Hilbert manifold

According to Falconer, one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension. 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.

Deterministic fractals

Hausdorff dimension (exact value)	Hausdorff dimension (approx.)	Name	Illustration	Remarks
Calculated	0.538	Feigenbaum attractor	File:Feigenbaum attractor.png	The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the logistic function for the critical parameter value ${\lambda }_{\mathrm{\infty }}=3.570$, where the period doubling is infinite. This dimension is the same for any differentiable and unimodal function.
$\frac{\log (2)}{\log (3)}$	0.6309	Cantor set	File:Cantor set in seven iterations.svg	Built by removing the central third at each iteration. Nowhere dense and not a countable set.
$\frac{\log (\phi )}{\log (2)}=\frac{\log (1+\sqrt{5})}{\log (2)}-1$	0.6942	Asymmetric Cantor set	File:AsymmCantor.png	The dimension is not $\frac{\log (2)}{\log (\frac{8}{3})}$, as would be expected from the generalized Cantor set with γ=1/4, which has the same length at each stage. Built by removing the second quarter at each iteration. Nowhere dense and not a countable set. $\phi =(1+\sqrt{5})/2$ (golden ratio).
$\frac{\log (5)}{\log (10)}$	0.69897	Real numbers with even digits	File:Even digits.png	Similar to a Cantor set.
$\log (1+\sqrt{2})$	0.88137	Spectrum of Fibonacci Hamiltonian		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.
$-\frac{\log (2)}{\log (\frac{1-\gamma }{2})}$	0<D<1	Generalized Cantor set	File:Generalized cantor set.png	Built by removing at the $m$th iteration the central interval of length $\gamma l_{m-1}$ from each remaining segment (of length $l_{m-1}=(1-\gamma {)}^{m-1}/2^{m-1}$). At $\gamma =1/3$ one obtains the usual Cantor set. Varying $\gamma$ between 0 and 1 yields any fractal dimension $0<D<1$.
$1$	1	Smith–Volterra–Cantor set	File:Smith-Volterra-Cantor set.svg	Built by removing a central interval of length $1/2^{2n}$ of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of ½.
$2+\frac{\log (1/2)}{\log (2)}=1$	1	Takagi or Blancmange curve	File:Takagi curve.png	Defined on the unit interval by $f(x)={\sum }_{n=0}^{\mathrm{\infty }}\frac{s(2^{n}x)}{2^n}$, where $s(x)$ is the sawtooth function. Special case of the Takahi-Landsberg curve: $f(x)={\sum }_{n=0}^{\mathrm{\infty }}{w}^{n}s(2^{n}x)$ with $w=1/2$. The Hausdorff dimension equals $2+log(w)/log(2)$ for $w$ in $[1/2,1]$. (Hunt cited by Mandelbrot).
Calculated	1.0812	Julia set z² + 1/4	File:Julia z2+0,25.png	Julia set for c = 1/4.
Solution s of $2|\alpha {|}^{3s}+|\alpha {|}^{4s}=1$	1.0933	Boundary of the Rauzy fractal	File:Rauzy fractal.png	Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: $1\mapsto 12$, $2\mapsto 13$ and $3\mapsto 1$. $\alpha$ is one of the conjugated roots of $z^3-z^2-z-1=0$.
$2\frac{\log (3)}{\log (7)}$	1.12915	contour of the Gosper island	File:Gosper Island 4.svg	Term used by Mandelbrot (1977). The Gosper island is the limit of the Gosper curve.
