View source for List of fractals by Hausdorff dimension

{{Short description|none}}
{{Use dmy dates|date=October 2019}}
According to [[Benoit Mandelbrot]], "A [[fractal]] is by definition a set for which the [[Hausdorff dimension|Hausdorff-Besicovitch dimension]] strictly exceeds the [[topological dimension]]."<ref>{{harvnb|Mandelbrot|1982|p=15}}</ref>
Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension.

==Deterministic fractals==
{| class="wikitable sortable"
|-
! Hausdorff dimension<br />(exact value) || Hausdorff dimension<br />(approx.) || Name || Illustration || width="40%" | Remarks
|-
| Calculated || align="right" | 0.538 || [[Logistic map|Feigenbaum attractor]] || align="center" |[[File:Feigenbaum attractor.png|150px]] || The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the [[logistic map]] for the critical parameter value <math>\lambda_\infty = 3.570</math>, where the period doubling is infinite. This dimension is the same for any differentiable and [[unimodal]] function.<ref>{{cite journal |first=Erik |last=Aurell |title=On the metric properties of the Feigenbaum attractor |journal=Journal of Statistical Physics |volume=47 |issue=3–4 |pages=439–458 |date=May 1987 |doi=10.1007/BF01007519 |bibcode=1987JSP....47..439A |s2cid=122213380 }}</ref>
|-
| <math>\log_3 2</math> || align="right" | 0.6309 || [[Cantor set]] || align="center" |[[File:Cantor set in seven iterations.svg|200px]] || Built by removing the central third at each iteration. [[Nowhere dense]] and not a [[countable set]].
|-
| <math>\frac{-\log 2}{\log\left(\displaystyle\frac{1-\gamma}{2}\right)}</math> || align="right" | 0<''D''<1 || 1D generalized symmetric Cantor set || align="center" |[[File:generalized cantor set.png|200px]] || Built by removing the central [[interval (mathematics)|interval]] of length <math>\gamma\,l_{n-1}</math> from each remaining interval of length <math>l_{n-1} = (1-\gamma)^{n-1}/2^{n-1}</math> at the ''n''th iteration. <math>\gamma=1/3</math> produces the usual middle-third [[Cantor set]]. Varying <math>\gamma</math> between 0 and 1 yields any fractal dimension <math>0<D<1</math>.<ref>{{cite journal |first1=A. Yu |last1=Cherny |first2=E.M. |last2=Anitas |first3=A.I. |last3=Kuklin |first4=M. |last4=Balasoiu |first5=V.A. |last5=Osipov |title=The scattering from generalized Cantor fractals |journal=J. Appl. Crystallogr. |volume=43 |issue= 4|pages=790–7 |year=2010 |doi=10.1107/S0021889810014184 |arxiv=0911.2497 |s2cid=94779870 }}</ref>
|-
| <math>\log_2 \varphi = \log_2(1+\sqrt{5}) - 1</math> || align="right" | 0.6942 || (1/4, 1/2) asymmetric [[Cantor set]] || align="center" |[[File:AsymmCantor.png|200px]] || Built by removing the second quarter at each iteration.<ref>{{Cite journal|author=Tsang, K. Y. |title=Dimensionality of Strange Attractors Determined Analytically |journal=Phys. Rev. Lett. |volume=57|issue=12|pages=1390–1393 |year=1986|pmid=10033437 |doi=10.1103/PhysRevLett.57.1390|bibcode=1986PhRvL..57.1390T}}</ref>
<math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| <math>\log_{10} 5 = 1 - \log_{10} 2</math> || align="right" | 0.69897 || [[Real number]]s whose base 10 digits are [[parity (mathematics)|even]] || align="center" |[[File:Even digits.png|200px]] || Similar to the [[Cantor set]].<ref name="Falconer">{{Cite book
  | last = Falconer | first = Kenneth | author-link=Kenneth Falconer (mathematician)
  | title = Fractal Geometry: Mathematical Foundations and Applications
  | publisher = John Wiley & Sons, Ltd.
  | year = 1990–2003
  | isbn =  978-0-470-84862-3
  | no-pp = true
  | page = xxv}}</ref>
|-
| <math>\log(1+\sqrt{2})</math> || align="right" | 0.88137 || Spectrum of Fibonacci Hamiltonian|| align="center" | || The study of the spectrum of the Fibonacci Hamiltonian proves upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that the spectrum converges to an explicit constant.<ref>{{cite journal |last1=Damanik |first1=D. |last2=Embree |first2=M. |last3=Gorodetski |first3=A. |first4=S. |last4=Tcheremchantse |title=The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian |journal=Commun. Math. Phys. |volume=280 |issue=2 |pages=499–516 |year=2008 |doi=10.1007/s00220-008-0451-3 |arxiv=0705.0338|bibcode=2008CMaPh.280..499D |s2cid=12245755 }}</ref>{{page needed|date=October 2018}}
|-
| <math>1</math> || align="right" | 1 || [[Smith–Volterra–Cantor set]] || align="center" |[[File:Smith-Volterra-Cantor set.svg|200px]] || Built by removing the central interval of length <math>2^{-2n}</math> from each remaining interval at the ''n''th iteration. Nowhere dense but has a [[Lebesgue measure]] of 1/2.
|-
| <math>2 + \log_2 \frac{1}{2} = 1</math> || align="right" | 1 || [[Takagi curve|Takagi or Blancmange curve]] || align="center" |[[File:Takagi curve.png|150px]] || Defined on the [[unit interval]] by <math>f(x) = \sum\nolimits_{n=0}^\infty 2^{-n}s(2^{n}x)</math>, where <math>s(x)</math> is the [[Triangle wave|triangle wave function]]. Not a fractal under Mandelbrot's definition, because its topological dimension is also <math>1</math>.<ref>{{Cite book| last = Vaz| first = Cristina | title = Noções Elementares Sobre Dimensão | isbn =  9788565054867 | year = 2019}}</ref> Special case of the Takahi-Landsberg curve: <math>f(x) = \sum\nolimits_{n=0}^\infty w^n s(2^n x)</math> with <math>w = 1/2</math>. The Hausdorff dimension equals <math>2 + \log_2 w</math> for <math>w</math> in <math>\left[1/2,1\right]</math>. (Hunt cited by Mandelbrot<ref>{{Cite book| last = Mandelbrot| first = Benoit | title = Gaussian self-affinity and Fractals | isbn =  978-0-387-98993-8 | year = 2002 | publisher = Springer }}</ref>).
|-
| Calculated|| align="right" | 1.0812 || [[Julia set]] z<sup>2</sup> + 1/4 || align="center" |[[File:Julia z2+0,25.png|100px]] || Julia set of ''f''(''z'') = ''z''<sup>2</sup> + 1/4.<ref name="McMullen"/>
|-
| Solution <math>s</math> of <math>2|\alpha|^{3s} + |\alpha|^{4s} = 1</math>|| align="right" | 1.0933 || Boundary of the [[Rauzy fractal]]|| align="center" |[[File:Rauzy fractal.png|150px]] || Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: <math>1\mapsto12</math>, <math>2\mapsto13</math> and <math>3\mapsto1</math>.<ref>Messaoudi, Ali. [http://matwbn.icm.edu.pl/ksiazki/aa/aa95/aa9531.pdf Frontième de numération complexe]", ''matwbn.icm.edu.pl''. {{in lang|fr}} Accessed: 27 October 2018.</ref>{{page needed|date=October 2018}}<ref>{{Citation | last1=Lothaire | first1=M. | author-link=M. Lothaire | title=Applied combinatorics on words | url=https://archive.org/details/appliedcombinato0000loth/page/525 | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-84802-2 | mr=2165687 | zbl=1133.68067 | year=2005 | volume=105 | page=[https://archive.org/details/appliedcombinato0000loth/page/525 525] }}</ref> <math>\alpha</math> is one of the conjugated roots of <math>z^3-z^2-z-1=0</math>.
