List of fractals by Hausdorff dimension

List of fractals by Hausdorff dimension
From Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension


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

[hide]

• 1 Deterministic fractals
• 2 Random and natural fractals
• 3 See also
• 4 Notes and references
• 5 Further reading
• 6 External links

Deterministic fractals[edit]

Hausdorff dimension (exact value)	Hausdorff dimension (approx.)	Name	Illustration	Remarks
Calculated	0.538	Feigenbaum attractor		The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of thelogistic function for the critical parameter value , where the period doubling is infinite. This dimension is the same for any differentiable andunimodal function.[2]
	0.6309	Cantor set		Built by removing the central third at each iteration.Nowhere dense and not a countable set.
	0.6942	Asymmetric Cantor set		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		Similar to the 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		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		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		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		Julia set for c = 1/4.[7]
Solution s of	1.0933	Boundary of the Rauzy fractal		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		Term used by Mandelbrot (1977).[10] The Gosper island is the limit of the Gosper curve.
Measured (box counting)	1.2	Dendrite Julia set		Julia set for parameters: Real = 0 and Imaginary = 1.
	1.2083	Fibonacci word fractal 60°		Build from the Fibonacci word. See also the standard Fibonacci word fractal. (golden ratio).
	1.2108	Boundary of the tame twindragon		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		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		Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
	1.2619	Koch curve		3 von Koch curves form the Koch snowflake or the anti-snowflake.
	1.2619	boundary of Terdragon curve		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		Cantor set in 2 dimensions.
	1.2619	2D L-system branch		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 set for c = −1.[13]
	1.3057	Apollonian gasket		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		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		Julia set for c = −0,123 + 0.745i.[16]
	1.4649	Vicsek fractal		Built by exchanging iteratively each square by a cross of 5 squares.
	1.4649	Quadratic von Koch curve (type 1)		One can recognize the pattern of the Vicsek fractal (above).
	1.49	Quadric cross
(conjectured exact)	1.5000	a 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.[1]
	1.5000	Quadratic von Koch curve (type 2)		Also called "Minkowski sausage".
	1.5236	Boundary of the Dragon curve		cf. Chang & Zhang.[17][18]
	1.5236	Boundary of thetwindragon curve		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		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		Also the triangle of Pascal modulo 2.
	1.5849	Sierpiński arrowhead curve		Same limit as the triangle (above) but built with a one-dimensional curve.
	1.5849	Boundary of the T-Square fractal		The dimension of the fractal itself (not the boundary) is [19]
	1.61803	a golden dragon		Built from two similarities of ratios and , with . Its dimension equals because . With (Golden number).
	1.6309	Pascal triangle modulo 3		For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram[20]).
	1.6309	Sierpinski Hexagon		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		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 3similarities of ratios 1/3, 1/2 and 2/3		Generalization : Providing the open set condition holds, the attractor of an iterated function system consisting of similarities of ratios , has Hausdorff dimension , solution of the equation coinciding with the iteration function of the Euclidean contraction factor: .[1]
	1.6826	Pascal triangle modulo 5		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		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		Built with ImageJ[23]
	1.7227	Pinwheel fractal		Built with Conway's Pinwheel tile.
	1.7712	Hexaflake		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°		Generalizing the von Koch curve with an angle achosen between 0 and 90°. The fractal dimension is then .
	1.8272	A self-affine fractal set		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		Built by exchanging iteratively each pentagon by a flake of 6 pentagons. (golden ratio).
solution of	1.8687	Monkeys tree		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		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		Cantor set in 3 dimensions.
	1.8928	Cartesian product of thevon Koch curve and theCantor set		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		Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
	1.974	Penrose tiling		See Ramachandrarao, Sinha & Sanyal.[25]
	2	Boundary of theMandelbrot set		The boundary and the set itself have the same dimension.[26]
	2	Julia set		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		Every Peano curve filling the plane has a Hausdorff dimension of 2.
	2	Hilbert curve
	2	Peano curve		And a family of curves built in a similar way, such as theWunderlich curves.
	2	Moore curve		Can be extended in 3 dimensions.
	2	Lebesgue curve or z-order curve		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		And its boundary has a fractal dimension of 1.5236270862.[29]
	2	Terdragon curve		L-system: F → F + F – F, angle = 120°.
	2	Gosper curve		Its boundary is the Gosper island.
Solution of	2	Curve filling the Koch snowflake		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		Each tetrahedron is replaced by 4 tetrahedra.
	2	H-fractal		Also the Mandelbrot tree which has a similar pattern.
	2	Pythagoras tree (fractal)		Every square generates two squares with a reduction ratio of .
	2	2D Greek cross fractal		Each segment is replaced by a cross formed by 4 segments.
Measured	2.01 ±0.01	Rössler attractor		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		For parameters v=40, =16 and b=4 . See McGuinness (1983)[32]
	2.3219	Fractal pyramid		Each square pyramid is replaced by 5 half-size square pyramids. (Different from the Sierpinski tetrahedron, which replaces each triangular pyramid with 4 half-size triangular pyramids).
	2.3296	Dodecahedron fractal		Each dodecahedron is replaced by 20 dodecahedra. (golden ratio).
	2.3347	3D quadratic Koch surface (type 1)		Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
	2.4739	Apollonian sphere packing		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)		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		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		Each icosahedron is replaced by 12 icosahedra. (golden ratio).
	2.5849	3D Greek cross fractal		Each segment is replaced by a cross formed by 6 segments.
	2.5849	Octahedron fractal		Each octahedron is replaced by 6 octahedra.
	2.5849	von Koch surface		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		And its surface has a fractal dimension of , which is the same as that by volume.
	3	3D Hilbert curve		A Hilbert curve extended to 3 dimensions.
	3	3D Lebesgue curve		A Lebesgue curve extended to 3 dimensions.
	3	3D Moore curve		A Moore curve extended to 3 dimensions.
	3	3D H-fractal		A H-fractal extended to 3 dimensions.[34]
(conjectured)	3 (to be confirmed)	Mandelbulb		Extension of the Mandelbrot set (power 8) in 3 dimensions[35][unreliable source?]

