Space-filling Curve

Illustration of Sierpiński’s space-filling curve in two dimensions. It is a mapping

$$\text{SFC}:{\mathbb{R}}^{+}\to [0;1]\times [0;1],$$

completely fill the unit square. More precisely this is true in the limit $k\to \mathrm{\infty }$, where $k$ is the order of the curve. The animation shows the curve for $k=0,1,2,3,4,5,6,7,8,9,10$. The length of the curve grows expenentially.