Measured (box counting)	1.2	Dendrite Julia set	File:Dendrite julia.png	Julia set for parameters: Real = 0 and Imaginary = 1.
$3\frac{\log (\phi )}{\log \left(\frac{3+\sqrt{13}}{2}\right)}$	1.2083	Fibonacci word fractal 60°	File:Fibo 60deg F18.png	Build from the Fibonacci word. See also the standard Fibonacci word fractal. $\phi =(1+\sqrt{5})/2$ (golden ratio).
$\begin{array}{cc}& \frac{2\log \left(\frac{\sqrt[3]{27-3\sqrt{78}}+\sqrt[3]{27+3\sqrt{78}}}{3}\right)}{\log (2)},\\ & {}^{\text{or root of}}\\ & 2^x-1=2^{(2-x)/2}\end{array}$	1.2108	Boundary of the tame twindragon	File:TameTwindragontile.png	One of the six 2-rep-tiles in the plane (can be tiled by two copies of itself, of equal size).
	1.26	Hénon map	File:Henon attractor.png	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.
$\frac{\log (4)}{\log (3)}$	1.2619	Koch curve	File:Koch curve.svg	3 von Koch curves form the Koch snowflake or the anti-snowflake.
$\frac{\log (4)}{\log (3)}$	1.2619	boundary of Terdragon curve	File:Terdragon boundary.png	L-system: same as dragon curve with angle = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle.
$\frac{\log (4)}{\log (3)}$	1.2619	2D Cantor dust	File:Carre cantor.gif	Cantor set in 2 dimensions.
$\frac{\log (4)}{\log (3)}$	1.2619	2D L-system branch	File:Onetwosix.png	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	1.2683	Julia set z2 − 1	File:Julia z2-1.png	Julia set for c = −1.
	1.3057	Apollonian gasket	File:Apollonian gasket.svg	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
	1.328	5 circles inversion fractal	File:Cicle inversion.svg	The limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an Apollonian packing. See
Calculated	1.3934	Douady rabbit	File:Douady rabbit.png	Julia set for c = −0,123 + 0.745i.
$\frac{\log (5)}{\log (3)}$	1.4649	Vicsek fractal	File:Box fractal.svg	Built by exchanging iteratively each square by a cross of 5 squares.
$\frac{\log (5)}{\log (3)}$	1.4649	Quadratic von Koch curve (type 1)	File:Quadratic Koch 2.png	One can recognize the pattern of the Vicsek fractal (above).
$\frac{\log (\frac{1}{3})}{\log (\sqrt{5})}$	1.49	Quadric cross	<cross>...</cross>File:Quadriccross.gif
$2-\frac{\log (\sqrt{2})}{\log (2)}=\frac{3}{2}$ (conjectured exact)	1.5000	a Weierstrass function: $f(x)={\sum }_{k=1}^{\mathrm{\infty }}\frac{\sin (2^{k}x)}{{\sqrt{2}}^k}$	File:Weierstrass functionAMD.png	The Hausdorff dimension of the Weierstrass function $f:[0,1]\to ℝ$ defined by $f(x)={\sum }_{k=1}^{\mathrm{\infty }}\frac{\sin (b^{k}x)}{a^k}$ with $1<a<2$ and $b>1$ has upper bound $2-\log (a)/\log (b)$. 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.
$\frac{\log (8)}{\log (4)}=\frac{3}{2}$	1.5000	Quadratic von Koch curve (type 2)	File:Quadratic Koch.png	Also called "Minkowski sausage".
$\frac{\log \left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)}{\log (2)}$	1.5236	Boundary of the Dragon curve	File:Boundary dragon curve.png	cf. Chang & Zhang.
