fractal dimension

{{Anchor|32seg}}

Image:32 segment fractal.jpg. The theoretical fractal dimension for this fractal is 5/3 ≈ 1.67; its empirical fractal dimension from box counting analysis is ±1%{{cite book

| first = Audrey

| last = Balay-Karperien

| title = Defining Microglial Morphology: Form, Function, and Fractal Dimension

| publisher = Charles Sturt University

| url = http://trove.nla.gov.au/work/162139699

| access-date = 9 July 2013

| year = 2004

| page = 86

}} using fractal analysis software.]]

A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale.{{rp|1}} Several types of fractal dimension can be measured theoretically and empirically (see Fig. 2). Fractal dimensions are used to characterize a broad spectrum of objects ranging from the abstract to practical phenomena, including turbulence,{{rp|97–104}} river networks,{{rp|246–247}} urban growth,{{Cite journal | last1 = Chen | first1 = Yanguang | title = Modeling Fractal Structure of City-Size Distributions Using Correlation Functions | doi = 10.1371/journal.pone.0024791 | journal = PLOS ONE | volume = 6 | issue = 9 | pages = e24791 | year = 2011 | pmid = 21949753 | pmc = 3176775|arxiv = 1104.4682 |bibcode = 2011PLoSO...624791C | doi-access = free}}{{cite web |url=http://library.thinkquest.org/26242/full/ap/ap.html |title=Applications |access-date=2007-10-21 |url-status=dead |archive-url=https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html |archive-date=2007-10-12}} human physiology,{{Cite journal | last1 = Popescu | first1 = D. P. | last2 = Flueraru | first2 = C. | last3 = Mao | first3 = Y. | last4 = Chang | first4 = S. | last5 = Sowa | first5 = M. G. | title = Signal attenuation and box-counting fractal analysis of optical coherence tomography images of arterial tissue | doi = 10.1364/boe.1.000268 | journal = Biomedical Optics Express | volume = 1 | issue = 1 | pages = 268–277 | year = 2010 | pmid = 21258464 | pmc = 3005165}}{{Cite journal | last1 = King | first1 = R. D. | last2 = George | first2 = A. T. | last3 = Jeon | first3 = T. | last4 = Hynan | first4 = L. S. | last5 = Youn | first5 = T. S. | last6 = Kennedy | first6 = D. N. | last7 = Dickerson | first7 = B. | author8 = the Alzheimer's Disease Neuroimaging Initiative | doi = 10.1007/s11682-008-9057-9 | title = Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis | journal = Brain Imaging and Behavior | volume = 3 | issue = 2 | pages = 154–166 | year = 2009 | pmid = 20740072| pmc =2927230 }} medicine, and market trends. The essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s,{{rp|19}} but the terms fractal and fractal dimension were coined by mathematician Benoit Mandelbrot in 1975.{{cite book | last = Falconer | first = Kenneth | title = Fractal Geometry | url = https://archive.org/details/fractalgeometrym00falc | url-access = limited | publisher = Wiley | year = 2003 | isbn = 978-0-470-84862-3 |page=[https://archive.org/details/fractalgeometrym00falc/page/n336 308]}}

{{cite book

| last = Sagan

| first = Hans

| title = Space-Filling Curves

| url = https://archive.org/details/spacefillingcurv00saga_539

| url-access = limited

| publisher = Springer-Verlag

| year = 1994

| isbn = 0-387-94265-3

| page=[https://archive.org/details/spacefillingcurv00saga_539/page/n170 156] }}

{{cite book

|author=Benoit B. Mandelbrot

|title=The fractal geometry of nature

|url=https://books.google.com/books?id=0R2LkE3N7-oC

|access-date=1 February 2012

|year=1983

|publisher=Macmillan

|isbn=978-0-7167-1186-5}}{{cite book

|editor1-last = Losa

|editor1-first= Gabriele A.

|editor2-last= Nonnenmacher

|editor2-first= Theo F.

|title=Fractals in biology and medicine

|url=https://books.google.com/books?id=t9l9GdAt95gC

|access-date=1 February 2012

|year=2005

|publisher=Springer

|isbn=978-3-7643-7172-2}}

{{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

| year = 1996

| isbn = 0-471-13938-6 }}{{cite book

| last1 = Albers

| last2 = Alexanderson | author2-link = Gerald L. Alexanderson

| title = Mathematical people : profiles and interviews

| url = https://archive.org/details/mathematicalpeop00djal

| url-access = limited

| publisher = AK Peters

| year = 2008

| isbn = 978-1-56881-340-0

| page = [https://archive.org/details/mathematicalpeop00djal/page/n242 214]

| chapter = Benoit Mandelbrot: In his own words}}

Fractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture. For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. Thus, it is 0 for sets describing points (0-dimensional sets); 1 for sets describing lines (1-dimensional sets having length only); 2 for sets describing surfaces (2-dimensional sets having length and width); and 3 for sets describing volumes (3-dimensional sets having length, width, and height). But this changes for fractal sets. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry.

