📂Dynamics

von Koch Curve

Definition ¹

The $K_{n+1}$ is defined as follows:

1. Divide every line segment of length $K_{n}$ into three equal parts of length $l$.
2. Add an equilateral triangle at the middle section with a side length of $l/3$.
3. Remove the overlapping part of the equilateral triangle and the $K_{n}$.

The limit $K \subset \mathbb{R}^{2}$ of such a set is defined as the von Koch Curve. $$ K := \lim_{n \to \infty} K_{n} $$

Explanation

The von Koch Curve is a renowned example of a fractal shape, known for its characteristic whereby $K_{n}$ increases by a factor of $4/3$ each time $n$ is incremented by 1. Consequently, if the length of $K_{0}$ is $1$, then the length of $K$ diverges to infinity as follows. $$ \lim_{n \to \infty} \left( {\frac{ 4 }{ 3 }} \right)^{n} \cdot 1 = \infty $$

See Also

• Cantor Set