Random and natural fractals[edit]

Hausdorff dimension (exact value)	Hausdorff dimension (approx.)	Name	Illustration	Remarks
1/2	0.5	Zeros of aWiener process		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 randomCantor setwith 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 : . ( is the expected value of ).[1]
Solution of	1.144...	von Koch curve with random interval		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		Values for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault[37] at theUniversity 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		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		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		(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		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 regularBrownianfunction (Wiener process)		Graph of a function f such that, for any two positive reals x andx+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		See J. Feder.[42]
Measured	1.55	Random walk with no self-intersection		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		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		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		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		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		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		From the 2005 results of the Sloan Digital Sky Survey.[45]
	2.33	Cauliflower		Every branch carries around 13 branches 3 times smaller.
	2.5	Balls of crumpled paper		When crumpling sheets of different sizes but made of the same type of paper and with the same aspect ratio (for example, different sizes in the ISO 216 A series), then the diameter of the balls so obtained elevated to a non-integer exponent between 2 and 3 will be approximately proportional to the area of the sheets from which the balls have been made.[46] Creases will form at all size scales (see Universality (dynamical systems)).
	2.50	3D DLA Cluster		In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.[41]
	2.50	Lichtenberg figure		Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.[41]
	2.5	regularBrowniansurface		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 Browniansurface of index follows the same definition but with a variance =, in that case its Hausdorff dimension = .[1]
Measured	2.52	3Dpercolationcluster		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		[47]
	2.79	Surface ofhuman brain		[48]
	2.97	Lung surface		The alveoli of a lung form a fractal surface close to 3.[41]
Calculated		Multiplicative cascade		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[edit]

Wikimedia Commons has media related to fractals.

• Fractal dimension
• Hausdorff dimension
• Scale invariance

Notes and references[edit]

