:List of fractals by Hausdorff dimension

According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."{{harvnb|Mandelbrot|1982|p=15}}

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

Hausdorff dimension (exact value) \|\| Hausdorff dimension (approx.) \|\| Name \|\| Illustration \|\| width="40%" \| Remarks
class="wikitable sortable"
Calculated	align="right" \| 0.538	Feigenbaum attractor	align="center" \|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 $\lambda_\infty = 3.570$ , where the period doubling is infinite. This dimension is the same for any differentiable and unimodal function.{{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 }}
$\log_3 2$	align="right" \| 0.6309	Cantor set	align="center" \|200px	Built by removing the central third at each iteration. Nowhere dense and not a countable set.
$\frac{-\log 2}{\log\left(\displaystyle\frac{1-\gamma}{2}\right)}$	align="right" \| 0<D<1	1D generalized symmetric Cantor set	align="center" \|200px	Built by removing the central interval of length $\gamma\,l_{n-1}$ from each remaining interval of length $l_{n-1} = (1-\gamma)^{n-1}/2^{n-1}$ at the nth iteration. $\gamma=1/3$ produces the usual middle-third Cantor set. Varying $\gamma$ between 0 and 1 yields any fractal dimension $0 . {{cite journal \|first1=A. Yu \|last1=Cherny \|first2=E.M. \|last2=Anitas \|first3=A.I. \|last3=Kuklin \|first4=M. \|last4=Balasoiu \|first5=V.A. \|last5=Osipov \|title=The scattering from generalized Cantor fractals \|journal=J. Appl. Crystallogr. \|volume=43 \|issue= 4\|pages=790–7 \|year=2010 \|doi=10.1107/S0021889810014184 \|arxiv=0911.2497 \|s2cid=94779870 }}$
$\log_2 \varphi = \log_2(1+\sqrt{5}) - 1$	align="right" \| 0.6942	(1/4, 1/2) asymmetric Cantor set	align="center" \|200px	Built by removing the second quarter at each iteration.{{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}} $\varphi = (1+\sqrt{5})/2$ (golden ratio).
$\log_{10} 5 = 1 - \log_{10} 2$	align="right" \| 0.69897	Real numbers whose base 10 digits are even	align="center" \|200px	Similar to the Cantor set.{{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}}
$\log(1+\sqrt{2})$	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.{{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 }}{{page needed\|date=October 2018}}
$1$	align="right" \| 1	Smith–Volterra–Cantor set	align="center" \|200px	Built by removing the central interval of length $2^{-2n}$ from each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of 1/2.
$2 + \log_2 \frac{1}{2} = 1$	align="right" \| 1	Takagi or Blancmange curve	align="center" \|150px	Defined on the unit interval by $f(x) = \sum\nolimits_{n=0}^\infty 2^{-n}s(2^{n}x)$ , where $s(x)$ is the triangle wave function. Not a fractal under Mandelbrot's definition, because its topological dimension is also $1$ .{{Cite book\| last = Vaz\| first = Cristina \| title = Noções Elementares Sobre Dimensão \| isbn = 9788565054867 \| year = 2019}} Special case of the Takahi-Landsberg curve: $f(x) = \sum\nolimits_{n=0}^\infty w^n s(2^n x)$ with $w = 1/2$ . The Hausdorff dimension equals $2 + \log_2 w$ for $w$ in $\left[1/2,1\right]$ . (Hunt cited by Mandelbrot{{Cite book\| last = Mandelbrot\| first = Benoit \| title = Gaussian self-affinity and Fractals \| isbn = 978-0-387-98993-8 \| year = 2002 \| publisher = Springer }}).
Calculated	align="right" \| 1.0812	Julia set z² + 1/4	align="center" \|100px	Julia set of f(z) = z² + 1/4.
Solution $s$ of $2\|\alpha\|^{3s} + \|\alpha\|^{4s} = 1$	align="right" \| 1.0933	Boundary of the Rauzy fractal	align="center" \|150px	Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: $1\mapsto12$ , $2\mapsto13$ and $3\mapsto1$ .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.{{page needed\|date=October 2018}}{{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] }} $\alpha$ is one of the conjugated roots of $z^3-z^2-z-1=0$ .
$2\log_7 3$	align="right" \| 1.12915	contour of the Gosper island	align="center" \|100px	Term used by Mandelbrot (1977).{{MathWorld \|id=GosperIsland \|title=Gosper Island \|access-date=27 October 2018}} The Gosper island is the limit of the Gosper curve.
