List of fractals by Hausdorff dimension

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A fractal is a geometric object whose Hausdorff dimension (δ) strictly exceeds its topological dimension. 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.

1 Deterministic fractals
2 Random and natural fractals
3 References
4 See also

[edit] Deterministic fractals

δ (exact value)	δ (value)	Name	Remarks
$\textstyle{\frac {ln(2)} {ln(\delta)}?}$	0.4498?	Logistic equation bifurcations	In the bifurcation diagram, when approaching the chaotic zone, a succession of period doubling appears, in a geometric progression tending to 1/δ. (δ=Feigenbaum constant=4.6692).
$\textstyle{\frac {ln(2)} {ln(3)}}$	0.6309	Cantor set	Built by withdrawing the central third at each iteration. Nowhere dense and not a countable set
$\textstyle{\frac {ln(6)} {ln(8)}}$	0.8617	Smith-Volterra-Cantor set	(In white on the figure) built by withdrawing the central quarter at each iteration. Nowhere dense but has a Lebesgue mesure of ½.
$\textstyle{\frac {ln(8)} {ln(7)}}$	1.0686	Gosper island
	1.26	Hénon map	The canonical Hénon map (with parameters a = 1.4 and b = 0.3) has Hausdorff dimension δ = 1.261 ± 0.003. Different parameters yield different δ values.
$\textstyle{\frac {ln(4)} {ln(3)}}$	1.2619	Von Koch curve	3 von Koch curves form the Koch snowflake or the anti-snowflake.
$\textstyle{\frac {ln(2)} {ln(\sqrt(3))}}$	1.2619	boundary of Terdragon curve, Fudgeflake	L-system: same as dragon curve with angle=30°. The Fudgeflake is based on 3 initial segments placed in a triangle.
$\textstyle{\frac {ln(4)} {ln(3)}}$	1.2619	2D Cantor dust	Cantor set in 2 dimensions.
	1.3057	Apollonian gasket
$\textstyle{\frac {ln(5)} {ln(3)}}$	1.4649	Box fractal	Built by exchanging iteratively each square by a cross of 5 squares.
$\textstyle{\frac {ln(5)} {ln(3)}}$	1.4649	Quadratic von Koch curve (type 1)	One can recognize the pattern of the box fractal (above).
$\textstyle{\frac {ln(8)} {ln(4)}}$	1.5000	Quadratic von Koch curve (type 2)	Also called " Minkowski saussage".
	1.5236	Dragon curve boundary	Cf Chang & Zhang^[1].
$\textstyle{\frac {ln(3)} {ln(2)}}$	1,5850	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.
$\textstyle{\frac {ln(3)} {ln(2)}}$	1.5850	Sierpinski triangle	It’s also the triangle of Pascal modulo 2.
$\textstyle{\frac {ln(3)} {ln(2)}}$	1.5850	Arrowhead Sierpinski curve	Same limit as the triangle (above) but built with a one-dimensionnal curve.
$\textstyle{1+log_3(2)}$	1.6309	Pascal triangle modulo 3	For a triangle modulo k, if k is prime, the fractal dimension is $\scriptstyle{1 + log_k(\frac{k+1}{2})}$ (Cf Stephen Wolfram ^[2])
$\textstyle{1+log_5(3)}$	1.6826	Pascal triangle modulo 5	For a triangle modulo k, if k is prime, the fractal dimension is $\scriptstyle{1 + log_k(\frac{k+1}{2})}$ (Cf Stephen Wolfram ^[3])
$\textstyle{\frac {ln(7)} {ln(3)}}$	1.7712	Hexaflake	Built by exchanging iterativelly 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).
$\textstyle{\frac {ln(4)} {ln(2(1+cos(85^\circ))}}$	1.7848	Von Koch curve 85°, Cesaro fractal	Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then $\scriptstyle{\frac{ln(4)}{ln(2(1+cos(a))}}$ . The Cesaro fractal is based on this pattern.
$\textstyle{\frac {ln(6)} {ln(1+\phi)}}$	1.8617	Pentaflake	Built by exchanging iterativelly each pentagon by a flake of 6 pentagons. $φ$ = golden number = $\scriptstyle{\frac{1+\sqrt{5}}{2}}$
$\textstyle{\frac {ln(8)} {ln(3)}}$	1.8928	Sierpinski carpet
$\textstyle{\frac {ln(8)} {ln(3)}}$	1.8928	3D Cantor dust	Cantor set in 3 dimensions.
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^[4]
$\textstyle{2}$	2	Mandelbrot set	Any plane object containing a disk has Hausdorff dimension δ = 2. However, note that the boundary of the Mandelbrot set also has Hausdorff dimension δ = 2.
$\textstyle{2}$	2	Sierpiński curve	Every peano curve filling the plane has a Hausdorff dimension of 2.
$\textstyle{2}$	2	Hilbert curve	Built in a similar way: the Moore curve
$\textstyle{2}$	2	Peano curve	And a familly of curves built in a similar way, such as the Wunderlich curves.
	2	Lebesgue curve or z-order curve	Unlike the previous ones this space-filling curve is differentiable almost everywhere.
$\textstyle{\frac {ln(2)} {ln(\sqrt{2})}}$	2	Dragon curve	And its boundary has a fractal dimension of 1,5236.
	2	Terdragon curve	L-System : F-> F+F-F. angle=120°.
$\textstyle{\frac {ln(4)} {ln(2)}}$	2	T-Square
$\textstyle{\frac {ln(4)} {ln(2)}}$	2	Gosper curve	Its boundary is the Gosper island.
$\textstyle{\frac {ln(4)} {ln(2)}}$	2	Sierpinski tetrahedron
$\textstyle{\frac {ln(4)} {ln(2)}}$	2	H-fractal	Also the « Mandelbrot tree » which has a similar pattern.
$\textstyle{\frac {ln(4)} {ln(2)}}$	2	2D greek cross fractal	Each segment is replaced by a cross formed by 4 segments.
	2.06	Lorenz attractor	For precise values of parameters.
$\textstyle{\frac {ln(20)} {ln(2+\phi)}}$	2.3296	Dodecaedron fractal	Each dodecaedron is replaced by 20 dodecaedrons.
$\textstyle{\frac {ln(13)} {ln(3)}}$	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 apollolian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert ^[5].
$\textstyle{\frac {ln(32)} {ln(4)}}$	2.50	3D quadratic Koch surface (type 2)	Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the first iteration.
$\textstyle{\frac {ln(16)} {ln(3)}}$	2.5237	Cantor hypercube	Cantor set in 4 dimensions. Generalization : in a space of dimension n, the cantor set has a hausdorff dimension of $\scriptstyle{n\frac{ln(2)}{ln(3)}}$
$\textstyle{\frac {ln(12)} {ln(1+\phi)}}$	2.5819	Icosaedron fractal	Each Icosaedron is replaced by 12 icosaedrons.
$\textstyle{\frac {ln(6)} {ln(2)}}$	2.5849	3D greek cross fractal	Each segment is replaced by a cross formed by 6 segments.
$\textstyle{\frac {ln(6)} {ln(2)}}$	2.5849	Octaedron fractal	Each octaedron is replaced by 6 octaedrons.
$\textstyle{\frac {ln(20)} {ln(3)}}$	2.7268	Menger sponge	And its surface has a fractal dimension of $\scriptstyle{\frac{ln(12)}{ln(3)} = 2.2618}$ .
$\textstyle{\frac {ln(8)} {ln(2)}}$	3	3D Hilbert curve	A Hilbert curve extended to 3 dimensions.