$\frac{\log \left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)}{\log (2)}$	1.5236	Boundary of the twindragon curve	File:Twindragontile.png	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).
$\frac{\log (3)}{\log (2)}$	1.5849	3-branches tree	File:Arbre 3 branches.pngFile:Arbre 3 branches2.png	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.
$\frac{\log (3)}{\log (2)}$	1.5849	Sierpinski triangle	File:Sierpinski8.svg	Also the triangle of Pascal modulo 2.
$\frac{\log (3)}{\log (2)}$	1.5849	Sierpiński arrowhead curve	File:PfeilspitzenFraktal.PNG	Same limit as the triangle (above) but built with a one-dimensional curve.
$\frac{\log (3)}{\log (2)}$	1.5849	Boundary of the T-Square fractal	File:T-Square fractal (evolution).png	The dimension of the fractal itself (not the boundary) is $\frac{\log (4)}{\log (2)}$
$\frac{\log \phi }{\log \sqrt[\phi ]{\phi }}=\phi$	1.61803	a golden dragon	File:Phi glito.png	Built from two similarities of ratios $r$ and $r^2$, with $r=1/{\phi }^{1/\phi }$. Its dimension equals $\phi$ because $(r^2{)}^{\phi }+r^{\phi }=1$. With $\phi =(1+\sqrt{5})/2$ (Golden number).
$1+\frac{\log 2}{\log 3}$	1.6309	Pascal triangle modulo 3	File:Pascal triangle modulo 3.png	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).
$\frac{\log (6)}{\log (3)}$	1.6309	Sierpinski Hexagon	File:Sierpinski hexagon.png	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.
$3\frac{\log (\phi )}{\log (1+\sqrt{2})}$	1.6379	Fibonacci word fractal	File:Fibonacci fractal F23 steps.png	Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F23 = 28657 segments). $\phi =(1+\sqrt{5})/2$ (golden ratio).
Solution of $(1/3{)}^s+(1/2{)}^s+(2/3{)}^s=1$	1.6402	Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3	File:IFS3sim3ratios.png	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 : ${\sum }_{k=1}^n{c}_k^s=1$.
$1+\frac{\log 3}{\log 5}$	1.6826	Pascal triangle modulo 5	File:Pascal triangle modulo 5.png	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)	1.7	Ikeda map attractor	File:Ikeda map a=1 b=0.9 k=0.4 p=6.jpg	For parameters a=1, b=0.9, k=0.4 and p=6 in the Ikeda map $z_{n+1}=a+b{z}_{n}exp[i[k-p/(1+\lfloor z_n{\rfloor }^2)]]$. It derives from a model of the plane-wave interactivity field in an optical ring laser. Different parameters yield different values.
$\frac{\log (50)}{\log (10)}$	1.7	50 segment quadric fractal	File:50seg.tif	Built with ImageJ
$\frac{4\log (2)}{\log (5)}$	1.7227	Pinwheel fractal	File:Pinwheel fractal.png	Built with Conway's Pinwheel tile.
$\frac{\log (7)}{\log (3)}$	1.7712	Hexaflake	File:Flocon hexagonal.gif	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).
$\frac{\log (4)}{\log (2(1+\cos (85^{\circ })))}$	1.7848	Von Koch curve 85°	File:Koch Curve 85degrees.png	Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then $\frac{\log (4)}{\log (2(1+\cos (a)))}\in [1,2]$.
$\frac{\log (3^{0.63}+2^{0.63})}{\log 2}$	1.8272	A self-affine fractal set	File:Self-affine set.png	Build iteratively from a $p\times q$ array on a square, with $p\le q$. Its Hausdorff dimension equals $\frac{\log \left(\sum _{k=1}^p{n}_k^a\right)}{\log p}$ with $a=\frac{\log p}{logq}$ 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.
