This scaling rule typifies conventional rules about geometry and dimension – referring to the examples above, it quantifies that D = 1 {\displaystyle D=1} for lines because N = 3 {\displaystyle N=3} when ε = 1 3 {\displaystyle \varepsilon ={\tfrac {1}{3}}} , and that D = 2 {\displaystyle D=2} for squares because N = 9 {\displaystyle N=9} when ε = 1 3 . The value of D for the Koch fractal discussed above, for instance, quantifies the pattern's inherent scaling, but does not uniquely describe nor provide enough information to reconstruct it. Unlike topological dimensions, the fractal index can take non- integer values, [18] indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles.

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illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first.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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The essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s, [5] :19 [15] but the terms fractal and fractal dimension were coined by mathematician Benoit Mandelbrot in 1975. 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. 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 dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture.

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. 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. 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. Every smaller piece is composed of an infinite number of scaled segments that look exactly like the first iteration.