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| According to [[Kenneth Falconer (mathematician)|Falconer]], one of the essential features of a [[fractal]] is that its [[Hausdorff dimension]] strictly exceeds its [[topological dimension]].<ref name="Falconer">{{Cite book
| | Irwin Butts is what my spouse loves to call me though I don't really like becoming known as like that. For a whilst she's been in South Dakota. To gather cash is a thing that I'm completely addicted to. Hiring is his occupation.<br><br>Here is my web-site; [http://www.crmidol.com/discussion/19701/how-tell-if-woman-interested-or-sexually-captivated-you http://www.crmidol.com/] |
| | last = Falconer | first = Kenneth | authorlink=Kenneth Falconer (mathematician)
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| | title = Fractal Geometry: Mathematical Foundations and Applications
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| | publisher = John Wiley & Sons, Ltd.
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| | year = 1990 & 2003
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| | isbn = 0-470-84862-6
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| | nopp = true
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| | page = xxv}}</ref>
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| Presented here is a list of fractals ordered by increasing Hausdorff dimension, with the purpose of visualizing what it means for a fractal to have a low or a high dimension.
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| ==Deterministic fractals==
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| {| border="0" cellpadding="4" rules="all" style="border: 1px solid #999; background-color:#FFFFFF"
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| ! Hausdorff dimension<br />(exact value) || Hausdorff dimension<br />(approx.) || Name || Illustration || width="40%" | Remarks
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| | 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 function]] for the critical parameter value <math>\scriptstyle{\lambda_\infty = 3.570}</math>, where the period doubling is infinite. This dimension is the same for any differentiable and [[unimodal]] function.<ref>[http://www.springerlink.com/content/j67u086652125p71/fulltext.pdf Fractal dimension of the Feigenbaum attractor]</ref>
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| | <math>\textstyle{\frac {\log(2)}{\log(3)}}</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]].
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| | <math>\textstyle{\frac {\log(\scriptstyle\varphi)}{\log(2)}=\frac{\log(1+\sqrt{5})}{\log(2)}-1}</math> || align="right" | 0.6942 || Asymmetric [[Cantor set]] || align="center" |[[File:AsymmCantor.png|200px]] || The dimension is not <math>\textstyle{\frac {\log(2)}{\log(\tfrac {8}{3})}}</math>, as would be expected from the generalized Cantor set with γ=1/4, which has the same length at each stage.<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 |url=http://prl.aps.org/abstract/PRL/v57/i12/p1390_1 |doi=10.1103/PhysRevLett.57.1390}}</ref>
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| Built by removing the second quarter at each iteration. [[Nowhere dense]] and not a [[countable set]].
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| <math>\scriptstyle\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
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| | <math>\textstyle{\frac {\log(5)}{\log(10)}}</math> || align="right" | 0.69897 || [[Real number]]s with even digits || align="center" |[[File:Even digits.png|200px]] || Similar to a [[Cantor set]].<ref name="Falconer"/>
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| | <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>[http://www.ruf.rice.edu/~dtd3/DEGT-FD.pdf Fractal dimension of the spectrum of the Fibonacci Hamiltonian]</ref>
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| | <math>\textstyle{-\frac{\log(2)}{\log(\frac{1-\gamma}{2})}}</math> || align="right" | 0<D<1 || Generalized Cantor set || align="center" |[[File:generalized cantor set.png|200px]] || Built by removing at the <math>m</math>th iteration the central interval of length <math>\gamma\,l_{m-1}</math> from each remaining segment (of length <math>\scriptstyle l_{m-1}=(1-\gamma)^{m-1}/2^{m-1}</math>). At <math>\scriptstyle\gamma=1/3</math> one obtains the usual [[Cantor set]]. Varying <math>\scriptstyle\gamma</math> between 0 and 1 yields any fractal dimension <math>\scriptstyle 0\,<\,D\,<\,1</math>.<ref>[http://arxiv.org/abs/0911.2497 The scattering from generalized Cantor fractals]</ref>
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| | <math>\textstyle{1}</math> || align="right" | 1 || [[Smith–Volterra–Cantor set]] || align="center" |[[File:Smith-Volterra-Cantor set.svg|200px]] || Built by removing a central interval of length <math>1/2^{2n}</math> of each remaining interval at the ''n''th iteration. Nowhere dense but has a [[Lebesgue measure]] of ½.