$\frac{\log (6)}{\log (1+\phi )}$	1.8617	Pentaflake	File:Penta plexity.png	Built by exchanging iteratively each pentagon by a flake of 6 pentagons. $\phi =(1+\sqrt{5})/2$ (golden ratio).
solution of $6(1/3{)}^s+5{(1/3\sqrt{3})}^s=1$	1.8687	Monkeys tree	File:Monkeytree.svg	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 $1/3\sqrt{3}$.
$\frac{\log (8)}{\log (3)}$	1.8928	Sierpinski carpet	File:Sierpinski carpet 6.png	Each face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1).
$\frac{\log (8)}{\log (3)}$	1.8928	3D Cantor dust	File:Cantor3D3.png	Cantor set in 3 dimensions.
$\frac{\log (4)}{\log (3)}+\frac{\log (2)}{\log (3)}=\frac{\log (8)}{\log (3)}$	1.8928	Cartesian product of the von Koch curve and the Cantor set	File:Koch Cantor cartesian product.png	Generalization : Let F×G be the cartesian product of two fractals sets F and G. Then $Di{m}_H(F\times G)=Di{m}_H(F)+Di{m}_H(G)$. See also the 2D Cantor dust and the Cantor cube.
Estimated	1.9340	Boundary of the Lévy C curve	File:LevyFractal.png	Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
	1.974	Penrose tiling	File:Pen0305c.gif	See Ramachandrarao, Sinha & Sanyal.
$2$	2	Boundary of the Mandelbrot set	File:Boundary mandelbrot set.png	The boundary and the set itself have the same dimension.
$2$	2	Julia set	File:Juliadim2.png	For determined values of c (including c belonging to the boundary of the Mandelbrot set), the Julia set has a dimension of 2.
$2$	2	Sierpiński curve	File:Sierpinski-Curve-3.png	Every Peano curve filling the plane has a Hausdorff dimension of 2.
$2$	2	Hilbert curve	File:Hilbert curve 3.svg
$2$	2	Peano curve	File:Peano curve.png	And a family of curves built in a similar way, such as the Wunderlich curves.
$2$	2	Moore curve	File:Moore-curve-stages-1-through-4.svg	Can be extended in 3 dimensions.
	2	Lebesgue curve or z-order curve	File:Z-order curve.png	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.
$\frac{\log (2)}{\log (\sqrt{2})}=2$	2	Dragon curve	File:Courbe du dragon.png	And its boundary has a fractal dimension of 1.5236270862.
	2	Terdragon curve	File:Terdragon curve.png	L-system: F → F + F – F, angle = 120°.
$\frac{\log (4)}{\log (2)}=2$	2	Gosper curve	File:Gosper curve 3.svg	Its boundary is the Gosper island.
Solution of $7(1/3{)}^s+6(1/3\sqrt{3}{)}^s=1$	2	Curve filling the Koch snowflake	File:Mandeltree.svg	Proposed by Mandelbrot in 1982, it fills the Koch snowflake. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio $1/3\sqrt{3}$.
$\frac{\log (4)}{\log (2)}=2$	2	Sierpiński tetrahedron	File:Tetraedre Sierpinski.png	Each tetrahedron is replaced by 4 tetrahedra.
$\frac{\log (4)}{\log (2)}=2$	2	H-fractal	File:H fractal2.png	Also the Mandelbrot tree which has a similar pattern.
$\frac{\log (2)}{\log (2/\sqrt{2})}=2$	2	Pythagoras tree (fractal)	File:PythagorasTree.png	Every square generates two squares with a reduction ratio of sqrt(2)/2.
