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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