Measured (box counting)	align="right" \| 1.2	Dendrite Julia set	align="center" \|150px	Julia set of f(z) = z² + i.
$\frac{3\log \varphi}{\log\left(\displaystyle\frac{3+\sqrt{13}}{2}\right)}$	align="right" \| 1.2083	Fibonacci word fractal 60°	align="center" \| 200px	Built from the Fibonacci word. See also the standard Fibonacci word fractal. $\varphi = (1+\sqrt{5})/2$ (golden ratio).
\| $\begin{align} & 2\log_2\left(\displaystyle\frac{\sqrt[3]{27-3\sqrt{78}}+\sqrt[3]{27+3\sqrt{78}}}{3}\right),\\ &\text{or root of }2^x-1=2^{(2-x)/2}\end{align}$	align="right" \| 1.2108	Boundary of the tame twindragon	align="center" \|150px	One of the six 2-rep-tiles in the plane (can be tiled by two copies of itself, of equal size).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.Duda, Jarek (March 2011). "[http://demonstrations.wolfram.com/TheBoundaryOfPeriodicIteratedFunctionSystems/ The Boundary of Periodic Iterated Function Systems]", Wolfram.com.
	align="right" \| 1.26	Hénon map	align="center" \|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.
$\log_3 4$	align="right" \| 1.2619	Triflake	align="center" \| 150px	Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
$\log_3 4$	align="right" \| 1.2619	Koch curve	align="center" \| 200px	3 Koch curves form the Koch snowflake or the anti-snowflake.
$\log_3 4$	align="right" \| 1.2619	boundary of Terdragon curve	align="center" \|150px	L-system: same as dragon curve with angle = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle.
$\log_3 4$	align="right" \| 1.2619	2D Cantor dust	align="center" \|100px	Cantor set in 2 dimensions.
$\log_3 4$	align="right" \| 1.2619	2D L-system branch	align="center" \|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 z² − 1	align="center" \|200px	Julia set of f(z) = z² − 1.
	align="right" \| 1.3057	Apollonian gasket	align="center" \|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 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.
	align="right" \| 1.328	5 circles inversion fractal	align="center" \|100px	The limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an Apollonian packing. See 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]
$\log_5 9$	align="right" \| 1.36521Mandelbrot, 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}}	Quadratic von Koch island using the type 1 curve as generator	align="center" \|150px	Also known as the Minkowski Sausage
Calculated	align="right" \| 1.3934	Douady rabbit	align="center" \|150px	Julia set of f(z) = -0.123 + 0.745i
$\log_3 5$	align="right" \| 1.4649	Vicsek fractal	align="center" \|100px	Built by exchanging iteratively each square by a cross of 5 squares.
$\log_3 5$	align="right" \| 1.4649	Quadratic von Koch curve (type 1)	align="center" \|150px	One can recognize the pattern of the Vicsek fractal (above).
$\log_{\sqrt{5}} \frac{10}{3}$	align="right" \| 1.4961	Quadric cross	align="center" \|{{anchor\|cross}}150px	File:Q Cross Fractal Generator.jpgBuilt by replacing each end segment with a cross segment scaled by a factor of 5^1/2, consisting of 3 1/3 new segments, as illustrated in the inset. Images generated with Fractal Generator for ImageJ.
$2 - \log_2 \sqrt{2} = \frac{3}{2}$	align="right" \| 1.5000	a Weierstrass function: $f(x) = \sum_{k=1}^\infty \frac{\sin(2^k x)}{\sqrt{2}^{\,k}}$	align="center" \|150px	The Hausdorff dimension of the graph of the Weierstrass function $f:[0,1] \to \mathbb{R}$ defined by $f(x) = \sum\nolimits_{k=1}^\infty a^k\sin(b^k x)$ with $1/b < a < 1$ and $b>1$ is $2 + \log_b a$ .{{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}}N. Zhang. The Hausdorff dimension of the graphs of fractal functions. (In Chinese). Master Thesis. Zhejiang University, 2018.
$\log_4 8 = \frac{3}{2}$	align="right" \| 1.5000	Quadratic von Koch curve (type 2)	align="center" \|150px	Also called "Minkowski sausage".
$\log_2\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)$	align="right" \| 1.5236	Boundary of the Dragon curve	align="center" \| 150px	cf. Chang & Zhang.[http://poignance.coiraweb.com/math/Fractals/Dragon/Bound.html Fractal dimension of the boundary of the dragon fractal]
