List of fractals by Hausdorff dimension

Benoit Mandelbrot has stated that "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, 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
Calculated0.538Feigenbaum attractorThe 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]
0.6309Cantor setBuilt by removing the central third at each iteration. Nowhere dense and not a countable set.
0.6942Asymmetric Cantor setThe 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).

0.69897Real numbers whose base 10 digits are evenSimilar to the Cantor set.[4]
0.88137Spectrum of Fibonacci HamiltonianThe 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.[5]
0<D<1Generalized Cantor setBuilt 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 .[6]
1Smith–Volterra–Cantor setBuilt by removing a central interval of length of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of ½.
1Takagi or Blancmange curveDefined on the unit interval by , where is the triangle wave function. Special case of the Takahi-Landsberg curve: with . The Hausdorff dimension equals for in . (Hunt cited by Mandelbrot[7]).
Calculated1.0812Julia set z² + 1/4Julia set for c = 1/4.[8]
Solution s of 1.0933Boundary of the Rauzy fractalFractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: , and .[9][10] is one of the conjugated roots of .
1.12915contour of the Gosper islandTerm used by Mandelbrot (1977).[11] The Gosper island is the limit of the Gosper curve.
Measured (box counting)1.2Dendrite Julia setJulia set for parameters: Real = 0 and Imaginary = 1.
1.2083Fibonacci word fractal 60°Build from the Fibonacci word. See also the standard Fibonacci word fractal.

(golden ratio).

1.2108Boundary of the tame twindragonOne of the six 2-rep-tiles in the plane (can be tiled by two copies of itself, of equal size).[12][13]
1.26Hénon mapThe 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.
1.2619TriflakeThree anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
1.2619Koch curve3 Koch curves form the Koch snowflake or the anti-snowflake.
1.2619boundary of Terdragon curveL-system: same as dragon curve with angle = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle.
1.26192D Cantor dustCantor set in 2 dimensions.
1.26192D L-system branchL-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.
Calculated1.2683Julia set z2  1Julia set for c = 1.[8]
1.3057Apollonian gasketStarting 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[8]
1.3285 circles inversion fractalThe limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an Apollonian packing. See[14]
1.36521[15]Quadratic von Koch island using the type 1 curve as generatorAlso known as the Minkowski Sausage
Calculated1.3934Douady rabbitJulia set for c = 0,123 + 0.745i.[8]
1.4649Vicsek fractalBuilt by exchanging iteratively each square by a cross of 5 squares.
1.4649Quadratic von Koch curve (type 1)One can recognize the pattern of the Vicsek fractal (above).
1.4961Quadric crossBuilt 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.

