# Knopp's Osgood Curve Construction

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.

Contributed by: Robert Dickau (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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.

## Permanent Citation