$\log_2\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)$	align="right" \| 1.5236	Boundary of the twindragon curve	align="center" \|150px	Can be built with two dragon curves. One of the six 2-rep-tiles in the plane (can be tiled by two copies of itself, of equal size).
$\log_2 3$	align="right" \| 1.5850	3-branches tree	align="center" \| 110px 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.
$\log_2 3$	align="right" \| 1.5850	Sierpinski triangle	align="center" \| 100px	Also the limiting shape of Pascal's triangle modulo 2.
$\log_2 3$	align="right" \| 1.5850	Sierpiński arrowhead curve	align="center" \| 100px	Same limit as the triangle (above) but built with a one-dimensional curve.
$\log_2 3$	align="right" \| 1.5850	Boundary of the T-square fractal	align="center" \| 200px	The dimension of the fractal itself (not the boundary) is $\log_2 4 = 2$
$\log_{\sqrt[\varphi]{\varphi}}(\varphi) = \varphi$	align="right" \| 1.61803	a golden dragon	align="center" \| 150px	Built from two similarities of ratios $r$ and $r^2$ , with $r = 1 / \varphi^{1/\varphi}$ . Its dimension equals $\varphi$ because $({r^2})^\varphi + r^\varphi = 1$ . $\varphi = (1+\sqrt{5})/2$ (golden ratio).
$1 + \log_3 2$	align="right" \| 1.6309	Pascal triangle modulo 3	align="center" \| 160px	For a triangle modulo k, if k is prime, the fractal dimension is $1 + \log_k\left(\frac{k+1}{2}\right)$ (cf. Stephen Wolfram{{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 }}).
$1 + \log_3 2$	align="right" \| 1.6309	Sierpinski Hexagon	align="center" \| 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.
$\frac{3\log \varphi}{\log(1+\sqrt{2})}$	align="right" \| 1.6379	Fibonacci word fractal	align="center" \| 150px	Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F₂₃ = 28657 segments).[http://hal.archives-ouvertes.fr/hal-00367972/en/ The Fibonacci word fractal] $\varphi = (1+\sqrt{5})/2$ (golden ratio).
Solution of $(1/3)^s + (1/2)^s + (2/3)^s = 1$	align="right" \| 1.6402	Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3	align="center" \| 200px	Generalization : Providing the open set condition holds, the attractor of an iterated function system consisting of $n$ similarities of ratios $c_n$ , has Hausdorff dimension $s$ , solution of the equation coinciding with the iteration function of the Euclidean contraction factor: $\sum\nolimits_{k=1}^n c_k^s = 1$ .
$\log_8 32 = \frac{5}{3}$	align="right" \| 1.6667	32-segment quadric fractal (1/8 scaling rule)	frameless see also: :File:32 Segment One Eighth Scale Quadric Fractal.jpg	File:32SegmentSmall.jpgBuilt 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.
$1 + \log_5 3$	align="right" \| 1.6826	Pascal triangle modulo 5	align="center" \| 160px	For a triangle modulo k, if k is prime, the fractal dimension is $1 + \log_k\left(\frac{k+1}{2}\right)$ (cf. Stephen Wolfram).
Measured (box-counting)	align="right" \| 1.7	Ikeda map attractor	align="center" \| 100px	For parameters a=1, b=0.9, k=0.4 and p=6 in the Ikeda map $z_{n+1} = a + bz_n \exp\left[i\left[k - p/\left(1 + \lfloor z_n \rfloor^2\right)\right]\right]$ . It derives from a model of the plane-wave interactivity field in an optical ring laser. Different parameters yield different values.{{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 }}
$1 + \log_{10} 5$	align="right" \| 1.6990	50 segment quadric fractal (1/10 scaling rule)	align="center" \| 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[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 }}..File:50SegmentSmall.jpg
$4\log_5 2$	align="right" \| 1.7227	Pinwheel fractal	align="center" \| 150px	Built with Conway's Pinwheel tile.
$\log_3 7$	align="right" \| 1.7712	Sphinx fractal	align="center" \| 150px	Built with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes.W. Trump, G. Huber, C. Knecht, R. Ziff, to be published
$\log_3 7$	align="right" \| 1.7712	Hexaflake	align="center" \| 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).
$\log_3 7$	align="right" \| 1.7712	Fractal H-I de Rivera	align="center" \| 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.
