Editing Fractal dimension (section)

== Examples ==
The concept of fractal dimension described in this article is a basic view of a complicated construct. The examples discussed here were chosen for clarity, and the scaling unit and ratios were known ahead of time. In practice, however, fractal dimensions can be determined using techniques that approximate scaling and detail from [[Limit (mathematics)|limits]] estimated from [[regression line]]s over [[log–log plot]]s of size vs scale. Several formal mathematical definitions of different types of fractal dimension are listed below. Although for compact sets with exact affine self-similarity all these dimensions coincide, in general they are not equivalent:
* [[Box-counting dimension]] is [[box counting|estimated]] as the exponent of a [[power law#Estimating the exponent from empirical data|power law]]:
*: <math>D_0 = \lim_{\varepsilon \to 0} \frac{\log N(\varepsilon)}{\log\frac{1}{\varepsilon}}.</math>
* [[Information dimension]] considers how the average [[information entropy|information]] needed to identify an occupied box scales with box size (<math>p</math> is a probability):
*: <math>D_1 = \lim_{\varepsilon \to 0} \frac{-\langle \log p_\varepsilon \rangle}{\log\frac{1}{\varepsilon}}.</math>
* [[Correlation dimension]] is based on <math>M</math> as the number of points used to generate a representation of a fractal and ''g''<sub>ε</sub>, the number of pairs of points closer than ε to each other:{{Citation needed|reason=Limit was incorrect, not sure if correction is|date=June 2017}}
*: <math>D_2 = \lim_{M \to \infty} \lim_{\varepsilon \to 0} \frac{\log (g_\varepsilon / M^2)}{\log \varepsilon}.</math> 
* Generalized, or Rényi dimensions: the box-counting, information, and correlation dimensions can be seen as special cases of a continuous spectrum of [[Rényi entropy|generalized dimensions]] of order α, defined by
*: <math>D_\alpha = \lim_{\varepsilon \to 0} \frac{\frac{1}{\alpha - 1} \log(\sum_i p_i^\alpha)}{\log\varepsilon}.</math>
* [[Higuchi dimension]]<ref>{{cite journal |first=T. |last=Higuchi |title=Approach to an irregular time-series on the basis of the fractal theory |journal=Physica D |volume=31 |issue=2 |pages=277–283 |year=1988 |doi=10.1016/0167-2789(88)90081-4 |bibcode=1988PhyD...31..277H }}</ref>
*: <math>D = \frac{d\log L(k)}{d \log k}.</math>
* [[Lyapunov dimension]]
* [[Multifractal]] dimensions: a special case of Rényi dimensions where scaling behaviour varies in different parts of the pattern.
* [[Uncertainty exponent]]
* [[Hausdorff dimension]]: For any subset <math>S</math> of a metric space <math>X</math> and <math>d \geq 0</math>, the ''d''-dimensional ''Hausdorff content'' of ''S'' is defined by <math display="block">
 C_H^d(S) := \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i > 0\Bigr\}.
</math> The Hausdorff dimension of ''S'' is defined by
*: <math>\dim_{\operatorname{H}}(X) := \inf\{d \ge 0: C_H^d(X) = 0\}.</math>
* [[Packing dimension]]
* [[Assouad dimension]]
* [[Local connected dimension]]<ref>{{Cite journal | last1 = Jelinek | first1 = A. | doi = 10.2147/OPTH.S1579 | last2 = Jelinek | first2 = H. F. | last3 = Leandro | first3 = J. J. | last4 = Soares | first4 = J. V. | last5 = Cesar Jr | first5 = R. M. | last6 = Luckie | first6 = A. | title = Automated detection of proliferative retinopathy in clinical practice | journal = Clinical Ophthalmology | pages = 109–122 | year = 2008 | pmid =  19668394 | pmc = 2698675 | volume=2 | issue=1 | doi-access = free }}</ref>
* Degree dimension describes the fractal nature of the degree distribution of graphs.<ref>{{Cite journal | last1 = Li | first1 = N. Z. | last2 = Britz | first2 = T. | title = On the scale-freeness of random colored substitution networks | journal = Proceedings of the American Mathematical Society | pages = 1377–1389 | year = 2024 | volume=152 | number=4 | doi = 10.1090/proc/16604 | arxiv = 2109.14463 }}</ref>
* [[Parabolic Hausdorff dimension]]