0:05:00.380,0:05:03.437
it's getting 4/3[br]the previous pass.

0:05:03.437,0:05:05.520
And so if you do that an[br]infinite number of times,

0:05:05.520,0:05:09.219
if you multiply any[br]number by 4/3 an

0:05:09.220,0:05:11.180
infinite number of[br]times, you're going

0:05:11.180,0:05:13.820
to get an infinite number[br]of infinite length.

0:05:13.820,0:05:16.274
So P infinity.

0:05:16.274,0:05:18.690
The perimeter, if you do this[br]an infinite number of times,

0:05:18.690,0:05:20.370
is infinite.

0:05:20.370,0:05:21.920
Now that by itself[br]is kind of cool,

0:05:21.920,0:05:24.461
just to think about something[br]that has an infinite perimeter.

0:05:24.461,0:05:28.192
But what's even neater is that[br]it actually has a finite area.

0:05:28.192,0:05:29.900
And when I say a finite[br]area, it actually

0:05:29.900,0:05:32.164
covers a bounded[br]amount of space.

0:05:32.164,0:05:34.080
And I could actually[br]draw a shape around this,

0:05:34.080,0:05:36.120
and this thing will[br]never expand beyond that.

0:05:36.120,0:05:37.620
And to think about[br]it, I'm not going

0:05:37.620,0:05:39.240
to do a really[br]formal proof, just

0:05:39.240,0:05:42.520
think about it, what happens[br]on any one of these sides.

0:05:42.520,0:05:44.630
So on that first pass we[br]have that this triangle

0:05:44.630,0:05:46.010
gets popped out.

0:05:46.010,0:05:49.340
And then, if you think about it,[br]if you just draw what happens,

0:05:49.340,0:05:51.681
the next iteration you draw[br]these two triangles right

0:05:51.681,0:05:52.180
over there.

0:05:52.180,0:05:54.270
And these two characters[br]right over there.

0:05:54.270,0:05:56.330
And then you put some[br]triangles over here,

0:05:56.330,0:05:58.520
and here, and here,[br]and here, and here.

0:05:58.520,0:05:59.700
So on and so forth.

0:05:59.700,0:06:01.960
But notice, you can keep[br]adding more and more.

0:06:01.960,0:06:04.630
You can add essentially an[br]infinite number of these bumps,

0:06:04.630,0:06:07.380
but you're never going to[br]go past this original point.

0:06:07.380,0:06:10.490
And the same thing is going[br]to be true on this side

0:06:10.490,0:06:11.540
right over here.

0:06:11.540,0:06:14.240
It's also going to be true[br]of this side over here.

0:06:14.240,0:06:16.920
Also going to be true[br]at this side over here.

0:06:16.920,0:06:19.680
Also going to be true[br]this side over there.

0:06:19.680,0:06:22.130
And then also going to be[br]true that side over there.

0:06:22.130,0:06:24.750
So even if you do this an[br]infinite number of times,

0:06:24.750,0:06:27.450
this shape, this Koch[br]snowflake will never

0:06:27.450,0:06:30.430
have a larger area than[br]this bounding hexagon.

0:06:30.430,0:06:33.190
Or which will never have a[br]larger area than a shape that

0:06:33.190,0:06:34.342
looks something like that.

0:06:34.342,0:06:35.925
I'm just kind of[br]drawing an arbitrary,

0:06:35.925,0:06:38.070
well I want to make it[br]outside of the hexagon,

0:06:38.070,0:06:40.849
I could put a circle[br]outside of it.

0:06:40.850,0:06:44.690
So this thing I drew in blue, or[br]this hexagon I drew in magenta,

0:06:44.690,0:06:47.050
those clearly have a fixed area.

0:06:47.050,0:06:49.290
And this Koch snowflake[br]will always be bounded.

0:06:49.290,0:06:52.070
Even though you can add these[br]bumps an infinite number

0:06:52.070,0:06:53.050
of times.

0:06:53.050,0:06:55.057
So a bunch of really[br]cool things here.

0:06:55.057,0:06:55.890
One, it's a fractal.

0:06:55.890,0:06:58.880
You can keep zooming in[br]and it'll look the same.

0:06:58.880,0:07:01.750
The other thing, infinite,[br]infinite perimeter,

0:07:01.750,0:07:04.787
and finite, finite area.

0:07:04.787,0:07:06.120
Now you might say, wait Sal, OK.

0:07:06.120,0:07:07.590
This is a very abstract thing.