Unlike topological dimensions, the fractal index can take non-integer values,{{Cite journal | last1 = Sharifi-Viand | first1 = A. | last2 = Mahjani | first2 = M. G. | last3 = Jafarian | first3 = M. | title = Investigation of anomalous diffusion and multifractal dimensions in polypyrrole film | doi = 10.1016/j.jelechem.2012.02.014 | journal = Journal of Electroanalytical Chemistry | volume = 671 | pages = 51–57 | year = 2012 }} indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does. For instance, a curve with a fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface. Similarly, a surface with fractal dimension of 2.1 fills space very much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume.{{rp|48}}See List of fractals by Hausdorff dimension for a graphic representation of different fractal dimensions. This general relationship can be seen in the two images of fractal curves in Fig. 2 and Fig. 3{{snd}} the 32-segment contour in Fig. 2, convoluted and space-filling, has a fractal dimension of 1.67, compared to the perceptibly less complex Koch curve in Fig. 3, which has a fractal dimension of approximately 1.2619.

{{anchor|kline}}

Image:blueklineani2.gif is a classic iterated fractal curve. It is made by starting from a line segment, and then iteratively scaling each segment by 1/3 into 4 new pieces laid end to end with 2 middle pieces leaning toward each other along an equilateral triangle, so that the whole new segment spans the distance between the endpoints of the original segment. The animation only shows a few iterations, but the theoretical curve is scaled in this way infinitely.]]

The relationship of an increasing fractal dimension with space-filling might be taken to mean fractal dimensions measure density, but that is not so; the two are not strictly correlated. Instead, a fractal dimension measures complexity, a concept related to certain key features of fractals: self-similarity and detail or irregularity.See {{slink|Fractal#Characteristics}}. These features are evident in the two examples of fractal curves. Both are curves with topological dimension of 1, so one might hope to be able to measure their length and derivative in the same way as with ordinary curves. But we cannot do either of these things, because fractal curves have complexity in the form of self-similarity and detail that ordinary curves lack. The self-similarity lies in the infinite scaling, and the detail in the defining elements of each set. The length between any two points on these curves is infinite, no matter how close together the two points are, which means that it is impossible to approximate the length of such a curve by partitioning the curve into many small segments.Helge von Koch, "On a continuous curve without tangents constructible from elementary geometry" In {{harvnb|Edgar|2004|pp=25–46}}. Every smaller piece is composed of an infinite number of scaled segments that look exactly like the first iteration. These are not rectifiable curves, meaning that they cannot be measured by being broken down into many segments approximating their respective lengths. They cannot be meaningfully characterized by finding their lengths and derivatives. However, their fractal dimensions can be determined, which shows that both fill space more than ordinary lines but less than surfaces, and allows them to be compared in this regard.

The two fractal curves described above show a type of self-similarity that is exact with a repeating unit of detail that is readily visualized. This sort of structure can be extended to other spaces (e.g., a fractal that extends the Koch curve into 3D space has a theoretical D = 2.5849). However, such neatly countable complexity is only one example of the self-similarity and detail that are present in fractals. The example of the coast line of Britain, for instance, exhibits self-similarity of an approximate pattern with approximate scaling.{{rp|26}} Overall, fractals show several types and degrees of self-similarity and detail that may not be easily visualized. These include, as examples, strange attractors, for which the detail has been described as in essence, smooth portions piling up,{{rp|49}} the Julia set, which can be seen to be complex swirls upon swirls, and heart rates, which are patterns of rough spikes repeated and scaled in time.{{Cite journal | last1 = Tan | first1 = Can Ozan | last2 = Cohen | first2 = Michael A. | last3 = Eckberg | first3 = Dwain L. | last4 = Taylor | first4 = J. Andrew | title = Fractal properties of human heart period variability: Physiological and methodological implications | doi = 10.1113/jphysiol.2009.169219 | journal = The Journal of Physiology | volume = 587 | issue = 15 | pages = 3929–3941 | year = 2009 | pmid = 19528254| pmc = 2746620}} Fractal complexity may not always be resolvable into easily grasped units of detail and scale without complex analytic methods, but it is still quantifiable through fractal dimensions.{{rp|197;262}}

