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# Koch snowflake fractal

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

Video transcript

so let's say this is an equilateral
triangle and what I want to do is make another shape out of this equilateral
triangle I'm gonna do that by taking each of the size of this triangle and
divide them into three equal sections into three equal sections so I'll try it
wasn't drawn super ideally but I think you get the point in the middle section
I want to construct another equilateral triangle the middle section right over
here I am going to construct another equilateral triangle is going to look
something like this then right over here and win it but another equilateral
triangle and so now I went from that equilateral triangle to something that's
looking like a star a star of David and then I'm gonna do it again so each of
the sides now and will divide into three equal sides and that middle segment I'm
gonna put an equilateral triangle I am going to put an equilateral triangles in
the middle segment I'm going to put an equilateral triangle every one of the sides so let me do it
right there right there that you get the idea but I want to make it clear let me
just so then like that and then like that like that then almost done for this
iteration this pass and look like that but I can do it again each of the segments I can do 1230 sides
and draw another equilateral triangles occurs there there there there there I
think you see where this is going and I could keep going on for ever and
for ever 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 every iteration we look at each side we divide into three
equal sides and the next iteration or three equal segments the next iteration
the middle segment we turn to another equilateral triangle this shape that
were 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 neil's
Fabian he'll go von coach 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 at any
scale you look at it so when you look at it at this scale so if you look at this
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 then if your resume and again you would see it
again and again so for actual anything that any scale and any level of zoom it
kind of looks roughly the same so that's why it's called a fraction of what is
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 coach snowflake were you were you were you you
keep an infinite number of times on every smaller little triangle you keep
adding another equilateral triangle on its side and to show that it has an
infant perimeter listens consider one side or year so let's say the side so
let's say we're starting right when we start with that original trial that side
and it has length and then we divided into three equal segments we divide into
three segments of those are going to be s over 30 over 30 over 30 over three and
S over three and the middle segment you
make an equilateral triangle in the middle segment you make an equilateral
triangles each of these sides are going to be a soldier three as over 33 and now
the the length of this new part I can call in line anymore get the length of this part right over
here this aside now doesn't have just a length of acid is now s over three times
for before was sooo three times three now you have 1234 segments that are s
over 30 now after one time after one piece after one time of doing this this this adding
triangles are new side after ad that bob is going to be four times over three or
four thirds s so if I original true if our original perimeter if I original
perimeter when it was just a triangle is P sub-zero after one pass after we had one
set of bombs that our perimeter is going to be so it's going to be for thirds
times the original one because each of the side you're ready before thirds
bigger now so this was made up of three sides now each of those sides of it
before thirds bigger so the new parameters can be for third time's that
then the second pass on it that's going to be for thirds times his first pass to
every pass you take its getting Four Thirds bigger it's getting I get a third
bigger on every it's getting Four Thirds the orig the the the the previous back
as if you do that an infinite number of times you multiply any any number by Four Thirds an
infinite number of times you're gonna get an infinite number of less infinite
length soapy infinity infinity the perimeter viewed as 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 better is that it
actually has a finite area when I see a fine idea 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 gonna do really formal proof just think about it
what happens on any one of these sides so that first pass we have that this
trial gets popped out and then if you think about it if you just drop 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 essentially an infinite number of these
bumps but you're never gonna 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 on this side over here also good to be true the side over here
also going to be true this side over there and they're also going to be true
that side of 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 I would never have a larger area than a shape
that looks something like that I'm just kind of drawing an arbitrary I wanna
make it outside of the hexagon I could put a circle outside of it so this thing
I do in blue or the hexagon I drew in magenta those clearly have a fixed area
and is coach snowflake will always be bounded even though you can add these
bumps an infinite number of times a bunch of really cool things you wanted
the fractal you can keep zooming in and look the same the other thing infinite infinite
perimeter and and and finite finite area that you might say it's ok 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 a fun thought experiment that people talk
about in the front the world and that's finding the perimeter of England or you
can actually do with any island and so England looks something like you know
I'm not an expert on you know look something like that but first you might
approximate the perimeter and measured this distance you might measure you
might measure this distance plus this distance plus this distance put that
distance was that distance the outlook it has a finite perimeter clearly has a
finite area but here look look that's has a finite perimeter but that's not as
good approximated little bit better than that
instead of doing it that rough you to make a bunch of smaller lines to make a
bunch of smaller lines you can hug the coast a little bit better you like okay
that's a much better approximation but then let's say you're in some piece of
Costa presume in if we zoom in enough zoom in enough the actual coastlines
gonna look something like this the actual coastline will have all of
these little Davidson it and essentially when you did that first when you did
this past you were just measuring you were just measuring that you like that's
not the perimeter of the coastline you have to do many many more side you 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 it's like a now
that is a good approximation for the perimeter but if your resume and on that
part of coastline even more electric turned out that it won't look exactly
like that it will actually come in and out like this maybe look something like
that so instead of having these rough lines that just measure it like that
you're going to go a little bit closer and hugs even tighter and you can really
keep on doing that until you get to the actual atomic level so the actual
coastline of of of a of a of an island or a continent training is actually
somewhat kind of fractal ish and sometimes it is you can kind of thing is
having an almost infinite perimeter obviously at some point of getting to
kind of the atomic level so it won't quite be the same but it's kind of the
same phenomena it's interesting thing to actually think about