Fractal dimension

Coastline of Britain measured using a 200 km scale

11.5 x 200 = 2300 km

Coastline of Britain measured using a 100 km scale

28 x 100 = 2800 km

Coastline of Britain measured using a 50 km scale

70 x 50 = 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 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 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 by 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.

^ Cite error: The named reference Falconer was invoked but never defined (see the help page).
^ Cite error: The named reference space filling was invoked but never defined (see the help page).
^ Vicsek, Tamás (1992). Fractal growth phenomena. World Scientific. p. 10. ISBN 978-981-02-0668-0.
^ Mandelbrot, B. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–8. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. S2CID 15662830. Archived from the original on 2021-10-19. Retrieved 2020-11-12.
^ Cite error: The named reference Mandelbrot1983 was invoked but never defined (see the help page).
^ Edgar, Gerald (2007). Measure, Topology, and Fractal Geometry. Springer. p. 7. ISBN 978-0-387-74749-1.
^ Harte, David (2001). Multifractals. Chapman & Hall. pp. 3–4. ISBN 978-1-58488-154-4.

[Falconer-1] Cite error: The named reference Falconer was invoked but never defined (see the help page).

[space_filling-2] Cite error: The named reference space filling was invoked but never defined (see the help page).

[vicsek-3] Vicsek, Tamás (1992). Fractal growth phenomena. World Scientific. p. 10. ISBN 978-981-02-0668-0.

[coastline-4] Mandelbrot, B. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–8. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. S2CID 15662830. Archived from the original on 2021-10-19. Retrieved 2020-11-12.

[Mandelbrot1983-5] Cite error: The named reference Mandelbrot1983 was invoked but never defined (see the help page).

[6] Edgar, Gerald (2007). Measure, Topology, and Fractal Geometry. Springer. p. 7. ISBN 978-0-387-74749-1.

[7] Harte, David (2001). Multifractals. Chapman & Hall. pp. 3–4. ISBN 978-1-58488-154-4.

[1]

[2]

[3]

[4]

[5]

[6]

[7]