$\frac{\log 4}{\log(2 + 2\cos(85^\circ))}$	align="right" \| 1.7848	Von Koch curve 85°	align="center" \| 150px	Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then $\frac{\log 4}{\log(2 + 2\cos a)} \in [1,2]$ .
$\log_2\left(3^{0.63} + 2^{0.63}\right)$	align="right" \| 1.8272	A self-affine fractal set	align="center" \| 200px	Build iteratively from a p-by-q array on a square, with $p \le q$ . Its Hausdorff dimension equals $\log_p\left(\sum\nolimits_{k=1}^p n_k^a\right)$ with $a = \log_q p$ and $n_k$ is the number of elements in the $k$ ^th column. The 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.
$\frac{\log 6}{\log(1+\varphi)}$	align="right" \| 1.8617	Pentaflake	align="center" \| 100px	Built by exchanging iteratively each pentagon by a flake of 6 pentagons. $\varphi = (1+\sqrt{5})/2$ (golden ratio).
solution of $6(1/3)^s + 5{(1/3\sqrt{3})}^s = 1$	align="right" \| 1.8687	Monkeys tree	align="center" \| 100px	This curve appeared in Benoit Mandelbrot's "Fractal geometry of Nature" (1983). It is based on 6 similarities of ratio $1/3$ and 5 similarities of ratio $1/3\sqrt{3}$ .[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 }}
$\log_3 8$	align="right" \| 1.8928	Sierpinski carpet	align="center" \| 100px	Each face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1).
$\log_3 8$	align="right" \| 1.8928	3D Cantor dust	align="center" \| 200px	Cantor set in 3 dimensions.
$\log_3 4 + \log_3 2 = \frac{\log 4}{\log 3}+\frac{\log 2}{\log 3} = \frac{\log 8}{\log 3}$	align="right" \| {{formatnum:1.8928}}	Cartesian product of the von Koch curve and the Cantor set	align="center" \| 150px	Generalization : Let F×G be the cartesian product of two fractals sets F and G. Then $\dim_H(F \times G) = \dim_H F + \dim_H G$ . See also the 2D Cantor dust and the Cantor cube.
$2\log_2 x$ where $x^9-3x^8+3x^7-3x^6+2x^5+4x^4-8x^3+$ $8x^2-16x+8 = 0$	align="right" \| 1.9340	Boundary of the Lévy C curve	align="center" \| 100px	Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
	align="right" \| 2	Penrose tiling	align="center" \|100px	See Ramachandrarao, Sinha & Sanyal.[http://www.iisc.ernet.in/currsci/aug102000/rc80.pdf Fractal dimension of a Penrose tiling]
$2$	align="right" \| 2	Boundary of the Mandelbrot set	align="center" \| 100px	The boundary and the set itself have the same Hausdorff dimension.{{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}}
$2$	align="right" \| 2	Julia set	align="center" \| 150px	For determined values of c (including c belonging to the boundary of the Mandelbrot set), the Julia set has a dimension of 2.
$2$	align="right" \| 2	Sierpiński curve	align="center" \| 100px	Every space-filling curve filling the plane has a Hausdorff dimension of 2.
$2$	align="right" \| 2	Hilbert curve	align="center" \| 100px
$2$	align="right" \| 2	Peano curve	align="center" \| 100px	And a family of curves built in a similar way, such as the Wunderlich curves.
$2$	align="right" \| 2	Moore curve	align="center" \| 100px	Can be extended in 3 dimensions.
	align="right" \| 2	Lebesgue curve or z-order curve	align="center" \| 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.[http://www.mathcurve.com/fractals/lebesgue/lebesgue.shtml Lebesgue curve variants]
$\log_{\sqrt{2}} 2 = 2$	align="right" \| 2	Dragon curve	align="center" \| 150px	And its boundary has a fractal dimension of 1.5236270862.{{cite arXiv \|first=Jarek \|last=Duda \|title=Complex base numeral systems \|date=2008 \|class=math.DS \|eprint=0712.1309v3}}
	align="right" \| 2	Terdragon curve	align="center" \| 150px	L-system: F → F + F – F, angle = 120°.
$\log_2 4 = 2$	align="right" \| 2	Gosper curve	align="center" \| 100px	Its boundary is the Gosper island.
Solution of $7({1/3})^s+6({1/3\sqrt{3}})^s=1$	align="right" \| 2	Curve filling the Koch snowflake	align="center" \| 100px	Proposed by Mandelbrot in 1982,{{cite book \|title=Penser les mathématiques \|last=Seuil \|isbn=2-02-006061-2 \|year=1982\|publisher=Seuil }} it fills the Koch snowflake. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio $1/3\sqrt{3}$ .
$\log_2 4 = 2$	align="right" \| 2	Sierpiński tetrahedron	align="center" \| 80px	Each tetrahedron is replaced by 4 tetrahedra.
$\log_2 4 = 2$	align="right" \| 2	H-fractal	align="center" \|150px	Also the Mandelbrot tree which has a similar pattern.