0:07:07.590,0:07:10.510
Things like this don't actually[br]exist in the real world.

0:07:10.510,0:07:12.849
And there's a fun[br]thought experiment

0:07:12.850,0:07:14.880
that people talk about[br]in the fractal world,

0:07:14.880,0:07:17.540
and that's finding the[br]perimeter of England.

0:07:17.540,0:07:19.390
Or you can actually[br]do it with any island.

0:07:19.390,0:07:21.020
And so England looks[br]something like--

0:07:21.020,0:07:22.750
and I'm not an[br]expert on, let's say

0:07:22.750,0:07:24.600
it looks something[br]like that-- so at first

0:07:24.600,0:07:26.099
you might approximate[br]the perimeter.

0:07:26.099,0:07:28.330
And you might measure[br]this distance,

0:07:28.330,0:07:32.240
you might measure this[br]distance, plus this distance,

0:07:32.240,0:07:34.970
plus this distance, plus that[br]distance, plus that distance,

0:07:34.970,0:07:36.051
plus that distance.

0:07:36.051,0:07:38.050
And you're like look, it[br]has a finite perimeter.

0:07:38.050,0:07:39.660
It clearly has a finite area.

0:07:39.660,0:07:42.070
But you're like, look, that[br]has a finite perimeter.

0:07:42.070,0:07:43.610
But you're like, no,[br]wait that's not as good.

0:07:43.610,0:07:45.610
You have to approximate it a[br]little bit better than that.

0:07:45.610,0:07:47.068
Instead of doing[br]it that rough, you

0:07:47.068,0:07:48.762
need to make a bunch[br]of smaller lines.

0:07:48.762,0:07:50.470
You need to make a[br]bunch of smaller lines

0:07:50.470,0:07:52.400
so you can hug the coast[br]a little bit better.

0:07:52.400,0:07:55.647
And you're like, OK, that's[br]a much better approximation.

0:07:55.647,0:07:57.730
But then, let's say you're[br]at some piece of coast,

0:07:57.730,0:08:02.620
if we zoom in enough,[br]the actual coast line

0:08:02.620,0:08:04.492
is going to look[br]something like this.

0:08:04.492,0:08:05.950
The actual coast[br]line will have all

0:08:05.950,0:08:08.240
of these little divots in it.

0:08:08.240,0:08:10.960
And essentially, when you did[br]that first, when did this pass,

0:08:10.960,0:08:13.292
you were just measuring that.

0:08:13.292,0:08:15.750
And you're like, that's not[br]the perimeter of the coastline.

0:08:15.750,0:08:17.791
You're going to have to[br]do many, many more sides.

0:08:17.791,0:08:22.300
You're going to do something[br]like this, to actually get

0:08:22.300,0:08:25.747
the perimeter of the coast line.

0:08:25.747,0:08:27.330
And you're just like,[br]hey, now that is

0:08:27.330,0:08:29.179
a good approximation[br]for the perimeter.

0:08:29.179,0:08:31.719
But if you were to zoom in on[br]that part of the coastline even

0:08:31.720,0:08:34.539
more, it'll actually turn out[br]that it won't look exactly

0:08:34.539,0:08:35.429
like that.

0:08:35.429,0:08:37.230
It'll actually come[br]in and out like this.

0:08:37.230,0:08:39.230
Maybe it'll look[br]something like that.

0:08:39.230,0:08:41.688
So instead of having these[br]rough lines that just measure it

0:08:41.688,0:08:43.605
like that, you're going[br]to say, oh wait, no, I

0:08:43.605,0:08:46.280
need to go a little bit closer[br]and hug it even tighter.

0:08:46.280,0:08:48.390
And you can really keep[br]on doing that until you

0:08:48.390,0:08:50.560
get to the actual atomic level.

0:08:50.560,0:08:55.300
So the actual coastline of[br]an island, or a continent,

0:08:55.300,0:08:58.640
or anything, is actually[br]somewhat kind of fractalish.

0:08:58.640,0:09:00.760
And it is, you can[br]kind of think of it

0:09:00.760,0:09:02.477
as having an almost[br]infinite perimeter.

0:09:02.477,0:09:04.060
Obviously at some[br]point you're getting

0:09:04.060,0:09:06.640
to kind of the atomic level,[br]so it won't quite be the same.

0:09:06.640,0:09:08.449
But it's kind of[br]the same phenomenon.

0:09:08.450,0:09:11.420
It's an interesting thing[br]to actually think about.