$\frac{\log (4)}{\log (2)}=2$	2	2D Greek cross fractal	File:Greek cross fractal stage 4.svg	Each segment is replaced by a cross formed by 4 segments.
Measured	2.01 ±0.01	Rössler attractor	File:Roessler attractor.png	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.
Measured	2.06 ±0.01	Lorenz attractor	File:Lorenz attractor.png	For parameters v=40,$\sigma$=16 and b=4 . See McGuinness (1983)
$\frac{\log (5)}{\log (2)}$	2.3219	Fractal pyramid	File:Fractal pyramid.jpg	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).
$\frac{\log (20)}{\log (2+\phi )}$	2.3296	Dodecahedron fractal	File:Dodecaedron fractal.jpg	Each dodecahedron is replaced by 20 dodecahedra. $\phi =(1+\sqrt{5})/2$ (golden ratio).
$\frac{\log (13)}{\log (3)}$	2.3347	3D quadratic Koch surface (type 1)	File:Quadratic Koch 3D (type1 stage2).png	Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
	2.4739	Apollonian sphere packing	File:Apollonian spheres2.png	The interstice left by the Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.
$\frac{\log (32)}{\log (4)}=\frac{5}{2}$	2.50	3D quadratic Koch surface (type 2)	File:Quadratic Koch 3D (type2 stage2).png	Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
$\frac{\log (16)}{\log (3)}$	2.5237	Cantor tesseract	no image available	Cantor set in 4 dimensions. Generalization: in a space of dimension n, the Cantor set has a Hausdorff dimension of $n\frac{\log (2)}{\log (3)}$.
$\frac{\log (\frac{\sqrt{7}}{6}-\frac{1}{3})}{\log (\sqrt{2}-1)}$	2.529	Jerusalem cube	File:Jerusalem Cube.jpg	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 $\sqrt{2}-1$.
$\frac{\log (12)}{\log (1+\phi )}$	2.5819	Icosahedron fractal	File:Icosaedron fractal.jpg	Each icosahedron is replaced by 12 icosahedra. $\phi =(1+\sqrt{5})/2$ (golden ratio).
$\frac{\log (6)}{\log (2)}$	2.5849	3D Greek cross fractal	File:Greek cross 3D 1 through 4.png	Each segment is replaced by a cross formed by 6 segments.
$\frac{\log (6)}{\log (2)}$	2.5849	Octahedron fractal	File:Octaedron fractal.jpg	Each octahedron is replaced by 6 octahedra.
$\frac{\log (6)}{\log (2)}$	2.5849	von Koch surface	File:Koch surface 3.png	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".
$\frac{\log (20)}{\log (3)}$	2.7268	Menger sponge	File:Menger.png	And its surface has a fractal dimension of $\frac{\log (20)}{\log (3)}=2.7268$, which is the same as that by volume.
$\frac{\log (8)}{\log (2)}=3$	3	3D Hilbert curve	File:Hilbert3d-step3.png	A Hilbert curve extended to 3 dimensions.
$\frac{\log (8)}{\log (2)}=3$	3	3D Lebesgue curve	File:Lebesgue-3d-step3.png	A Lebesgue curve extended to 3 dimensions.
$\frac{\log (8)}{\log (2)}=3$	3	3D Moore curve	File:Moore3d-step3.png	A Moore curve extended to 3 dimensions.
$\frac{\log (8)}{\log (2)}=3$	3	3D H-fractal	File:3D H-fractal.png	A H-fractal extended to 3 dimensions.
$3$ (to be confirmed)	3 (to be confirmed)	Mandelbulb	File:Mandelbulb 5 iterations.png	Extension of the Mandelbrot set (power 8) in 3 dimensionsTemplate:Verify credibility