The terms fractal dimension and fractal were coined by Mandelbrot in 1975, about a decade after he published his paper on self-similarity in the coastline of Britain. Various historical authorities credit him with also synthesizing centuries of complicated theoretical mathematics and engineering work and applying them in a new way to study complex geometries that defied description in usual linear terms. The earliest roots of what Mandelbrot synthesized as the fractal dimension have been traced clearly back to writings about nondifferentiable, infinitely self-similar functions, which are important in the mathematical definition of fractals, around the time that calculus was discovered in the mid-1600s.{{rp|405}} There was a lull in the published work on such functions for a time after that, then a renewal starting in the late 1800s with the publishing of mathematical functions and sets that are today called canonical fractals (such as the eponymous works of von Koch, Sierpiński, and Julia), but at the time of their formulation were often considered antithetical mathematical "monsters".

{{cite book

| editor-last = Edgar

| editor-first = Gerald

| title = Classics on Fractals

| publisher = Westview Press

| year = 2004| isbn = 978-0-8133-4153-8 }}

{{cite web

|title=A History of Fractal Geometry

|work=MacTutor History of Mathematics

|author=Trochet, Holly

|archive-url=https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html

|archive-date=12 March 2012

|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/fractals.html

|year=2009

|url-status=dead

}}

These works were accompanied by perhaps the most pivotal point in the development of the concept of a fractal dimension through the work of Hausdorff in the early 1900s who defined a "fractional" dimension that has come to be named after him and is frequently invoked in defining modern fractals.{{rp|44}}

{{cite book

| last = Mandelbrot

| first = Benoit

| title = Fractals and Chaos

| publisher = Springer

| year = 2004

| isbn = 978-0-387-20158-0

| quote = A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension

| page= 38}}{{cite book

| last = Gordon

| first = Nigel

| title = Introducing fractal geometry

| publisher = Icon

| location = Duxford

| year = 2000

| isbn = 978-1-84046-123-7

| page = [https://archive.org/details/introducingfract0000lesm/page/71 71]

| url = https://archive.org/details/introducingfract0000lesm/page/71

}}

See Fractal history for more information

{{anchor|calculations}}

{{anchor|unity}}

[[Image:Fractaldimensionexample-2.png|right|thumb|alt=Lines, squares, and cubes.|Figure 4. Traditional notions of geometry for defining scaling and dimension.

$1$, $1^2\; =\; 1$, $1^3\; =\; 1;$

$2$, $2^2\; =\; 4$, $2^3\; =\; 8;$

$3$, $3^2\; =\; 9$, $3^3\; =\; 27.$Appignanesi, Richard; ed. (2006). Introducing Fractal Geometry, p. 28. Icon. {{ISBN|978-1840467-13-0}}.]]

The concept of a fractal dimension rests in unconventional views of scaling and dimension.{{cite book |author=Iannaccone, Khokha |year=1996 |title=Fractal Geometry in Biological Systems |publisher=CRC Press |isbn=978-0-8493-7636-8}} As Fig. 4 illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick, then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. This holds in 2 dimensions as well. If one measures the area of a square, then measures again with a box of side length 1/3 the size of the original, one will find 9 times as many squares as with the first measure. Such familiar scaling relationships can be defined mathematically by the general scaling rule in equation {{EqNote|1}}, where the variable $N$ stands for the number of measurement units (sticks, squares, etc.), $\backslash varepsilon$ for the scaling factor, and $D$ for the fractal dimension:

{{NumBlk|:|$N\; =\; \backslash varepsilon^\{-D\}.$|{{EquationRef|1}}}}

This scaling rule typifies conventional rules about geometry and dimension{{snd}} referring to the examples above, it quantifies that $D\; =\; 1$ for lines because $N\; =\; 3$ when $\backslash varepsilon\; =\; 1/3$, and that $D\; =\; 2$ for squares because $N\; =\; 9$ when $\backslash varepsilon\; =\; 1/3.$

{{anchor|koch}}

Image:KochFlake.svgs of the Koch snowflake, which has a Hausdorff dimension of approximately 1.2619.]]

