# Space-filling Curve

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

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

completely fill the unit square. More precisely this is true in the limit $$k \rightarrow \infty$$, 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.