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