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 
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. Notice that 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  Note that the dimension is not .^{[3]}
Built by removing the second quarter at each iteration. Nowhere dense and not a countable set. (golden ratio). 

0.69897  Real numbers with even digits  Similar to a Cantor set^{[1]}.  
0.88137  Spectrum of Fibonacci Hamiltonian  The study 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]}  
1  Smith–Volterra–Cantor set  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.  
1.0000  Takagi or Blancmange curve  Defined on the unit interval by , where s(x) is the sawtooth function. Special case of the TakahiLandsberg curve: with . The Hausdorff dimension equals 2 + log(w) / log(2) for w in . (Hunt cited by Mandelbrot ^{[5]} ).  
1.0686  contour of the Gosper island  
Calculated  1.0812  Julia set z² + 1/4  Julia set for c = 1/4. ^{[6]}  
Solution s of 2  α  ^{3s} +  α  ^{4s} = 1  1.0933  Boundary of the Rauzy fractal  Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: , and . ^{[7]}.α is one of the conjugated roots of z^{3} − z^{2} − z − 1 = 0.  
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). 

1.26  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.  
1.2619  Koch curve  3 von 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.  
Calculated  1.2683  Julia set z^{2} − 1  Julia set for c = −1. ^{[8]}  
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 ^{[9]}  
Calculated (Box counting)  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 ^{[10]}  
Calculated  1.3934  Douady rabbit  Julia set for c = −0,123 + 0.745i. ^{[11]}  
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).  
(conjectured exact)  1.5000  a Weierstrass function:  The Hausdorff dimension of the Weierstrass function defined by with 1 < a < 2 and b > 1 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  Dragon curve boundary  cf. Chang & Zhang.^{[12]}^{[13]}  
1.585  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.  
1.585  Sierpinski triangle  Also the triangle of Pascal modulo 2.  
1.585  Sierpiński arrowhead curve  Same limit as the triangle (above) but built with a onedimensional curve.  
1.61803  a golden dragon  Built from two similarities of ratios r and r^{2}, 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^{[14]}).  
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). ^{[15]}.
(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 n similarities of ratios c_{n}, has Hausdorff dimension s, solution of the equation : ^{[1]}.  
1.6826  Pascal triangle modulo 5  For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram^{[14]}).  
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 n_{k} is the number of elements in the k^{th} column. The boxcountig 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 1 / 3 and 5 similarities of ratio ^{[16]}.  
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.  
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^{[17]}.  
2  Boundary of the Mandelbrot set  The boundary and the set itself have the same dimension ^{[18]}.  
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. ^{[19]}.  
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.^{[20]}  
2  Dragon curve  And its boundary has a fractal dimension of 1.5236270862^{[21]}.  
2  Terdragon curve  Lsystem: F → F + F – F, angle = 120°.  
2  TSquare  
2  Gosper curve  Its boundary is the Gosper island.  
2  Sierpiński tetrahedron  Each tetrahedron is replaced by 4 tetrahedra.  
2  Hfractal  Also the « Mandelbrot tree » which has a similar pattern.  
2  Pythagoras tree  Every square generates 2 squares with a reduction ratio of sqrt(2)/2.  
2  2D Greek cross fractal  Each segment is replaced by a cross formed by 4 segments.  
2.06  Lorenz attractor  For parameters v=40,σ=16 and b=4 . see McGuinness (1983)^{[22]}  
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 apollolian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.^{[23]}  
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 .  
2.5819  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 triangle is replaced by 6 triangles, twice smaller.  
2.7268  Menger sponge  And its surface has a fractal dimension of .  
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. 
Hausdorf dimension (exact value) 
Hausdorf dimension (approx.) 
Name  Illustration  Remarks 

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 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(X) is the expected value of X).^{[1]}  
Solution of s + 1 = 12 * 2 ^{− (s + 1)} − 6 * 3 ^{− (s + 1)}  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  1.25  Coastline of Great Britain  Fractal dimension of the west coast of Great Britain, as measured by Lewis Fry Richardson and cited by Benoît Mandelbrot.^{[24]}  
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]}  
1.33  Boundary of Brownian motion  (cf. Lawler, Schramm, Werner).^{[25]}  
1.33  2D polymer  Similar to the brownian motion in 2D with non selfintersection. ^{[26]}.  
1.33  Percolation front in 2D, Corrosion front in 2D  Fractal dimension of the percolationbyinvasion front, at the percolation threshold (59.3%). It’s also the fractal dimension of a stopped corrosion front ^{[26]}.  
1.40  Clusters of clusters 2D  When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4. ^{[26]}  
1.5  Graph of a regular Brownian function  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 α follows the same definition but with a variance = h^{2α}, in that case its Hausdorff dimension =2 − α^{[1]}.  
Measured  1.52  Coastline of Norway  See J. Feder. ^{[27]}  
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 ^{[26]}.  
1.70  2D DLA Cluster  In 2 dimensions, clusters formed by diffusionlimited aggregation, have a fractal dimension of around 1.70 ^{[26]}.  
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]}.  
1.8958  2D percolation cluster  Under the percolation threshold (59.3%) the percolationbyinvasion cluster has a fractal dimension of 91/48 ^{[26]}. Beyond that threshold, le 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. See reference ^{[28]}  
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. [1] 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 ^{[26]}.  
2.50  Lichtenberg figure  Their appearance and growth appear to be related to the process of diffusionlimited aggregation or DLA ^{[26]}.  
2.5  regular Brownian surface  A function , gives the height of a point (x,y) such that, for two given positive increments h and k, then has a centered Gaussian distribution with variance = . Generalization : The fractional Brownian surface of index α follows the same definition but with a variance = (h^{2} + k^{2})^{α}, in that case its Hausdorff dimension = 3 − α^{[1]}.  
Measured  2.66  Broccoli  ^{[29]}  
2.79  Surface of human brain  ^{[30]}  
2.97  Lung surface  The alveoli of a lung form a fractal surface close to 3 ^{[26]}.  
Calculated  3  Quantum string drifting randomly  Hausdorff dimension of a quantum string whose representative point randomly drifts through loop space.^{[31]}  
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^{[32]}. 
