# Fractal dimension

## Ratio providing a statistical index of complexity variation with scale / From Wikipedia, the free encyclopedia

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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 has also been mythologized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently and in a fractal dimension, i.e. one that does not have to be an integer.[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 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.