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List of fractals by Hausdorff dimension: Wikis

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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.

Contents

Deterministic fractals

Hausdorff dimension
(exact value)
Hausdorff dimension
(approx.)
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 \scriptstyle{\lambda_\infty = 3.570}, where the period doubling is infinite. Notice that this dimension is the same for any differentiable and unimodal function.[2]
\textstyle{\frac {\log(2)}{\log(3)}} 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.
\textstyle{\frac {\log(\scriptstyle\varphi)}{\log(2)}=\frac{\log(1+\sqrt{5})}{\log(2)}-1} 0.6942 Asymmetric Cantor set AsymmCantor.png Note that the dimension is not \textstyle{\frac {\log(3)}{\log(4)}}.[3]

Built by removing the second quarter at each iteration. Nowhere dense and not a countable set. \scriptstyle\varphi = (1+\sqrt{5})/2 (golden ratio).

\textstyle{\frac {\log(5)}{\log(10)}} 0.69897 Real numbers with even digits Even digits.png Similar to a Cantor set[1].
 \log{(1+\sqrt{2})} 0.88137 Spectrum of Fibonacci Hamiltonian The study 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]
\textstyle{1} 1 Smith–Volterra–Cantor set Smith-Volterra-Cantor set.svg Built by removing a central interval of length 1 / 22n of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of ½.
\textstyle{2+\frac {\log(1/2)} {\log(2)}=1} 1.0000 Takagi or Blancmange curve Takagi curve.png Defined on the unit interval by \textstyle{f(x) = \sum_{n=0}^\infty {s(2^{n}x)\over 2^n}}, where s(x) is the sawtooth function. Special case of the Takahi-Landsberg curve: \textstyle{f(x) = \sum_{n=0}^\infty {w^n s(2^{n}x)}} with \scriptstyle{w = 1/2}. The Hausdorff dimension equals 2 + log(w) / log(2) for w in \scriptstyle{\left[ 1/2,1\right]}. (Hunt cited by Mandelbrot [5] ).
\textstyle{\frac {\log(8)} {\log(7)}} 1.0686 contour of the Gosper island Gosper Island 3.svg
Calculated 1.0812 Julia set z² + 1/4 Julia z2+0,25.png Julia set for c = 1/4. [6]
Solution s of 2 | α | 3s + | α | 4s = 1 1.0933 Boundary of the Rauzy fractal Rauzy fractal.png Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: \scriptstyle{1\mapsto12}, \scriptstyle{2\mapsto13} and \scriptstyle{3}\mapsto1. [7].α is one of the conjugated roots of z3z2z − 1 = 0.
Measured (box counting) 1.2 Dendrite Julia set Dendrite julia.png Julia set for parameters: Real = 0 and Imaginary = 1.
\textstyle{3\frac{\log(\varphi)}{\log \left(\frac{3+\sqrt{13}}{2}\right)}} 1.2083 Fibonacci word fractal 60° Fibo 60deg F18.png Build from the Fibonacci word. See also the standard Fibonacci word fractal.

\scriptstyle\varphi = (1+\sqrt{5})/2 (golden ratio).

