According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."[1]
Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension.
The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the logistic map for the critical parameter value , where the period doubling is infinite. This dimension is the same for any differentiable and unimodal function.[2]
Built by removing the central interval of length from each remaining interval of length at the nth iteration. produces the usual middle-third Cantor set. Varying between 0 and 1 yields any fractal dimension .[3]
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.[6][page needed]
Defined on the unit interval by , where is the triangle wave function. Not a fractal under Mandelbrot's definition, because its topological dimension is also .[7] Special case of the Takahi-Landsberg curve: with . The Hausdorff dimension equals for in . (Hunt cited by Mandelbrot[8]).
Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: , and .[10][page needed][11] is one of the conjugated roots of .
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.
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.
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]
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).[13]
1.5850
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.
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: .[5]
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.
For parameters a=1, b=0.9, k=0.4 and p=6 in the Ikeda map . It derives from a model of the plane-wave 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].
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).
1.7712
Fractal H-I de Rivera
Starting from a unit square dividing its dimensions into three equal parts to form nine self-similar squares with the first square, two middle squares (the one that is above and the one below the central square) are removed in each of the seven squares not eliminated the process is repeated, so it continues indefinitely.
Build iteratively from a p-by-q array on a square, with . Its Hausdorff dimension equals [5] with and is the number of elements in the 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.
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 .[25]
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.[28]
Each triangle is replaced by 6 triangles, of which 4 identical triangles form a diamond based pyramid and the remaining two remain flat with lengths and relative to the pyramid triangles. The dimension is a parameter, self-intersection occurs for values greater than 2.3.[33]
2.3219
Fractal pyramid
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).
Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the first (blue block), second (plus green blocks), third (plus yellow blocks) and fourth (plus clear blocks) iterations.
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 .
Start with a 6-sided polyhedron whose faces are isosceles triangles with sides of ratio 2:2:3 . Replace each polyhedron with 3 copies of itself, 2/3 smaller.[35]
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: (where is the expected value of ).[5]
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)[40]
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.[5]
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.[43]
Graph of a function such that, for any two positive reals and , the difference of their images has the centered gaussian distribution with variance . Generalization: the fractional Brownian motion of index follows the same definition but with a variance , in that case its Hausdorff dimension equals .[5]
In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.[43]
1.7381
Fractal percolation with 75% probability
The fractal percolation model is constructed by the progressive replacement of each square by a 3-by-3 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 .[5]
1.75
2D percolation cluster hull
The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk,[45] or by Schramm-Loewner Evolution.
In a square lattice, under the site percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48.[43][46] Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings".
From the 2005 results of the Sloan Digital Sky Survey.[47]
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.[48] Creases will form at all size scales (see Universality (dynamical systems)).
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 equals .[5]
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.[46] Beyond that threshold, the cluster is infinite.
San-Hoon Kim used a direct scanning method and a mathematical analysis of the cross section of a cauliflower to conclude that the fractal dimension of it is ~2.8.[49]
2.97
Lung surface
The alveoli of a lung form a fractal surface close to 3.[43]
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.[51]
^Cherny, A. Yu; Anitas, E.M.; Kuklin, A.I.; Balasoiu, M.; Osipov, V.A. (2010). "The scattering from generalized Cantor fractals". J. Appl. Crystallogr. 43 (4): 790–7. arXiv:0911.2497. doi:10.1107/S0021889810014184. S2CID94779870.
^Peter Mörters, Yuval Peres, "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.
^ abHutzler, S. (2013). "Fractal Ireland". Science Spin. 58: 19–20. Retrieved 15 November 2016.
(See contents page, archived 26 July 2013)
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