$$

Dimension of the Cantor set

Question

One way to define the Cantor set is as follows:

$$\begin{array}{rl}{C}_0& =[0,1]\\ C_1& =\frac{1}{3}{C}_0\cup (\frac{2}{3}+\frac{1}{3}{C}_0)\\ C_2& =\frac{1}{3}{C}_1\cup (\frac{2}{3}+\frac{1}{3}{C}_1)\\ & ⋮\\ C_n& =\frac{1}{3}{C}_{n-1}\cup (\frac{2}{3}+\frac{1}{3}{C}_{n-1})\end{array}$$

Finally, define

$C=\bigcap _{n=0}^{\mathrm{\infty }}{C}_n$. See an illustration here.

Comparing

$3C$ and

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

Experiments

You need: handout

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
   $0.0{\bar{2}}_3$ 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