Fractal Examples Listed by Hausdorff Dimension
According to Falconer, one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension.^{[1]} 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.
Contents
Deterministic fractals
Hausdorff dimension (exact value)  Hausdorff dimension (approx.)  Name  Illustration  Remarks 

Calculated  0.538  Feigenbaum attractor  The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the logistic function for the critical parameter value , where the period doubling is infinite. This dimension is the same for any differentiable and unimodal function.^{[2]}  
0.6309  Cantor set  Built by removing the central third at each iteration. Nowhere dense and not a countable set.  
0.6942  Asymmetric Cantor set  The dimension is not , which is the generalized Cantor set with γ=1/4, which has the same length at each stage.^{[3]}
Built by removing the second quarter at each iteration. Nowhere dense and not a countable set. (golden cut).  
0.69897  Real numbers whose base 10 digits are even  Similar to the Cantor set.^{[1]}  
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.^{[4]}  
0<D<1  Generalized Cantor set  Built by removing at the ^{th} iteration the central interval of length from each remaining segment (of length ). At one obtains the usual Cantor set. Varying between 0 and 1 yields any fractal dimension .^{[5]}  
1  Smith–Volterra–Cantor set  Built by removing a central interval of length of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of ½.  
1  Takagi or Blancmange curve  Defined on the unit interval by , where is the sawtooth function. Special case of the TakahiLandsberg curve: with w = 1 / 2 {\displaystyle w=1/2} . The Hausdorff dimension equals for w {\displaystyle w} in . (Hunt cited by Mandelbrot^{[6]}).  
Calculated  1.0812  Julia set z² + 1/4  Julia set for c = 1/4.^{[7]}  
Solution s of 2  α  3 s +  α  4 s = 1 {\displaystyle 2\alpha ^{3s}+\alpha ^{4s}=1}  1.0933  Boundary of the Rauzy fractal  Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: , and .^{[8]}^{[9]} is one of the conjugated roots of .  
1.12915  contour of the Gosper island  Term used by Mandelbrot (1977).^{[10]} The Gosper island is the limit of the Gosper curve.  
Measured (box counting)  1.2  Dendrite Julia set  Julia set for parameters: Real = 0 and Imaginary = 1.  
1.2083  Fibonacci word fractal 60°  Build from the Fibonacci word. See also the standard Fibonacci word fractal. (golden ratio).  
1.2108  Boundary of the tame twindragon  One of the six 2reptiles in the plane (can be tiled by two copies of itself, of equal size).^{[11]}^{[12]}  
1.26  Hénon map  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.  
1.261859507  Triflake  Three antisnowflakes arranged in a way that a kochsnowflake forms in between the antisnowflakes.  
1.2619  Koch curve  3 Koch curves form the Koch snowflake or the antisnowflake.  
1.2619  boundary of Terdragon curve  Lsystem: same as dragon curve with angle = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle.  
1.2619  2D Cantor dust  Cantor set in 2 dimensions.  
1.2619  2D Lsystem branch  LSystems branching pattern having 4 new pieces scaled by 1/3. Generating the pattern using statistical instead of exact selfsimilarity yields the same fractal dimension.  
Calculated  1.2683  Julia set z^{2} − 1  Julia set for c = −1.^{[13]}  
1.3057  Apollonian gasket  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^{[14]}  
1.328  5 circles inversion fractal  The limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an Apollonian packing. See^{[15]}  
Calculated  1.3934  Douady rabbit  Julia set for c = −0,123 + 0.745i.^{[16]}  
1.4649  Vicsek fractal  Built by exchanging iteratively each square by a cross of 5 squares.  
1.4649  Quadratic von Koch curve (type 1)  One can recognize the pattern of the Vicsek fractal (above).  
1.4961  Quadric cross 
Images generated with Fractal Generator for ImageJ.  
