Benoit B.Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.
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 illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is 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, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.
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 line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.
The Mandelbrot set is the set of complex numbers for which the function does not diverge to infinity when iterated from , i.e., for which the sequence , , etc., remains bounded in absolute value.
In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.
In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It has also been mythologized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently and in a fractal dimension, i.e. one that does not have to be an integer.
Lewis Fry Richardson, FRS was an English mathematician, physicist, meteorologist, psychologist, and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them. He is also noted for his pioneering work concerning fractals and a method for solving a system of linear equations known as modified Richardson iteration.
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in or, more generally, in any metric space.
This is a list of fractal topics, by Wikipedia page, See also list of dynamical systems and differential equations topics.
In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set in a Euclidean space , or more generally in a metric space . It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand.
"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoit Mandelbrot, first published in Science on 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals.
Arc length is the distance between two points along a section of a curve.
In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set , the "density" of A is 0 or 1 at almost every point in . Additionally, the "density" of A is 1 at almost every point in A. Intuitively, this means that the "edge" of A, the set of points in A whose "neighborhood" is partially in A and partially outside of A, is negligible.
The coastline of the United Kingdom is formed by a variety of natural features including islands, bays, headlands and peninsulas. It consists of the coastline of the island of Great Britain and the north-east coast of the island of Ireland, as well as many much smaller islands. Much of the coastline is accessible and quite varied in geography and habitats. Large stretches have been designated areas of natural beauty, notably the Jurassic Coast and various stretches referred to as heritage coast. They are both very long, spreading through the mainland.
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.
In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of curvature in geometry. In the form presented below, the concept was introduced in 1995 by the mathematician Mark S. Melnikov; accordingly, it may be referred to as the Melnikov curvature or Menger-Melnikov curvature. Melnikov and Verdera (1995) established a powerful connection between the curvature of measures and the Cauchy kernel.
Lacunarity, from the Latin lacuna, meaning "gap" or "lake", is a specialized term in geometry referring to a measure of how patterns, especially fractals, fill space, where patterns having more or larger gaps generally have higher lacunarity. Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity. This is illustrated in Figure 1 showing three fractal patterns. When rotated 90°, the first two fairly homogeneous patterns do not appear to change, but the third more heterogeneous figure does change and has correspondingly higher lacunarity. The earliest reference to the term in geometry is usually attributed to Benoit Mandelbrot, who, in 1983 or perhaps as early as 1977, introduced it as, in essence, an adjunct to fractal analysis. Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis in particular.
The Minkowski sausage or Minkowski curve is a fractal first proposed by and named for Hermann Minkowski as well as its casual resemblance to a sausage or sausage links. The initiator is a line segment and the generator is a broken line of eight parts one fourth the length.
