### 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.

## Deterministic fractals

Hausdorff dimension
(exact value)
Hausdorff dimension
(approx.)
NameIllustrationRemarks
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 Takahi-Landsberg 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 2-rep-tiles 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 anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
1.2619 Koch curve 3 Koch curves form the Koch snowflake or the anti-snowflake.
1.2619 boundary of Terdragon curve L-system: 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 L-system branch 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 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).

 The quadric cross is made by scaling the 3-segment generator unit by 51/2 then adding 3 full scaled units, one to each original segment, plus a third of a scaled unit (blue) to increase the length of the pedestal of the starting 3-segment unit (purple). Built by replacing each end segment with a cross segment scaled by a factor of 51/2, consisting of 3 1/3 new segments, as illustrated in the inset. Images generated with Fractal Generator for ImageJ. 2 − log 2 ⁡ ( 2 ) = 3 2 {\displaystyle 2-\log _{2}({\sqrt {2}})={\frac {3}{2}}} (conjectured exact) 1.5000 a Weierstrass function: f ( x ) = ∑ k = 1 ∞ sin ⁡ ( 2 k x ) 2 k {\displaystyle \displaystyle f(x)=\sum _{k=1}^{\infty }{\frac {\sin(2^{k}x)}{{\sqrt {2}}^{k}}}} The Hausdorff dimension of the Weierstrass function f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\to \mathbb {R} } defined by f ( x ) = ∑ k = 1 ∞ a − k sin ⁡ ( b k x ) {\displaystyle f(x)=\sum _{k=1}^{\infty }a^{-k}\sin(b^{k}x)} with 1 < a < 2 {\displaystyle 1 1 {\displaystyle b>1} has upper bound 2 − log b ⁡ ( a ) {\displaystyle 2-\log _{b}(a)} . 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] log 4 ⁡ ( 8 ) = 3 2 {\displaystyle \log _{4}(8)={\frac {3}{2}}} 1.5000 Quadratic von Koch curve (type 2) Also called "Minkowski sausage". log 2 ⁡ ( 1 + 73 − 6 87 3 + 73 + 6 87 3 3 ) {\displaystyle \log _{2}\left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87}}}}+{\sqrt[{3}]{73+6{\sqrt {87}}}}}{3}}\right)} 1.5236 Boundary of the Dragon curve cf. Chang & Zhang.[17][18] log 2 ⁡ ( 1 + 73 − 6 87 3 + 73 + 6 87 3 3 ) {\displaystyle \log _{2}\left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87}}}}+{\sqrt[{3}]{73+6{\sqrt {87}}}}}{3}}\right)} 1.5236 Boundary of the twindragon curve 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).[11] log 2 ⁡ ( 3 ) {\displaystyle \log _{2}(3)} 1.5849 3-branches 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 2-branches tree has a fractal dimension of only 1. log 2 ⁡ ( 3 ) {\displaystyle \log _{2}(3)} 1.5849 Sierpinski triangle Also the triangle of Pascal modulo 2. log 2 ⁡ ( 3 ) {\displaystyle \log _{2}(3)} 1.5849 Sierpiński arrowhead curve Same limit as the triangle (above) but built with a one-dimensional curve. log 2 ⁡ ( 3 ) {\displaystyle \log _{2}(3)} 1.5849 Boundary of the T-Square fractal The dimension of the fractal itself (not the boundary) is log 2 ⁡ ( 4 ) = 2 {\displaystyle \log _{2}(4)=2} [19] log φ φ ⁡ ( φ ) = φ {\displaystyle \log _{\sqrt[{\varphi }]{\varphi }}(\varphi )=\varphi } 1.61803 a golden dragon Built from two similarities of ratios r {\displaystyle r} and r 2 {\displaystyle r^{2}} , with r = 1 / φ 1 / φ {\displaystyle r=1/\varphi ^{1/\varphi }} . Its dimension equals φ {\displaystyle \varphi } because ( r 2 ) φ + r φ = 1 {\displaystyle ({r^{2}})^{\varphi }+r^{\varphi }=1} . With φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} (Golden number). 1 + log 3 ⁡ ( 2 ) {\displaystyle 1+\log _{3}(2)} 1.6309 Pascal triangle modulo 3 For a triangle modulo k, if k is prime, the fractal dimension is 1 + log k ⁡ ( k + 1 2 ) {\displaystyle \scriptstyle {1+\log _{k}\left({\frac {k+1}{2}}\right)}} (cf. Stephen Wolfram[20]). 1 + log 3 ⁡ ( 2 ) {\displaystyle 1+\log _{3}(2)} 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. 3 log ⁡ ( φ ) log ⁡ ( 1 + 2 ) {\displaystyle 3{\frac {\log(\varphi )}{\log(1+{\sqrt {2}})}}} 1.6379 Fibonacci word fractal Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F23 = 28657 segments).[21] φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} (golden ratio). Solution of ( 1 / 3 ) s + ( 1 / 2 ) s + ( 2 / 3 ) s = 1 {\displaystyle (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 Generalization : Providing the open set condition holds, the attractor of an iterated function system consisting of n {\displaystyle n} similarities of ratios c n {\displaystyle c_{n}} , has Hausdorff dimension s {\displaystyle s} , solution of the equation coinciding with the iteration function of the Euclidean contraction factor: ∑ k = 1 n c k s = 1 {\displaystyle \sum _{k=1}^{n}c_{k}^{s}=1} .[1] log 8 ⁡ ( 32 ) = 5 3 {\displaystyle \log _{8}(32)={\frac {5}{3}}} 1.6667 32-segment quadric fractal (1/8 scaling rule) [2]
 Generator for 32 segment 1/8 scale quadric fractal. Built by scaling the 32 segment generator (see inset) by 1/8 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 32/log 8 = 1.6667. Images generated with Fractal Generator for ImageJ. 1 + log 5 ⁡ ( 3 ) {\displaystyle 1+\log _{5}(3)} 1.6826 Pascal triangle modulo 5 For a triangle modulo k, if k is prime, the fractal dimension is 1 + log k ⁡ ( k + 1 2 ) {\displaystyle \scriptstyle {1+\log _{k}\left({\frac {k+1}{2}}\right)}} (cf. Stephen Wolfram[20]). Measured (box-counting) 1.7 Ikeda map attractor 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 + ⌊ z n ⌋ 2 ) ] ] {\displaystyle 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.[22] 1 + log 10 ⁡ ( 5 ) {\displaystyle 1+\log _{10}(5)} 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].

## Random and natural fractals

Hausdorff dimension
(exact value)
Hausdorff dimension
(approx.)
NameIllustrationRemarks
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 self-intersection.[42]
1.333 Percolation front in 2D, Corrosion front in 2D 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.[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 self-intersection Self-avoiding random walk in a square lattice, with a « go-back » routine for avoiding dead ends.
1.66 3D polymer   Similar to the brownian motion in a cubic lattice, but without self-intersection.[42]
1.70 2D DLA Cluster In 2 dimensions, clusters formed by diffusion-limited 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 sub-squares, each sub-square 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 hull-generating walk,[44] or by Schramm-Loewner Evolution.
1.8958 2D percolation cluster In a square lattice, under the site percolation threshold (59.3%) the percolation-by-invasion 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 non-integer 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 diffusion-limited 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 diffusion-limited 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 percolation-by-invasion 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]

## Notes and references

1. [Meakin (1987)]

• 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).