Fractal Dimension

A fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently than the space it is embedded in; a fractal dimension does not have to be an integer.

The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoît Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. 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. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale.

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 a rectifiable curve: 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 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. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which in this case is a number between one and two.