The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to have one length, when remeasured using a new stick scaled by 1/3 of the old may be 4 times as many scaled sticks long rather than the expected 3 (see Fig. 5). In this case, $N\; =\; 4$ when $\backslash varepsilon\; =\; 1/3,$ and the value of $D$ can be found by rearranging equation {{EqNote|1}}:

{{NumBlk|:|$\backslash log\_\backslash varepsilon\; N\; =\; -D\; =\; \backslash frac\{\backslash log\; N\}\{\backslash log\; \backslash varepsilon\}.$|{{EquationRef|2}}}}

That is, for a fractal described by $N\; =\; 4$, when $\backslash varepsilon\; =\; 1/3$, such as the Koch snowflake, $D\; =\; 1.26185\backslash ldots$, a non-integer value suggesting that the fractal has a dimension not equal to the space it resides in.

Of note, images shown in this page are not true fractals because the scaling described by $D$ cannot continue past the point of their smallest component, a pixel. However, the theoretical patterns that the images represent have no discrete pixel-like pieces, but rather are composed of an infinite number of infinitely scaled segments and do indeed have the claimed fractal dimensions.

{{anchor|statistical koch like scaling image}}

File:onetwosix.png branching fractals that are made by producing 4 new parts for every 1/3 scaling, thus having the same theoretical $D$ as the Koch curve, and for which the empirical box counting $D$ has been demonstrated with 2% accuracy.]]

As is the case with dimensions determined for lines, squares, and cubes, fractal dimensions are general descriptors that do not uniquely define patterns.{{cite book |author=Vicsek, Tamás |title=Fluctuations and scaling in biology |publisher=Oxford University Press |year=2001 |isbn=0-19-850790-9 }} The value of D for the Koch fractal discussed above, for instance, quantifies the pattern's inherent scaling, but does not uniquely describe nor provide enough information to reconstruct it. Many fractal structures or patterns could be constructed that have the same scaling relationship but are dramatically different from the Koch curve, as is illustrated in Fig. 6.

{{anchor|specific definitions}}

For examples of how fractal patterns can be constructed, see Fractal, Sierpinski triangle, Mandelbrot set, Diffusion-limited aggregation, L-system.

File:Wiki df figure.png

The concept of fractality is applied increasingly in the field of surface science, providing a bridge between surface characteristics and functional properties.{{Citation |last=Pfeifer |first=Peter |chapter=Fractals in Surface Science: Scattering and Thermodynamics of Adsorbed Films |date=1988 |volume=10 |pages=283–305 |editor-last=Vanselow |editor-first=Ralf |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-73902-6_10 |isbn=9783642739040 |editor2-last=Howe |editor2-first=Russell |title=Chemistry and Physics of Solid Surfaces VII |series=Springer Series in Surface Sciences}}. Numerous surface descriptors are used to interpret the structure of nominally flat surfaces, which often exhibit self-affine features across multiple length-scales. Mean surface roughness, usually denoted R_A, is the most commonly applied surface descriptor, however, numerous other descriptors including mean slope, root-mean-square roughness (R_RMS) and others are regularly applied. It is found, however, that many physical surface phenomena cannot readily be interpreted with reference to such descriptors, thus fractal dimension is increasingly applied to establish correlations between surface structure in terms of scaling behavior and performance.{{Cite journal |last1=Milanese |first1=Enrico |last2=Brink |first2=Tobias |last3=Aghababaei |first3=Ramin |last4=Molinari |first4=Jean-François |date=December 2019 |title=Emergence of self-affine surfaces during adhesive wear |journal=Nature Communications |volume=10 |issue=1 |pages=1116 |doi=10.1038/s41467-019-09127-8 |issn=2041-1723 |pmc=6408517 |pmid=30850605 |bibcode=2019NatCo..10.1116M}} The fractal dimensions of surfaces have been employed to explain and better understand phenomena in areas of contact mechanics,[https://www.researchgate.net/publication/318345969_Contact_stiffness_of_multiscale_surfaces_by_truncation_analysis Contact stiffness of multiscale surfaces], In the International Journal of Mechanical Sciences (2017), 131. frictional behavior,[https://www.researchgate.net/publication/283675011_Static_friction_at_fractal_interfaces Static Friction at Fractal Interfaces], Tribology International (2016), vol. 93. electrical contact resistance{{cite journal |first1=Zhai |last1=Chongpu |first2=Hanaor |last2=Dorian |first3=Proust |last3=Gwénaëlle |first4=Gan |last4=Yixiang |title=Stress-Dependent Electrical Contact Resistance at Fractal Rough Surfaces |journal=Journal of Engineering Mechanics |volume=143 |issue=3 |pages=B4015001 |year=2017 |doi=10.1061/(ASCE)EM.1943-7889.0000967 }} and transparent conducting oxides.{{Cite journal |last1=Kalvani |first1=Payam Rajabi |last2=Jahangiri |first2=Ali Reza |last3=Shapouri |first3=Samaneh |last4=Sari |first4=Amirhossein |last5=Jalili |first5=Yousef Seyed |date=August 2019 |title=Multimode AFM analysis of aluminum-doped zinc oxide thin films sputtered under various substrate temperatures for optoelectronic applications |journal=Superlattices and Microstructures |volume=132 |pages=106173 |doi=10.1016/j.spmi.2019.106173 |s2cid=198468676 }}

