Is there a rigorous mathematical definition of the Koch curve?

Question

Is there a rigorous mathematical definition of the Koch curve? Wikipedia says that mathematics is not given a rigorous formal definition of a fractal in general. And also I have not found a strict definition of the Koch curve. Everywhere write "go to the limit of the curves." But what is the formal definition of the limit of the curves?

Answer 1

The rigorous definition means that you have to use a consistent parameterization for each step in the construct. What we do to achieve this is to keep the first third and last third of each segment intact and replace the middle third. So in complex plane we start with the parameterization:

$$\gamma_0(t) = t$$

Then the step is done by replacing the middle segments. This is most easily done by adding a perpendicular offset. To do this we just put

$$\gamma_{n+1}(t) = \gamma_n(t) + i\gamma_n'(t)\phi(3^nt)3^{-n}$$

where the derivative is taken to be one sided (we will anyway have to rely on $\phi(3^nt)=0$ in the corner ponts of $\gamma$) and $\phi$ is a function for the perpendicular offset:

$$\phi(t) = \begin{cases} \sqrt{3}(t-n-1/3) & \mbox{if } n+1/3 \le t \le n+1/2 \\ -\sqrt{3}(t-n-2/3) & \mbox{if } n+1/2 < t \le n+2/3 \\ 0 & \mbox{otherwise} \end{cases}$$

Now we have a sequence of functions $\gamma_n$ and have to make some things certain as we are to define the Koch curve as $\gamma(t) = \lim_{n\to\infty}\gamma_n(t)$.

First of all we have to show that $\gamma_n(t)$ is continuous which should be rather straight forward. We use that $\gamma_n'(t)$ is continuous (actually constant) whenever $\phi(3^nt)\ne0$.

We also should prove that $|\gamma_{n+1}(t)-\gamma_n(t)|<(2/3)^n$ which would guarantee that $\gamma(t) = \lim_{n\to\infty}\gamma_n(t)$ is defined at all, but also that it converges uniformly and therefore $\gamma(t)$ is continuous as well.

$$\gamma_{n+1}(t)-\gamma_n(t) = i\gamma_n'(t)\phi(3^nt)3^{-n}$$

in order to estimate this we need an estimate of the derivative:

$$|\gamma_{n+1}'(t)|= |\gamma_n'(t) + i\gamma_n''(t)\phi(3^nt)3^{-n} + i\gamma_n'(t)\phi'(3^nt)| = |\gamma_n'(t) (1 + i\phi'(3^nt))| < 2|\gamma_n'(t)|$$

This means that $|\gamma_n'(t)|<2^n$ inserting it

$$|\gamma_{n+1}(t)-\gamma_n(t)| = |i\gamma_n'(t)\phi(3^nt)3^{-n}| < 2^n|\phi(3^nt)|3^{-n} < (2/3)^n$$