Knopp's Osgood Curve Construction
Starting with a triangle, remove a triangle-shaped region in such a way that two triangles remain, where the ratio of the removed triangle area to the original triangle area is . Repeating the process on the two remaining triangles—removing a proportion of area from each—creates four triangles, and further repetitions double the number of remaining triangles. By carefully choosing the proportions of areas removed , you can generate a set of points with any desired Lebesgue measure between 0 and 1. The construction is due to Knopp, a refinement of previous attempts by Sierpinski and Osgood.
Snapshot 1: when is the constant , the limit of the iteration is the Koch curve
Snapshots 2, 3: when converges, the Lebesgue measure of the resulting point set is greater than 0; with , by selecting a value of one can achieve any desired Lebesgue measure in
H. Sagan, Space-Filling Curves, New York: Springer-Verlag, 1994 pp. 136–140.