List of Fractals By Hausdorff Dimension - Random and Natural Fractals

Random and Natural Fractals

Hausdorf dimension
(exact value)		 Hausdorf 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.
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 ).
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).
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.
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.
1.333 Boundary of Brownian motion (cf. Mandelbrot, Lawler, Schramm, Werner).
1.333 2D polymer Similar to the brownian motion in 2D with non self-intersection.
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.
1.40 Clusters of clusters 2D When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.
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 =.
Measured 1.52 Coastline of Norway See J. Feder.
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.
1.70 2D DLA Cluster In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.
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 .
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, 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. 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.
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. 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.
2.50 Lichtenberg figure Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.
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 = .
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. Beyond that threshold, the cluster is infinite.
Measured 2.66 Broccoli
2.79 Surface of human brain
2.97 Lung surface The alveoli of a lung form a fractal surface close to 3.
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.

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