(conjectured exact)  1.5000  a Weierstrass function:  The Hausdorff dimension of the Weierstrass function defined by with and has upper bound . 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.^{[1]}  
1.5000  Quadratic von Koch curve (type 2)  Also called "Minkowski sausage".  
1.5236  Boundary of the Dragon curve  cf. Chang & Zhang.^{[17]}^{[18]}  
1.5236  Boundary of the twindragon curve  Can be built with two dragon curves. One of the six 2reptiles in the plane (can be tiled by two copies of itself, of equal size).^{[11]}  
1.5849  3branches tree  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 2branches tree has a fractal dimension of only 1.  
1.5849  Sierpinski triangle  Also the triangle of Pascal modulo 2.  
1.5849  Sierpiński arrowhead curve  Same limit as the triangle (above) but built with a onedimensional curve.  
1.5849  Boundary of the TSquare fractal  The dimension of the fractal itself (not the boundary) is log 2 ( 4 ) = 2 {\displaystyle \log _{2}(4)=2} ^{[19]}  
1.61803  a golden dragon  Built from two similarities of ratios and , with . Its dimension equals because . With (Golden number).  
1.6309  Pascal triangle modulo 3  For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram^{[20]}).  
1.6309  Sierpinski Hexagon  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.  
1.6379  Fibonacci word fractal  Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F_{23} = 28657 segments).^{[21]} (golden ratio).  
Solution of  1.6402  Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3  Generalization : Providing the open set condition holds, the attractor of an iterated function system consisting of similarities of ratios , has Hausdorff dimension , solution of the equation coinciding with the iteration function of the Euclidean contraction factor: .^{[1]}  
1.6667  32segment quadric fractal (1/8 scaling rule)  [2] 


1.6826  Pascal triangle modulo 5  For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram^{[20]}).  
Measured (boxcounting)  1.7  Ikeda map attractor  For parameters a=1, b=0.9, k=0.4 and p=6 in the Ikeda map . It derives from a model of the planewave interactivity field in an optical ring laser. Different parameters yield different values.^{[22]}  
1.6990  50 segment quadric fractal (1/10 scaling rule)  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^{[23]}. 
 
1.7227  Pinwheel fractal  Built with Conway's Pinwheel tile.  
1.7712  Hexaflake  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).  
1.7848  Von Koch curve 85°  Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then .  
1.8272  A selfaffine fractal set  Build iteratively from a array on a square, with . Its Hausdorff dimension equals ^{[1]} with and is the number of elements in the ^{th} column. The boxcounting dimension yields a different formula, therefore, a different value. Unlike selfsimilar sets, the Hausdorff dimension of selfaffine sets depends on the position of the iterated elements and there is no formula, so far, for the general case.  
1.8617  Pentaflake  Built by exchanging iteratively each pentagon by a flake of 6 pentagons. (golden ratio).  
solution of  1.8687  Monkeys tree  This curve appeared in Benoit Mandelbrot's "Fractal geometry of Nature" (1983). It is based on 6 similarities of ratio and 5 similarities of ratio .^{[24]}  
log 3 ( 8 ) {\displaystyle \log _{3}(8)}  1.8928  Sierpinski carpet  Each face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1).  
1.8928  3D Cantor dust  Cantor set in 3 dimensions.  
1.8928  Cartesian product of the von Koch curve and the Cantor set  Generalization : Let F×G be the cartesian product of two fractals sets F and G. Then .^{[1]} See also the 2D Cantor dust and the Cantor cube.  
Estimated  1.9340  Boundary of the Lévy C curve  Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.  
