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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
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