1.26 Hénon map Henon attractor.png 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.
\textstyle{\frac {\log(4)} {\log(3)}} 1.2619 Koch curve Koch curve.svg 3 von Koch curves form the Koch snowflake or the anti-snowflake.
\textstyle{\frac {\log(4)} {\log(3)}} 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.
\textstyle{\frac {\log(4)} {\log(3)}} 1.2619 2D Cantor dust Carre cantor.gif Cantor set in 2 dimensions.
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 [9]
Calculated (Box counting) 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 [10]
Calculated 1.3934 Douady rabbit Douady rabbit.png Julia set for c = −0,123 + 0.745i. [11]
\textstyle{\frac {\log(5)} {\log(3)}} 1.4649 Vicsek fractal Box fractal.png Built by exchanging iteratively each square by a cross of 5 squares.
\textstyle{\frac {\log(5)} {\log(3)}} 1.4649 Quadratic von Koch curve (type 1) Quadratic Koch 2.png One can recognize the pattern of the Vicsek fractal (above).
 \textstyle{2 -\frac{\log(\sqrt{2})}{\log(2)}=\frac{3}{2}} (conjectured exact) 1.5000 a Weierstrass function: \textstyle{f(x)=\sum_{k=1}^\infty \frac {sin(2^k x)} {\sqrt{2}^k}} Weierstrass functionAMD.png The Hausdorff dimension of the Weierstrass function \scriptstyle{f : [0,1] \to \mathbb{R}} defined by \textstyle{f(x)=\sum_{k=1}^\infty \frac {sin(b^k x)} {a^k}} with 1 < a < 2 and b > 1 has upper bound \scriptstyle{2 -\log(a)/\log(b)}. 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]
\textstyle{\frac {\log(8)} {\log(4)} = \frac{3}{2}} 1.5000 Quadratic von Koch curve (type 2) Quadratic Koch.png Also called "Minkowski sausage".
\textstyle{\frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)} {\log(2)}} 1.5236 Dragon curve boundary Boundary dragon curve.png cf. Chang & Zhang.[12][13]
\textstyle{\frac {\log(3)} {\log(2)}} 1.585 3-branches tree Arbre 3 branches.pngArbre 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.
\textstyle{\frac {\log(3)} {\log(2)}} 1.585 Sierpinski triangle SierpinskiTriangle.PNG Also the triangle of Pascal modulo 2.
\textstyle{\frac {\log(3)} {\log(2)}} 1.585 Sierpiński arrowhead curve Pfeilspitzen fraktal.png Same limit as the triangle (above) but built with a one-dimensional curve.
\textstyle{\frac{\log{\varphi}}{\log{\sqrt[\varphi]{\varphi}}}=\varphi} 1.61803 a golden dragon Phi glito.png Built from two similarities of ratios r and r2, with \scriptstyle{r = 1 / \varphi^{1/\varphi}}. Its dimension equals \scriptstyle{\varphi} because \scriptstyle{({r^2})^\varphi+r^\varphi = 1}. With \scriptstyle\varphi = (1+\sqrt{5})/2 (Golden number).
\textstyle{1+\frac{\log 2}{\log 3}} 1.6309 Pascal triangle modulo 3 Pascal triangle modulo 3.png For a triangle modulo k, if k is prime, the fractal dimension is \scriptstyle{1 + \log_k\left(\frac{k+1}{2}\right)} (cf. Stephen Wolfram[14]).
\textstyle{3\frac{\log(\varphi)}{\log (1+\sqrt{2})}} 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). [15].

\scriptstyle\varphi = (1+\sqrt{5})/2 (golden ratio).

Solution of \scriptstyle{(1/3)^s + (1/2)^s + (2/3)^s = 1} 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 n similarities of ratios cn, has Hausdorff dimension s, solution of the equation : \scriptstyle{\sum_{k=1}^n c_k^s = 1} [1].
\textstyle{1+\frac{\log 3}{\log 5}} 1.6826 Pascal triangle modulo 5 Pascal triangle modulo 5.png For a triangle modulo k, if k is prime, the fractal dimension is \scriptstyle{1 + \log_k\left(\frac{k+1}{2}\right)} (cf. Stephen Wolfram[14]).
\textstyle{\frac {4 \log(2)} {\log(5)}} 1.7227 Pinwheel fractal Pinwheel fractal.png Built with Conway's Pinwheel tile.
\textstyle{\frac {\log(7)} {\log(3)}} 1.7712 Hexaflake Flocon hexagonal.gif 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).
\textstyle{\frac {\log(4)} {\log(2(1+\cos(85^\circ)))}} 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 \scriptstyle{\frac{\log(4)}{\log(2(1+\cos(a)))}} \in [1,2].
\textstyle{\frac{\log{(3^{0.63}+2^{0.63})}} {\log{2}}} 1.8272 A self-affine fractal set Self-affine set.png Build iteratively from a \scriptstyle{p \times q} array on a square, with \scriptstyle{p \le q}. Its Hausdorff dimension equals \scriptstyle{\frac{\log{\left (\sum_{k=1}^p n_k^a \right )}} {\log{p}}}[1] with \scriptstyle{a=\frac{\log{ p}}{log{ q}}} and nk is the number of elements in the kth column. The box-countig 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.
\textstyle{\frac {\log(6)} {\log(1+\varphi)}} 1.8617 Pentaflake Penta plexity.png Built by exchanging iteratively each pentagon by a flake of 6 pentagons.