$\frac{\log 2}{\log(2/\sqrt{2})} = 2$	align="right" \| {{formatnum:2}}	Pythagoras tree (fractal)	align="center" \|150px	Every square generates two squares with a reduction ratio of $1/\sqrt{2}$ .
$\log_2 4 = 2$	align="right" \| 2	2D Greek cross fractal	align="center" \|100px	Each segment is replaced by a cross formed by 4 segments.
Measured	align="right" \| 2.01 ± 0.01	Rössler attractor	align="center" \| 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.[http://www.ocf.berkeley.edu/~trose/rossler.html Fractals and the Rössler attractor]
Measured	align="right" \| 2.06 ± 0.01	Lorenz attractor	align="center" \|100px	For parameters $\rho=40$ , $\sigma=16$ and $\beta=4$ . See McGuinness (1983){{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 }}
$4 + c^D + d^D = (c+d)^D$	align="right" \|2<D<2.3	Pyramid surface	align="center" \|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 $c$ and $d$ relative to the pyramid triangles. The dimension is a parameter, self-intersection occurs for values greater than 2.3.{{Cite journal\|last=Lowe\|first=Thomas\|date=24 October 2016\|title=Three Variable Dimension Surfaces\|url=https://www.researchgate.net/publication/309391846\|journal=ResearchGate}}
$\log_2 5$	align="right" \| 2.3219	Fractal pyramid	align="center" \|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).
$\frac{\log 20}{\log(2+\varphi)}$	align="right" \| 2.3296	Dodecahedron fractal	align="center" \|100px	Each dodecahedron is replaced by 20 dodecahedra. $\varphi = (1+\sqrt{5})/2$ (golden ratio).
$\log_3 13$	align="right" \| 2.3347	3D quadratic Koch surface (type 1)	align="center" \|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" \|100px	The interstice left by the Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.[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 }}
$\log_4 32 = \frac{5}{2}$	align="right" \| 2.50	3D quadratic Koch surface (type 2)	align="center" \|150px	Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
$\frac{\log\left(\frac{\sqrt{7}}{6}-\frac{1}{3}\right)}{\log(\sqrt2-1)}$	align="right" \| 2.529	Jerusalem cube	align="center" \| 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 $\sqrt{2}-1$ .
$\frac{\log 12}{\log(1+\varphi)}$	align="right" \| 2.5819	Icosahedron fractal	align="center" \|100px	Each icosahedron is replaced by 12 icosahedra. $\varphi = (1+\sqrt{5})/2$ (golden ratio).
$1 + \log_2 3$	align="right" \| 2.5849	3D Greek cross fractal	align="center" \|200px	Each segment is replaced by a cross formed by 6 segments.
$1 + \log_2 3$	align="right" \| 2.5849	Octahedron fractal	align="center" \|100px	Each octahedron is replaced by 6 octahedra.
$1 + \log_2 3$	align="right" \| 2.5849	von Koch surface	align="center" \|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".
$\frac{\log 3}{\log(3/2)}$	align="right" \| 2.7095	Von Koch in 3D	align="center" \| 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.{{cite web \|url=https://www.researchgate.net/publication/262600735 \|last=Baird \|first=Eric \|date=2014 \|title=The Koch curve in three dimensions \|via=ResearchGate}}
$\log_3 20$	align="right" \| 2.7268	Menger sponge	align="center" \| 100px	And its surface has a fractal dimension of $\log_3 20$ , which is the same as that by volume.
$\log_2 8 = 3$	align="right" \| 3	3D Hilbert curve	align="center" \| 100px	A Hilbert curve extended to 3 dimensions.
$\log_2 8 = 3$	align="right" \| 3	3D Lebesgue curve	align="center" \| 100px	A Lebesgue curve extended to 3 dimensions.
$\log_2 8 = 3$	align="right" \| 3	3D Moore curve	align="center" \| 100px	A Moore curve extended to 3 dimensions.
$\log_2 8 = 3$	align="right" \| 3	3D H-fractal	align="center" \| 120px	A H-fractal extended to 3 dimensions.{{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 }}
$3$ (conjectured)	align="right" \| {{formatnum:3}} (to be confirmed)	Mandelbulb	align="center" \|100px	Extension of the Mandelbrot set (power 9) in 3 dimensions[http://www.fractalforums.com/theory/hausdorff-dimension-of-the-mandelbulb/15/ Hausdorff dimension of the Mandelbulb]{{Unreliable source?\|date=September 2011}}