|-
| <math>2\log_7 3</math> || align="right" | 1.12915 || contour of the [[Gosper island]] || align="center" |[[File:Gosper Island 4.svg|100px]] || Term used by Mandelbrot (1977).<ref>{{MathWorld |id=GosperIsland |title=Gosper Island |access-date=27 October 2018}}</ref> The Gosper island is the limit of the [[Gosper curve]].
|-
| Measured (box counting) || align="right" | 1.2 || Dendrite [[Julia set]] || align="center" |[[File:Dendrite julia.png|150px]] || Julia set of ''f''(''z'') = ''z''<sup>2</sup> + i.
|-
| <math>\frac{3\log \varphi}{\log\left(\displaystyle\frac{3+\sqrt{13}}{2}\right)}</math> || align="right" | 1.2083 || [[Fibonacci word fractal|Fibonacci word fractal 60°]] || align="center" | [[File:Fibo 60deg F18.png|200px]] || Build from the [[Fibonacci word]]. See also the standard Fibonacci word fractal.
<math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| |<math>\begin{align} & 2\log_2\left(\displaystyle\frac{\sqrt[3]{27-3\sqrt{78}}+\sqrt[3]{27+3\sqrt{78}}}{3}\right),\\ &\text{or root of }2^x-1=2^{(2-x)/2}\end{align}</math>|| align="right" | 1.2108 || Boundary of the tame twindragon || align="center" |[[File:TameTwindragontile.png|150px]] || One of the six 2-[[rep-tile]]s in the plane (can be tiled by two copies of itself, of equal size).<ref name="2-reptiles">Ngai, Sirvent, Veerman, and Wang (October 2000). "[https://doi.org/10.1023%2FA%3A1005206301454 On 2-Reptiles in the Plane 1999]", ''Geometriae Dedicata'', Volume 82. Accessed: 29 October 2018.</ref><ref name="Boundary">Duda, Jarek (March 2011). "[http://demonstrations.wolfram.com/TheBoundaryOfPeriodicIteratedFunctionSystems/ The Boundary of Periodic Iterated Function Systems]", ''Wolfram.com''.</ref>
|-
| || align="right" | 1.26 || [[Hénon map]] || align="center" |[[File:Henon.jpg|100px]] || The canonical [[Hénon map]] (with parameters ''a''&nbsp;=&nbsp;1.4 and ''b''&nbsp;=&nbsp;0.3) has Hausdorff dimension 1.261&nbsp;±&nbsp;0.003. Different parameters yield different dimension values.
|-
| <math>\log_3 4</math> || align="right" | 1.2619 || [[Triflake]] || align="center" | [[File:Triflake.png|150px]] || Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
|-
| <math>\log_3 4</math> || align="right" | 1.2619 || [[Koch curve]] || align="center" | [[File:Koch curve.svg|200px]] || 3 Koch curves form the Koch snowflake or the anti-snowflake.
|-
| <math>\log_3 4</math> || align="right" | 1.2619 || boundary of [[Dragon curve|Terdragon curve]] || align="center" |[[File:Terdragon boundary.png|150px]] || L-system: same as dragon curve with angle&nbsp;=&nbsp;30°. The Fudgeflake is based on 3 initial segments placed in a triangle.
|-
| <math>\log_3 4</math> || align="right" | 1.2619 || 2D [[Cantor dust]] || align="center" |[[File:Carre cantor.gif|100px]] || Cantor set in 2 dimensions.
|-
| <math>\log_3 4</math> || align="right" | 1.2619 || 2D [[L-system]] branch || align="center" |[[File:Onetwosix.png|200px]] || L-Systems branching pattern having 4 new pieces scaled by 1/3. Generating the pattern using statistical instead of exact self-similarity yields the same fractal dimension.
|-
| Calculated|| align="right" | 1.2683 || [[Julia set]] z<sup>2</sup>&nbsp;&minus;&nbsp;1 || align="center" |[[File:Julia z2-1.png|200px]] || Julia set of ''f''(''z'') = ''z''<sup>2</sup> − 1.<ref name="McMullen"/>
|-
| || align="right" | 1.3057 || [[Apollonian gasket]] || align="center" |[[File:Apollonian gasket.svg|120px]] || Starting with 3 tangent circles, repeatedly packing new circles into the complementary interstices. Also the limit set generated by reflections in 4 mutually tangent circles. See <ref name="McMullen">McMullen, Curtis T. (3 October 1997). "[http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf Hausdorff dimension and conformal dynamics III: Computation of dimension]", ''Abel.Math.Harvard.edu''. Accessed: 27 October 2018.</ref>
|-
| || align="right" | 1.328 || 5 [[Circle inversion|circles inversion]] fractal || align="center" |[[File:Cicle inversion.svg|100px]] || The limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an Apollonian packing. See <ref>Chang, Angel and Zhang, Tianrong. {{Cite web |url=http://classes.yale.edu/fractals/CircInvFrac/CircDim/CircDim2.html |title=On the Fractal Structure of the Boundary of Dragon Curve |access-date=9 February 2019 |archive-url=https://web.archive.org/web/20110614063904/http://classes.yale.edu/Fractals/CircInvFrac/CircDim/CircDim2.html |archive-date=14 June 2011 |url-status=bot: unknown }} [https://stanford.edu/~angelx/pubs/dragonbound.pdf pdf]</ref>
|-
| <math>\log_5 9</math>|| align="right" | 1.36521<ref>Mandelbrot, B. B. (1983). ''The Fractal Geometry of Nature'', p.48. New York: W. H. Freeman. {{ISBN|9780716711865}}. Cited in: {{MathWorld |id=MinkowskiSausage |title=Minkowski Sausage |access-date=22 September 2019}}</ref> || [[Koch island|Quadratic von Koch island]] using the type 1 curve as generator|| align="center" |[[File:Karperienflakeani2.gif|150px]] || Also known as the [[Minkowski Sausage]]
|-
| Calculated|| align="right" | 1.3934 || [[Douady rabbit]] || align="center" |[[File:Douady rabbit.png|150px]] || Julia set of ''f''(''z'') = -0.123 + 0.745i<ref name="McMullen"/>
|-
| <math>\log_3 5</math>|| align="right" | 1.4649 || [[Vicsek fractal]] || align="center" |[[File:Box fractal.svg|100px]] || Built by exchanging iteratively each square by a cross of 5 squares.
|-
| <math>\log_3 5</math>|| align="right" | 1.4649 || [[Koch curve|Quadratic von Koch curve (type 1)]]|| align="center" |[[File:Quadratic Koch 2.svg|150px]] || One can recognize the pattern of the Vicsek fractal (above).