(conjectured exact)1.5000a Weierstrass function: 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.[4]
1.5000Quadratic von Koch curve (type 2)Also called "Minkowski sausage".
1.5236Boundary of the Dragon curvecf. Chang & Zhang.[16][13]
1.5236Boundary of the twindragon curveCan 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).[12]
1.58493-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.
1.5849Sierpinski triangleAlso the triangle of Pascal modulo 2.
1.5849Sierpiński arrowhead curveSame limit as the triangle (above) but built with a one-dimensional curve.
1.5849Boundary of the T-square fractalThe dimension of the fractal itself (not the boundary) is
1.61803a golden dragonBuilt from two similarities of ratios and , with . Its dimension equals because . With (Golden number).
1.6309Pascal triangle modulo 3For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram[17]).
1.6309Sierpinski HexagonBuilt 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.
1.6379Fibonacci word fractalFractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F23 = 28657 segments).[18] (golden ratio).
Solution of 1.6402Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3Generalization : 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: .[4]
1.666732-segment quadric fractal (1/8 scaling rule) 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.
1.6826Pascal triangle modulo 5For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram[17]).
Measured (box-counting)1.7Ikeda map attractorFor 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.[19]
1.699050 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[20].
1.7227Pinwheel fractalBuilt with Conway's Pinwheel tile.
1.7712Sphinx fractalBuilt with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes.[21]
1.7712HexaflakeBuilt 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).
log (7)/log (3) 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.
1.7848Von Koch curve 85°Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then .
1.8272A self-affine fractal setBuild iteratively from a array on a square, with . Its Hausdorff dimension equals [4] 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.
1.8617PentaflakeBuilt by exchanging iteratively each pentagon by a flake of 6 pentagons. (golden ratio).
solution of 1.8687Monkeys treeThis curve appeared in Benoit Mandelbrot's "Fractal geometry of Nature" (1983). It is based on 6 similarities of ratio and 5 similarities of ratio .[22]
1.8928Sierpinski carpetEach face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1).
1.89283D Cantor dustCantor set in 3 dimensions.
1.8928Cartesian product of the von Koch curve and the Cantor setGeneralization : Let F×G be the cartesian product of two fractals sets F and G. Then .[4] See also the 2D Cantor dust and the Cantor cube.
where 1.9340Boundary of the Lévy C curveEstimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
2Penrose tilingSee Ramachandrarao, Sinha & Sanyal.[23]
2Boundary of the Mandelbrot setThe boundary and the set itself have the same Hausdorff dimension.[24]
2Julia setFor determined values of c (including c belonging to the boundary of the Mandelbrot set), the Julia set has a dimension of 2.[24]
2Sierpiński curveEvery Peano curve filling the plane has a Hausdorff dimension of 2.
2Hilbert curve
2Peano curveAnd a family of curves built in a similar way, such as the Wunderlich curves.
2Moore curveCan be extended in 3 dimensions.
2Lebesgue curve or z-order curveUnlike 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.[25]
2Dragon curveAnd its boundary has a fractal dimension of 1.5236270862.[26]
2Terdragon curveL-system: F  F + F  F, angle = 120°.
2Gosper curveIts boundary is the Gosper island.
Solution of 2Curve filling the Koch snowflakeProposed by Mandelbrot in 1982,[27] it fills the Koch snowflake. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio .
2Sierpiński tetrahedronEach tetrahedron is replaced by 4 tetrahedra.
2H-fractalAlso the Mandelbrot tree which has a similar pattern.
2Pythagoras tree (fractal)Every square generates two squares with a reduction ratio of .
22D Greek cross fractalEach segment is replaced by a cross formed by 4 segments.
Measured2.01 ±0.01Rössler attractorThe 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.[28]
Measured2.06 ±0.01Lorenz attractorFor parameters ,=16 and . See McGuinness (1983)[29]
2.3219Fractal pyramidEach 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).
2.3296Dodecahedron fractalEach dodecahedron is replaced by 20 dodecahedra. (golden ratio).
2<D<2.3Pyramid surfaceEach 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.[30]
2.33473D quadratic Koch surface (type 1)Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
2.4739Apollonian sphere packingThe interstice left by the Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.[31]
2.503D quadratic Koch surface (type 2)Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
2.529Jerusalem cubeThe 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 .
2.5819Icosahedron fractalEach icosahedron is replaced by 12 icosahedra. (golden ratio).
2.58493D Greek cross fractalEach segment is replaced by a cross formed by 6 segments.
2.5849Octahedron fractalEach octahedron is replaced by 6 octahedra.
2.5849von Koch surfaceEach 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".

2.7095Von Koch in 3DStart 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.[32]
2.7268Menger spongeAnd its surface has a fractal dimension of , which is the same as that by volume.
33D Hilbert curveA Hilbert curve extended to 3 dimensions.
33D Lebesgue curveA Lebesgue curve extended to 3 dimensions.
33D Moore curveA Moore curve extended to 3 dimensions.
33D H-fractalA H-fractal extended to 3 dimensions.[33]
(conjectured)3 (to be confirmed)MandelbulbExtension of the Mandelbrot set (power 8) in 3 dimensions[34]

Random and natural fractals

Hausdorff dimension
(exact value)
Hausdorff dimension
(approx.)
NameIllustrationRemarks
1/20.5Zeros of a Wiener processThe zeros of a Wiener process (Brownian motion) are a nowhere dense set of Lebesgue measure 0 with a fractal structure.[4][35]
Solution of where and 0.7499a 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: (where is the expected value of ).[4]
Solution of 1.144...von Koch curve with random intervalThe length of the middle interval is a random variable with uniform distribution on the interval (0,1/3).[4]
Measured1.22±0.02Coastline of IrelandValues for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault[36] at the University of Ulster and Theoretical Physics students at Trinity College, Dublin, under the supervision of S. Hutzler.[37]

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)[37]

