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.