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| <math>\log_{\sqrt{5}} \frac{10}{3}</math>|| align="right" | 1.4961 ||Quadric cross || align="center" |{{anchor|cross}}[[File:Quadriccross.gif|150px]] ||[[File:Q Cross Fractal Generator.jpg|thumb|The quadric cross is made by scaling the 3-segment generator unit by 5<sup>1/2</sup> then adding 3 full scaled units, one to each original segment, plus a third of a scaled unit (blue) to increase the length of the pedestal of the starting 3-segment unit (purple).]]Built by replacing each end segment with a cross segment scaled by a factor of 5<sup>1/2</sup>, consisting of 3 1/3 new segments, as illustrated in the inset.

''Images generated with Fractal Generator for ImageJ.''  
|-
|<math>2 - \log_2 \sqrt{2} = \frac{3}{2}</math>|| align="right" | 1.5000 || a [[Weierstrass function]]: <math>f(x) = \sum_{k=1}^\infty \frac{\sin(2^k x)}{\sqrt{2}^{\,k}}</math> || align="center" |[[File:Weierstrass functionAMD.png|150px]] || The Hausdorff dimension of the [[graph of a function|graph]] of the Weierstrass function <math>f:[0,1] \to \mathbb{R}</math> defined by <math>f(x) = \sum\nolimits_{k=1}^\infty a^k\sin(b^k x)</math> with <math>1/b < a < 1</math> and <math>b>1</math> is <math>2 + \log_b a</math>.<ref>{{Cite journal|last=Shen|first=Weixiao|date=2018|title=Hausdorff dimension of the graphs of the classical Weierstrass functions|journal=Mathematische Zeitschrift|language=en|volume=289|issue=1–2|pages=223–266|doi=10.1007/s00209-017-1949-1|issn=0025-5874|arxiv=1505.03986|s2cid=118844077}}</ref><ref>N. Zhang. The Hausdorff dimension of the graphs of fractal functions. (In Chinese). Master Thesis. Zhejiang University, 2018.</ref>
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|<math>\log_4 8 = \frac{3}{2}</math>|| align="right" | 1.5000 || [[Minkowski Sausage|Quadratic von Koch curve (type 2)]] || align="center" |[[File:Quadratic Koch.svg|150px]] || Also called "Minkowski sausage".
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|<math>\log_2\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)</math> || align="right" | 1.5236 || Boundary of the [[Dragon curve]] || align="center" | [[File:Boundary dragon curve.png|150px]]|| cf. Chang & Zhang.<ref>[http://poignance.coiraweb.com/math/Fractals/Dragon/Bound.html Fractal dimension of the boundary of the dragon fractal]</ref><ref name="Boundary"/>
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|<math>\log_2\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)</math> || align="right" | 1.5236 || Boundary of the [[Dragon curve|twindragon curve]]|| align="center" |[[File:Twindragontile.png|150px]] || Can be built with two dragon curves. One of the six 2-[[rep-tile]]s in the plane (can be tiled by two copies of itself, of equal size).<ref name="2-reptiles"/>
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| <math>\log_2 3</math> || align="right" | 1.5850 || 3-branches tree || align="center" | [[File:Arbre 3 branches.png|110px]] [[File:Arbre 3 branches2.png|110px]] || Each branch carries 3 branches (here 90° and 60°). The fractal dimension of the entire tree is the fractal dimension of the terminal branches. NB: the 2-branches tree has a fractal dimension of only 1.
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| <math>\log_2 3</math> || align="right" | 1.5850 || [[Sierpinski triangle]] || align="center" | [[File:Sierpinski8.svg|100px]] || Also the limiting shape of [[Pascal's triangle]] [[modular arithmetic|modulo]] 2.
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| <math>\log_2 3</math> || align="right" | 1.5850 || [[Sierpiński arrowhead curve]] || align="center" | [[File:PfeilspitzenFraktal.PNG|100px]] || Same limit as the triangle (above) but built with a one-dimensional curve.
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| <math>\log_2 3</math> || align="right" | 1.5850 || Boundary of the [[T-square (fractal)|T-square]] fractal || align="center" | [[File:T-Square fractal (evolution).png|200px]] || The dimension of the fractal itself (not the boundary) is <math>\log_2 4 = 2</math>
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| <math>\log_{\sqrt[\varphi]{\varphi}}(\varphi) = \varphi</math> || align="right" | 1.61803 || a golden [[dragon curve|dragon]] || align="center" | [[File:Phi glito.png|150px]] || Built from two similarities of ratios <math>r</math> and <math>r^2</math>, with <math>r = 1 / \varphi^{1/\varphi}</math>. Its dimension equals <math>\varphi</math> because <math>({r^2})^\varphi + r^\varphi = 1</math>.
<math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
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| <math>1 + \log_3 2</math> || align="right" | 1.6309 || [[Pascal triangle]] modulo 3 || align="center" | [[File:Pascal triangle modulo 3.png|160px]] || For a triangle modulo ''k'', if ''k'' is [[prime number|prime]], the fractal dimension is <math>1 + \log_k\left(\frac{k+1}{2}\right)</math> (cf. [[Stephen Wolfram]]<ref name="stephenwolfram.com">{{Cite web |url=http://www.stephenwolfram.com/publications/articles/ca/84-geometry/1/text.html |title=Fractal dimension of the Pascal triangle modulo k |access-date=2 October 2006 |archive-date=15 October 2012 |archive-url=https://web.archive.org/web/20121015154035/http://www.stephenwolfram.com/publications/articles/ca/84-geometry/1/text.html |url-status=dead }}</ref>).
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| <math>1 + \log_3 2</math> || align="right" | 1.6309 || [[N-flake#Hexaflake|Sierpinski Hexagon]] || align="center" | [[File:Sierpinski hexagon 4th Iteration.svg|150px]] || Built in the manner of the [[Sierpinski carpet]], on an hexagonal grid, with 6 similitudes of ratio 1/3. The [[Koch snowflake]] is present at all scales.
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| <math>\frac{3\log \varphi}{\log(1+\sqrt{2})}</math> || align="right" | 1.6379 || [[Fibonacci word fractal]] || align="center" | [[File:Fibonacci fractal F23 steps.png|150px]] || Fractal based on the [[Fibonacci word]] (or Rabbit sequence) Sloane A005614. Illustration : [[Fractal curve]] after 23 steps (''F''<sub>23</sub>&nbsp;=&nbsp;28657 segments).<ref name="AMD">[http://hal.archives-ouvertes.fr/hal-00367972/en/ The Fibonacci word fractal]</ref>
<math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
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| Solution of <math>(1/3)^s + (1/2)^s + (2/3)^s = 1</math> || align="right" | 1.6402 || Attractor of [[Iterated function system|IFS]] with 3 [[Similarity (geometry)|similarities]] of ratios 1/3, 1/2 and 2/3 || align="center" | [[File:IFS3sim3ratios.png|200px]] || Generalization : Providing the [[open set condition]] holds, the attractor of an [[iterated function system]] consisting of <math>n</math> similarities of ratios <math>c_n</math>, has Hausdorff dimension <math>s</math>, solution of the equation coinciding with the iteration function of the Euclidean contraction factor: <math>\sum\nolimits_{k=1}^n c_k^s = 1</math>.<ref name="Falconer"/>
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|<math>\log_8 32 = \frac{5}{3}</math> || align="right" | 1.6667 || 32-segment quadric fractal (1/8 scaling rule)
||[[File:8 scale fractal.png|frameless]] see also: [[:File:32 Segment One Eighth Scale Quadric Fractal.jpg]]
||[[File:32SegmentSmall.jpg|thumb|Generator for 32 segment 1/8 scale quadric fractal.]]Built by scaling the 32 segment generator (see inset) by 1/8 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 32/log 8 = 1.6667. ''Images generated with Fractal Generator for ImageJ.''
