Iterated Function Systems

Koch Curve

The Koch Curve is one of the earliest examples of a fractal, but was named for Helge von Koch as it appeared in his paper, “On a Continuous Curve Without Tangents, Constructible from Elementary Geometry”. It is made from repetitively adding triangles with $\frac{1}{3}$ the length of every line to every such line. It is generated by (equal probability):

$f_1(x)=\begin{bmatrix} \frac{1}{3} & 0 \\0 & \frac{1}{3} \end{bmatrix}*x$

$f_2(x)=\begin{bmatrix} \frac{1}{6} & \frac{-\sqrt{3}}{6} \\ \frac{\sqrt{3}}{6} & \frac{1}{6} \end{bmatrix}*x + \begin{bmatrix} \frac{1}{3} \\ 0 \end{bmatrix}$

$f_3(x)=\begin{bmatrix} \frac{1}{6} & \frac{\sqrt{3}}{6} \\ \frac{-\sqrt{3}}{6} & \frac{1}{6} \end{bmatrix}*x + \begin{bmatrix} \frac{1}{2} \\ \frac{\sqrt{3}}{6} \end{bmatrix}$