$fractal dimension$

$fractal dimension$

Similarly, a surface with fractal dimension of 2.1 fills space very much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume. For instance, a curve with a fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface. Unlike topological dimensions, the fractal index can take non- integer values, indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry. Thus, it is 0 for sets describing points (0-dimensional sets) 1 for sets describing lines (1-dimensional sets having length only) 2 for sets describing surfaces (2-dimensional sets having length and width) and 3 for sets describing volumes (3-dimensional sets having length, width, and height). For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. įractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture. The essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s, : 19 but the terms fractal and fractal dimension were coined by mathematician Benoit Mandelbrot in 1975. Fractal dimensions are used to characterize a broad spectrum of objects ranging from the abstract to practical phenomena, including turbulence, : 97–104 river networks, : 246–247 urban growth, human physiology, medicine, and market trends. : 1 Several types of fractal dimension can be measured theoretically and empirically ( see Fig. The theoretical fractal dimension for this fractal is 5/3 ≈ 1.67 its empirical fractal dimension from box counting analysis is ☑% using fractal analysis software.Ī fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale. A 32-segment quadric fractal scaled and viewed through boxes of different sizes. 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.įigure 2. 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. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. 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. One non-trivial example is the fractal dimension of a Koch snowflake.

$fractal dimension$

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.' 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. 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. 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. 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. 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. 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.

$fractal dimension$

A fractal pattern changes with the scale at which it is measured. 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.

$fractal dimension$

As the length of the measuring stick is scaled smaller and smaller, the total length of the coastline measured increases (See Coastline paradox).

$fractal dimension$