Editing Fractal dimension (section)

{{short description|Ratio providing a statistical index of complexity variation with scale}}
{{anchor|coastline}}
{{multiple image
   | width     = 100
   | footer    = 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]]).
   | align     = right
   | image1    = britain-fractal-coastline-200km.png
   | alt1      = Coastline of Britain measured using a 200 km scale
   | caption1  = 11.5 × 200 km = 2300 km
   | image2    = britain-fractal-coastline-100km.png
   | alt2      = Coastline of Britain measured using a 100 km scale
   | caption2  = 28 × 100 km = 2800 km
   | image3    = britain-fractal-coastline-50km.png
   | alt3      = Coastline of Britain measured using a 50 km scale
   | caption3  = 70 × 50 km = 3500 km
}}
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 [[Scaling (geometry)|scale]] at which it is measured. 
It is also a measure of the [[Space-filling curve|space-filling]] capacity of a pattern and tells how a fractal scales differently, in a fractal (non-integer) dimension.<ref name="Falconer" /><ref name="space filling"/><ref name="vicsek">{{cite book | last = Vicsek | first = Tamás | title = Fractal growth phenomena | publisher = World Scientific | year = 1992 | isbn = 978-981-02-0668-0 | page=10}}</ref>

The main idea of "fractured" [[Hausdorff dimension|dimensions]] has a long history in mathematics, but the term itself was brought to the fore by [[Benoit Mandelbrot]] based on [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|his 1967 paper]] on [[self-similarity]] in which he discussed ''fractional dimensions''.<ref name="coastline">{{Cite journal | last1 = Mandelbrot | first1 = B. | title = How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension | doi = 10.1126/science.156.3775.636 | journal = Science | volume = 156 | issue = 3775 | pages = 636–638 | year = 1967 | pmid = 17837158 | bibcode = 1967Sci...156..636M | s2cid = 15662830 | url = http://ena.lp.edu.ua:8080/handle/ntb/52473 | access-date = 2020-11-12 | archive-date = 2021-10-19 | archive-url = https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473 | url-status = dead }}</ref> 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 [[#coastline|Fig.&nbsp;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.<ref name="Mandelbrot1983"/> There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale, see {{slink|#Examples}} below.

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".<ref>{{cite book |first=Gerald |last=Edgar |title=Measure, Topology, and Fractal Geometry |url=https://books.google.com/books?id=dk2vruTv0_gC&pg=PR7 |date=2007 |publisher=Springer |isbn=978-0-387-74749-1 |pages=7}}</ref>

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 [[Arc length|rectifiable]]: the length of the curve between any two points on the Koch snowflake is [[Arc length#Curves with infinite length|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.<ref>{{cite book | last = Harte | first = David | title = Multifractals | url = https://archive.org/details/multifractalsthe00hart_175 | url-access = limited | publisher = Chapman & Hall | year = 2001 | isbn = 978-1-58488-154-4 |pages=[https://archive.org/details/multifractalsthe00hart_175/page/n55 3]–4}}</ref> 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.