List of fractals by Hausdorff dimension

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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
Name Illustration Remarks
Calculated 0.538 Feigenbaum attractor 150px 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 200px Built by removing the central third at each iteration. Nowhere dense and not a countable set.
0.6942 Asymmetric Cantor set 200px The dimension is not , as would be expected from 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 ratio).

0.69897 Real numbers whose base 10 digits are even 200px Similar to a 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 200px 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 200px 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 150px Defined on the unit interval by , where is the sawtooth function. Special case of the Takahi-Landsberg curve: with . The Hausdorff dimension equals for in . (Hunt cited by Mandelbrot[6]).
Calculated 1.0812 Julia set z² + 1/4 100px Julia set for c = 1/4.[7]
Solution s of 1.0933 Boundary of the Rauzy fractal 150px 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 100px Term used by Mandelbrot (1977).[10] The Gosper island is the limit of the Gosper curve.
Measured (box counting) 1.2 Dendrite Julia set 150px Julia set for parameters: Real = 0 and Imaginary = 1.
1.2083 Fibonacci word fractal 60° 200px Build from the Fibonacci word. See also the standard Fibonacci word fractal.

(golden ratio).

1.2108 Boundary of the tame twindragon 150px 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 100px 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 150px Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
1.2619 Koch curve 200px 3 von Koch curves form the Koch snowflake or the anti-snowflake.
1.2619 boundary of Terdragon curve 150px 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 100px Cantor set in 2 dimensions.
1.2619 2D L-system branch 200px 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 200px Julia set for c = −1.[13]
1.3057 Apollonian gasket 100px 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 100px 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 150px Julia set for c = −0,123 + 0.745i.[16]
1.4649 Vicsek fractal 100px Built by exchanging iteratively each square by a cross of 5 squares.
1.4649 Quadratic von Koch curve (type 1) 150px One can recognize the pattern of the Vicsek fractal (above).
1.49 Quadric cross {{safesubst:#invoke:anchor|main}}150px
(conjectured exact) 1.5000 a Weierstrass function: 150px 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) 150px Also called "Minkowski sausage".
1.5236 Boundary of the Dragon curve 150px cf. Chang & Zhang.[17][18]
1.5236 Boundary of the twindragon curve 150px 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]
1.5849 3-branches tree 110px110px 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.
1.5849 Sierpinski triangle 100px Also the triangle of Pascal modulo 2.
1.5849 Sierpiński arrowhead curve 100px Same limit as the triangle (above) but built with a one-dimensional curve.
1.5849 Boundary of the T-Square fractal 200px The dimension of the fractal itself (not the boundary) is [19]
1.61803 a golden dragon 150px Built from two similarities of ratios and , with . Its dimension equals because . With (Golden number).
1.6309 Pascal triangle modulo 3 160px For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram[20]).
1.6309 Sierpinski Hexagon 150px 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 150px Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F23 = 28657 segments).[21] (golden ratio).
Solution of 1.6402 Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3 200px 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]
1.6826 Pascal triangle modulo 5 160px For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram[20]).
Measured (box-counting) 1.7 Ikeda map attractor 100px 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]
1.7 50 segment quadric fractal 150px Built with ImageJ[23]
1.7227 Pinwheel fractal 150px Built with Conway's Pinwheel tile.
1.7712 Hexaflake 100px 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° 150px Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then .
1.8272 A self-affine fractal set 200px 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 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.
1.8617 Pentaflake 100px Built by exchanging iteratively each pentagon by a flake of 6 pentagons. (golden ratio).
solution of 1.8687 Monkeys tree 100px 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]
1.8928 Sierpinski carpet 100px 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 200px Cantor set in 3 dimensions.
1.8928 Cartesian product of the von Koch curve and the Cantor set 150px 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 100px Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
1.974 Penrose tiling 100px See Ramachandrarao, Sinha & Sanyal.[25]
2 Boundary of the Mandelbrot set 100px The boundary and the set itself have the same dimension.[26]
2 Julia set 150px 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 100px Every Peano curve filling the plane has a Hausdorff dimension of 2.
2 Hilbert curve 100px
2 Peano curve 100px And a family of curves built in a similar way, such as the Wunderlich curves.
2 Moore curve 100px Can be extended in 3 dimensions.
2 Lebesgue curve or z-order curve 100px 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]
2 Dragon curve 150px And its boundary has a fractal dimension of 1.5236270862.[29]
2 Terdragon curve 150px L-system: F → F + F – F, angle = 120°.
2 Gosper curve 100px Its boundary is the Gosper island.
Solution of 2 Curve filling the Koch snowflake 100px 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 .
2 Sierpiński tetrahedron 80px Each tetrahedron is replaced by 4 tetrahedra.
2 H-fractal 150px Also the Mandelbrot tree which has a similar pattern.
2 Pythagoras tree (fractal) 150px Every square generates two squares with a reduction ratio of .
2 2D Greek cross fractal 100px Each segment is replaced by a cross formed by 4 segments.
Measured 2.01 ±0.01 Rössler attractor 100px 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 100px For parameters v=40,=16 and b=4 . See McGuinness (1983)[32]
2.3219 Fractal pyramid 100px 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).
2.3296 Dodecahedron fractal 100px Each dodecahedron is replaced by 20 dodecahedra. (golden ratio).
2.3347 3D quadratic Koch surface (type 1) 150px Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
2.4739 Apollonian sphere packing 100px 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) 150px 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 150px 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 .
2.5819 Icosahedron fractal 100px Each icosahedron is replaced by 12 icosahedra. (golden ratio).
2.5849 3D Greek cross fractal 200px Each segment is replaced by a cross formed by 6 segments.
2.5849 Octahedron fractal 100px Each octahedron is replaced by 6 octahedra.
2.5849 von Koch surface 150px 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".