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| <math>1 + \log_5 3</math> || align="right" | 1.6826 || [[Pascal triangle]] modulo 5 || align="center" | [[File:Pascal triangle modulo 5.png|160px]] || For a triangle modulo ''k'', if ''k'' is prime, the fractal dimension is <math>1 + \log_k\left(\frac{k+1}{2}\right)</math> (cf. [[Stephen Wolfram]]<ref name="stephenwolfram.com"/>).
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| Measured (box-counting) || align="right" | 1.7 || [[Ikeda map]] attractor || align="center" | [[File:Ikeda map a=1 b=0.9 k=0.4 p=6.jpg|100px]] || For parameters ''a''=1, ''b''=0.9, ''k''=0.4 and ''p''=6 in the Ikeda map <math>z_{n+1} = a + bz_n \exp\left[i\left[k - p/\left(1 + \lfloor z_n \rfloor^2\right)\right]\right]</math>. It derives from a model of the plane-wave interactivity field in an optical ring laser. Different parameters yield different values.<ref>{{cite journal |first=James |last=Theiler |title=Estimating fractal dimension |journal=J. Opt. Soc. Am. A |volume=7 |issue=6 |pages=1055–73 |year=1990 |doi=10.1364/JOSAA.7.001055 |bibcode=1990JOSAA...7.1055T |url=http://public.lanl.gov/jt/Papers/est-fractal-dim.pdf }}</ref>
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| <math>1 + \log_{10} 5</math> || align="right" | 1.6990 || 50 segment quadric fractal (1/10 scaling rule) || align="center" | [[File:50seg.tif|150px]] || Built by scaling the 50 segment generator (see inset) by 1/10 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator.  The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 50/log 10 = 1.6990. ''Images generated with Fractal Generator for ImageJ''<ref>[http://rsb.info.nih.gov/ij/plugins/fractal-generator.html Fractal Generator for ImageJ] {{webarchive|url=https://web.archive.org/web/20120320124725/http://rsb.info.nih.gov/ij/plugins/fractal-generator.html |date=20 March 2012 }}.</ref>''.''[[File:50SegmentSmall.jpg|thumb|Generator for 50 Segment Fractal.]]
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| <math>4\log_5 2</math> || align="right" | 1.7227 || [[Pinwheel tiling|Pinwheel fractal]] || align="center" | [[File:Pinwheel fractal.png|150px]] || Built with Conway's Pinwheel tile.
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| <math>\log_3 7</math> || align="right" | 1.7712 || [[Sphinx tiling|Sphinx fractal]] || align="center" | [[File:Sphinx rep-tile fractal.gif|150px]] || Built with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes.<ref>W. Trump, G. Huber, C. Knecht, R. Ziff, to be published</ref>
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| <math>\log_3 7</math> || align="right" | 1.7712 || [[Hexaflake]] || align="center" | [[File:HexaFlake_5th_Iteration_Center.svg|100px]] || Built by exchanging iteratively each hexagon by a flake of 7 hexagons. Its boundary is the von Koch flake and contains an infinity of Koch snowflakes (black or white).
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| <math>\log_3 7</math> || align="right" | 1.7712 || Fractal H-I de Rivera || align="center" | [[File:Fractal H-I de Rivera.jpg|100px]] || Starting from a unit square dividing its dimensions into three equal parts to form nine self-similar squares with the first square, two middle squares (the one that is above and the one below the central square) are removed in each of the seven squares not eliminated the process is repeated, so it continues indefinitely.
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| <math>\frac{\log 4}{\log(2 + 2\cos(85^\circ))}</math> || align="right" | 1.7848 || [[Koch curve|Von Koch curve 85°]] || align="center" | [[File:Koch Curve 85degrees.png|150px]] || Generalizing the von Koch curve with an angle ''a'' chosen between 0 and 90°. The fractal dimension is then <math>\frac{\log 4}{\log(2 + 2\cos a)} \in [1,2]</math>.
|-
| <math>\log_2\left(3^{0.63} + 2^{0.63}\right)</math> || align="right" | 1.8272 || A self-[[affine transformation|affine]] fractal set || align="center" | [[File:Self-affine set.png|200px]] || Build iteratively from a ''p''-by-''q'' array on a square, with <math>p \le q</math>. Its Hausdorff dimension equals <math>\log_p\left(\sum\nolimits_{k=1}^p n_k^a\right)</math><ref name="Falconer"/> with <math>a = \log_q p</math> and <math>n_k</math> is the number of elements in the <math>k</math><sup>th</sup> column. The [[Minkowski–Bouligand dimension|box-counting dimension]] yields a different formula, therefore, a different value. Unlike self-similar sets, the Hausdorff dimension of self-affine sets depends on the position of the iterated elements and there is no formula, so far, for the general case.
|-
| <math>\frac{\log 6}{\log(1+\varphi)}</math> || align="right" | 1.8617 || [[N-flake#Pentaflake|Pentaflake]] || align="center" | [[File:Pentaflake-C 3rd Iteration Blue.svg|100px]] || Built by exchanging iteratively each pentagon by a flake of 6 pentagons.
<math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| solution of <math>6(1/3)^s + 5{(1/3\sqrt{3})}^s = 1</math> || align="right" | 1.8687 || Monkeys tree || align="center" | [[File:Monkeytree.svg|100px]] || This curve appeared in [[Benoit Mandelbrot]]'s "Fractal geometry of Nature" (1983). It is based on 6 similarities of ratio <math>1/3</math> and 5 similarities of ratio <math>1/3\sqrt{3}</math>.<ref>[http://www.coaauw.org/boulder-eyh/eyh_fractal.html Monkeys tree fractal curve] {{webarchive|url=https://archive.today/20020921135308/http://www.coaauw.org/boulder-eyh/eyh_fractal.html |date=21 September 2002 }}</ref>
|-
| <math>\log_3 8</math> || align="right" | 1.8928 || [[Sierpinski carpet]] || align="center" | [[File:Sierpinski carpet 6.png|100px]] || Each face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1).
|-
| <math>\log_3 8</math> || align="right" | 1.8928 || 3D [[Cantor dust]] || align="center" | [[File:Cantor3D3.png|200px]]|| Cantor set in 3 dimensions.
|-
| <math>\log_3 4 + \log_3 2 = \frac{\log 4}{\log 3}+\frac{\log 2}{\log 3} = \frac{\log 8}{\log 3}</math> || align="right" | {{formatnum:1.8928}} || Cartesian product of the [[von Koch curve]] and the [[Cantor set]] || align="center" | [[File:Koch Cantor cartesian product.png|150px]]|| Generalization : Let ''F''×''G'' be the cartesian product of two fractals sets ''F'' and ''G''. Then <math>\dim_H(F \times G) = \dim_H F + \dim_H G</math>.<ref name="Falconer"/> See also the 2D [[Cantor dust]] and the [[Cantor cube]].
