## Cantor Set

In this task we will investigate an interesting mathematical object called the Cantor Set. It is a simple example of a fractal with some pretty weird properties. Here is how we construct it: Draw a black interval on the number line from 0 to 1 and call this set $C_0$. Create a new set called $C_1$ by removing the “middle third” of the interval, i.e. all the numbers between $\frac13$ and $\frac23$. So $C_1$ consists of two black intervals of length one-third. We continue by removing the middle third of each of the two remaining intervals and calling this set $C_2$. We then take out the middle third of each remaining black interval to create $C_3$ and so on (see diagram below).

- How many black intervals are in $C_0$, $C_1$, $C_2$, $C_3$, $\ldots$ , $C_{10}$?
- Add up the total length of the pieces that are removed at each stage. We can think of this as the total length of the ”gaps” between the black intervals. What is the total length of the gaps in $C_0$, $C_1$, $C_2$, $C_3$, $\ldots$, $C_{10}$?
- If we continued to remove more and more middle thirds, how much length of the original interval would we eventually remove? Are there any points that would be left?

*Note:* As a thought experiment, imagine we could continue the middle third removal forever. There are still some numbers that would never get removed like $0$, $\frac13$, $\frac23$, $1$ etc. The set of numbers that will never be removed is called the Cantor Set and it has some amazing properties. For example, there are infinitely many numbers in the Cantor Set (even uncountably many numbers), but it contains no intervals of numbers and its total length is zero.