This video was sponsored by curiositystream home to over 2 500 documentaries and non-fiction titles for curious minds the cantor set is weird Infinite size zero length and a fractional dimension are all going to show up here Plus, this is an introduction to some of the mathematics behind fractals which three blue one brown has done a cool video on And this is kind of like an intro to that We're gonna see similar ideas here, but unfortunately won't be as pretty anyways, what is the cantor set? Well, it's just a bunch of numbers that lie between 0 and 1 inclusive But which ones? to find that out and construct the set what we do is Take this set of real numbers again between zero and one including the endpoints and this will be called c naught Then we're going to remove the middle one. Third of this line segment Excluding the endpoints as then we keep one third and two thirds in this next set, which will be called c1 Then keep going remove the middle one third from each of these two line segments while leaving in the endpoints So we produced four smaller line segments This pattern would then go on forever The cantor set is simply the set of all numbers that are never removed as we do this forever That may take some time to digest but now for certain numbers it's a little more obvious whether they are Or are not in the set For example one half is not in the cantor set Yes, it's in c naught but then it gets removed and the cantor set is the numbers that are never removed This means.46 is also not in the set that's gone after c naught as well Point 2 isn't there either it survives the first cut but it's gone after that Same goes for 0.21 So now which numbers are in the set Well zero is in there We included the end points in c naught and zero never gets removed since it's never in the middle. Third of any interval Same goes for one that must be in the cantor set as well Fact it shouldn't be too hard to see that all of the endpoints that are created in each new set Must be in the final cantor set Look at one third it becomes an end point in c1 And from there it will never be removed since it can't be in the middle of an interval anymore 1 9 becomes an end point in c2 and that will never be removed. So it's in the cantor set as well 127th is in there Two-thirds and so on you'll notice a pattern of only having denominators with an integer power of three in them So now we know that the cantor set is actually infinite in the first set we have two endpoints in the next set we have 4 Then 8 16 and we just keep multiplying by 2 forever. All those points won't be removed. Meaning they're all in the cantor set So the cantor set is infinite It is not obvious at all whether there are non-end points in the final set though But let's look at the second item on our list Because it seems like we're removing a lot of numbers as we go down these sets But just how much? Well, the first set has a length of 1 of course But that's not the length of the cantor set since the second set removes a length of one-third All those numbers can't be in the final set Then the next set has a length of c1 or one minus a third minus two of those one-ninth chunks And the next set removes 4 of those 127 chunks and this keeps going Now realize this series here of just the negative numbers is Geometric, every term is being multiplied by two-thirds to get to the next one-third times two-thirds is two-ninths Times two-thirds again is four-twenty-sevenths And using the infinite series formula, you'll find all the negatives sum to negative one Then we have one plus that negative one. So the length of the cantor set is zero Just as a reminder the set of real numbers from 0 to 1 Is also just a bunch of points infinitely many just like the cantor set But this interval obviously has non-zero length while the cantor set does not So are the infinitely many points in here More infinite somehow than the cantor set Well, let's see Let's look at a point in the cantor set such as zero You should agree that after the first cut our number or any number in the set will be either on the left or the right line segment Can't be in the middle for sure because that's gone And zero is obviously on the left side But after the next cut we find our number must be in the left portion of that first left cut Or on the right portion of that first cut Or it's in the left of the right side or the right of the right Those are the only options and 0 is in the left of the left after another cut we break all those options up into their own left right section and 0 Is now in the left of the left of the left If we keep going we'd find the number 0 can be written as Lll forever it's like an instruction to find where the point will eventually be Here, let's do the same thing with one third one third is on the left after that first cut so its first letter is l Then it's on the right of the next split. So the next letter is r Then the right of the next and then the right of the next and so on so one third can be written as l r rr forever 2 9 would be l R. Then l's forever One is all r's And in fact every number in the cantor set can be written as an infinite list of ls and rs If I went l r l r l r forever Well that refers to the left third then the right one after the split then the next left one and then right And forever it converges to some point So the cantor set is every single combination of infinite l's and r's you could possibly have Or we could replace l with zero and r with 1 So the cantor set is really every infinitely long binary number But again, those binary numbers are nothing more than instructions telling us where that element is as we move down our sets There's no way we could have a finite binary number in the set either since a finite sequence of ones and zeros like zero zero or Ll would correspond to an interval that would lose its middle third in the next split So we actually just discovered something pretty interesting about this set The cantor set is not just infinite It's really infinite or the actual term. It's uncountably infinite. It's a bigger infinity Than some others and here's what I mean by that See I can have a list of all positive integers I can start at one Go to two then three and continue this pattern and i'll never miss one Same goes for all integers because I can start at zero then go to one negative 1 2 negative 2 and so on These are infinite sets But they're what we call countable sets because they can be