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| | <math>\textstyle{2+\frac {\log(1/2)} {\log(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>\textstyle{f(x) = \sum_{n=0}^\infty {s(2^{n}x)\over 2^n}}</math>, where <math>s(x)</math> is the sawtooth function. Special case of the Takahi-Landsberg curve: <math>\textstyle{f(x) = \sum_{n=0}^\infty {w^n s(2^{n}x)}}</math> with <math>\scriptstyle{w = 1/2}</math>. The Hausdorff dimension equals <math>2+log(w)/log(2)</math> for <math>w</math> in <math>\scriptstyle{\left[ 1/2,1\right]}</math>. (Hunt cited by Mandelbrot<ref>{{Cite book| last = Mandelbrot| first = Benoit | title = Gaussian self-affinity and Fractals | isbn = 0-387-98993-5 }}</ref>).
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| | Calculated|| align="right" | 1.0812 || [[Julia set]] z² + 1/4 || align="center" |[[File:Julia z2+0,25.png|100px]] || Julia set for ''c'' = 1/4.<ref>[http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf fractal dimension of the Julia set for c = 1/4]</ref>
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| | Solution s 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>\scriptstyle{1\mapsto12}</math>, <math>\scriptstyle{2\mapsto13}</math> and <math>\scriptstyle{3}\mapsto1</math>.<ref>[http://matwbn.icm.edu.pl/ksiazki/aa/aa95/aa9531.pdf Boundary of the Rauzy fractal]</ref><ref>{{Citation | last1=Lothaire | first1=M. | authorlink = M. Lothaire | title=Applied combinatorics on words | url=http://www-igm.univ-mlv.fr/~berstel/Lothaire/AppliedCW/AppCWContents.html | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-84802-2; 978-0-521-84802-2 | mr=2165687 | zbl=1133.68067 | year=2005 | volume=105 | page=525}}</ref> <math>\alpha</math> is one of the conjugated roots of <math>z^3-z^2-z-1=0</math>.
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| | <math>\textstyle{2\frac {\log(3)} {\log(7)}}</math> || align="right" | 1.12915 || contour of the [[Gosper island]] || align="center" |[[File:Gosper Island 4.svg|100px]] || Term used by Mandelbrot (1977).<ref>[http://mathworld.wolfram.com/GosperIsland.html Gosper island on Mathworld]</ref> The Gosper island is the limit of the [[Gosper curve]].
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| | Measured (box counting) || align="right" | 1.2 || Dendrite [[Julia set]] || align="center" |[[File:Dendrite julia.png|150px]] || Julia set for parameters: Real = 0 and Imaginary = 1.
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| | <math>\textstyle{3\frac{\log(\varphi)}{\log \left(\frac{3+\sqrt{13}}{2}\right)}}</math> || align="right" | 1.2083 || [[Fibonacci word|Fibonacci word fractal 60°]] || align="center" | [[File:Fibo 60deg F18.png|200px]] || Build from the [[Fibonacci word]]. See also the standard Fibonacci word fractal.
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| <math>\scriptstyle\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
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| | |<math>\begin{align}&\textstyle{\frac{2\log\left(\frac{\sqrt[3]{27-3\sqrt{78}}+\sqrt[3]{27+3\sqrt{78}}}{3}\right)}{\log(2)}},\\ &^{\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">[http://www.springerlink.com/content/t5630411w45638g0/ On 2-reptiles in the plane, Ngai, 1999]</ref><ref>[http://demonstrations.wolfram.com/TheBoundaryOfPeriodicIteratedFunctionSystems/ Recurrent construction of the boundary of the dragon curve (for n=2, D=1)]</ref>
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| | || align="right" | 1.26 || [[Hénon map]] || align="center" |[[File:Henon attractor.png|100px]] || The canonical [[Hénon map]] (with parameters ''a'' = 1.4 and ''b'' = 0.3) has Hausdorff dimension 1.261 ± 0.003. Different parameters yield different dimension values.
