Main content

## Geometry (all content)

### Course: Geometry (all content) > Unit 8

Lesson 8: Koch snowflake fractal# Koch snowflake fractal

A shape that has an infinite perimeter but finite area. Created by Sal Khan.

## Want to join the conversation?

- "6:13" is there a way for me to calculate the area of the koch triangle, i mean it is finite right?(74 votes)
- the area of a Koch snowflake is 8/5 of the area of the original triangle - http://en.wikipedia.org/wiki/Koch_snowflake#Properties(63 votes)

- So, is infinite, but it is in a limited area, so it's infinite, but at the same time it's finite?

Edit: Wouldn't it eventually get so small that you couldn't make it any bigger?

Edit: So if it can get smaller and smaller forever, doesn't it have a infinite perimeter and an infinite area?

Edit: My brain hurts.(42 votes)- It's perimeter is infinite. Its area is confinable and approaches a real number which could be rational (but doesn't have to be), but it is a real number.

You can't actually say exactly how big it is, but you can say what it is smaller than and will never be bigger than.(40 votes)

- Would it be true to say that no object of this shape can possibly exist because any object made of matter would find a limit when the size of the triangle at the next layer of triangles is smaller than the size of the elementary particles it is made of?(12 votes)
- This is math, not the physical world, so there is no limit, you can carry on to infinity.(16 votes)

- As Sal says on this video the perimeter of this koch snowflake is infinite.

One really intriguing question popped out of my mind.

Are not all irrational numbers like pi based on some simple recursive formula as fractals do.

So we could be able to make a clear definition to irrational numbers by fractals.

What is the fractal formula for pi??(10 votes)- That's a very fun thing to explore. Vi has an interesting video on the subject.

https://www.khanacademy.org/math/vi-hart/v/fractal-fractions

I have a feeling that a purely fractal fraction like this couldn't give you pi without tinkering with the formula as you get deeper, but you should explore and see if I'm wrong!(6 votes)

- Does this Koch Snowflake Fractal goes on forever till infinity?(4 votes)
- Yes, that is the basic property of any fractal ( infinity! )(10 votes)

- When will this be used in a job or real life scenario?(0 votes)
- It certainly could be used (and is used) in art installations, graphics and design, also fabric design. Because it is used in these fields, it must be used in computer programming, because to represent the shapes, you have to know the mathematics that would allow you to produce the shapes.

On another level, no one will hire you just because you know how to construct a Snowflake Fractal, or calculate its area or perimeter (well, maybe someone who needs to produce a design would), but there are many applications that might benefit from knowledge of Koch snowflakes. Notice that Sal mentioned a little of this late in the video where he was comparing the coasts of real land masses to fractal shapes. I have seen the mathematics of fractals used in biology as a way of modeling very complex body surfaces (say, of lizards covered with clunky scales). Mathematical modeling is an important way of understanding the reason why a condition exists.

I would say, it pays to learn the concepts regardless of whether they seem useful to you, because you will keep your mind sharp and you may find a better way to predict drought, for example, based on some mathematical concept that you happened to learn. It happens all the time.(2 votes)

- In the video, where you first made your two marks on each side of the triangle, can you make the smaller triangles going inward instead of outward, and having it still be considered a fractal?(5 votes)
- Of course you can! And yes, it would still be a fractal.

I made a program of it:

http://www.khanacademy.org/cs/ingrown-koch-snowflake/1248111109

You can change the value of "iteration" to make it more or less complex.(5 votes)

- If we consider the 10000000000th iteration then side of the triangle will be s/(3^1000000000)=0.000000000033s. Which is a very small value. So how does this small value contribute to the perimeter of the Koch triangle??(4 votes)
- The fact that the perimiter is increasing for every iteration does not imply that the perimeter is final. However, in this case, the perimeter increases "fast enough" with every iteration which leads to an infinite length.(1 vote)

- does the shape have to be a triangle?(2 votes)
- There are thousands of fractals. Some take triangles as base ( like Koch Snowflake or Sierpinski Triangle), but other can use squares ( Vicsek fractal, Sierpinsky carpet), and, in fact, you don't have to be based on any shape, a beautiful example of that are Julia's Sets and Mandelbrot's Set ( which are based on complex numbers but generate mysteriously elaborate shapes ).

Thera are many more examples, but the most fun thing is to try to come up with your own fractal patterns!(6 votes)

- I once read that fractals have a dimension that is not an integer, like 2,5. The explanation that was given, was to look what would happen if you double the length of your figure. For example, if you double a line, its length doubles (2^1), so it has dimension 1. If you double the sides of a square, the area becomes 4 times as big (2^2), so it has dimension 2. If you doubles the side of a cube, its volume becomes 8 times as big (2^3), so it has dimension 3.

So, using this definition of 'dimension', fractals would have a dimension that is not an integer. Can anyone please elaborate that theory for me?(3 votes)

## Video transcript

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. 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. So I could just there,
there, there, there. I think you see
where this is going. 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. 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. A Koch snowflake,
and it was first described by this gentleman
right over here, who is a Swedish mathematician,
Niels Fabian Helge von 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. 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. 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. And to show that it has
an infinite perimeter, let's just consider
one side over here. 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. Let's say it has length s. And then we divide it
into three equal segments. So those are going to
be s/3, s/3-- actually, let me write it this way. s/3, s/3, and s/3. And in the middle segment, you
make an equilateral triangle. So each of these sides
are going to be s/3. s/3, s/3. 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. Now you have 1, 2, 3, 4
segments that are s/3. 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. So if our original
perimeter when it was just a triangle is p sub 0. 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. 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.