Knopp's Osgood Curve Construction

r

0.3333

iterations j

1

r

j

=

r

2

r

/

2

j

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

r

1

. Repeating the process on the two remaining triangles—removing a proportion of area

r

2

from each—creates four triangles, and further repetitions double the number of remaining triangles. By carefully choosing the proportions of areas removed

r

j

, 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.