Osgood curve

Both examples have positive area in parts of the curve, but zero area in other parts; this flaw was corrected by Knopp (1917), who found a curve that has positive area in every neighborhood of each of its points, based on an earlier construction of Wacław Sierpiński.

[4] For instance, Knopp's construction involves recursively splitting triangles into pairs of smaller triangles, meeting at a shared vertex, by removing triangular wedges.

When each level of this construction removes the same fraction of the area of its triangles, the result is a Cesàro fractal such as the Koch snowflake.

Instead, reducing the fraction of area removed per level, rapidly enough to leave a constant fraction of the area unremoved, produces an Osgood curve.

[3] Another way to construct an Osgood curve is to form a two-dimensional version of the Smith–Volterra–Cantor set, a totally disconnected point set with non-zero area, and then apply the Denjoy–Riesz theorem according to which every bounded and totally disconnected subset of the plane is a subset of a Jordan curve.

Example of an Osgood curve, constructed by recursively removing wedges from triangles. The wedge angles shrink exponentially, as does the fraction of area removed in each level, leaving nonzero area in the final curve.