If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Koch snowflake fractal

## 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 into three equal sections so my equilateral triangle wasn't drawn super ideally but I think we'll get the point and the middle section I want to construct another equilateral triangle so the middle section right over here I am going 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 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 that middle segment I'm going to put an equilateral triangle I am going to put in equilateral triangles in the middle segment I am 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 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 will look like that then I can do it again each of the segment's I can divide into three equal sides and draw another equilateral triangle so I guess there 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 every set 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 coach snowflake and I'm sure I'm mispronouncing the coach part the coach snowflake and was first described by this gentleman right over here who was a Swedish mathematician Niels Fabian Helga von ko I'm sure I'm mispronouncing it and this is 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 looks 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 if R actual is anything that is 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 actually make the Coache snowflake where you were you where you you keep an infinite number of times on every smaller little triangle here you keep adding other 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 and let's say it has length s and then we divide it into three equal segments we divide into three equal segments so those are going to be s over 3 s over 3 so let me write it this way s over 3 s over 3 and s over 3 and in the middle segment you make an equilateral triangle in the middle segment you make an equilateral triangle so each of these sides are going to be s over 3 s over 3 s over 3 and now the 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 over 3 times 4 before it was s over 3 times 3 now you have one two three four segments that are s over three so now after one time after one pace after one time of doing this this this adding triangles our new side after we had that bump is going to be 4 times s over 3 or it equals 4/3 s-so if our original true if our original perimeter if our original perimeter when it was just a triangle is P Sub Zero after one pass after we add one set of bumps then our perimeter is going to be so it's 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 is going to be 4/3 times that and then the then when you take a second pass on it there's going to be 4/3 times this first pass so every pass you take it's getting 4/3 bigger it's getting I guess 1/3 bigger on every it's getting for Thursday previous pass and so if you do that an infinite number of times if you multiply any any number by 4/3 an infinite number of times you are going to get an infinite number of infinite length so P infinity 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 abounded amount of space that 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 on this side right over here it's also going to be true on 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 coach snowflake will never have a larger area than this bounding hexagon or which will never have a lot 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 can 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 coach 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 end and finite finite area now you might say wait Sal okay this is a very abstract thing things like this don't actually exist in the real world and there's a fun thought of expert on third 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 you know I'm not an expert on it you know 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 you might measure this distance Plus this distance Plus this distance Plus that distance Plus that distance Plus that distance you're like look it has a finite perimeter it clearly has a finite area but here look look that has a finite perimeter but you're like nah 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 line so you can hug the coast a little bit better and you're like okay that's a much better approximation but then let's say you're at some piece of coast if we zoom in if we zoom in enough if we zoom in enough the actual coastline is going to look something like this the actual coastline will have all of these little divots in it and essentially when you did that first when you did this pass you were just measuring you were just measuring that and you're like that's not the perimeter of the cosine you're going to have to do many many more side you're going have to do something like this you're going to have to do something like this to actually get the perimeter to actually get the perimeter of the coastline and you're like hey now that is a good approximation for the perimeter but if you were to zoom in on that part of 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 now I need to go a little bit closer and get 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 fractal ish and it's some cut it is you can kind of think it is 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 interesting thing to actually think about