|-
|<math>2\log_2 x</math> where <math>x^9-3x^8+3x^7-3x^6+2x^5+4x^4-8x^3+</math><math>8x^2-16x+8 = 0</math>|| align="right" | 1.9340 || Boundary of the [[Lévy C curve]] || align="center" | [[File:LevyFractal.png|100px]] || Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
|-
| || align="right" | 2 || [[Penrose tiling]] || align="center" |[[File:pen0305c.gif|100px]] || See Ramachandrarao, Sinha & Sanyal.<ref>[http://www.iisc.ernet.in/currsci/aug102000/rc80.pdf Fractal dimension of a Penrose tiling]</ref>
|-
| <math>2</math> || align="right" | 2 || Boundary of the [[Mandelbrot set]] || align="center" | [[File:Boundary mandelbrot set.png|100px]] || The boundary and the set itself have the same Hausdorff dimension.<ref name=Shishikura91>{{cite arXiv |first=Mitsuhiro |last=Shishikura |title=The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets |date=1991 |eprint=math/9201282}}</ref>
|-
| <math>2</math> || align="right" | 2 || [[Julia set]] || align="center" | [[File:Juliadim2.png|150px]] || For determined values of ''c'' (including ''c'' [[Misiurewicz point|belonging to the boundary]] of the Mandelbrot set), the Julia set has a dimension of 2.<ref name=Shishikura91/>
|-
| <math>2</math> || align="right" | 2 || [[Sierpiński curve]] || align="center" | [[File:Sierpinski-Curve-3.png|100px]] || Every [[space-filling curve]] filling the plane has a Hausdorff dimension of 2.
|-
| <math>2</math> || align="right" | 2 || [[Hilbert curve]] || align="center" | [[File:Hilbert curve 3.svg|100px]]||
|-
| <math>2</math> || align="right" | 2 || [[Peano curve]] || align="center" | [[File:Peano curve.png|100px]]|| And a family of curves built in a similar way, such as the [[Wunderlich curves]].
|-
| <math>2</math> || align="right" | 2 || [[Moore curve]] || align="center" | [[File:Moore-curve-stages-1-through-4.svg|100px]]|| Can be extended in 3 dimensions.
|-
| || align="right" | 2 || [[z-order (curve)|Lebesgue curve or z-order curve]] || align="center" | [[File:z-order curve.png|100px]]|| Unlike the previous ones this space-filling curve is differentiable almost everywhere. Another type can be defined in 2D. Like the Hilbert Curve it can be extended in 3D.<ref>[http://www.mathcurve.com/fractals/lebesgue/lebesgue.shtml Lebesgue curve variants]</ref>
|-
| <math>\log_{\sqrt{2}} 2 = 2</math> || align="right" | 2 || [[Dragon curve]] || align="center" | [[File:Courbe du dragon.png|150px]]|| And its boundary has a fractal dimension of 1.5236270862.<ref>{{cite arXiv |first=Jarek |last=Duda |title=Complex base numeral systems |date=2008 |class=math.DS |eprint=0712.1309v3}}</ref>
|-
| || align="right" | 2 || [[Dragon curve|Terdragon curve]] || align="center" | [[File:Terdragon curve.png|150px]]|| L-system: ''F''&nbsp;→&nbsp;''F''&nbsp;+&nbsp;F&nbsp;&ndash;&nbsp;F, angle&nbsp;=&nbsp;120°.
|-
| <math>\log_2 4 = 2</math> || align="right" | 2 || [[Gosper curve]] || align="center" | [[File:Gosper curve 3.svg|100px]]|| Its boundary is the Gosper island.
|-
| Solution of <math>7({1/3})^s+6({1/3\sqrt{3}})^s=1</math> || align="right" | 2 || Curve filling the [[Koch snowflake]] || align="center" | [[File:Mandeltree.svg|100px]]|| Proposed by Mandelbrot in 1982,<ref>{{cite book |title=Penser les mathématiques |last=Seuil |isbn=2-02-006061-2 |year=1982|publisher=Seuil }}</ref> it fills the [[Koch snowflake]]. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio <math>1/3\sqrt{3}</math>.
|-
| <math>\log_2 4 = 2</math> || align="right" | 2 || [[Sierpiński triangle|Sierpiński tetrahedron]] || align="center" | [[File:Tetraedre Sierpinski.png|80px]]|| Each [[tetrahedron]] is replaced by 4 tetrahedra.
|-
| <math>\log_2 4 = 2</math> || align="right" | 2 || [[H-fractal]] || align="center" |[[File:H fractal2.png|150px]]|| Also the [[Mandelbrot tree]] which has a similar pattern.
|-
| <math>\frac{\log 2}{\log(2/\sqrt{2})} = 2</math> || align="right" | {{formatnum:2}} || [[Pythagoras tree (fractal)]] || align="center" |[[File:PythagorasTree.png|150px]]|| Every square generates two squares with a reduction ratio of <math>1/\sqrt{2}</math>.
|-
| <math>\log_2 4 = 2</math> || align="right" | 2 || [[Iterated function system#Example: Greek cross fractal|2D Greek cross fractal]] || align="center" |[[File:Greek cross fractal stage 4.svg|100px]] || Each segment is replaced by a cross formed by 4 segments.
|-
| Measured || align="right" | 2.01 ± 0.01|| [[Rössler attractor]] || align="center" | [[File:Roessler attractor.png|100px]] || The fractal dimension of the Rössler attractor is slightly above 2. For ''a''=0.1, ''b''=0.1 and ''c''=14 it has been estimated between 2.01 and 2.02.<ref>[http://www.ocf.berkeley.edu/~trose/rossler.html Fractals and the Rössler attractor]</ref>
|-
| Measured || align="right" | 2.06 ± 0.01|| [[Lorenz attractor]] || align="center" |[[File:Lorenz attractor.png|100px]] || For parameters <math>\rho=40</math>, <math>\sigma=16</math> and <math>\beta=4</math> . See McGuinness (1983)<ref>{{cite journal |first=M.J. |last=McGuinness |title=The fractal dimension of the Lorenz attractor |journal=Physics Letters |volume=99A |pages=5–9 |year=1983 |issue=1 |doi=10.1016/0375-9601(83)90052-X |bibcode=1983PhLA...99....5M }}</ref>
|-
|<math>4 + c^D + d^D = (c+d)^D</math>|| align="right" |2<''D''<2.3 || Pyramid surface || align="center" |[[File:Pyramid surface fractal.png|200px]]|| Each triangle is replaced by 6 triangles, of which 4 identical triangles form a diamond based pyramid and the remaining two remain flat with lengths <math>c</math> and <math>d</math> relative to the pyramid triangles. The dimension is a parameter, self-intersection occurs for values greater than 2.3.<ref>{{Cite journal|last=Lowe|first=Thomas|date=24 October 2016|title=Three Variable Dimension Surfaces|url=https://www.researchgate.net/publication/309391846|journal=ResearchGate}}</ref>
|-
| <math>\log_2 5</math>|| align="right" | 2.3219 || Fractal pyramid || align="center" |[[File:Fractal pyramid.jpg|100px]]|| Each [[square pyramid]] is replaced by 5 half-size square pyramids. (Different from the Sierpinski tetrahedron, which replaces each [[triangular pyramid]] with 4 half-size triangular pyramids).
|-
| <math>\frac{\log 20}{\log(2+\varphi)}</math>|| align="right" | 2.3296 || [[N-flake#Dodecahedron flake|Dodecahedron fractal]]|| align="center" |[[File:Dodecaedron fractal.jpg|100px]]|| Each [[dodecahedron]] is replaced by 20 dodecahedra.