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| | <math>\textstyle{\frac {\log(4)} {\log(3)}}</math> || align="right" | 1.2619 || [[Koch curve]] || align="center" | [[File:Koch curve.svg|200px]] || 3 von Koch curves form the Koch snowflake or the anti-snowflake.
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| | <math>\textstyle{\frac {\log(4)} {\log(3)}}</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 = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle.
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| | <math>\textstyle{\frac {\log(4)} {\log(3)}}</math> || align="right" | 1.2619 || 2D [[Cantor dust]] || align="center" |[[File:Carre cantor.gif|100px]] || Cantor set in 2 dimensions.
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| | <math>\textstyle{\frac {\log(4)} {\log(3)}}</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.
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| | Calculated|| align="right" | 1.2683 || [[Julia set]] z<sup>2</sup> − 1 || align="center" |[[File:Julia z2-1.png|200px]] || Julia set for ''c'' = −1.<ref>[http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf fractal dimension of the z²-1 Julia set]</ref>
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| | || align="right" | 1.3057 || [[Apollonian gasket]] || align="center" |[[File:Apollonian gasket.svg|100px]] || 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>[http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf fractal dimension of the apollonian gasket]</ref>
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| | || 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>[http://classes.yale.edu/Fractals/CircInvFrac/CircDim/CircDim2.html fractal dimension of the 5 circles inversion fractal]</ref>
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| | Calculated|| align="right" | 1.3934 || [[Douady rabbit]] || align="center" |[[File:Douady rabbit.png|150px]] || Julia set for ''c'' = −0,123 + 0.745i.<ref>[http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf fractal dimension of the Douady rabbit]</ref>
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| | <math>\textstyle{\frac {\log(5)} {\log(3)}}</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.
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| | <math>\textstyle{\frac {\log(5)} {\log(3)}}</math>|| align="right" | 1.4649 || [[Koch curve|Quadratic von Koch curve (type 1)]]|| align="center" |[[File:Quadratic Koch 2.png|150px]] || One can recognize the pattern of the Vicsek fractal (above).
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| | <math>\textstyle{\frac {\log(\frac{1}{3})} {\log(\sqrt{5})}}</math>|| align="right" | 1.49 ||Quadric cross || align="center" |{{anchor|cross}}[[File:Quadriccross.gif|150px]] ||
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| |<math> \textstyle{2 -\frac{\log(\sqrt{2})}{\log(2)}=\frac{3}{2}}</math> (conjectured exact)|| align="right" | 1.5000 || a [[Weierstrass function]]: <math>\textstyle{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 Weierstrass function <math>\scriptstyle{f : [0,1] \to \mathbb{R}}</math> defined by <math>\textstyle{f(x)=\sum_{k=1}^\infty \frac {\sin(b^k x)} {a^k}}</math> with <math>1<a<2</math> and <math>b>1</math> has upper bound <math>\scriptstyle{2 -\log(a)/\log(b)}</math>. It is believed to be the exact value. The same result can be established when, instead of the sine function, we use other periodic functions, like cosine.<ref name="Falconer" />
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| |<math>\textstyle{\frac {\log(8)} {\log(4)} = \frac{3}{2}}</math>|| align="right" | 1.5000 || [[Koch curve|Quadratic von Koch curve (type 2)]] || align="center" |[[File:Quadratic Koch.png|150px]] || Also called "Minkowski sausage".
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| |<math>\textstyle{\frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)}
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| {\log(2)}}</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>[http://demonstrations.wolfram.com/TheBoundaryOfPeriodicIteratedFunctionSystems/ Recurrent construction of the boundary of the dragon curve (for n=2, D=2)]</ref>
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| |<math>\textstyle{\frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)}
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| {\log(2)}}</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>\textstyle{\frac {\log(3)} {\log(2)}}</math> || align="right" | 1.5849 || 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>\textstyle{\frac {\log(3)} {\log(2)}}</math> || align="right" | 1.5849 || [[Sierpinski triangle]] || align="center" | [[File:Sierpinski8.svg|100px]] || Also the triangle of Pascal modulo 2.