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 limits estimated from regression lines over log–log plots 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 estimated as the exponent of a power law:
• : $D\_0\; =\; \backslash lim\_\{\backslash varepsilon\; \backslash to\; 0\}\; \backslash frac\{\backslash log\; N(\backslash varepsilon)\}\{\backslash log\backslash frac\{1\}\{\backslash varepsilon\}\}.$
• Information dimension considers how the average information needed to identify an occupied box scales with box size ($p$ is a probability):
• : $D\_1\; =\; \backslash lim\_\{\backslash varepsilon\; \backslash to\; 0\}\; \backslash frac\{-\backslash langle\; \backslash log\; p\_\backslash varepsilon\; \backslash rangle\}\{\backslash log\backslash frac\{1\}\{\backslash varepsilon\}\}.$
• Correlation dimension is based on $M$ as the number of points used to generate a representation of a fractal and g_ε, 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}}
• : $D\_2\; =\; \backslash lim\_\{M\; \backslash to\; \backslash infty\}\; \backslash lim\_\{\backslash varepsilon\; \backslash to\; 0\}\; \backslash frac\{\backslash log\; (g\_\backslash varepsilon\; /\; M^2)\}\{\backslash log\; \backslash varepsilon\}.$
• Generalized, or Rényi dimensions: the box-counting, information, and correlation dimensions can be seen as special cases of a continuous spectrum of generalized dimensions of order α, defined by
• : $D\_\backslash alpha\; =\; \backslash lim\_\{\backslash varepsilon\; \backslash to\; 0\}\; \backslash frac\{\backslash frac\{1\}\{\backslash alpha\; -\; 1\}\; \backslash log(\backslash sum\_i\; p\_i^\backslash alpha)\}\{\backslash log\backslash varepsilon\}.$
• Higuchi dimension{{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 }}
• : $D\; =\; \backslash frac\{d\backslash log\; L(k)\}\{d\; \backslash log\; k\}.$
• 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 $S$ of a metric space $X$ and $d\; \backslash geq\; 0$, the d-dimensional Hausdorff content of S is defined by $$$$

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\}.

The Hausdorff dimension of S is defined by

• : $\backslash dim\_\{\backslash operatorname\{H\}\}(X)\; :=\; \backslash inf\backslash \{d\; \backslash ge\; 0:\; C\_H^d(X)\; =\; 0\backslash \}.$
• Packing dimension
• Assouad dimension
• Local connected dimension{{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 }}
• Degree dimension describes the fractal nature of the degree distribution of graphs.{{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 }}
• Parabolic Hausdorff dimension

Many real-world phenomena exhibit limited or statistical fractal properties and fractal dimensions that have been estimated from sampled data using computer-based fractal analysis techniques.