1.974  Penrose tiling  See Ramachandrarao, Sinha & Sanyal.^{[25]}  
2  Boundary of the Mandelbrot set  The boundary and the set itself have the same dimension.^{[26]}  
2  Julia set  For determined values of c (including c belonging to the boundary of the Mandelbrot set), the Julia set has a dimension of 2.^{[27]}  
2  Sierpiński curve  Every Peano curve filling the plane has a Hausdorff dimension of 2.  
2  Hilbert curve  
2  Peano curve  And a family of curves built in a similar way, such as the Wunderlich curves.  
2  Moore curve  Can be extended in 3 dimensions.  
2  Lebesgue curve or zorder curve  Unlike the previous ones this spacefilling curve is differentiable almost everywhere. Another type can be defined in 2D. Like the Hilbert Curve it can be extended in 3D.^{[28]}  
2  Dragon curve  And its boundary has a fractal dimension of 1.5236270862.^{[29]}  
2  Terdragon curve  Lsystem: F → F + F – F, angle = 120°.  
2  Gosper curve  Its boundary is the Gosper island.  
Solution of  2  Curve filling the Koch snowflake  Proposed by Mandelbrot in 1982,^{[30]} it fills the Koch snowflake. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio 1 / 3 3 {\displaystyle 1/3{\sqrt {3}}} .  
2  Sierpiński tetrahedron  Each tetrahedron is replaced by 4 tetrahedra.  
2  Hfractal  Also the Mandelbrot tree which has a similar pattern.  
2  Pythagoras tree (fractal)  Every square generates two squares with a reduction ratio of .  
2  2D Greek cross fractal  Each segment is replaced by a cross formed by 4 segments.  
Measured  2.01 ±0.01  Rössler attractor  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.^{[31]}  
Measured  2.06 ±0.01  Lorenz attractor  For parameters ,=16 and . See McGuinness (1983)^{[32]}  
2.3219  Fractal pyramid  Each square pyramid is replaced by 5 halfsize square pyramids. (Different from the Sierpinski tetrahedron, which replaces each triangular pyramid with 4 halfsize triangular pyramids).  
2.3296  Dodecahedron fractal  Each dodecahedron is replaced by 20 dodecahedra. (golden ratio).  
2.3347  3D quadratic Koch surface (type 1)  Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.  
2.4739  Apollonian sphere packing  The interstice left by the Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.^{[33]}  
2.50  3D quadratic Koch surface (type 2)  Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.  
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 .  
2.529  Jerusalem cube  The iteration n is built with 8 cubes of iteration n1 (at the corners) and 12 cubes of iteration n2 (linking the corners). The contraction ratio is .  
2.5819  Icosahedron fractal  Each icosahedron is replaced by 12 icosahedra. (golden ratio).  
2.5849  3D Greek cross fractal  Each segment is replaced by a cross formed by 6 segments.  
2.5849  Octahedron fractal  Each octahedron is replaced by 6 octahedra.  
2.5849  von Koch surface  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".  
2.7095  Von Koch in 3D  Start with a 6sided polyhedron whose faces are isosceles triangles with sides of ratio 2:2:3 . Replace each polyhedron with 3 copies of itself, 2/3 smaller.^{[34]}  
2.7268  Menger sponge  And its surface has a fractal dimension of , which is the same as that by volume.  
3  3D Hilbert curve  A Hilbert curve extended to 3 dimensions.  
3  3D Lebesgue curve  A Lebesgue curve extended to 3 dimensions.  
3  3D Moore curve  A Moore curve extended to 3 dimensions.  