Measured1.25Coastline of Great BritainFractal dimension of the west coast of Great Britain, as measured by Lewis Fry Richardson and cited by Benoît Mandelbrot.[38]
1.2619von Koch curve with random orientationOne 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.[4]
1.333Boundary of Brownian motion(cf. Mandelbrot, Lawler, Schramm, Werner).[39]
1.3332D polymerSimilar to the brownian motion in 2D with non-self-intersection.[40]
1.333Percolation front in 2D, Corrosion front in 2DFractal 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.[40]
1.40Clusters of clusters 2DWhen limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.[40]
1.5Graph of a regular Brownian function (Wiener process)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 .[4]
Measured1.52Coastline of NorwaySee J. Feder.[41]
Measured1.55Random walk with no self-intersectionSelf-avoiding random walk in a square lattice, with a "go-back" routine for avoiding dead ends.
1.663D polymerSimilar to the brownian motion in a cubic lattice, but without self-intersection.[40]
1.702D DLA ClusterIn 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.[40]
1.7381Fractal percolation with 75% probabilityThe fractal percolation model is constructed by the progressive replacement of each square by a 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 .[4]
7/41.752D percolation cluster hullThe hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk,[42] or by Schramm-Loewner Evolution.
1.89582D percolation clusterIn a square lattice, under the site percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48.[40][43] Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings".
2Brownian motionOr random walk. The Hausdorff dimensions equals 2 in 2D, in 3D and in all greater dimensions (K.Falconer "The geometry of fractal sets").
MeasuredAround 2Distribution of galaxy clustersFrom the 2005 results of the Sloan Digital Sky Survey.[44]
2.5Balls of crumpled paperWhen 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.[45] Creases will form at all size scales (see Universality (dynamical systems)).
2.503D DLA ClusterIn 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.[40]
2.50Lichtenberg figureTheir appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.[40]
2.5regular Brownian surfaceA 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 .[4]
Measured2.523D percolation clusterIn a cubic lattice, at the site percolation threshold (31.1%), the 3D percolation-by-invasion cluster has a fractal dimension of around 2.52.[43] Beyond that threshold, the cluster is infinite.
Measured and calculated~2.7The surface of BroccoliSan-Hoon Kim used a direct scanning method and a cross section analysis of a broccoli to conclude that the fractal dimension of it is ~2.7.[46]
2.79Surface of human brain[47]
Measured and calculated~2.8CauliflowerSan-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.[46]
2.97Lung surfaceThe alveoli of a lung form a fractal surface close to 3.[40]
CalculatedMultiplicative cascadeThis 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.[48]

See also

Notes and references

  1. Mandelbrot 1982, p. 15
  2. Aurell, Erik (May 1987). "On the metric properties of the Feigenbaum attractor". Journal of Statistical Physics. 47 (3–4): 439–458. doi:10.1007/BF01007519.
  3. Tsang, K. Y. (1986). "Dimensionality of Strange Attractors Determined Analytically". Phys. Rev. Lett. 57 (12): 1390–1393. Bibcode:1986PhRvL..57.1390T. doi:10.1103/PhysRevLett.57.1390. PMID 10033437.
  4. Falconer, Kenneth (1990–2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd. xxv. ISBN 978-0-470-84862-3.
  5. Damanik, D.; Embree, M.; Gorodetski, A.; Tcheremchantse, S. (2008). "The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian". Commun. Math. Phys. 280 (2): 499–516. arXiv:0705.0338. Bibcode:2008CMaPh.280..499D. doi:10.1007/s00220-008-0451-3.
  6. Cherny, A. Yu; Anitas, E.M.; Kuklin, A.I.; Balasoiu, M.; Osipov, V.A. (2010). "The scattering from generalized Cantor fractals". J. Appl. Cryst. 43 (4): 790–7. arXiv:0911.2497. doi:10.1107/S0021889810014184.
  7. Mandelbrot, Benoit (2002). Gaussian self-affinity and Fractals. ISBN 978-0-387-98993-8.
  8. McMullen, Curtis T. (3 October 1997). "Hausdorff dimension and conformal dynamics III: Computation of dimension", Abel.Math.Harvard.edu. Accessed: 27 October 2018.
  9. Messaoudi, Ali. Frontième de numération complexe", matwbn.icm.edu.pl. (in French) Accessed: 27 October 2018.
  10. Lothaire, M. (2005), Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, 105, Cambridge University Press, p. 525, ISBN 978-0-521-84802-2, MR 2165687, Zbl 1133.68067
  11. Weisstein, Eric W. "Gosper Island". MathWorld. Retrieved 27 October 2018.
  12. Ngai, Sirvent, Veerman, and Wang (October 2000). "On 2-Reptiles in the Plane 1999", Geometriae Dedicata, Volume 82. Accessed: 29 October 2018.
  13. Duda, Jarek (March 2011). "The Boundary of Periodic Iterated Function Systems", Wolfram.com.
  14. Chang, Angel and Zhang, Tianrong. On the Fractal Structure of the Boundary of Dragon Curve at the Wayback Machine (archived 22 September 2015) pdf
  15. Mandelbrot, B. B. (1983). The Fractal Geometry of Nature, p.48. New York: W. H. Freeman. ISBN 9780716711865. Cited in: Weisstein, Eric W. "Minkowski Sausage". MathWorld. Retrieved 22 September 2019.
  16. Fractal dimension of the boundary of the dragon fractal
  17. Fractal dimension of the Pascal triangle modulo k
  18. The Fibonacci word fractal
  19. Theiler, James (1990). "Estimating fractal dimension" (PDF). J. Opt. Soc. Am. A. 7 (6): 1055–73. doi:10.1364/JOSAA.7.001055.
  20. Fractal Generator for ImageJ Archived 20 March 2012 at the Wayback Machine.
  21. W. Trump, G. Huber, C. Knecht, R. Ziff, to be published
  22. Monkeys tree fractal curve Archived 21 September 2002 at Archive.today
  23. Fractal dimension of a Penrose tiling
  24. Shishikura, Mitsuhiro (1991). "The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets". arXiv:math/9201282. Cite journal requires |journal= (help)
  25. Lebesgue curve variants
  26. Duda, Jarek (2008). "Complex base numeral systems". arXiv:0712.1309v3. Bibcode:2007arXiv0712.1309D. Cite journal requires |journal= (help)
  27. Seuil (1982). Penser les mathématiques. ISBN 2-02-006061-2.
  28. Fractals and the Rössler attractor
  29. McGuinness, M.J. (1983). "The fractal dimension of the Lorenz attractor". Physics Letters. 99A: 5–9. doi:10.1016/0375-9601(83)90052-X.
  30. Lowe, Thomas (24 October 2016). "Three Variable Dimension Surfaces". ResearchGate.
  31. The Fractal dimension of the apollonian sphere packing Archived 6 May 2016 at the Wayback Machine
  32. Hou, B.; Xie, H.; Wen, W.; Sheng, P. (2008). "Three-dimensional metallic fractals and their photonic crystal characteristics" (PDF). Phys. Rev. B. 77 (12): 125113. Bibcode:2008PhRvB..77l5113H. doi:10.1103/PhysRevB.77.125113.
  33. Hausdorff dimension of the Mandelbulb
  34. Peter Mörters, Yuval Peres, Oded Schramm, "Brownian Motion", Cambridge University Press, 2010
  35. 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.
  36. Hutzler, S. (2013). "Fractal Ireland". Science Spin. 58: 19–20. Retrieved 15 November 2016. (See contents page, archived 26 July 2013)
  37. How long is the coast of Britain? Statistical self-similarity and fractional dimension, B. Mandelbrot
  38. Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin (2001). "The Dimension of the Planar Brownian Frontier is 4/3". Math. Res. Lett. 8 (4): 401–411. arXiv:math/0010165. Bibcode:2000math.....10165L. doi:10.4310/MRL.2001.v8.n4.a1.
  39. Sapoval, Bernard (2001). Universalités et fractales. Flammarion-Champs. ISBN 2-08-081466-4.
  40. Feder, J., "Fractals,", Plenum Press, New York, (1988).
  41. Hull-generating walks
  42. M Sahini; M Sahimi (2003). Applications Of Percolation Theory. CRC Press. ISBN 978-0-203-22153-2.
  43. Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey
  44. "Power Law Relations". Yale. Archived from the original on 28 June 2010. Retrieved 29 July 2010. Cite journal requires |journal= (help)
  45. Kim, Sang-Hoon (2 February 2008). "Fractal dimensions of a green broccoli and a white cauliflower". arXiv:cond-mat/0411597.
  46. Fractal dimension of the surface of the human brain
  47. [Meakin (1987)]

Further reading

  • Mandelbrot, Benoît (1982). The Fractal Geometry of Nature. W.H. Freeman. ISBN 0-7167-1186-9.
  • Peitgen, Heinz-Otto (1988). Saupe, Dietmar (ed.). The Science of Fractal Images. Springer Verlag. ISBN 0-387-96608-0.
  • Barnsley, Michael F. (1 January 1993). Fractals Everywhere. Morgan Kaufmann. ISBN 0-12-079061-0.
  • Sapoval, Bernard; Mandelbrot, Benoît B. (2001). Universalités et fractales: jeux d'enfant ou délits d'initié?. Flammarion-Champs. ISBN 2-08-081466-4.
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