\scriptstyle\varphi = (1+\sqrt{5})/2 (golden ratio).

solution of \scriptstyle{6(1/3)^s+5{(1/3\sqrt{3})}^s=1} 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 1 / 3 and 5 similarities of ratio \scriptstyle{1/{3\sqrt{3}}}[16].
\textstyle{\frac {\log(8)} {\log(3)}} 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).
\textstyle{\frac {\log(8)} {\log(3)}} 1.8928 3D Cantor dust Cantor3D3.png Cantor set in 3 dimensions.
Estimated 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.
1.974 Penrose tiling Pen0305c.gif See Ramachandrarao, Sinha & Sanyal[17].
\textstyle{2} 2 Boundary of the Mandelbrot set Boundary mandelbrot set.png The boundary and the set itself have the same dimension [18].
\textstyle{2} 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. [19].
\textstyle{2} 2 Sierpiński curve Sierpinski-Curve-3.png Every Peano curve filling the plane has a Hausdorff dimension of 2.
\textstyle{2} 2 Hilbert curve Hilbert curve 3.svg
\textstyle{2} 2 Peano curve Peano curve.png And a family of curves built in a similar way, such as the Wunderlich curves.
\textstyle{2} 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.[20]
\textstyle{\frac {\log(2)} {\log(\sqrt{2})} = 2} 2 Dragon curve Courbe du dragon.png And its boundary has a fractal dimension of 1.5236270862[21].
2 Terdragon curve Terdragon curve.png L-system: F → F + F – F, angle = 120°.
\textstyle{\frac {\log(4)} {\log(2)} = 2} 2 T-Square T-Square fractal (evolution).png
\textstyle{\frac {\log(4)} {\log(2)} = 2} 2 Gosper curve Gosper curve 3.svg Its boundary is the Gosper island.
\textstyle{\frac {\log(4)} {\log(2)} = 2} 2 Sierpiński tetrahedron Tetraedre Sierpinski.png Each tetrahedron is replaced by 4 tetrahedra.
\textstyle{\frac {\log(4)} {\log(2)} = 2} 2 H-fractal H fractal2.png Also the « Mandelbrot tree » which has a similar pattern.
\textstyle{\frac {\log(2)} {\log(2/\sqrt{2})} = 2} 2 Pythagoras tree PythagorasTree.png Every square generates 2 squares with a reduction ratio of sqrt(2)/2.
\textstyle{\frac {\log(4)} {\log(2)} = 2} 2 2D Greek cross fractal Greek cross fractal stage 4.svg Each segment is replaced by a cross formed by 4 segments.
2.06 Lorenz attractor Lorenz attractor.png For parameters v=40,σ=16 and b=4 . see McGuinness (1983)[22]
\textstyle{\frac {\log(20)} {\log(2+\varphi)}} 2.3296 Dodecahedron fractal Dodecaedron fractal.jpg Each dodecahedron is replaced by 20 dodecahedra.

\scriptstyle\varphi = (1+\sqrt{5})/2 (golden ratio).

\textstyle{\frac {\log(13)} {\log(3)}} 2.3347 3D quadratic Koch surface (type 1) Quadratic Koch 3D (type1 stage2).png Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
2.4739 Apollonian sphere packing Apollonian spheres2.png The interstice left by the apollolian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.[23]
\textstyle{\frac {\log(32)} {\log(4)} = \frac{5}{2}} 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.
\textstyle{\frac {\log(16)} {\log(3)}} 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 \scriptstyle{n\frac{\log(2)}{\log(3)}}.
\textstyle{\frac {\log(12)} {\log(1+\varphi)}} 2.5819 Icosahedron fractal Icosaedron fractal.jpg Each icosahedron is replaced by 12 icosahedra. \scriptstyle\varphi = (1+\sqrt{5})/2 (golden ratio).
\textstyle{\frac {\log(6)} {\log(2)}} 2.5849 3D Greek cross fractal Greek cross 3D 1 through 4.png Each segment is replaced by a cross formed by 6 segments.
\textstyle{\frac {\log(6)} {\log(2)}} 2.5849 Octahedron fractal Octaedron fractal.jpg Each octahedron is replaced by 6 octahedra.
\textstyle{\frac {\log(6)} {\log(2)}} 2.5849 von Koch surface Koch surface 3.png Each equilateral triangle is replaced by 6 triangles, twice smaller.
\textstyle{\frac {\log(20)} {\log(3)}} 2.7268 Menger sponge Menger.png And its surface has a fractal dimension of \scriptstyle{\frac{\log(12)}{\log(3)} = 2.2618}.
\textstyle{\frac {\log(8)} {\log(2)} = 3} 3 3D Hilbert curve Hilbert3d-step3.png A Hilbert curve extended to 3 dimensions.
\textstyle{\frac {\log(8)} {\log(2)} = 3} 3 3D Lebesgue curve Lebesgue-3d-step3.png A Lebesgue curve extended to 3 dimensions.
\textstyle{\frac {\log(8)} {\log(2)} = 3} 3 3D Moore curve Moore3d-step3.png A Moore curve extended to 3 dimensions.

Random and natural fractals

Hausdorf dimension
(exact value)
Hausdorf dimension
(approx.)
Name Illustration Remarks
Solution of \scriptstyle{E(C_1^s + C_2^s)=1} where \scriptstyle{E(C_1)=0.5} and \scriptstyle{E(C_2)=0.3} 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 C1, a variable percentage of the length of the original interval. Same for the right interval, with a random variable C2. Its Hausdorff Dimension s satisfies : \scriptstyle{E(C_1^s + C_2^s)=1}. (E(X) is the expected value of X).[1]
Solution of s + 1 = 12 * 2 − (s + 1) − 6 * 3 − (s + 1) 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). [1]
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.[24]
\textstyle{\frac {\log(4)} {\log(3)}} 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.[1]
\textstyle{\frac {4}{3}} 1.33 Boundary of Brownian motion Front mouvt brownien.png (cf. Lawler, Schramm, Werner).[25]
\textstyle{\frac {4}{3}} 1.33 2D polymer Similar to the brownian motion in 2D with non self-intersection. [26].
\textstyle{\frac {4}{3}} 1.33 Percolation front in 2D, Corrosion front in 2D Front de percolation.png Fractal dimension of the percolation-by-invasion front, at the percolation threshold (59.3%). It’s also the fractal dimension of a stopped corrosion front [26].
1.40 Clusters of clusters 2D When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4. [26]
\textstyle{2-\frac{1}{2}} 1.5 Graph of a regular Brownian function Wiener process zoom.png Graph of a function f such that, for any two positive reals x and x+h, the difference of their images f(x + h) − f(x) has the centered gaussian distribution with variance = h. Generalization : The fractional Brownian motion of index α follows the same definition but with a variance = h, in that case its Hausdorff dimension =2 − α[1].
Measured 1.52 Coastline of Norway Norway municipalities.png See J. Feder. [27]
Measured 1.55 Random walk with no self-intersection Polymer 2D.png Self-avoiding random walk in a square lattice, with a « go-back » routine for avoiding dead ends.
\textstyle{\frac {5} {3}} 1.66 3D polymer Similar to the brownian motion in a cubic lattice, but without self-intersection [26].
1.70 2D DLA Cluster Agregation limitee par diffusion.png In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70 [26].
\textstyle{\frac {\log(9*0.75)} {\log(3)}} 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 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 \textstyle{\frac {\log(9p)} {\log(3)}}[1].
\textstyle{\frac {91} {48}} 1.8958 2D percolation cluster Amas de percolation.png Under the percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48 [26]. Beyond that threshold, le cluster is infinite and 91/48 becomes the fractal dimension of the « clearings ».
\textstyle{\frac {\log(2)} {\log(\sqrt{2})} = 2} 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. See reference [28]
\textstyle{\frac {\log(13)} {\log(3)}} 2.33 Cauliflower Blumenkohl-1.jpg Every branch carries around 13 branches 3 times smaller.
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. [1] Creases will form at all size scales (see Universality (dynamical systems)).
2.50 3D DLA Cluster 3D diffusion-limited aggregation2.jpg In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50 [26].
2.50 Lichtenberg figure PlanePair2.jpg Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA [26].
\textstyle{3-\frac{1}{2}} 2.5 regular Brownian surface Brownian surface.png A function \scriptstyle{f:\mathbb{R}^2 -> \mathbb{R}}, gives the height of a point (x,y) such that, for two given positive increments h and k, then \scriptstyle{f(x+h,y+k)-f(x,y)} has a centered Gaussian distribution with variance = \scriptstyle{\sqrt{h^2+k^2}}. Generalization : The fractional Brownian surface of index α follows the same definition but with a variance = (h2 + k2)α, in that case its Hausdorff dimension = 3 − α[1].
Measured 2.66 Broccoli Broccoli DSC00862.png [29]
2.79 Surface of human brain Cerebellum NIH.png [30]
2.97 Lung surface Thorax Lung 3d (2).jpg The alveoli of a lung form a fractal surface close to 3 [26].
Calculated 3 Quantum string drifting randomly Point&string.png Hausdorff dimension of a quantum string whose representative point randomly drifts through loop space.[31]
Calculated \textstyle{\in(0,2)} 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[32].

See also

Notes

  1. ^ a b c d e f g h i j k Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd.. xxv. ISBN 0-470-84862-6.  
  2. ^ Fractal dimension of the Feigenbaum attractor
  3. ^ Tsang, K. Y. (1986). "Dimensionality of Strange Attractors Determined Analytically". Phys. Rev. Lett. 57: 1390-1393.  
  4. ^ Fractal dimension of the spectrum of the Fibonacci Hamiltonian
  5. ^ Mandelbrot, Benoit. Gaussian self-affinity and Fractals. ISBN 0-387-98993-5.  
  6. ^ fractal dimension of the Julia set for c = 1/4
  7. ^ Boundary of the Rauzy fractal
  8. ^ fractal dimension of the z²-1 Julia set
  9. ^ fractal dimension of the apollonian gasket
  10. ^ fractal dimension of the 5 circles inversion fractal
  11. ^ fractal dimension of the Douady rabbit
  12. ^ Fractal dimension of the boundary of the dragon fractal
  13. ^ Recurrent construction of the boundary of the dragon curve
  14. ^ a b Fractal dimension of the Pascal triangle modulo k
  15. ^ The Fibonacci word fractal
  16. ^ Monkeys tree fractal curve
  17. ^ Fractal dimension of a Penrose tiling
  18. ^ Fractal dimension of the boundary of the Mandelbrot set
  19. ^ Fractal dimension of certain Julia sets
  20. ^ Lebesgue curve variants
  21. ^ Complex base numeral systems
  22. ^ The fractal dimension of the Lorenz attractor, Mc Guinness (1983)
  23. ^ Fractal dimension of the apollonian sphere packing
  24. ^ How long is the coast of Britain? Statistical self-similarity and fractional dimension, B. Mandelbrot
  25. ^ Fractal dimension of the brownian motion boundary
  26. ^ a b c d e f g h i Bernard Sapoval "Universalités et fractales", Flammarion-Champs (2001), ISBN=2-080-81466-4
  27. ^ Feder, J., "Fractals,", Plenum Press, New York, (1988).
  28. ^ Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey
  29. ^ Fractal dimension of the broccoli
  30. ^ Fractal dimension of the surface of the human brain
  31. ^ The Hausdorf dimension of a quantum string
  32. ^ [Meakin (1987)]

References

  • 1Kenneth Falconer, Fractal Geometry, John Wiley & Son Ltd; ISBN 0-471-92287-0 (March 1990)

Further reading and references

  • 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 2080814664 (2001).

External links

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