3  3D Hfractal  A Hfractal extended to 3 dimensions.^{[35]}  
(conjectured)  3 (to be confirmed)  Mandelbulb  Extension of the Mandelbrot set (power 8) in 3 dimensions^{[36]}^{[unreliable source?]} 
Random and natural fractals
Hausdorff dimension (exact value)  Hausdorff dimension (approx.)  Name  Illustration  Remarks 

1/2  0.5  Zeros of a Wiener process  The zeros of a Wiener process (Brownian motion) are a nowhere dense set of Lebesgue measure 0 with a fractal structure.^{[1]}^{[37]}  
Solution of where and  0.7499  a random Cantor set with 50%  30%  Generalization : At each iteration, the length of the left interval is defined with a random variable , a variable percentage of the length of the original interval. Same for the right interval, with a random variable . Its Hausdorff Dimension satisfies : . ( is the expected value of ).^{[1]}  
Solution of  1.144...  von Koch curve with random interval  The length of the middle interval is a random variable with uniform distribution on the interval (0,1/3).^{[1]}  
Measured  1.22±0.02  Coastline of Ireland  Values for the fractal dimension
of the entire coast of Ireland were determined by McCartney, Abernethy and Gault^{[38]} at the University of Ulster and Theoretical Physics students
at Trinity College, Dublin, under the supervision of S. Hutzler.^{[39]}
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)^{[39]}  
Measured  1.25  Coastline of Great Britain  Fractal dimension of the west coast of Great Britain, as measured by Lewis Fry Richardson and cited by Benoît Mandelbrot.^{[40]}  
1.2619  von Koch curve with random orientation  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.^{[1]}  
1.333  Boundary of Brownian motion  (cf. Mandelbrot, Lawler, Schramm, Werner).^{[41]}  
1.333  2D polymer  Similar to the brownian motion in 2D with non selfintersection.^{[42]}  
1.333  Percolation front in 2D, Corrosion front in 2D  Fractal dimension of the percolationbyinvasion front (accessible perimeter), at the percolation threshold (59.3%). It's also the fractal dimension of a stopped corrosion front.^{[42]}  
1.40  Clusters of clusters 2D  When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.^{[42]}  
1.5  Graph of a regular Brownian function (Wiener process)  Graph of a function f such that, for any two positive reals x and x+h, the difference of their images has the centered gaussian distribution with variance = h. Generalization : The fractional Brownian motion of index follows the same definition but with a variance , in that case its Hausdorff dimension =.^{[1]}  
Measured  1.52  Coastline of Norway  See J. Feder.^{[43]}  
Measured  1.55  Random walk with no selfintersection  Selfavoiding random walk in a square lattice, with a « goback » routine for avoiding dead ends.  
1.66  3D polymer  Similar to the brownian motion in a cubic lattice, but without selfintersection.^{[42]}  
1.70  2D DLA Cluster  In 2 dimensions, clusters formed by diffusionlimited aggregation, have a fractal dimension of around 1.70.^{[42]}  
1.7381  Fractal percolation with 75% probability  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 subsquares, each subsquare being retained with probability p. The "almost sure" Hausdorff dimension equals .^{[1]}  
7/4  1.75  2D percolation cluster hull  The hull or boundary of a percolation cluster. Can also be generated by a hullgenerating walk,^{[44]} or by SchrammLoewner Evolution.  
1.8958  2D percolation cluster  In a square lattice, under the site percolation threshold (59.3%) the percolationbyinvasion cluster has a fractal dimension of 91/48.^{[42]}^{[45]} Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings".  
2  Brownian motion  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  From the 2005 results of the Sloan Digital Sky Survey.^{[46]}  
2.33  Cauliflower  Every branch carries around 13 branches 3 times smaller.  
2.5  Balls of crumpled paper  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 noninteger exponent between 2 and 3 will be approximately proportional to the area of the sheets from which the balls have been made.^{[47]} Creases will form at all size scales (see Universality (dynamical systems)).  
2.50  3D DLA Cluster  In 3 dimensions, clusters formed by diffusionlimited aggregation, have a fractal dimension of around 2.50.^{[42]}  
2.50  Lichtenberg figure  Their appearance and growth appear to be related to the process of diffusionlimited aggregation or DLA.^{[42]}  
2.5  regular Brownian surface  A function , gives the height of a point such that, for two given positive increments and , then has a centered Gaussian distribution with variance = . Generalization : The fractional Brownian surface of index follows the same definition but with a variance = , in that case its Hausdorff dimension = .^{[1]}  
Measured  2.52  3D percolation cluster  In a cubic lattice, at the site percolation threshold (31.1%), the 3D percolationbyinvasion cluster has a fractal dimension of around 2.52.^{[45]} Beyond that threshold, the cluster is infinite.  
Measured  2.66  Broccoli  ^{[48]}  
2.79  Surface of human brain  ^{[49]}  
2.97  Lung surface  The alveoli of a lung form a fractal surface close to 3.^{[42]}  
Calculated  Multiplicative cascade  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.^{[50]} 
See also
Wikimedia Commons has media related to fractals. 
