Editing Fractal (section)

==Definition and characteristics==
One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented [[Shape|geometric shape]] that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole";<ref name="Mandelbrot1983" /> this is generally helpful but limited. Authors disagree on the exact definition of ''fractal'', but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in.<ref name="Mandelbrot1983" /><ref name="Gouyet">{{cite book | last=Gouyet | first=Jean-François | title=Physics and fractal structures | publisher=Masson Springer | location=Paris/New York | year=1996 | isbn=978-0-387-94153-0 }}</ref><ref name="Falconer">{{Cite book | last=Falconer | first=Kenneth | title=Fractal Geometry: Mathematical Foundations and Applications | publisher=John Wiley & Sons | year=2003 | pages=xxv | isbn= 978-0-470-84862-3 | no-pp=true }}</ref><ref name="vicsek" /><ref>{{cite book | last=Edgar | first=Gerald | title=Measure, topology, and fractal geometry | publisher=Springer-Verlag | location=New York | year=2008 | isbn=978-0-387-74748-4 |page=1 }}</ref>

One point agreed on is that fractal patterns are characterized by [[fractal dimension]]s, but whereas these numbers quantify [[complexity]] (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns.<ref>{{cite book |last=Karperien |first=Audrey |title=Defining microglial morphology: Form, Function, and Fractal Dimension |publisher=Charles Sturt University |year= 2004 |doi=10.13140/2.1.2815.9048 }}</ref> In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose [[Hausdorff–Besicovitch dimension]] is greater than its [[Lebesgue covering dimension|topological dimension]].<ref name="Mandelbrot quote" /> However, this requirement is not met by [[space-filling curve]]s such as the [[Hilbert curve]].<ref group=notes name="space filling note" />

Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to [[Kenneth Falconer (mathematician)|Falconer]], fractals should be only generally characterized by a [[wikt:gestalt|gestalt]] of the following features;<ref name="Falconer" />

* Self-similarity, which may include:
:* Exact self-similarity: identical at all scales, such as the [[#koch|Koch snowflake]]
:* Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the [[Mandelbrot set]]'s satellites are approximations of the entire set, but not exact copies.
:* Statistical self-similarity: repeats a pattern [[stochastic]]ally so numerical or statistical measures are preserved across scales; e.g., [[#random|randomly generated fractals]] like the well-known example of the [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|coastline of Britain]] for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines fractals like the Koch snowflake.<ref name="vicsek" />
:* Qualitative self-similarity: as in a time series<ref name="time series">{{cite book | last=Peters | first=Edgar | title=Chaos and order in the capital markets : a new view of cycles, prices, and market volatility | publisher=Wiley | location=New York | year=1996 | isbn=978-0-471-13938-6 }}</ref>
:* [[Multifractal]] scaling: characterized by more than one fractal dimension or scaling rule
* Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have [[emergent properties]]<ref>{{cite book | first1=John |last1=Spencer |first2=Michael S. C. |last2=Thomas |first3=James L. |last3=McClelland |title=Toward a unified theory of development : connectionism and dynamic systems theory re-considered |publisher=Oxford University Press |location=Oxford/New York |year=2009 |isbn=978-0-19-530059-8 }}</ref> (related to the next criterion in this list).
* Irregularity locally and globally that cannot easily be described in the language of traditional [[Euclidean geometry]] other than as the limit of a [[recursion|recursively]] defined sequence of stages. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls";<ref name="Mandelbrot Chaos" />''see [[#algorithms|Common techniques for generating fractals]]''.

As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, and is easily described in Euclidean language without a need for recursion.<ref name="Mandelbrot1983" /><ref name="vicsek" />