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| | <math>\textstyle{\frac {\log(3)} {\log(2)}}</math> || align="right" | 1.5849 || [[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>\textstyle{\frac {\log(3)} {\log(2)}}</math> || align="right" | 1.5849 || 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>\textstyle{\frac {\log(4)} {\log(2)}}</math> <ref>https://en.wikipedia.org/wiki/T-Square_(fractal)</ref>
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| | <math>\textstyle{\frac{\log{\varphi}}{\log{\sqrt[\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>\scriptstyle{r = 1 / \varphi^{1/\varphi}}</math>. Its dimension equals <math>\scriptstyle{\varphi}</math> because <math>\scriptstyle{({r^2})^\varphi+r^\varphi = 1}</math>. With <math>\scriptstyle\varphi = (1+\sqrt{5})/2</math> ([[Golden ratio|Golden number]]).
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| | <math>\textstyle{1+\frac{\log 2}{\log 3}}</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, the fractal dimension is <math>\scriptstyle{1 + \log_k\left(\frac{k+1}{2}\right)}</math> (cf. [[Stephen Wolfram]]<ref name="stephenwolfram.com">[http://www.stephenwolfram.com/publications/articles/ca/84-geometry/1/text.html Fractal dimension of the Pascal triangle modulo k]</ref>).
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| | <math>\textstyle{\frac{\log(6)}{\log (3)}}</math> || align="right" | 1.6309 || [[N-flake#Hexaflake|Sierpinski Hexagon]] || align="center" | [[File:Sierpinski hexagon.png|100px]] || 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>\textstyle{3\frac{\log(\varphi)}{\log (1+\sqrt{2})}}</math> || align="right" | 1.6379 || [[Fibonacci word|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> = 28657 segments).<ref name="AMD">[http://hal.archives-ouvertes.fr/hal-00367972/en/ The Fibonacci word fractal]</ref> <math>\scriptstyle\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
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| | Solution of <math>\scriptstyle{(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 : <math>\scriptstyle{\sum_{k=1}^n c_k^s = 1}</math>.<ref name="Falconer"/>
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| | <math>\textstyle{1+\frac{\log 3}{\log 5}}</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>\scriptstyle{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>\scriptstyle {z_{n+1} = a + bz_n exp[i[k - p/(1 + \lfloor z_n \rfloor^2)]]} </math>. It derives from a model of the plane-wave interactivity field in an optical ring laser. Different parameters yield different values.<ref>[http://public.lanl.gov/jt/Papers/est-fractal-dim.pdf Estimating Fractal dimension]</ref>
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| | <math>\textstyle{\frac {\log(50)} {\log(10)}}</math> || align="right" | 1.7 || 50 segment quadric fractal || align="center" | [[File:50seg.tif|150px]] || Built with ImageJ<ref>[http://rsb.info.nih.gov/ij/plugins/fractal-generator.html Fractal Generator for ImageJ].</ref>
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| | <math>\textstyle{\frac {4 \log(2)} {\log(5)}}</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>\textstyle{\frac {\log(7)} {\log(3)}}</math> || align="right" | 1.7712 || [[Hexaflake]] || align="center" | [[File:Flocon hexagonal.gif|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>\textstyle{\frac {\log(4)} {\log(2(1+\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>\scriptstyle{\frac{\log(4)}{\log(2(1+\cos(a)))}} \in [1,2]</math>.