Random and natural fractals

Hausdorff dimension (exact value)	Hausdorff dimension (approx.)	Name	Illustration	Remarks
1/2	0.5	Zeros of a Wiener process	File:Wiener process set of zeros.gif	The zeros of a Wiener process (Brownian motion) are a nowhere dense set of Lebesgue measure 0 with a fractal structure.
Solution of $E(C_1^s+C_2^s)=1$ where $E(C_1)=0.5$ and $E(C_2)=0.3$	0.7499	a random Cantor set with 50% - 30%	File:Random Cantor set.png	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 : $E(C_1^s+C_2^s)=1$. ($E(X)$ is the expected value of $X$).
Solution of $s+1=12\ast 2^{-(s+1)}-6\ast 3^{-(s+1)}$	1.144...	von Koch curve with random interval	File:Random interval koch.png	The length of the middle interval is a random variable with uniform distribution on the interval (0,1/3).
Measured	1.22±0.02	Coastline of Ireland	File:Ireland (MODIS).jpg	Values for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault at the University of Ulster and Theoretical Physics students at Trinity College, Dublin, under the supervision of S. Hutzler. 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	1.25	Coastline of Great Britain	File:Britain-fractal-coastline-combined.jpg	Fractal dimension of the west coast of Great Britain, as measured by Lewis Fry Richardson and cited by Benoît Mandelbrot.
$\frac{\log (4)}{\log (3)}$	1.2619	von Koch curve with random orientation	File:Random orientation koch.png	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.
$\frac{4}{3}$	1.333	Boundary of Brownian motion	File:Front mouvt brownien.png	(cf. Mandelbrot, Lawler, Schramm, Werner).
$\frac{4}{3}$	1.333	2D polymer		Similar to the brownian motion in 2D with non self-intersection.
$\frac{4}{3}$	1.333	Percolation front in 2D, Corrosion front in 2D	File:Front de percolation.png	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.
	1.40	Clusters of clusters 2D		When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.
$2-\frac{1}{2}$	1.5	Graph of a regular Brownian function (Wiener process)	File:Wiener process zoom.png	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 =$2-\alpha$.
Measured	1.52	Coastline of Norway	File:Norway municipalities.png	See J. Feder.
Measured	1.55	Random walk with no self-intersection	File:Polymer 2D.png	Self-avoiding random walk in a square lattice, with a « go-back » routine for avoiding dead ends.
$\frac{5}{3}$	1.66	3D polymer		Similar to the brownian motion in a cubic lattice, but without self-intersection.
	1.70	2D DLA Cluster	File:Agregation limitee par diffusion.png	In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.
$\frac{\log (9\ast 0.75)}{\log (3)}$	1.7381	Fractal percolation with 75% probability	File:Fractal percolation 75.png	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 $\frac{\log (9p)}{\log (3)}$.
7/4	1.75	2D percolation cluster hull	File:PercolationHull.png	The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk, or by Schramm-Loewner Evolution.
$\frac{91}{48}$	1.8958	2D percolation cluster	File:Amas de percolation.png	In a square lattice, under the site percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48. Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the « clearings ».
$\frac{\log (2)}{\log (\sqrt{2})}=2$	2	Brownian motion	File:Mouvt brownien2.png	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	Around 2	Distribution of galaxy clusters	File:Abell 1835 Hubble.jpg	From the 2005 results of the Sloan Digital Sky Survey.
$\frac{\log (13)}{\log (3)}$	2.33	Cauliflower	File:Blumenkohl-1.jpg	Every branch carries around 13 branches 3 times smaller.
	2.5	Balls of crumpled paper	File:Paperball.png	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. Creases will form at all size scales (see Universality (dynamical systems)).
	2.50	3D DLA Cluster	File:3D diffusion-limited aggregation2.jpg	In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.
	2.50	Lichtenberg figure	File:PlanePair2.jpg	Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.
$3-\frac{1}{2}$	2.5	regular Brownian surface	File:Brownian surface.png	A function $f:{ℝ}^2->ℝ$, 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 = $3-\alpha$.
Measured	2.52	3D percolation cluster	File:3Dpercolation.png	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	2.66	Broccoli	File:Broccoli DSC00862.png
	2.79	Surface of human brain	File:Cerebellum NIH.png
	2.97	Lung surface	File:Thorax Lung 3d (2).jpg	The alveoli of a lung form a fractal surface close to 3.
Calculated	$\in (0,2)$	Multiplicative cascade	File:3fractals2.jpg	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.

See also

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• Fractal dimension
• Hausdorff dimension
• Scale invariance

Notes and references

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• Benoît Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman & Co; ISBN 0-7167-1186-9 (September 1982).
• Heinz-Otto Peitgen, The Science of Fractal Images, Dietmar Saupe (editor), Springer Verlag, ISBN 0-387-96608-0 (August 1988)
• Michael F. Barnsley, Fractals Everywhere, Morgan Kaufmann; ISBN 0-12-079061-0
• Bernard Sapoval, « Universalités et fractales », collection Champs, Flammarion. ISBN 2-08-081466-4 (2001).

External links

• The fractals on Mathworld
• Other fractals on Paul Bourke's website
• Soler's Gallery
• Fractals on mathcurve.com
• 1000fractales.free.fr - Project gathering fractals created with various softwares
• Fractals unleashed

30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí.