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 (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

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 A shape or figure is a graphics, graphical representation of an obje ...

. 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 r ...

, 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, space ...

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 of ...

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 that a ...

or a Jordan arc) 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 wit ...

. 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, of a ...

two, like

s. However, they cannot be space-filling curves: by

Netto's theorem In mathematical analysis, Netto's theorem states that continuous bijections of smooth manifolds preserve dimension. That is, there does not exist a continuous bijection between two smooth manifolds of different dimension. It is named after Eugen N ...

, 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 Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and t ...

. Knopp's example has the additional advantage that its area can be made arbitrarily close to the area of its

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

.; , 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 illu ...

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 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 Curv ...

. 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 In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volt ...

, 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 In topology, the Denjoy–Riesz theorem states that every compact set of totally disconnected points in the Euclidean plane can be covered by a continuous image of the unit interval, without self-intersections (a Jordan arc). Definitions and sta ...

according to which every bounded and totally disconnected subset of the plane is a subset of a Jordan curve.

Definition and properties

History

Construction

Notes

References

External links