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Fractal dimension - Wikipedia

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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 appli

Fractal dimension - Wikipedia Jump to content From Wikipedia, the free encyclopedia Real-valued number of spatial dimensions 11.5 × 200 km = 2300 km 28 × 100 km = 2800 km 70 × 50 km = 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 geometric measure theory , fractal dimensions enable consistent statistical indexes of complexity in patterns . Since fractal patterns can be scale -variant, measuring space-filling capacity should be possible in non-integer (fractal) dimensions. [

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