-
-
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.
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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.
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So I could just there,
there, there, there.
-
I think you see
where this is going.
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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.
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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.
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A Koch snowflake,
and it was first
-
described by this gentleman
right over here, who
-
is a Swedish mathematician,
Niels Fabian Helge von
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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.
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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.
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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.
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And to show that it has
an infinite perimeter,
-
let's just consider
one side over here.
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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.
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Let's say it has length s.
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And then we divide it
into three equal segments.
-
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So those are going to
be s/3, s/3-- actually,
-
let me write it this way.
-
s/3, s/3, and s/3.
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And in the middle segment, you
make an equilateral triangle.
-
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So each of these sides
are going to be s/3.
-
s/3, s/3.
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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.
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Now you have 1, 2, 3, 4
segments that are s/3.
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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.
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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 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.
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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.