2.7268 Menger sponge 100px And its surface has a fractal dimension of , which is the same as that by volume.
3 3D Hilbert curve 100px A Hilbert curve extended to 3 dimensions.
3 3D Lebesgue curve 100px A Lebesgue curve extended to 3 dimensions.
3 3D Moore curve 100px A Moore curve extended to 3 dimensions.
3 3D H-fractal 120px A H-fractal extended to 3 dimensions.[34]
(conjectured) 3 (to be confirmed) Mandelbulb 100px Extension of the Mandelbrot set (power 8) in 3 dimensions[35]Template:Verify credibility

Random and natural fractals

Hausdorff dimension
(exact value)
Hausdorff dimension
Name Illustration Remarks
1/2 0.5 Zeros of a Wiener process 150px The zeros of a Wiener process (Brownian motion) are a nowhere dense set of Lebesgue measure 0 with a fractal structure.[1][36]
Solution of where and 0.7499 a random Cantor set with 50% - 30% 150px 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 200px 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 150px Values for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault[37] at the University of Ulster and Theoretical Physics students at Trinity College, Dublin, under the supervision of S. Hutzler.[38]

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)[38]

Measured 1.25 Coastline of Great Britain 200px Fractal dimension of the west coast of Great Britain, as measured by Lewis Fry Richardson and cited by Benoît Mandelbrot.[39]
1.2619 von Koch curve with random orientation 200px 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 150px (cf. Mandelbrot, Lawler, Schramm, Werner).[40]
1.333 2D polymer Similar to the brownian motion in 2D with non self-intersection.[41]
1.333 Percolation front in 2D, Corrosion front in 2D 150px 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.[41]
1.40 Clusters of clusters 2D When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.[41]
1.5 Graph of a regular Brownian function (Wiener process) 150px 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 100px See J. Feder.[42]
Measured 1.55 Random walk with no self-intersection 150px 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.[41]
1.70 2D DLA Cluster 150px In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.[41]
1.7381 Fractal percolation with 75% probability 150px 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 150px The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk,[43] or by Schramm-Loewner Evolution.
1.8958 2D percolation cluster 150px In a square lattice, under the site percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48.[41][44] Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the « clearings ».
2 Brownian motion 150px 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 100px From the 2005 results of the Sloan Digital Sky Survey.[45]
2.33 Cauliflower 100px Every branch carries around 13 branches 3 times smaller.
2.5 Balls of crumpled paper 100px 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.[46] Creases will form at all size scales (see Universality (dynamical systems)).
2.50 3D DLA Cluster 150px In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.[41]
2.50 Lichtenberg figure 100px Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.[41]