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| | <math>\textstyle{\frac{\log{(3^{0.63}+2^{0.63})}} {\log{2}}}</math> || align="right" | 1.8272 || A self-[[affine transformation|affine]] fractal set || align="center" | [[File:Self-affine set.png|200px]] || Build iteratively from a <math>\scriptstyle{p \times q}</math> array on a square, with <math>\scriptstyle{p \le q}</math>. Its Hausdorff dimension equals <math>\scriptstyle{\frac{\log{\left (\sum_{k=1}^p n_k^a \right )}} {\log{p}}}</math><ref name="Falconer"/> with <math>\scriptstyle{a=\frac{\log{ p}}{log{ q}}} </math> and <math>n_k</math> is the number of elements in the <math>k^{th}</math> 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.
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| | <math>\textstyle{\frac {\log(6)} {\log(1+\varphi)}}</math> || align="right" | 1.8617 || [[N-flake#Pentaflake|Pentaflake]] || align="center" | [[File:Penta plexity.png|100px]] || Built by exchanging iteratively each pentagon by a flake of 6 pentagons.
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| <math>\scriptstyle\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
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| | solution of <math>\scriptstyle{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>\scriptstyle{1/{3\sqrt{3}}}</math>.<ref>[http://www.coaauw.org/boulder-eyh/eyh_fractal.html Monkeys tree fractal curve]</ref>
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| | <math>\textstyle{\frac {\log(8)} {\log(3)}}</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).
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| | <math>\textstyle{\frac {\log(8)} {\log(3)}}</math> || align="right" | 1.8928 || 3D [[Cantor dust]] || align="center" | [[File:Cantor3D3.png|200px]]|| Cantor set in 3 dimensions.
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| | <math>\textstyle{\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]].
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| |Estimated || 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.
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| | || align="right" | 1.974 || [[Penrose tiling]] || align="center" |[[File:pen0305c.gif|100px]] || See Ramachandrarao, Sinha & Sanyal.<ref>[http://www.ias.ac.in/currsci/aug102000/rc80.pdf Fractal dimension of a Penrose tiling]</ref>
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| | <math>\textstyle{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 dimension.<ref>[http://arxiv.org/abs/math/9201282 Fractal dimension of the boundary of the Mandelbrot set]</ref>
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| | <math>\textstyle{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>[http://arxiv.org/abs/math/9201282 Fractal dimension of certain Julia sets]</ref>
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| | <math>\textstyle{2}</math> || align="right" | 2 || [[Sierpiński curve]] || align="center" | [[File:Sierpinski-Curve-3.png|100px]] || Every [[Peano curve]] filling the plane has a Hausdorff dimension of 2.
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| | <math>\textstyle{2}</math> || align="right" | 2 || [[Hilbert curve]] || align="center" | [[File:Hilbert curve 3.svg|100px]]||
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| | <math>\textstyle{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]].
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| | <math>\textstyle{2}</math> || align="right" | 2 || [[Moore curve]] || align="center" | [[File:Moore-curve-stages-1-through-4.svg|100px]]|| Can be extended in 3 dimensions.
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| | || 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>
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| | <math>\textstyle{\frac {\log(2)} {\log(\sqrt{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>[http://arxiv.org/pdf/0712.1309 Complex base numeral systems]</ref>
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| | || align="right" | 2 || [[Dragon curve|Terdragon curve]] || align="center" | [[File:Terdragon curve.png|150px]]|| L-system: ''F'' → ''F'' + F – F, angle = 120°.
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| | <math>\textstyle{\frac {\log(4)} {\log(2)} = 2}</math> || align="right" | 2 || [[Gosper curve]] || align="center" | [[File:Gosper curve 3.svg|100px]]|| Its boundary is the Gosper island.
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| | Solution of <math>\scriptstyle{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>"Penser les mathématiques", Seuil ISBN 2-02-006061-2 (1982)</ref> it fills the [[Koch snowflake]]. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio <math>\scriptstyle{1/3\sqrt{3}}</math>.
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| | <math>\textstyle{\frac {\log(4)} {\log(2)} = 2}</math> || align="right" | 2 || [[Sierpiński triangle|Sierpiński tetrahedron]] || align="center" | [[File:Tetraedre Sierpinski.png|80px]]|| Each [[tetrahedron]] is replaced by 4 tetrahedra.