Practically, measurements of fractal dimension are affected by various methodological issues and are sensitive to numerical or experimental noise and limitations in the amount of data. Nonetheless, the field is rapidly growing as estimated fractal dimensions for statistically self-similar phenomena may have many practical applications in various fields, including astronomy,{{Cite journal |last1=Caicedo-Ortiz |first1=H. E. |last2=Santiago-Cortes |first2=E. |last3=López-Bonilla |first3=J. |last4=Castañeda |first4=H. O. |year=2015 |title=Fractal dimension and turbulence in Giant HII Regions |journal=Journal of Physics: Conference Series |volume=582 |issue=1 |pages=1–5 |doi=10.1088/1742-6596/582/1/012049 |arxiv=1501.04911 |bibcode=2015JPhCS.582a2049C |doi-access=free}} acoustics,{{cite journal |title=A Mathematical Approach to Correlating Objective Spectro-Temporal Features of Non-linguistic Sounds With Their Subjective Perceptions in Humans |year=2019 |pmid=31417350 |last1=Burns |first1=T. |last2=Rajan |first2=R. |journal=Frontiers in Neuroscience |volume=13 |page=794 |doi=10.3389/fnins.2019.00794 |pmc=6685481 |doi-access=free}}{{Cite journal |last1=Maragos |first1=P. |last2=Potamianos |first2=A. |year=1999 |title=Fractal dimensions of speech sounds: Computation and application to automatic speech recognition |journal=The Journal of the Acoustical Society of America |volume=105 |issue=3 |pages=1925–1932 |bibcode=1999ASAJ..105.1925M |doi=10.1121/1.426738 |pmid=10089613}} architecture,{{cite book |last1=Ostwald |first1=Michael J. |last2=Vaughan |first2=Josephine |last3=Tucker |first3=Chris |title=Characteristic visual complexity: Fractal dimensions in the architecture of frank lloyd wright and le corbusier. In: Architecture and Mathematics from Antiquity to the Future: Volume II: The 1500s to the Future |date=2015 |publisher=Springer International Publishing |isbn=978-331900143-2 |pages=339–354 |url=https://www.researchgate.net/publication/271197753}}{{Cite book |last=Bovill |first=Carl |title=Fractal geometry in architecture and design |date=1996 |publisher=Birkhäuser |isbn=978-0-8176-3795-8 |series=Design science collection |location=Boston}}{{cite journal |last1=Vaughan |first1=Josephine |last2=Ostwald |first2=Michael j. |date=2014 |title=Measuring the significance of façade transparency in Australian regionalist architecture: A computational analysis of 10 designs by Glenn Murcutt |journal=Architectural Science Review |volume=57 |issue=4 |pages=249–259 |doi=10.1080/00038628.2014.940273 |hdl=1959.13/1293729|hdl-access=free }} geology and earth sciences,{{Cite journal |last=Avşar |first=Elif |date=2020-09-01 |title=Contribution of fractal dimension theory into the uniaxial compressive strength prediction of a volcanic welded bimrock |url=https://doi.org/10.1007/s10064-020-01778-y |journal=Bulletin of Engineering Geology and the Environment |language=en |volume=79 |issue=7 |pages=3605–3619 |doi=10.1007/s10064-020-01778-y |bibcode=2020BuEGE..79.3605A |s2cid=214648440 |issn=1435-9537}} diagnostic imaging,{{Cite journal

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An alternative to a direct measurement is considering a mathematical model that resembles formation of a real-world fractal object. In this case, a validation can also be done by comparing other than fractal properties implied by the model, with measured data. In colloidal physics, systems composed of particles with various fractal dimensions arise. To describe these systems, it is convenient to speak about a distribution of fractal dimensions and, eventually, a time evolution of the latter: a process that is driven by a complex interplay between aggregation and coalescence.{{Cite journal | doi = 10.1002/mats.201300140 | title = Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems | journal = Macromolecular Theory and Simulations | volume = 23 | issue = 3 | pages = 170–181 | year = 2014 | last1 = Kryven | first1 = I. | last2 = Lazzari | first2 = S. | last3 = Storti | first3 = G. | url = http://dare.uva.nl/personal/pure/en/publications/population-balance-modeling-of-aggregation-and-coalescence-in-colloidal-systems(05340e78-b40a-4bd0-ac6a-7f044fff1617).html}}

• {{annotated link|List of fractals by Hausdorff dimension}}
• {{annotated link|Lacunarity}}
• {{annotated link|Fractal derivative}}

{{reflist|group="notes"}}

{{reflist}}

• {{cite book |first1=Benoit B. |last1=Mandelbrot |author-link=Benoit Mandelbrot |first2=Richard L. |last2=Hudson |title=The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward |url=https://books.google.com/books?id=zg91TAIs6bgC |date=2010 |publisher=Profile Books |isbn=978-1-84765-155-6}}

{{commons category}}

• [http://www.trusoft-international.com TruSoft's Benoit], fractal analysis software product calculates fractal dimensions and hurst exponents.
• [http://www.stevec.org/fracdim/ A Java Applet to Compute Fractal Dimensions]
• [http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm Introduction to Fractal Analysis]
• {{cite web|last=Bowley|first=Roger|title=Fractal Dimension|url=http://www.sixtysymbols.com/videos/fractal.htm|work=Sixty Symbols|publisher=Brady Haran for the University of Nottingham|year=2009}}
• "[https://www.youtube.com/watch?v=gB9n2gHsHN4 Fractals are typically not self-similar"]. 3Blue1Brown.

{{Fractals}}

{{Dimension topics}}

Category:Chaos theory

Category:Dynamical systems

Category:Dimension theory

Category:Fractals