listed like this They start somewhere and following some rule or pattern will never miss a number All infinite binary numbers though cannot be listed. That's an uncountable set If you tried to make that list, even though it's infinite You could give it to me and I could find an infinitely long binary number not on your list Vsauce and numberphile and plenty of other youtubers have talked about this concept before Usually in regards to the set of real numbers, but the idea is the exact same with infinite binary numbers My number would just start off by taking your first number and flipping the first digit So this number will not be the same as your first one because they have different first digits Then i'd take your second number and flip the second digit And same goes for the third and its third digit and fourth and this pattern would continue Thus my number is different than all of yours It can't match any of them because one of the digits is guaranteed to be flipped from yours Thus I have found an infinitely long binary number that is not on your list. So your list is not complete This tells us the cantor set is uncountable which means it's a bigger infinity than the integers for example, those are countable In fact, the cantor set is just as big or it has the same cardinality as the set of real numbers And that's weird Actually for me, that's probably the least intuitive part of this video In the last video we saw two infinite sets are the same size If you can map every element in one set to every element in the other without overlap and without missing anything Hence, the positive integers and the positive even integers are surprisingly the same size And the set of all real numbers between 0 and 1 is the same size as the set of all real numbers Since there exists a mapping that goes from each element between 0 and 1 to something different in the real numbers And now we know the cantor set is also just as big as all real numbers Meaning it's also just as big as all real numbers from zero to one where this all started So we remove infinitely many things in order to obtain the cantor set yet. The size never changes along the way Before when I asked whether the top set was a bigger infinity since it has non-zero length The answer was no The cantor set is just as big And thus everything in it can be mapped to something different from zero to one without missing anything This hopefully kind of shows you how strange infinity is The cantor set seems like it should be smaller since we keep removing more But the size remains uncountable And when you zoom into the cantor set you get a sense of that idea of infinity with this fractal-like behavior Okay, that was a lot of math for those who need a break here's some epic music followed by a possible ad And we're back so the fractional dimension part, how does that come in Well, let's just categorize integer dimensions first We say a line segment is one dimensional a square is two dimensional and a cube is three dimensional But how are we defining dimension here? Well, here's one way if I multiply the length of this line segment by three I get three line segments, obviously Multiplying the length by three left me a three of the original thing But if I multiply all the sides of a square by three I get a square that is nine times bigger Multiplying all lengths by three led to a bigger square that consists of nine or three squared of the smaller squares And for the cube if I multiply all sides by 3 I get another cube that is 27 times bigger Multiplying all lengths by 3 left me with 27 or 3 cubed original cubes Notice that the exponent in each case matches the dimension of that object So we can say this is what defines the dimension of something When you multiply all sides by any positive number not just three the entire shape gets bigger By a factor of that number to some power and that power is the dimension Now I chose three because that's what we're going to use for the cantor set We're going to triple all those intervals starting with c naught So instead of going from 0 to 1 we're going from 0 to 3 Then after doing the middle third split we get two intervals one that goes from zero to one And the other that goes from two to three But notice this is just what the real cantor set starts with Once we continue that same pattern the left portion is just going to create the same cantor set Well, the right is going to create another cantor set that is shifted But it will definitely be the same size So for all those simple shapes tripling the lengths led to something that was 3 9 or 27 times bigger But for the cantor set tripling all those intervals led to something that is two times bigger. We got two cantor sets basically But we have to write this number 2 as a power of 3 just like above And 2 equals 3 to the power of natural log of 2 divided by natural log of 3 or roughly 0.631 That is the dimension of the cantor set This tells us that scaling all those intervals which are used to make the cantor set doesn't change the overall size as simply as more familiar shapes And the same thing can be seen with fractals as you scale those up and down That's why this all relates to the mathematics of fractals I'll link all these related videos down below But if you want to explore some more real world applications of patterns and chaos And you can do so over at curiositystream the sponsor of this video This here is the same documentary mentioned in the last video the secret life of chaos That covers patterns found in pure mathematics such as fractals But also how patterns and chaos apply to computer algorithms population growth and just the universe around us Another series they have related to all this though is nature's mathematics which covers patterns and mathematical beauty found in nature Like is the case with the fibonacci sequence? So there's a lot to explore on this platform for the math and science enthusiasts out there Curiosity stream is available on a variety of platforms worldwide and it only comes out to 299 per month But if you sign up by using the link below you'll get your first month's membership completely free So no risk in giving it a try and with this you'll have unlimited access to top documentaries that i'm sure many of you will enjoy And with that i'm going to end that video there Thanks, as always my supporters on patreon social media links to follow me are down below and i'll see you guys in the next video You