2.5 regular Brownian surface 150px 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 225px 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.[44] Beyond that threshold, the cluster is infinite.
Measured 2.66 Broccoli 100px [47]
2.79 Surface of human brain 100px [48]
2.97 Lung surface 100px The alveoli of a lung form a fractal surface close to 3.[41]
Calculated Multiplicative cascade 150px 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.[49]

See also


Notes and references

  1. 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. Fractal dimension of the Feigenbaum attractor
  3. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  4. Fractal dimension of the spectrum of the Fibonacci Hamiltonian
  5. The scattering from generalized Cantor fractals
  6. {{#invoke:citation/CS1|citation |CitationClass=book }}
  7. fractal dimension of the Julia set for c = 1/4
  8. Boundary of the Rauzy fractal
  9. {{#invoke:citation/CS1|citation |CitationClass=citation }}
  10. Gosper island on Mathworld
  11. 11.0 11.1 On 2-reptiles in the plane, Ngai, 1999
  12. Recurrent construction of the boundary of the dragon curve (for n=2, D=1)
  13. fractal dimension of the z²-1 Julia set
  14. fractal dimension of the apollonian gasket
  15. fractal dimension of the 5 circles inversion fractal
  16. fractal dimension of the Douady rabbit
  17. Fractal dimension of the boundary of the dragon fractal
  18. Recurrent construction of the boundary of the dragon curve (for n=2, D=2)
  19. T-Square (fractal)
  20. 20.0 20.1 Fractal dimension of the Pascal triangle modulo k
  21. The Fibonacci word fractal
  22. Estimating Fractal dimension
  23. Fractal Generator for ImageJ.
  24. Monkeys tree fractal curve
  25. Fractal dimension of a Penrose tiling
  26. Fractal dimension of the boundary of the Mandelbrot set
  27. Fractal dimension of certain Julia sets
  28. Lebesgue curve variants
  29. Complex base numeral systems
  30. "Penser les mathématiques", Seuil ISBN 2-02-006061-2 (1982)
  31. Fractals and the Rössler attractor
  32. The fractal dimension of the Lorenz attractor, Mc Guinness (1983)
  33. Fractal dimension of the apollonian sphere packing
  34. Template:Cite paper
  35. Hausdorff dimension of the Mandelbulb
  36. Peter Mörters, Yuval Peres, Oded Schramm, "Brownian Motion", Cambridge University Press, 2010
  37. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  38. 38.0 38.1 Hutzler, S. (2013). Fractal Ireland. Science Spin, 58, 19-20.
  39. How long is the coast of Britain? Statistical self-similarity and fractional dimension, B. Mandelbrot
  40. Fractal dimension of the brownian motion boundary
  41. 41.0 41.1 41.2 41.3 41.4 41.5 41.6 41.7 41.8 Bernard Sapoval "Universalités et fractales", Flammarion-Champs (2001), ISBN=2-08-081466-4
  42. Feder, J., "Fractals,", Plenum Press, New York, (1988).
  43. Hull-generating walks
  44. 44.0 44.1 "Applications of percolation" theory by Muhammad Sahimi (1994)
  45. Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey
  46. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  47. Fractal dimension of the broccoli
  48. Fractal dimension of the surface of the human brain
  49. [Meakin (1987)]

Further reading

  • 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).

External links

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