<math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| <math>\log_3 13</math> || align="right" | 2.3347 || [[Koch curve|3D quadratic Koch surface (type 1)]] || align="center" |[[Image:koch_quadratic_3d_fractal.svg|150px]]|| Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the first (blue block), second (plus green blocks), third (plus yellow blocks) and fourth (plus clear blocks) iterations.
|-
| || align="right" | 2.4739 || [[Apollonian sphere packing]] || align="center" |[[File:Apollonian spheres2.png|100px]] || The interstice left by the Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.<ref>[http://www.scivis.ethz.ch/publications/pdf/1994/borkovec1994fractal.pdf The Fractal dimension of the apollonian sphere packing] {{webarchive|url=https://web.archive.org/web/20160506190118/http://www.scivis.ethz.ch/publications/pdf/1994/borkovec1994fractal.pdf |date=6 May 2016 }}</ref>
|-
| <math>\log_4 32 = \frac{5}{2}</math> || align="right" | 2.50 || [[Koch curve|3D quadratic Koch surface (type 2)]] || align="center" |[[File:Quadratic Koch 3D (type2 stage2).png|150px]]|| Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
|-
| <math>\frac{\log\left(\frac{\sqrt{7}}{6}-\frac{1}{3}\right)}{\log(\sqrt2-1)}</math> || align="right" | 2.529 || [[Jerusalem cube]] || align="center" | [[File:Jerusalem Cube.jpg|150px]] || The iteration ''n'' is built with 8 cubes of iteration ''n''−1 (at the corners) and 12 cubes of iteration n-2 (linking the corners). The contraction ratio is <math>\sqrt{2}-1</math>. 
|-
| <math>\frac{\log 12}{\log(1+\varphi)}</math> || align="right" | 2.5819 || [[N-flake#Icosahedron flake|Icosahedron fractal]] || align="center" |[[File:Icosaedron fractal.jpg|100px]]|| Each [[icosahedron]] is replaced by 12 icosahedra.
<math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| <math>1 + \log_2 3</math> || align="right" | 2.5849 || [[Iterated function system#Example: Greek cross fractal|3D Greek cross fractal]] || align="center" |[[File:Greek cross 3D 1 through 4.png|200px]]|| Each segment is replaced by a cross formed by 6 segments.
|-
| <math>1 + \log_2 3</math> || align="right" | 2.5849 || [[N-flake#Octahedron flake|Octahedron fractal]] || align="center" |[[File:Octaedron fractal.jpg|100px]]|| Each [[octahedron]] is replaced by 6 octahedra.
|-
| <math>1 + \log_2 3</math> || align="right" | 2.5849 || [[Koch curve|von Koch surface]] || align="center" |[[File:Koch surface 3.png|150px]]|| Each equilateral triangular face is cut into 4 equal triangles.
Using the central triangle as the base, form a tetrahedron.  Replace the triangular base with the tetrahedral "tent". 
|-
| <math>\frac{\log 3}{\log(3/2)}</math> || align="right" | 2.7095 || [[Koch curve|Von Koch in 3D]] || align="center" | [[File:Koch Curve in Three Dimensions ("Delta" fractal).jpg|100px]] || Start with a 6-sided polyhedron whose faces are isosceles triangles with sides of ratio 2:2:3 . Replace each polyhedron with 3 copies of itself, 2/3 smaller.<ref>{{cite web |url=https://www.researchgate.net/publication/262600735 |last=Baird |first=Eric |date=2014 |title=The Koch curve in three dimensions |via=[[ResearchGate]]}}</ref>
|-
| <math>\log_3 20</math> || align="right" | 2.7268 || [[Menger sponge]] || align="center" | [[File:Menger.png|100px]] || And its surface has a fractal dimension of <math>\log_3 20</math>, which is the same as that by volume.
|-
| <math>\log_2 8 = 3</math> || align="right" | 3 || [[Hilbert curve|3D Hilbert curve]] || align="center" | [[File:Hilbert3d-step3.png|100px]]|| A Hilbert curve extended to 3 dimensions.
|-
| <math>\log_2 8 = 3</math> || align="right" | 3 || [[z-order (curve)|3D Lebesgue curve]] || align="center" | [[File:Lebesgue-3d-step3.png|100px]]|| A Lebesgue curve extended to 3 dimensions.
|-
| <math>\log_2 8 = 3</math> || align="right" | 3 || [[Moore curve|3D Moore curve]] || align="center" | [[File:Moore3d-step3.png|100px]]|| A Moore curve extended to 3 dimensions.
|-
| <math>\log_2 8 = 3</math> || align="right" | 3 || 3D [[H-fractal]] || align="center" | [[File:3D H-fractal.png|120px]]|| A H-fractal extended to 3 dimensions.<ref>{{cite journal |first1=B. |last1=Hou |first2=H. |last2=Xie |first3=W. |last3=Wen |first4=P. |last4=Sheng | year = 2008
 | title = Three-dimensional metallic fractals and their photonic crystal characteristics |journal= Phys. Rev. B |volume=77 |issue=12 |page=125113
 |bibcode=2008PhRvB..77l5113H |doi=10.1103/PhysRevB.77.125113 |url=http://repository.ust.hk/ir/bitstream/1783.1-25969/1/PhysRevB.77.125113.pdf }}</ref>
|-
| <math>3</math> ([[conjecture]]d) || align="right" | {{formatnum:3}} (to be confirmed) || [[Mandelbulb]] || align="center" |[[File:Mandelbulb 1,024GP Overview 20211110 002 ALT.png|100px]]|| Extension of the Mandelbrot set (power 9) in 3 dimensions<ref>[http://www.fractalforums.com/theory/hausdorff-dimension-of-the-mandelbulb/15/ Hausdorff dimension of the Mandelbulb]</ref>{{Unreliable source?|date=September 2011}}
|}

==Random and natural fractals==
{| class="wikitable sortable"
|-
! Hausdorff dimension<br />(exact value) || Hausdorff dimension<br />(approx.) || Name || Illustration || width="40%" | Remarks
|-
|<math>\frac{1}{2}</math> || align="right" | 0.5 || Zeros of a [[Wiener process]] || align="center" |[[File:Wiener process set of zeros.gif|150px]] || The zeros of a Wiener process (Brownian motion) are a [[nowhere dense set]] of [[Lebesgue measure]] 0 with a fractal structure.<ref name="Falconer"/><ref>Peter Mörters, Yuval Peres, "Brownian Motion", Cambridge University Press, 2010</ref>
|-
| Solution of <math>E(C_1^s + C_2^s)=1</math> where <math>E(C_1)=0.5</math> and <math>E(C_2)=0.3</math>|| align="right" | 0.7499 || a random [[Cantor set]] with 50% - 30% || align="center" |[[File:Random Cantor set.png|150px]] || Generalization: at each iteration, the length of the left interval is defined with a random variable <math>C_1</math>, a variable percentage of the length of the original interval. Same for the right interval, with a random variable <math>C_2</math>. Its Hausdorff Dimension <math>s</math> satisfies: <math>E(C_1^s + C_2^s)=1</math> (where <math>E(X)</math> is the [[expected value]] of <math>X</math>).<ref name="Falconer"/>
|-
|Solution of <math>s+1 = 12\cdot2^{-(s+1)}-6\cdot3^{-(s+1)}</math>||align="right"|1.144...||[[von Koch curve]] with random interval||align="center"| [[File:Random interval koch.png|200px]] || The length of the middle interval is a random variable with uniform distribution on the interval (0,1/3).<ref name="Falconer"/>
|-