Random and natural fractals

Hausdorff dimension (exact value) \|\| Hausdorff dimension (approx.) \|\| Name \|\| Illustration \|\| width="40%" \| Remarks
class="wikitable sortable"
$\frac{1}{2}$	align="right" \| 0.5	Zeros of a Wiener process	align="center" \|150px	The zeros of a Wiener process (Brownian motion) are a nowhere dense set of Lebesgue measure 0 with a fractal structure.Peter Mörters, Yuval Peres, "Brownian Motion", Cambridge University Press, 2010
Solution of $E(C_1^s + C_2^s)=1$ where $E(C_1)=0.5$ and $E(C_2)=0.3$	align="right" \| 0.7499	a random Cantor set with 50% - 30%	align="center" \|150px	Generalization: at each iteration, the length of the left interval is defined with a random variable $C_1$ , a variable percentage of the length of the original interval. Same for the right interval, with a random variable $C_2$ . Its Hausdorff Dimension $s$ satisfies: $E(C_1^s + C_2^s)=1$ (where $E(X)$ is the expected value of $X$ ).
Solution of $s+1 = 12\cdot2^{-(s+1)}-6\cdot3^{-(s+1)}$	align="right"\|1.144...	von Koch curve with random interval	align="center"\| 200px	The length of the middle interval is a random variable with uniform distribution on the interval (0,1/3).
Measured	align="right"\|1.22 ± 0.02	Coastline of Ireland	align="center"\| 150px	Values for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault{{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}} at the University of Ulster and Theoretical Physics students at Trinity College, Dublin, under the supervision of 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) 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)
Measured	align="right"\|1.25	Coastline of Great Britain	align="center"\| 200px	Fractal dimension of the west coast of Great Britain, as measured by Lewis Fry Richardson and cited by Benoît Mandelbrot.[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
$\frac{\log 4}{\log 3}$	align="right" \| 1.2619	von Koch curve with random orientation	align="center" \| 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.
$\frac{4}{3}$	align="right" \| 1.333	Boundary of Brownian motion	align="center" \|150px	(cf. Mandelbrot, Lawler, Schramm, Werner).{{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 }}
$\frac{4}{3}$	align="right" \| 1.333	Polymer in 2D	align="center" \|	Similar to the Brownian motion in 2D with non-self-intersection.{{cite book \|first=Bernard \|last=Sapoval \|title=Universalités et fractales \|publisher=Flammarion-Champs \|year=2001 \|isbn=2-08-081466-4 }}
$\frac{4}{3}$	align="right" \| 1.333	Percolation front in 2D, Corrosion front in 2D	align="center" \| 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.
	align="right" \| 1.40	Clusters of clusters 2D	align="center" \|	When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.
$2-\frac{1}{2}$	align="right" \| 1.5	Graph of a regular Brownian function (Wiener process)	align="center" \| 150px	Graph of a function $f$ such that, for any two positive reals $x$ and $x+h$ , the difference of their images $f(x+h)-f(x)$ has the centered gaussian distribution with variance $h$ . Generalization: the fractional Brownian motion of index $\alpha$ follows the same definition but with a variance $h^{2\alpha}$ , in that case its Hausdorff dimension equals $2-\alpha$ .
Measured	align="right" \| 1.52	Coastline of Norway	align="center" \|100px	See J. Feder.Feder, J., "Fractals", Plenum Press, New York, (1988).
Measured	align="right" \| 1.55	Self-avoiding walk	align="center" \| 150px	Random walk in a square lattice that avoids visiting the same place twice, with a "go-back" routine for avoiding dead ends.