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| | <math>\textstyle{\frac {\log(4)} {\log(2)} = 2}</math> || align="right" | 2 || [[H-fractal]] || align="center" |[[File:H fractal2.png|150px]]|| Also the [[Mandelbrot tree]] which has a similar pattern.
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| | <math>\textstyle{\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 sqrt(2)/2.
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| | <math>\textstyle{\frac {\log(4)} {\log(2)} = 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.
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| | 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>
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| | Measured || align="right" | 2.06 ±0.01|| [[Lorenz attractor]] || align="center" |[[File:Lorenz attractor.png|100px]] || For parameters v=40,<math>\sigma</math>=16 and b=4 . See McGuinness (1983)<ref>[http://adsabs.harvard.edu/abs/1983PhLA...99....5M The fractal dimension of the Lorenz attractor, Mc Guinness (1983)]</ref>
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| | <math>\textstyle{\frac {\log(5)} {\log(2)}}</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).
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| | <math>\textstyle{\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.
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| <math>\scriptstyle\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
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| | <math>\textstyle{\frac {\log(13)} {\log(3)}}</math> || align="right" | 2.3347 || [[Koch curve|3D quadratic Koch surface (type 1)]] || align="center" |[[File:Quadratic Koch 3D (type1 stage2).png|150px]]|| Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
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| | || 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://graphics.ethz.ch/~peikert/papers/apollonian.pdf Fractal dimension of the apollonian sphere packing]</ref>
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| | <math>\textstyle{\frac {\log(32)} {\log(4)} = \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.
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| | <math>\textstyle{\frac {\log(16)} {\log(3)}}</math> || align="right" | 2.5237 || [[Cantor tesseract]] || align="center" | no image available || Cantor set in 4 dimensions. Generalization: in a space of dimension ''n'', the Cantor set has a Hausdorff dimension of <math>\scriptstyle{n\frac{\log(2)}{\log(3)}}</math>.
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| | <math>\textstyle{\frac{\log(\frac{\sqrt7}6-\frac{1}3)}{\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>\scriptstyle \sqrt {2} - 1</math>.
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| | <math>\textstyle{\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>\scriptstyle\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
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| | <math>\textstyle{\frac {\log(6)} {\log(2)}}</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.
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| | <math>\textstyle{\frac {\log(6)} {\log(2)}}</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.
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| | <math>\textstyle{\frac {\log(6)} {\log(2)}}</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.
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| Using the central triangle as the base, form a tetrahedron. Replace the triangular base with the tetrahedral "tent".
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| | <math>\textstyle{\frac {\log(20)} {\log(3)}}</math> || align="right" | 2.7268 || [[Menger sponge]] || align="center" | [[File:Menger.png|100px]] || And its surface has a fractal dimension of <math>\scriptstyle{\frac{\log(20)}{\log(3)} = 2.7268}</math>, which is the same as that by volume.
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| | <math>\textstyle{\frac {\log(8)} {\log(2)} = 3}</math> || align="right" | 3 || [[Hilbert curve|3D Hilbert curve]] || align="center" | [[File:Hilbert3d-step3.png|100px]]|| A Hilbert curve extended to 3 dimensions.
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| | <math>\textstyle{\frac {\log(8)} {\log(2)} = 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.
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| | <math>\textstyle{\frac {\log(8)} {\log(2)} = 3}</math> || align="right" | 3 || [[Moore curve|3D Moore curve]] || align="center" | [[File:Moore3d-step3.png|100px]]|| A Moore curve extended to 3 dimensions.