|Measured||align="right"|1.22 ± 0.02||Coastline of [[Ireland]]||align="center"| [[File:Ireland (MODIS).jpg|150px]] || Values for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault<ref>{{cite journal|last1=McCartney|first1=Mark|first2=Gavin |last2=Abernethya |first3=Lisa |last3=Gaulta|title=The Divider Dimension of the Irish Coast|journal=Irish Geography|date=24 June 2010|volume=43|issue=3|pages=277–284|doi=10.1080/00750778.2011.582632}}</ref> at the [[University of Ulster]] and [[Theoretical Physics]] students at [[Trinity College, Dublin]], under the supervision of S. Hutzler.<ref name="S.Hutzler">{{cite journal|last1=Hutzler |first1=S. |title=Fractal Ireland |journal=Science Spin |date=2013 |volume=58 |pages=19–20 |url=https://issuu.com/spin35/docs/spin_58_all |access-date=15 November 2016 }}
(See [https://web.archive.org/web/20130726164417/http://www.sciencespin.com/magazine/archive/2013/05/ contents page], archived 26 July 2013)</ref>

Note that there are marked differences between Ireland's ragged west coast (fractal dimension of about 1.26) and the much smoother east coast (fractal dimension 1.10)<ref name="S.Hutzler" />
|-
|Measured||align="right"|1.25||[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|Coastline of Great Britain]]||align="center"| [[File:Britain-fractal-coastline-combined.jpg|200px]] || Fractal dimension of the west coast of Great Britain, as measured by [[Lewis Fry Richardson]] and cited by [[Benoît Mandelbrot]].<ref>[http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf How long is the coast of Britain? Statistical self-similarity and fractional dimension], B. Mandelbrot</ref>
|-
| <math>\frac{\log 4}{\log 3}</math> || align="right" | 1.2619 || [[von Koch curve]] with random orientation || align="center" | [[File:Random orientation koch.png|200px]] || One introduces here an element of randomness which does not affect the dimension, by choosing, at each iteration, to place the equilateral triangle above or below the curve.<ref name="Falconer"/>
|-
|<math>\frac{4}{3}</math> || align="right" | 1.333 || Boundary of Brownian motion || align="center" |[[File:Front mouvt brownien.png|150px]] || (cf. Mandelbrot, [[Gregory Lawler|Lawler]], [[Oded Schramm|Schramm]], [[Wendelin Werner|Werner]]).<ref>{{cite journal |first1=Gregory F. |last1=Lawler |first2=Oded |last2=Schramm |first3=Wendelin |last3=Werner |title=The Dimension of the Planar Brownian Frontier is 4/3 |journal=Math. Res. Lett. |volume=8 |issue=4 |pages=401–411 |year=2001 |arxiv=math/0010165|bibcode=2000math.....10165L |doi=10.4310/MRL.2001.v8.n4.a1 |s2cid=5877745 }}</ref>
|-
|<math>\frac{4}{3}</math> || align="right" | 1.333 || [[Polymer]] in 2D || align="center" | || Similar to the Brownian motion in 2D with non-self-intersection.<ref name="sapoval">{{cite book |first=Bernard |last=Sapoval |title=Universalités et fractales |publisher=Flammarion-Champs |year=2001 |isbn=2-08-081466-4 }}</ref>
|-
|<math>\frac{4}{3}</math> || align="right" | 1.333 || [[Percolation]] front in 2D, [[Corrosion]] front in 2D || align="center" | [[File:Front de percolation.png|150px]] || Fractal dimension of the percolation-by-invasion front (accessible perimeter), at the [[percolation threshold]] (59.3%). It's also the fractal dimension of a stopped corrosion front.<ref name="sapoval" />
|-
| || align="right" | 1.40 || [[diffusion-limited aggregation|Clusters of clusters 2D]] || align="center" | || When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.<ref name="sapoval" />
|-
| <math>2-\frac{1}{2}</math>|| align="right" | 1.5|| Graph of a regular [[Fractional Brownian motion|Brownian]] function ([[Wiener process]]) || align="center" | [[File:Wiener process zoom.png|150px]] || Graph of a function <math>f</math> such that, for any two positive reals <math>x</math> and <math>x+h</math>, the difference of their images <math>f(x+h)-f(x)</math> has the centered gaussian distribution with variance <math>h</math>. Generalization: the [[fractional Brownian motion]] of index <math>\alpha</math> follows the same definition but with a variance <math>h^{2\alpha}</math>, in that case its Hausdorff dimension equals <math>2-\alpha</math>.<ref name="Falconer" />
|-
| Measured|| align="right" | 1.52|| [[Fjord|Coastline of Norway]] || align="center" |[[File:Norway municipalities 2020 blank.svg|100px]] || See J. Feder.<ref>Feder, J., "Fractals", Plenum Press, New York, (1988).</ref>
|-
| Measured|| align="right" | 1.55 || [[Self-avoiding walk]] || align="center" | [[File:Polymer 2D.png|150px]]|| Random walk in a square lattice that avoids visiting the same place twice, with a "go-back" routine for avoiding dead ends.
|-
| <math>\frac{5}{3}</math>|| align="right" | 1.66|| Polymer in 3D || align="center" | || Similar to the Brownian motion in a cubic lattice, but without self-intersection.<ref name="sapoval" />
|-
| || align="right" | 1.70 || [[Diffusion-limited aggregation|2D DLA Cluster]] || align="center" | [[File:Aggregation limitee par diffusion.png|150px]]|| In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.<ref name="sapoval" />
|-
| <math>\frac{\log(9\cdot0.75)}{\log 3}</math>|| align="right" | 1.7381|| Fractal percolation with 75% probability|| align="center" |[[File:Fractal percolation 75.png|150px]] || The fractal percolation model is constructed by the progressive replacement of each square by a 3-by-3 grid in which is placed a random collection of sub-squares, each sub-square being retained with probability ''p''. The "almost sure" Hausdorff dimension equals <math>\frac{\log(9p)}{\log 3}</math>.<ref name="Falconer" />
|-
| <math>\frac{7}{4}</math> || align="right" | 1.75 || 2D percolation cluster hull || align="center" | [[File:PercolationHull.png|150px]]|| The hull or boundary of a percolation cluster.  Can also be generated by a hull-generating walk,<ref>[http://deepblue.lib.umich.edu/handle/2027.42/27787 Hull-generating walks]</ref> or by Schramm-Loewner Evolution.
|-
| <math>\frac{91}{48}</math> || align="right" | 1.8958 || [[2D percolation cluster]] || align="center" | [[File:Amas de percolation.png|150px]] || In a square lattice, under the site [[percolation threshold]] (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48.<ref name="sapoval" /><ref name="Sahimi">{{cite book|author1=M Sahini|author2=M Sahimi|title=Applications Of Percolation Theory|url=https://books.google.com/books?id=MJwqsbWBc-YC|year=2003|publisher=CRC Press|isbn=978-0-203-22153-2}}</ref> Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings".
|-
| <math>\frac{\log 2}{\log \sqrt{2}} = 2</math> || align="right" | 2 || [[Brownian motion]] || align="center" | [[File:Mouvt brownien2.png|150px]]|| Or random walk. The Hausdorff dimensions equals 2 in 2D, in 3D and in all greater dimensions (K.Falconer "The geometry of fractal sets").