Notes and references
 Falconer, Kenneth (1990–2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd. xxv. ISBN 0470848626.
 Fractal dimension of the Feigenbaum attractor
 Tsang, K. Y. (1986). "Dimensionality of Strange Attractors Determined Analytically". Phys. Rev. Lett. 57 (12): 1390–1393. doi:10.1103/PhysRevLett.57.1390. PMID 10033437.
 Fractal dimension of the spectrum of the Fibonacci Hamiltonian
 The scattering from generalized Cantor fractals
 Mandelbrot, Benoit. Gaussian selfaffinity and Fractals. ISBN 0387989935.
 fractal dimension of the Julia set for c = 1/4
 Boundary of the Rauzy fractal
 Lothaire, M. (2005), Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, 105, Cambridge University Press, p. 525, ISBN 9780521848022, MR 2165687, Zbl 1133.68067, ISBN 9780521848022
 Gosper island on Mathworld
 On 2reptiles in the plane, Ngai, 1999
 Recurrent construction of the boundary of the dragon curve (for n=2, D=1)
 fractal dimension of the z²1 Julia set
 fractal dimension of the apollonian gasket
 fractal dimension of the 5 circles inversion fractal
 fractal dimension of the Douady rabbit
 Fractal dimension of the boundary of the dragon fractal
 Recurrent construction of the boundary of the dragon curve (for n=2, D=2)
 TSquare (fractal)
 Fractal dimension of the Pascal triangle modulo k
 The Fibonacci word fractal
 Estimating Fractal dimension
 Fractal Generator for ImageJ.
 Monkeys tree fractal curve
 Fractal dimension of a Penrose tiling
 Fractal dimension of the boundary of the Mandelbrot set
 Fractal dimension of certain Julia sets
 Lebesgue curve variants
 Complex base numeral systems
 "Penser les mathématiques", Seuil ISBN 2020060612 (1982)
 Fractals and the Rössler attractor
 The fractal dimension of the Lorenz attractor, Mc Guinness (1983)
 The Fractal dimension of the apollonian sphere packing
 [1]
 B. Hou; H. Xie; W. Wen & P. Sheng (2008). "Threedimensional metallic fractals and their photonic crystal characteristics". Phys. Rev. B 77, 125113.
 Hausdorff dimension of the Mandelbulb
 Peter Mörters, Yuval Peres, Oded Schramm, "Brownian Motion", Cambridge University Press, 2010
 McCartney, Mark; Abernethya, Gavin; Gaulta, Lisa (24 June 2010). "The Divider Dimension of the Irish Coast". Irish Geography. 43 (3): 277–284. doi:10.1080/00750778.2011.582632. Retrieved 4 December 2014.
 Hutzler, S. (2013). Fractal Ireland. Science Spin, 58, 1920.
 How long is the coast of Britain? Statistical selfsimilarity and fractional dimension, B. Mandelbrot
 Fractal dimension of the brownian motion boundary
 Bernard Sapoval "Universalités et fractales", FlammarionChamps (2001), ISBN 2080814664
 Feder, J., "Fractals,", Plenum Press, New York, (1988).
 Hullgenerating walks
 Applications of percolation theory by Muhammad Sahimi (1994)
 Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey
 "Power Law Relations". Yale. Retrieved 29 July 2010
 Fractal dimension of the broccoli
 Fractal dimension of the surface of the human brain
 [Meakin (1987)]
Further reading
 Benoît Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman & Co; ISBN 0716711869 (September 1982).
 HeinzOtto Peitgen, The Science of Fractal Images, Dietmar Saupe (editor), Springer Verlag, ISBN 0387966080 (August 1988)
 Michael F. Barnsley, Fractals Everywhere, Morgan Kaufmann; ISBN 0120790610
 Bernard Sapoval, « Universalités et fractales », collection Champs, Flammarion. ISBN 2080814664 (2001).