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| | <math>\textstyle{\frac {\log(8)} {\log(2)} = 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 paper | author = B. Hou, H. Xie, W. Wen, and P. Sheng | year = 2008
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| | title = Three-dimensional metallic fractals and their photonic crystal characteristics | publisher = Phys. Rev. B '''77''', 125113
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| | url = http://prb.aps.org/abstract/PRB/v77/i12/e125113}}</ref>
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| | <math>\textstyle{3}</math> (to be confirmed) || align="right" | {{formatnum:3}} (to be confirmed) || [[Mandelbulb]] || align="center" |[[File:Mandelbulb 5 iterations.png|100px]]|| Extension of the Mandelbrot set (power 8) in 3 dimensions<ref>[http://www.fractalforums.com/theory/hausdorff-dimension-of-the-mandelbulb/15/ Hausdorff dimension of the Mandelbulb]</ref>{{Verify credibility|date=September 2011}}
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| |}
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| ==Random and natural fractals==
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| {| border="0" cellpadding="4" rules="all" style="border: 1px solid #999; background-color:#FFFFFF"
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| |- align="center" bgcolor="#cccccc"
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| ! Hausdorff dimension<br />(exact value) || Hausdorff dimension<br />(approx.) || Name || Illustration || width="40%" | Remarks
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| |1/2 || 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, Oded Schramm, "Brownian Motion", Cambridge University Press, 2010</ref>
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| | Solution of <math>\scriptstyle{E(C_1^s + C_2^s)=1}</math> where <math>\scriptstyle{E(C_1)=0.5}</math> and <math>\scriptstyle{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>\scriptstyle{E(C_1^s + C_2^s)=1}</math>. (<math>E(X)</math> is the [[expected value]] of <math>X</math>).<ref name="Falconer"/>
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| |Solution of <math>s+1=12*2^{-(s+1)}-6*3^{-(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"/>
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| |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>McCartney M., Abernethy G., and Gault L. (2010). ''The Divider Dimension of the Irish Coast''. Irish Geography, '''43''', 277-284.</ref> at the [[University of Ulster]] and [[Theoretical Physics]] students at [[Trinity College, Dublin]], under the supervision of S. Hutzler.<ref name="S.Hutzler">Hutzler, S. (2013). [http://www.sciencespin.com/magazine/archive/2013/05/ ''Fractal Ireland'']. Science Spin, 58, 19-20.</ref>
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| 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" />
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| |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>
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| | <math>\textstyle{\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"/>
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| |<math>\textstyle{\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>[http://arxiv.org/abs/math/0010165 Fractal dimension of the brownian motion boundary]</ref>
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| |<math>\textstyle{\frac {4}{3}}</math> || align="right" | 1.333 || [[2D polymer]] || align="center" | || Similar to the brownian motion in 2D with non self-intersection.<ref name="sapoval">Bernard Sapoval "Universalités et fractales", Flammarion-Champs (2001), ISBN=2-08-081466-4</ref>
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| |<math>\textstyle{\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" />
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| | || 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" />
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| | <math>\textstyle{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 ''f'' such that, for any two positive reals ''x'' and ''x+h'', the difference of their images <math>f(x+h)-f(x)</math> has the centered gaussian distribution with variance ''= h''. 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 =<math>2-\alpha</math>'''.<ref name="Falconer" />
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| | Measured|| align="right" | 1.52|| [[Fjord|Coastline of Norway]] || align="center" |[[File:Norway municipalities.png|100px]] || See J. Feder.<ref>Feder, J., "Fractals,", Plenum Press, New York, (1988).</ref>
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| | Measured|| align="right" | 1.55 || [[Random walk with no self-intersection]] || align="center" | [[File:Polymer 2D.png|150px]]|| Self-avoiding random walk in a square lattice, with a « go-back » routine for avoiding dead ends.
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| | <math>\textstyle{\frac {5} {3}}</math>|| align="right" | 1.66|| [[3D polymer]] || align="center" | || Similar to the brownian motion in a cubic lattice, but without self-intersection.<ref name="sapoval" />
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| | || align="right" | 1.70 || [[Diffusion-limited aggregation|2D DLA Cluster]] || align="center" | [[File:Agregation 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" />
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| | <math>\textstyle{\frac {\log(9*0.75)} {\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 3x3 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>\textstyle{\frac {\log(9p)} {\log(3)}}</math>.<ref name="Falconer" />
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| | 7/4 || 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.