1. ^ Jump up to:^{a b c d e f g h i j k l m} Falconer, Kenneth (1990–2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd. xxv. ISBN 0-470-84862-6.
2. Jump up^ Fractal dimension of the Feigenbaum attractor
3. Jump up^ Tsang, K. Y. (1986). "Dimensionality of Strange Attractors Determined Analytically".Phys. Rev. Lett. 57 (12): 1390–1393. doi:10.1103/PhysRevLett.57.1390.PMID 10033437.
4. Jump up^ Fractal dimension of the spectrum of the Fibonacci Hamiltonian
5. Jump up^ The scattering from generalized Cantor fractals
6. Jump up^ Mandelbrot, Benoit. Gaussian self-affinity and Fractals. ISBN 0-387-98993-5.
7. Jump up^ fractal dimension of the Julia set for c = 1/4
8. Jump up^ Boundary of the Rauzy fractal
9. Jump up^ 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, ISBN 978-0-521-84802-2
10. Jump up^ Gosper island on Mathworld
11. ^ Jump up to:^{a b} On 2-reptiles in the plane, Ngai, 1999
12. Jump up^ Recurrent construction of the boundary of the dragon curve (for n=2, D=1)
13. Jump up^ fractal dimension of the z²-1 Julia set
14. Jump up^ fractal dimension of the apollonian gasket
15. Jump up^ fractal dimension of the 5 circles inversion fractal
16. Jump up^ fractal dimension of the Douady rabbit
17. Jump up^ Fractal dimension of the boundary of the dragon fractal
18. Jump up^ Recurrent construction of the boundary of the dragon curve (for n=2, D=2)
19. Jump up^ T-Square (fractal)
20. ^ Jump up to:^{a b} Fractal dimension of the Pascal triangle modulo k
21. Jump up^ The Fibonacci word fractal
22. Jump up^ Estimating Fractal dimension
23. Jump up^ Fractal Generator for ImageJ.
24. Jump up^ Monkeys tree fractal curve
25. Jump up^ Fractal dimension of a Penrose tiling
26. Jump up^ Fractal dimension of the boundary of the Mandelbrot set
27. Jump up^ Fractal dimension of certain Julia sets
28. Jump up^ Lebesgue curve variants
29. Jump up^ Complex base numeral systems
30. Jump up^ "Penser les mathématiques", Seuil ISBN 2-02-006061-2 (1982)
31. Jump up^ Fractals and the Rössler attractor
32. Jump up^ The fractal dimension of the Lorenz attractor, Mc Guinness (1983)
33. Jump up^ Fractal dimension of the apollonian sphere packing
34. Jump up^ B. Hou, H. Xie, W. Wen, and P. Sheng (2008). "Three-dimensional metallic fractals and their photonic crystal characteristics". Phys. Rev. B 77, 125113.
35. Jump up^ Hausdorff dimension of the Mandelbulb
36. Jump up^ Peter Mörters, Yuval Peres, Oded Schramm, "Brownian Motion", Cambridge University Press, 2010
37. Jump up^ 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. Retrieved 4 December 2014.
38. ^ Jump up to:^{a b} Hutzler, S. (2013). Fractal Ireland. Science Spin, 58, 19-20.
39. Jump up^ How long is the coast of Britain? Statistical self-similarity and fractional dimension, B. Mandelbrot
40. Jump up^ Fractal dimension of the brownian motion boundary
41. ^ Jump up to:^{a b c d e f g h i} Bernard Sapoval "Universalités et fractales", Flammarion-Champs (2001), ISBN=2-08-081466-4
42. Jump up^ Feder, J., "Fractals,", Plenum Press, New York, (1988).
43. Jump up^ Hull-generating walks
44. ^ Jump up to:^{a b} "Applications of percolation" theory by Muhammad Sahimi (1994)
45. Jump up^ Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey
46. Jump up^ "Power Law Relations". Yale. Retrieved 29 July 2010
47. Jump up^ Fractal dimension of the broccoli
48. Jump up^ Fractal dimension of the surface of the human brain
49. Jump up^ [Meakin (1987)]

Further reading[edit]

• 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[edit]

• The fractals on Mathworld
• Other fractals on Paul Bourke's website
• Soler's Gallery
• Fractals on mathcurve.com
• 1000fractales.free.fr - Project gathering fractals created with various software
• Fractals unleashed

[hide] v t e Fractals   Characteristics Fractal dimension Hausdorff dimension Topological dimension Recursion Self-similarity   Iterated function system Barnsley fern Cantor set Dragon curve Koch snowflake Menger sponge Sierpinski carpet Sierpinski triangle Space-filling curve T-square   Strange attractor Multifractal system   L-system Space-filling curve   Escape-time fractals Burning Ship fractal Julia set Lyapunov fractal Mandelbrot set Nova fractal   Random fractals

http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension