List of fractals by Hausdorff dimension

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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[edit]

Hausdorff dimension
(exact value)
Hausdorff dimension
Name Illustration Remarks
Calculated 0.538 Feigenbaum attractor Feigenbaum attractor.png 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 Cantor set in seven iterations.svg Built by removing the central third at each iteration. Nowhere dense and not a countable set.
0.6942 Asymmetric Cantor set AsymmCantor.png The 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).

0.69897 Real numbers whose base 10 digits are even Even digits.png Similar to the Cantor set.[4]
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.[5][page needed]
0<D<1 Generalized Cantor set Generalized cantor set.png 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 .[6]
1 Smith–Volterra–Cantor set Smith-Volterra-Cantor set.svg 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 Takagi curve.png Defined 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]).
Calculated 1.0812 Julia set z² + 1/4 Julia z2+0,25.png Julia set for c = 1/4.[8]
Solution s of 1.0933 Boundary of the Rauzy fractal Rauzy fractal.png Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: , and .[9][page needed][10] is one of the conjugated roots of .
1.12915 contour of the Gosper island Gosper Island 4.svg Term used by Mandelbrot (1977).[11] The Gosper island is the limit of the Gosper curve.
Measured (box counting) 1.2 Dendrite Julia set Dendrite julia.png Julia set for parameters: Real = 0 and Imaginary = 1.
1.2083 Fibonacci word fractal 60° Fibo 60deg F18.png Build from the Fibonacci word. See also the standard Fibonacci word fractal.

(golden ratio).

1.2108 Boundary of the tame twindragon TameTwindragontile.png One of the six 2-rep-tiles in the plane (can be tiled by two copies of itself, of equal size).[12][13]
1.26 Hénon map Henon.jpg 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 Triflake Triflake.png Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
1.2619 Koch curve Koch curve.svg 3 Koch curves form the Koch snowflake or the anti-snowflake.
1.2619 boundary of Terdragon curve Terdragon boundary.png 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 Carre cantor.gif Cantor set in 2 dimensions.
1.2619 2D L-system branch Onetwosix.png 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 Julia z2-1.png Julia set for c = −1.[8]
1.3057 Apollonian gasket Apollonian gasket.svg 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[8]
1.328 5 circles inversion fractal Cicle inversion.svg The 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 generator Karperienflakeani2.gif Also known as the Minkowski Sausage
Calculated 1.3934 Douady rabbit Douady rabbit.png Julia set for c = −0,123 + 0.745i.[8]
1.4649 Vicsek fractal Box fractal.svg Built by exchanging iteratively each square by a cross of 5 squares.
1.4649 Quadratic von Koch curve (type 1) Quadratic Koch 2.svg One can recognize the pattern of the Vicsek fractal (above).
1.4961 Quadric cross Quadriccross.gif
The quadric cross is made by scaling the 3-segment generator unit by 51/2 then adding 3 full scaled units, one to each original segment, plus a third of a scaled unit (blue) to increase the length of the pedestal of the starting 3-segment unit (purple).
Built 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.

1.5000 a Weierstrass function: Weierstrass functionAMD.png The Hausdorff dimension of the Weierstrass function defined by with and is .[16][17]
1.5000 Quadratic von Koch curve (type 2) Quadratic Koch.svg Also called "Minkowski sausage".
1.5236 Boundary of the Dragon curve Boundary dragon curve.png cf. Chang & Zhang.[18][13]
1.5236 Boundary of the twindragon curve Twindragontile.png 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).[12]
1.5850 3-branches tree Arbre 3 branches.png Arbre 3 branches2.png 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.
1.5850 Sierpinski triangle Sierpinski8.svg Also the triangle of Pascal modulo 2.
1.5850 Sierpiński arrowhead curve PfeilspitzenFraktal.PNG Same limit as the triangle (above) but built with a one-dimensional curve.
1.5850 Boundary of the T-square fractal T-Square fractal (evolution).png The dimension of the fractal itself (not the boundary) is
1.61803 a golden dragon Phi glito.png Built from two similarities of ratios and , with . Its dimension equals because . With (Golden number).
1.6309 Pascal triangle modulo 3 Pascal triangle modulo 3.png For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram[19]).
1.6309 Sierpinski Hexagon Sierpinski hexagon 4th Iteration.svg 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 Fibonacci fractal F23 steps.png Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F23 = 28657 segments).[20] (golden ratio).
Solution of 1.6402 Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3 IFS3sim3ratios.png 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: .[4]
1.6667 32-segment quadric fractal (1/8 scaling rule) 8 scale fractal.png see also: File:32 Segment One Eighth Scale Quadric Fractal.jpg
Generator for 32 segment 1/8 scale quadric fractal.
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.
1.6826 Pascal triangle modulo 5 Pascal triangle modulo 5.png For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram[19]).
Measured (box-counting) 1.7 Ikeda map attractor Ikeda map a=1 b=0.9 k=0.4 p=6.jpg 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.[21]
1.6990 50 segment quadric fractal (1/10 scaling rule) 50seg.tif 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[22].
Generator for 50 Segment Fractal.
1.7227 Pinwheel fractal Pinwheel fractal.png Built with Conway's Pinwheel tile.
1.7712 Sphinx fractal Sphinx rep-tile fractal.gif Built with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes.[23]
1.7712 Hexaflake HexaFlake 5th Iteration Center.svg 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 Fractal H-I de Rivera.jpg 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.
1.7848 Von Koch curve 85° Koch Curve 85degrees.png 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 Self-affine set.png Build 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.
1.8617 Pentaflake Pentaflake-C 3rd Iteration Blue.svg Built by exchanging iteratively each pentagon by a flake of 6 pentagons. (golden ratio).
solution of 1.8687 Monkeys tree Monkeytree.svg 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 Sierpinski carpet 6.png 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 Cantor3D3.png Cantor set in 3 dimensions.
1.8928 Cartesian product of the von Koch curve and the Cantor set Koch Cantor cartesian product.png Generalization : 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 1.9340 Boundary of the Lévy C curve LevyFractal.png Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
2 Penrose tiling Pen0305c.gif See Ramachandrarao, Sinha & Sanyal.[25]
2 Boundary of the Mandelbrot set Boundary mandelbrot set.png The boundary and the set itself have the same Hausdorff dimension.[26]
2 Julia set Juliadim2.png For determined values of c (including c belonging to the boundary of the Mandelbrot set), the Julia set has a dimension of 2.[26]
2 Sierpiński curve Sierpinski-Curve-3.png Every Peano curve filling the plane has a Hausdorff dimension of 2.
2 Hilbert curve Hilbert curve 3.svg
2 Peano curve Peano curve.png And a family of curves built in a similar way, such as the Wunderlich curves.
2 Moore curve Moore-curve-stages-1-through-4.svg Can be extended in 3 dimensions.
2 Lebesgue curve or z-order curve Z-order curve.png 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.[27]
2 Dragon curve Courbe du dragon.png And its boundary has a fractal dimension of 1.5236270862.[28]
2 Terdragon curve Terdragon curve.png L-system: F → F + F – F, angle = 120°.
2 Gosper curve Gosper curve 3.svg Its boundary is the Gosper island.
Solution of 2 Curve filling the Koch snowflake Mandeltree.svg Proposed by Mandelbrot in 1982,[29] it fills the Koch snowflake. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio .
2 Sierpiński tetrahedron Tetraedre Sierpinski.png Each tetrahedron is replaced by 4 tetrahedra.
2 H-fractal H fractal2.png Also the Mandelbrot tree which has a similar pattern.
2 Pythagoras tree (fractal) PythagorasTree.png Every square generates two squares with a reduction ratio of .
2 2D Greek cross fractal Greek cross fractal stage 4.svg Each segment is replaced by a cross formed by 4 segments.
Measured 2.01 ±0.01 Rössler attractor Roessler attractor.png 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.[30]
Measured 2.06 ±0.01 Lorenz attractor Lorenz attractor.png For parameters ,=16 and . See McGuinness (1983)[31]
2<D<2.3 Pyramid surface Pyramid surface fractal.png 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.[32]
2.3219 Fractal pyramid Fractal pyramid.jpg 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 Dodecaedron fractal.jpg Each dodecahedron is replaced by 20 dodecahedra. (golden ratio).
2.3347 3D quadratic Koch surface (type 1) Koch quadratic 3d fractal.svg 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.
2.4739 Apollonian sphere packing Apollonian spheres2.png 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) Quadratic Koch 3D (type2 stage2).png Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
2.529 Jerusalem cube Jerusalem Cube.jpg 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 Icosaedron fractal.jpg Each icosahedron is replaced by 12 icosahedra. (golden ratio).
2.5849 3D Greek cross fractal Greek cross 3D 1 through 4.png Each segment is replaced by a cross formed by 6 segments.
2.5849 Octahedron fractal Octaedron fractal.jpg Each octahedron is replaced by 6 octahedra.
2.5849 von Koch surface Koch surface 3.png 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".

2.7095 Von Koch in 3D Koch Curve in Three Dimensions ("Delta" fractal).jpg 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.[34]
2.7268 Menger sponge Menger.png And its surface has a fractal dimension of , which is the same as that by volume.
3 3D Hilbert curve Hilbert3d-step3.png A Hilbert curve extended to 3 dimensions.
3 3D Lebesgue curve Lebesgue-3d-step3.png A Lebesgue curve extended to 3 dimensions.
3 3D Moore curve Moore3d-step3.png A Moore curve extended to 3 dimensions.
3 3D H-fractal 3D H-fractal.png A H-fractal extended to 3 dimensions.[35]
(conjectured) 3 (to be confirmed) Mandelbulb Mandelbulb 1,6GP Overview 20211110 002.png Extension of the Mandelbrot set (power 9) in 3 dimensions[36][unreliable source?]

Random and natural fractals[edit]

Hausdorff dimension
(exact value)
Hausdorff dimension
Name Illustration Remarks
1/2 0.5 Zeros of a Wiener process Wiener process set of zeros.gif The zeros of a Wiener process (Brownian motion) are a nowhere dense set of Lebesgue measure 0 with a fractal structure.[4][37]
Solution of where and 0.7499 a random Cantor set with 50% - 30% Random Cantor set.png 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 interval Random interval koch.png The length of the middle interval is a random variable with uniform distribution on the interval (0,1/3).[4]
Measured 1.22±0.02 Coastline of Ireland Ireland (MODIS).jpg Values for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault[38] at the University of Ulster and Theoretical Physics students at Trinity College, Dublin, under the supervision of S. Hutzler.[39]

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)[39]

Measured 1.25 Coastline of Great Britain Britain-fractal-coastline-combined.jpg Fractal dimension of the west coast of Great Britain, as measured by Lewis Fry Richardson and cited by Benoît Mandelbrot.[40]
1.2619 von Koch curve with random orientation Random orientation koch.png 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.[4]
1.333 Boundary of Brownian motion Front mouvt brownien.png (cf. Mandelbrot, Lawler, Schramm, Werner).[41]
1.333 Polymer in 2D Similar to the Brownian motion in 2D with non-self-intersection.[42]
1.333 Percolation front in 2D, Corrosion front in 2D Front de percolation.png 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.[42]
1.40 Clusters of clusters 2D When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.[42]
1.5 Graph of a regular Brownian function (Wiener process) Wiener process zoom.png 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]
Measured 1.52 Coastline of Norway Norway municipalities 2020 blank.svg See J. Feder.[43]
Measured 1.55 Self-avoiding walk Polymer 2D.png Random walk in a square lattice that avoids visiting the same place twice, with a "go-back" routine for avoiding dead ends.
1.66 Polymer in 3D Similar to the Brownian motion in a cubic lattice, but without self-intersection.[42]
1.70 2D DLA Cluster Aggregation limitee par diffusion.png In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.[42]
1.7381 Fractal percolation with 75% probability Fractal percolation 75.png The 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]
7/4 1.75 2D percolation cluster hull PercolationHull.png The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk,[44] or by Schramm-Loewner Evolution.
1.8958 2D percolation cluster Amas de percolation.png In a square lattice, under the site percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48.[42][45] Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings".
2 Brownian motion Mouvt brownien2.png 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 Abell 1835 Hubble.jpg From the 2005 results of the Sloan Digital Sky Survey.[46]
2.5 Balls of crumpled paper Paperball.png 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.[47] Creases will form at all size scales (see Universality (dynamical systems)).
2.50 3D DLA Cluster 3D DLA.jpg In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.[42]
2.50 Lichtenberg figure PlanePair2.jpg Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.[42]
2.5 regular Brownian surface Brownian surface.png 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 .[4]
Measured 2.52 3D percolation cluster 3Dpercolation.png 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.[45] Beyond that threshold, the cluster is infinite.
Measured and calculated ~2.7 The surface of Broccoli Broccoli DSC00862.png San-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.[48]
2.79 Surface of human brain Cerebellum NIH.png [49][failed verification]
Measured and calculated ~2.8 Cauliflower Blumenkohl-1.jpg 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.[48]
2.97 Lung surface Thorax Lung 3d (2).jpg The alveoli of a lung form a fractal surface close to 3.[42]
Calculated Multiplicative cascade 3fractals2.jpg 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.[50][full citation needed]

See also[edit]

Notes and references[edit]

  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. Bibcode:1987JSP....47..439A. doi:10.1007/BF01007519. S2CID 122213380.
  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. ^ a b c d e f g h i j k 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. S2CID 12245755.
  6. ^ 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. S2CID 94779870.
  7. ^ Mandelbrot, Benoit (2002). Gaussian self-affinity and Fractals. ISBN 978-0-387-98993-8.
  8. ^ a b c d McMullen, Curtis T. (3 October 1997). "Hausdorff dimension and conformal dynamics III: Computation of dimension", Accessed: 27 October 2018.
  9. ^ Messaoudi, Ali. Frontième de numération complexe", (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. ^ a b Ngai, Sirvent, Veerman, and Wang (October 2000). "On 2-Reptiles in the Plane 1999", Geometriae Dedicata, Volume 82. Accessed: 29 October 2018.
  13. ^ a b Duda, Jarek (March 2011). "The Boundary of Periodic Iterated Function Systems",
  14. ^ Chang, Angel and Zhang, Tianrong. "On the Fractal Structure of the Boundary of Dragon Curve". Archived from the original on 14 June 2011. Retrieved 9 February 2019.CS1 maint: bot: original URL status unknown (link) 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. ^ Shen, Weixiao (2018). "Hausdorff dimension of the graphs of the classical Weierstrass functions". Mathematische Zeitschrift. 289 (1–2): 223–266. arXiv:1505.03986. doi:10.1007/s00209-017-1949-1. ISSN 0025-5874. S2CID 118844077.
  17. ^ N. Zhang. The Hausdorff dimension of the graphs of fractal functions. (In Chinese). Master Thesis. Zhejiang University, 2018.
  18. ^ Fractal dimension of the boundary of the dragon fractal
  19. ^ a b Fractal dimension of the Pascal triangle modulo k
  20. ^ The Fibonacci word fractal
  21. ^ Theiler, James (1990). "Estimating fractal dimension" (PDF). J. Opt. Soc. Am. A. 7 (6): 1055–73. Bibcode:1990JOSAA...7.1055T. doi:10.1364/JOSAA.7.001055.
  22. ^ Fractal Generator for ImageJ Archived 20 March 2012 at the Wayback Machine.
  23. ^ W. Trump, G. Huber, C. Knecht, R. Ziff, to be published
  24. ^ Monkeys tree fractal curve Archived 21 September 2002 at
  25. ^ Fractal dimension of a Penrose tiling
  26. ^ a b Shishikura, Mitsuhiro (1991). "The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets". arXiv:math/9201282.
  27. ^ Lebesgue curve variants
  28. ^ Duda, Jarek (2008). "Complex base numeral systems". arXiv:0712.1309v3 [math.DS].
  29. ^ Seuil (1982). Penser les mathématiques. ISBN 2-02-006061-2.
  30. ^ Fractals and the Rössler attractor
  31. ^ McGuinness, M.J. (1983). "The fractal dimension of the Lorenz attractor". Physics Letters. 99A (1): 5–9. Bibcode:1983PhLA...99....5M. doi:10.1016/0375-9601(83)90052-X.
  32. ^ Lowe, Thomas (24 October 2016). "Three Variable Dimension Surfaces". ResearchGate.
  33. ^ The Fractal dimension of the apollonian sphere packing Archived 6 May 2016 at the Wayback Machine
  34. ^ [1]
  35. ^ 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.
  36. ^ Hausdorff dimension of the Mandelbulb
  37. ^ Peter Mörters, Yuval Peres, Oded Schramm, "Brownian Motion", Cambridge University Press, 2010
  38. ^ 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.
  39. ^ a b Hutzler, S. (2013). "Fractal Ireland". Science Spin. 58: 19–20. Retrieved 15 November 2016. (See contents page, archived 26 July 2013)
  40. ^ How long is the coast of Britain? Statistical self-similarity and fractional dimension, B. Mandelbrot
  41. ^ 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. S2CID 5877745.
  42. ^ a b c d e f g h i Sapoval, Bernard (2001). Universalités et fractales. Flammarion-Champs. ISBN 2-08-081466-4.
  43. ^ Feder, J., "Fractals,", Plenum Press, New York, (1988).
  44. ^ Hull-generating walks
  45. ^ a b M Sahini; M Sahimi (2003). Applications Of Percolation Theory. CRC Press. ISBN 978-0-203-22153-2.
  46. ^ Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey
  47. ^ "Power Law Relations". Yale. Archived from the original on 28 June 2010. Retrieved 29 July 2010. Cite journal requires |journal= (help)
  48. ^ a b Kim, Sang-Hoon (2 February 2008). "Fractal dimensions of a green broccoli and a white cauliflower". arXiv:cond-mat/0411597.
  49. ^ Fractal dimension of the surface of the human brain
  50. ^ [Meakin (1987)]

Further reading[edit]

External links[edit]