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| | <math>\textstyle{\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">[http://books.google.fr/books?id=MJwqsbWBc-YC&dq=applications+of+percolation+theory&source=gbs_navlinks_s "Applications of percolation" theory by Muhammad Sahimi (1994)]</ref> Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the « clearings ».
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| | <math>\textstyle{\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").
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| | 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>[http://arxiv.org/abs/astro-ph/0501583v2 Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey]</ref>
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| | <math>\textstyle{\frac {\log(13)} {\log(3)}}</math> || align="right" | 2.33 || [[Cauliflower]] || align="center" | [[File:Blumenkohl-1.jpg|100px]]|| Every branch carries around 13 branches 3 times smaller.
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| | || 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 | first= | last= | date= | accessdate=29 July 2010 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> Creases will form at all size scales (see [[Universality (dynamical systems)]]).
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| | || align="right" | 2.50 || [[diffusion-limited aggregation|3D DLA Cluster]] || align="center" | [[File:3D diffusion-limited aggregation2.jpg|100px]] || In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.<ref name="sapoval" />
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| | || 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" />
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| | <math>\textstyle{3-\frac{1}{2}}</math>|| align="right" | 2.5|| regular [[Brownian motion|Brownian]] surface|| align="center" | [[File:Brownian surface.png|150px]] || A function <math>\scriptstyle{f:\mathbb{R}^2 -> \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>\scriptstyle{f(x+h,y+k)-f(x,y)}</math> has a centered Gaussian distribution with variance = <math>\scriptstyle{\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 = <math>3-\alpha</math>'''.<ref name="Falconer" />
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| | 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.
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| | Measured|| align="right" | 2.66 || [[Broccoli]] || align="center" | [[File:Broccoli DSC00862.png|100px]] ||<ref>[http://hypertextbook.com/facts/2002/broccoli.shtml Fractal dimension of the broccoli]</ref>
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| | || align="right" | 2.79 || Surface of [[Cerebral cortex|human brain]] || align="center" | [[File:Cerebellum NIH.png|100px]] ||<ref>[http://www.heise.de/tr/artikel/54311/2/0 Fractal dimension of the surface of the human brain]</ref>
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| | || 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" />
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| | Calculated || align="right" | <math>\textstyle{\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>[Meakin (1987)]</ref>
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| |}
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| ==See also==
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| {{Commons|Fractal|fractals}}
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| * [[Fractal dimension]]
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| * [[Hausdorff dimension]]
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| * [[Scale invariance]]
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| ==Notes and references==
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| {{Reflist|2}}
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| ==Further reading==
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| * Benoît Mandelbrot, ''The Fractal Geometry of Nature'', W. H. Freeman & Co; ISBN 0-7167-1186-9 (September 1982).
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| * Heinz-Otto Peitgen, ''The Science of Fractal Images'', Dietmar Saupe (editor), Springer Verlag, ISBN 0-387-96608-0 (August 1988)
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| * Michael F. Barnsley, ''Fractals Everywhere'', Morgan Kaufmann; ISBN 0-12-079061-0
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| * Bernard Sapoval, « Universalités et fractales », collection Champs, Flammarion. ISBN 2-08-081466-4 (2001).
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| ==External links==
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| * [http://mathworld.wolfram.com/search/?query=fractal The fractals on Mathworld]
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| * [http://local.wasp.uwa.edu.au/~pbourke/fractals/ Other fractals on Paul Bourke's website]
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| * [http://soler7.com/Fractals/FractalsSite.html Soler's Gallery]
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| * [http://www.mathcurve.com/fractals/fractals.shtml Fractals on mathcurve.com]
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| * [http://1000fractales.free.fr/index.htm 1000fractales.free.fr - Project gathering fractals created with various softwares]
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| * [http://library.thinkquest.org/26242/full/index.html Fractals unleashed]
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| {{Use dmy dates|date=September 2010}}
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| {{Fractals}}
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| {{DEFAULTSORT:List Of Fractals By Hausdorff Dimension}}
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| [[Category:Fractals]]
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| [[Category:Fractal curves]]
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| [[Category:Mathematics-related lists|Fractals by Hausdorff dimension]]
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