Ratio providing a statistical index of complexity variation with scale
11.5 x 200 = 2300 km
28 x 100 = 2800 km
70 x 50 = 3500 km
Figure 1. As the length of the measuring stick is scaled smaller and smaller, the total length of the coastline measured increases (See Coastline paradox).
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 is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension.[1][2][3]
The main idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions.[4] In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick.[5] There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale: see the section Examples.
Ultimately, the term fractal dimension became the phrase with which Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word fractal, a term he created. After several iterations over years, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants."[6]
One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means rectifiable: the length of the curve between any two points on the Koch snowflake is infinite. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively by thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional.[7] Therefore, its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which is often a number between one and two; in the case of the Koch snowflake, it is approximately 1.2619.
^Cite error: The named reference Falconer was invoked but never defined (see the help page).
^Cite error: The named reference space filling was invoked but never defined (see the help page).
^Vicsek, Tamás (1992). Fractal growth phenomena. World Scientific. p. 10. ISBN 978-981-02-0668-0.
^Mandelbrot, B. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–8. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. S2CID 15662830. Archived from the original on 2021-10-19. Retrieved 2020-11-12.
^Cite error: The named reference Mandelbrot1983 was invoked but never defined (see the help page).
^Edgar, Gerald (2007). Measure, Topology, and Fractal Geometry. Springer. p. 7. ISBN 978-0-387-74749-1.
^Harte, David (2001). Multifractals. Chapman & Hall. pp. 3–4. ISBN 978-1-58488-154-4.
fractaldimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern...
mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractaldimension strictly exceeding...
Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here...
In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractaldimension, that was introduced in 1918 by mathematician Felix...
as a general rule, fractal curves have different dimensions, also see fractaldimension and list of fractals by Hausdorff dimension. Starting in the 1950s...
This results from the fractal curve–like properties of coastlines; i.e., the fact that a coastline typically has a fractaldimension. Although the "paradox...
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractaldimension and other fractal characteristics...
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...
calculate the 'overall' fractaldimension of a real landscape can result in measures of negative fractaldimension, or of fractaldimension above 3. In particular...
\sim l_{B}^{d_{B}}} , A network can be classified as fractal or not and the fractaldimension can be found. For example, the WWW, the human brain, metabolic...
impedance. In general, the fractaldimension of a fractal antenna is a poor predictor of its performance and application. Not all fractal antennas work well for...
dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension...
universe shows a definite fractal aspect over a fairly wide range of scale, with a fractaldimension of about 2. The fractaldimension of a homogeneous 3D object...
are the fractal dimensions of the stable and unstable manifolds, respectively and N=2 is the dimensionality of the system. The fractaldimension of the...
(1859–1937) by Benoît Mandelbrot (1924–2010). H is directly related to fractaldimension, D, and is a measure of a data series' "mild" or "wild" randomness...
The FractalDimension of Architecture is a book that applies the mathematical concept of fractaldimension to the analysis of the architecture of buildings...
kind of three dimensionalfractal. Fractal Expressionism is a term used to differentiate traditional visual art that incorporates fractal elements such...
referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating...
chaotic mixing is a process by which flow tracers develop into complex fractals under the action of a fluid flow. The flow is characterized by an exponential...
intensities. When box counting is done to determine a fractaldimension known as the box counting dimension, the information recorded is usually either yes...
multifractal system is a generalization of a fractal system in which a single exponent (the fractaldimension) is not enough to describe its dynamics; instead...
In fractal geometry, the Higuchi dimension (or Higuchi fractaldimension (HFD)) is an approximate value for the box-counting dimension of the graph of...
Global Information
