In mathematics, an **Osgood curve** is a non-self-intersecting curve (either a Jordan curve or a Jordan arc) of positive area.^{ [1] } More formally, these are curves in the Euclidean plane with positive two-dimensional Lebesgue measure.

The first examples of Osgood curves were found by William FoggOsgood ( 1903 ) and HenriLebesgue ( 1903 ). 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. Knopp's example has the additional advantage that its area can be controlled to be any desired fraction of the area of its convex hull.^{ [2] }

Although most space-filling curves are not Osgood curves (they have positive area but often include infinitely many self-intersections, failing to be Jordan curves) it is possible to modify the recursive construction of space-filling curves or other fractal curves to obtain an Osgood curve.^{ [3] } For instance, Knopp's construction involves recursively splitting triangles into pairs of smaller triangles, meeting at a shared vertex, by removing triangular wedges. When the removed wedges at each level of this construction cover the same fraction of the area of their triangles, the result is a Cesàro fractal such as the Koch snowflake, but removing wedges whose areas shrink more rapidly produces an Osgood curve.^{ [2] }

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.^{ [4] }

- ↑ Radó (1948).
- 1 2 Knopp (1917); Sagan (1994), Section 8.3, The Osgood Curves of Sierpínski and Knopp, pp. 136–140.
- ↑ Knopp (1917); Lance & Thomas (1991); Sagan (1993)).
- ↑ Balcerzak & Kharazishvili (1999).

In mathematics, a **fractal** is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension. Fractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set. Fractals exhibit similar patterns at increasingly small scales called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

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 with the standard measure of length, area, or volume. In general, it is also called ** n-dimensional volume**,

The **Sierpiński carpet** is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is one generalization of the Cantor set to two dimensions; another is the Cantor dust.

The **Sierpiński triangle**, also called the **Sierpiński gasket** or **Sierpiński sieve**, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.

The **Koch snowflake** 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 Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch.

In mathematics, a **curve** is an object similar to a line, but that does not have to be straight.

In mathematics, the **Menger sponge** is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

In topology, a **Jordan curve**, sometimes called a *plane simple closed curve*, is a non-self-intersecting continuous loop in the plane. The **Jordan curve theorem** asserts that every Jordan 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 points, so that every continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. While the statement of this theorem seems to be intuitively obvious, it takes some ingenuity to prove it by elementary means. *"Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."*. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.

In mathematical analysis, a **space-filling curve** is a curve whose range contains the entire 2-dimensional unit square. Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called *Peano curves*, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.

This is a **list of fractal topics**, by Wikipedia page, See also list of dynamical systems and differential equations topics.

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 **ℝ** that is nowhere dense, yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line, and Volterra introduced a similar example in 1881. The Cantor set as we know it today followed in 1883. The Smith-Volterra-Cantor set is topologically equivalent to the middle-thirds Cantor set.

A **fractal curve** is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length.

**William Fogg Osgood** was an American mathematician, born in Boston.

In mathematics the **Vicsek fractal**, also known as **Vicsek snowflake** or **box fractal**, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, particularly in cellular phones.

**Space-filling trees** are geometric constructions that are analogous to space-filling curves, but have a branching, tree-like structure and are rooted. A space-filling tree is defined by an incremental process that results in a tree for which every point in the space has a finite-length path that converges to it. In contrast to space-filling curves, individual paths in the tree are short, allowing any part of the space to be quickly reached from the root. The simplest examples of space-filling trees have a regular, self-similar, fractal structure, but can be generalized to non-regular and even randomized/Monte-Carlo variants. Space-filling trees have interesting parallels in nature, including fluid distribution systems, vascular networks, and fractal plant growth, and many interesting connections to L-systems in computer science.

An ** n-flake**,

In the geometry of tessellations, a **rep-tile** or **reptile** is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of *Scientific American*. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in *Mathematics Magazine*.

In plane geometry the **Blaschke–Lebesgue theorem** states that the Reuleaux triangle has the least area of all curves of given constant width. In the form that every curve of a given width has area at least as large as the Reuleaux triangle, it is also known as the **Blaschke–Lebesgue inequality**. It is named after Wilhelm Blaschke and Henri Lebesgue, who published it separately in the early 20th century.

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.

- Balcerzak, M.; Kharazishvili, A. (1999), "On uncountable unions and intersections of measurable sets",
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*Archiv der Mathematik und Physik*,**26**: 103–115. - Lance, Timothy; Thomas, Edward (1991), "Arcs with positive measure and a space-filling curve",
*American Mathematical Monthly*,**98**(2): 124–127, doi:10.2307/2323941, JSTOR 2323941, MR 1089456 . - Lebesgue, H. (1903), "Sur le problème des aires",
*Bulletin de la Société Mathématique de France*(in French),**31**: 197–203, doi: 10.24033/bsmf.694 - Osgood, William F. (1903), "A Jordan Curve of Positive Area",
*Transactions of the American Mathematical Society*,**4**(1): 107–112, doi: 10.1090/S0002-9947-1903-1500628-5 , ISSN 0002-9947, JFM 34.0533.02, JSTOR 1986455, MR 1500628 . - Radó, Tibor (1948),
*Length and Area*, American Mathematical Society Colloquium Publications, vol. 30, American Mathematical Society, New York, p. 157, ISBN 9780821846216, MR 0024511 . - Sagan, Hans (1993), "A geometrization of Lebesgue's space-filling curve",
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