Take the diagonal of a square, as shown in the first diagram. Replace this by the nine diagonals of smaller squares, as in the second diagram (drawn here in a way that indicates the order in which the diagonals are to be traced out). Now replace each straight section of this by the nine diagonals of even smaller squares, to obtain the result shown in the third diagram. The Peano curve is the curve obtained when this process is continued indefinitely. It has the remarkable property that it passes through every point of the square, and it is therefore described as space-filling.

First iteration

Second iteration

Third iteration