|-
| Measured || align="right" | Around 2 || Distribution of [[galaxy cluster]]s || align="center" | [[File:Abell 1835 Hubble.jpg|100px]]|| From the 2005 results of the Sloan Digital Sky Survey.<ref>[https://arxiv.org/abs/astro-ph/0501583v2 Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey]</ref>
|-
| || align="right" | 2.5 || Balls of crumpled paper || align="center" | [[File:Paperball.png|100px]] || When crumpling sheets of different sizes but made of the same type of paper and with the same aspect ratio (for example, different sizes in the [[ISO 216]] A series), then the diameter of the balls so obtained elevated to a non-integer exponent between 2 and 3 will be approximately proportional to the area of the sheets from which the balls have been made.<ref>{{Cite journal | publisher=Yale | url=http://classes.yale.edu/fractals/FracAndDim/BoxDim/PowerLaw/CrumpledPaper.html | title=Power Law Relations | access-date=29 July 2010 | url-status=dead | archive-url=https://web.archive.org/web/20100628020140/http://classes.yale.edu/fractals/FracAndDim/BoxDim/PowerLaw/CrumpledPaper.html | archive-date=28 June 2010 | df=dmy-all }}</ref> Creases will form at all size scales (see [[Universality (dynamical systems)]]).
|-
| || align="right" | 2.50 || [[diffusion-limited aggregation|3D DLA Cluster]] || align="center" | [[File:3D DLA.jpg|150px]] || In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.<ref name="sapoval" />
|-
| || align="right" | 2.50 || [[Lichtenberg figure]] || align="center" | [[File:PlanePair2.jpg|100px]] || Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.<ref name="sapoval" />
|-
| <math>3-\frac{1}{2}</math>|| align="right" | 2.5|| regular [[Brownian surface]]|| align="center" | [[File:Brownian surface.png|150px]] || A function <math>f:\mathbb{R}^2 \to \mathbb{R}</math>, gives the height of a point <math>(x,y)</math> such that, for two given positive increments <math>h</math> and <math>k</math>, then <math>f(x+h,y+k)-f(x,y)</math> has a centered Gaussian distribution with variance <math>\sqrt{h^2 + k^2}</math>. Generalization: the [[fractional Brownian motion|fractional Brownian]] surface of index <math>\alpha</math> follows the same definition but with a variance <math>(h^2+k^2)^\alpha</math>, in that case its Hausdorff dimension equals <math>3-\alpha</math>.<ref name="Falconer" />
|-
| Measured || align="right" | 2.52 || 3D [[Percolation theory|percolation]] cluster || align="center" |[[File:3Dpercolation.png|225px]] || In a cubic lattice, at the site [[percolation threshold]] (31.1%), the 3D percolation-by-invasion cluster has a fractal dimension of around 2.52.<ref name="Sahimi"/> Beyond that threshold, the cluster is infinite.
|-
|Measured and calculated|| align="right" |~2.7|| The surface of [[Broccoli]] || align="center" | [[File:Broccoli DSC00862.png|100px]] ||San-Hoon Kim used a direct scanning method and a cross section analysis of a broccoli to conclude that the fractal dimension of it is ~2.7.<ref name=":0">{{Cite arXiv |eprint=cond-mat/0411597|title=Fractal dimensions of a green broccoli and a white cauliflower|last=Kim|first=Sang-Hoon|date=2 February 2008}}</ref>
|-
| Measured || align="right" | ~2.8 || Surface of [[Cerebral cortex|human brain]] || align="center" | [[File:Cerebellum NIH.png|100px]]|| Measured with segmented three-dimensional high-resolution magnetic resonance images<ref>{{cite journal |first1=Valerij G. |last1=Kiselev |first2=Klaus R. |last2=Hahn |first3=Dorothee P. |last3=Auer |title=Is the brain cortex a fractal? |journal=NeuroImage |volume=20 |issue=3 |pages=1765–1774 |year=2003 |doi=10.1016/S1053-8119(03)00380-X|pmid=14642486 |s2cid=14240006 }}</ref>
|-
|Measured and calculated|| align="right" |~2.8|| [[Cauliflower]] || align="center" | [[File:Blumenkohl-1.jpg|100px]]||San-Hoon Kim used a direct scanning method and a mathematical analysis of the cross section of a cauliflower to conclude that the fractal dimension of it is ~2.8.<ref name=":0" />
|-
| || align="right" | 2.97 || Lung surface || align="center" |[[File:Thorax Lung 3d (2).jpg|100px]] || The alveoli of a lung form a fractal surface close to 3.<ref name="sapoval" />
|-
| Calculated || align="right" | <math>\in (0,2)</math> || [[Multiplicative cascade]] || align="center" | [[File:3fractals2.jpg|150px]] || This is an example of a [[multifractal]] distribution. However, by choosing its parameters in a particular way we can force the distribution to become a monofractal.<ref>{{cite journal |first1=Paul |last1=Meakin |title=Diffusion-limited aggregation on multifractal lattices: A model for fluid-fluid displacement in porous media |journal=Physical Review A |volume=36 |issue=6 |pages=2833–2837 |year=1987 |doi=10.1103/PhysRevA.36.2833|pmid=9899187 |bibcode=1987PhRvA..36.2833M }}</ref>
|}

==See also==
{{Commons|Fractal|fractals}}
* [[Fractal dimension]]
* [[Hausdorff dimension]]
* [[Scale invariance]]

==Notes and references==
{{Reflist|30em}}

==Further reading==
*{{cite book |first=Benoît |last=Mandelbrot |title=The Fractal Geometry of Nature |publisher=W.H. Freeman |year=1982 |isbn=0-7167-1186-9 |url-access=registration |url=https://archive.org/details/fractalgeometryo00beno }}
*{{cite book |first=Heinz-Otto |last=Peitgen |editor-first=Dietmar |editor-last=Saupe |title=The Science of Fractal Images |publisher=Springer Verlag |year=1988 |isbn=0-387-96608-0 |url-access=registration |url=https://archive.org/details/scienceoffractal0000unse }}
*{{cite book |first=Michael F. |last=Barnsley |title=Fractals Everywhere |publisher=Morgan Kaufmann |isbn=0-12-079061-0 |date=1 January 1993 }}
*{{cite book |first1=Bernard |last1=Sapoval |first2=Benoît B. |last2=Mandelbrot |title=Universalités et fractales: jeux d'enfant ou délits d'initié? |publisher=Flammarion-Champs |year=2001 |isbn=2-08-081466-4 }}

==External links==
* [http://mathworld.wolfram.com/search/?query=fractal The fractals on Mathworld]
* [https://web.archive.org/web/20060905203033/http://local.wasp.uwa.edu.au/~pbourke/fractals/ Other fractals on Paul Bourke's website]
* [http://soler7.com/Fractals/FractalsSite.html Soler's Gallery]
* [http://www.mathcurve.com/fractals/fractals.shtml Fractals on mathcurve.com]
* [http://1000fractales.free.fr/index.htm 1000fractales.free.fr - Project gathering fractals created with various software]
* [https://web.archive.org/web/20060923100014/http://library.thinkquest.org/26242/full/index.html Fractals unleashed]
* [https://ifstile.com IFStile - software that computes the dimension of the boundary of self-affine tiles]

{{Fractals}}

{{DEFAULTSORT:Fractals By Hausdorff Dimension}}
[[Category:Fractals|Hausdorff Dimension]]
[[Category:Fractal curves|Hausdorff Dimension]]
[[Category:Mathematics-related lists]]