$\frac{5}{3}$	align="right" \| 1.66	Polymer in 3D	align="center" \|	Similar to the Brownian motion in a cubic lattice, but without self-intersection.
	align="right" \| 1.70	2D DLA Cluster	align="center" \| 150px	In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.
$\frac{\log(9\cdot0.75)}{\log 3}$	align="right" \| 1.7381	Fractal percolation with 75% probability	align="center" \|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 $\frac{\log(9p)}{\log 3}$ .
$\frac{7}{4}$	align="right" \| 1.75	2D percolation cluster hull	align="center" \| 150px	The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk,[http://deepblue.lib.umich.edu/handle/2027.42/27787 Hull-generating walks] or by Schramm-Loewner Evolution.
$\frac{91}{48}$	align="right" \| 1.8958	2D percolation cluster	align="center" \| 150px	In a square lattice, under the site percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48.{{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}} Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings".
$\frac{\log 2}{\log \sqrt{2}} = 2$	align="right" \| 2	Brownian motion	align="center" \| 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 clusters	align="center" \| 100px	From the 2005 results of the Sloan Digital Sky Survey.[https://arxiv.org/abs/astro-ph/0501583v2 Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey]
	align="right" \| 2.5	Balls of crumpled paper	align="center" \| 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.{{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 }} Creases will form at all size scales (see Universality (dynamical systems)).
	align="right" \| 2.50	3D DLA Cluster	align="center" \| 150px	In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.
	align="right" \| 2.50	Lichtenberg figure	align="center" \| 100px	Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.
$3-\frac{1}{2}$	align="right" \| 2.5	regular Brownian surface	align="center" \| 150px	A function $f:\mathbb{R}^2 \to \mathbb{R}$ , gives the height of a point $(x,y)$ such that, for two given positive increments $h$ and $k$ , then $f(x+h,y+k)-f(x,y)$ has a centered Gaussian distribution with variance $\sqrt{h^2 + k^2}$ . Generalization: the fractional Brownian surface of index $\alpha$ follows the same definition but with a variance $(h^2+k^2)^\alpha$ , in that case its Hausdorff dimension equals $3-\alpha$ .
Measured	align="right" \| 2.52	3D percolation cluster	align="center" \|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. Beyond that threshold, the cluster is infinite.
Measured and calculated	align="right" \|~2.7	The surface of Broccoli	align="center" \| 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.{{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}}
Measured	align="right" \| ~2.8	Surface of human brain	align="center" \| 100px	Measured with segmented three-dimensional high-resolution magnetic resonance images{{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 }}
Measured and calculated	align="right" \|~2.8	Cauliflower	align="center" \| 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.
	align="right" \| 2.97	Lung surface	align="center" \|100px	The alveoli of a lung form a fractal surface close to 3.
Calculated	align="right" \| $\in (0,2)$	Multiplicative cascade	align="center" \| 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.{{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 }}

Notes and references

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]

Hausdorff Dimension

Category:Mathematics-related lists

:List of fractals by Hausdorff dimension

Deterministic fractals

Random and natural fractals

See also

Notes and references

Further reading

External links