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# Dimension of the Cantor set

## Question

One way to define the Cantor set is as follows:

$$
\begin{aligned}
C_0 &= [0,1] \\
C_1 &= \frac{1}{3}C_0 \cup (\frac{2}{3} + \frac{1}{3}C_0) \\
C_2 &= \frac{1}{3}C_1 \cup (\frac{2}{3} + \frac{1}{3}C_1) \\
~ & \vdots \\
C_n &= \frac{1}{3}C_{n-1} \cup (\frac{2}{3} + \frac{1}{3}C_{n-1}) \\
\end{aligned}
$$


Finally, define $C = \bigcap_{n=0}^\infty C_n$. See an illustration [here](https://www.math.nsysu.edu.tw/~chlin/math-runway/cantor-set.pdf).

Comparing $3C$ and $C$, what would you guess the ratio of their lengths is?

## Experiments

You need: [handout](https://www.math.nsysu.edu.tw/~chlin/math-runway/cantor-set.pdf)

1. On the handout, mark the endpoints of each segment in ternary number.
2. Is $0.2_3$ in $C$?
3. Is $0.11_3$ in $C$?
4. Try to draw $3C$.
5. Compare it with $C$.

## Intuition

If $I$ is an interval in $\mathbb{R}$, then $3C$ has $3$ times the length of $I$.

If $I$ is a square in $\mathbb{R}^2$, then $3C$ has $3^2$ times the area of $I$.

If $I$ is a cube in $\mathbb{R}^3$, then $3C$ has $3^3$ times the volumn of $I$.

If $C$ has dimension $d$, then $3C$ supposedly has $3^d$ times the "size" of $C$. As $3^d = 2$, it suggest that the dimension of $C$ is $\log_3^2 \sim 0.631\cdots$. This number is called the _fractal dimension_ .

## More questions to think about

1. Is $0.0\overline{2}_3$ in $C$?
2. Is $0.2123_3$ in $C$?
3. Is $0.\overline{20}$ in $C$?
4. Describe $C$ in terms of ternary numbers.
5. What is the "length" of $C$?

## Resources

1. [YouTube: Fractals are typically not self-similar by 3Blue1Brown](https://youtu.be/gB9n2gHsHN4?si=BxkTOqtUP2pxt4n3)