mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, an Osgood curve is a non-self-intersecting

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

that has positive

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...

. Despite its area, it is not possible for such a curve to cover a

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

, distinguishing them from

space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, spa ...

s. Osgood curves are named after

William Fogg Osgood William Fogg Osgood (March 10, 1864, Boston – July 22, 1943, Belmont, Massachusetts) was an American mathematician. Education and career In 1886, he graduated from Harvard, where, after studying at the universities of Göttingen (1887–188 ...

Definition and properties

A curve in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

is defined to be an Osgood curve when it is non-self-intersecting (that is, it is either a

Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition tha ...

or a

Jordan arc In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior p ...

) and it has positive area. More formally, it must have positive two-dimensional

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...

. Osgood curves have

Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, o ...

two, like

s. However, they cannot be space-filling curves: by Netto's theorem, covering all of the points of the plane, or of any convex subset of the plane, would lead to self-intersections.

History

The first examples of Osgood curves were found by and . Both examples have positive area in parts of the curve, but zero area in other parts; this flaw was corrected by , who found a curve that has positive area in every neighborhood of each of its points, based on an earlier construction of Wacław Sierpiński. Knopp's example has the additional advantage that its area can be made arbitrarily close to the area of its convex hull.; , Section 8.3, The Osgood Curves of Sierpínski and Knopp
pp. 136–140

Construction

It is possible to modify the recursive construction of certain

fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as il ...

s and space-filling curves to obtain an Osgood curve.; ; 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 In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham. The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all speci ...

such as the

Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Cur ...

. Instead, reducing the fraction of area removed per level, rapidly enough to leave a constant fraction of the area unremoved, produces an Osgood curve. Another way to construct an Osgood curve is to form a two-dimensional version of the Smith–Volterra–Cantor set, a

totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...

point set with non-zero area, and then apply the Denjoy–Riesz theorem according to which every

bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...

and totally disconnected subset of the plane is a subset of a Jordan curve.

Notes

References

*. *. *. * *. *. *. *.

External links

* {{DEFAULTSORT:Osgood Curve Plane curves Area