[edit] Random and natural fractals

δ (exact value)	δ (value)	Name	Remarks
Measured	1.24	Coastline of Great Britain
$\textstyle{\frac {4}{3}}$	1.33	Boundary of Brownian motion	(Cf Wenderlin Werner)^[6].
$\textstyle{\frac {4}{3}}$	1.33	2D Polymer	Similar to the brownian motion in 2D with non self-intersection. (Cf Sapoval).
$\textstyle{\frac {4}{3}}$	1.33	Percolation front in 2D, Corrosion front in 2D	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 (Cf Sapoval).
	1.40	Clusters of clusters 2D	When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4. (Cf Sapoval)
Measured	1.52	Coastline of Norway
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.
$\textstyle{\frac {5} {3}}$	1.66	3D Polymer	Similar to the brownian motion in a cubic lattice, but without self-intersection (Cf Sapoval).
	1.70	2D DLA Cluster	In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70 (Cf Sapoval).
$\textstyle{\frac {91} {48}}$	1.8958	2D Percolation cluster	Under the percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48 (Cf Sapoval). Beyond that threshold, le cluster is infinite and 91/48 becomes the fractal dimension of the « clearings ».
$\textstyle{\frac {ln(2)} {ln(\sqrt{2})}}$	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").
$\textstyle{\frac {ln(13)} {ln(3)}}$	2.33	Cauliflower	Every branch carries around 13 branches 3 times smaller.
	2.4 ± 0.2	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. ^[7] 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 (Cf Sapoval).
	2.97	Lung surface	The alveoli of a lung form a fractal surface close to 3 (Cf Sapoval).

[edit] References

^ Fractal dimension of the boundary of the dragon fractal
^ Fractal dimension of the Pascal triangle modulo k
^ Fractal dimension of the Pascal triangle modulo k
^ Fractal dimension of a penrose tiling
^ Fractal dimension of the apollonian sphere packing
^ Fractal dimension of the brownian motion boundary
^ Filipponi, A. (2005), "Una sequenza di palle di carta" in Introduzione alla fisica, Zanichelli, ISBN 88-08-07073-5, pp. 64–68 (in [[Italian (language)|]])

[edit] See also

[edit] Bibliography

¹Kenneth Falconer, Fractal Geometry, John Wiley & Son Ltd; ISBN 0471922870 (mars 1990)
Benoît Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman & Co; ISBN 0716711869 (septembre 1982).
Heinz-Otto Peitgen, The Science of Fractal Images, Dietmar Saupe (éditeur), Springer Verlag, ISBN 0387966080 (août 1988)
Michael F. Barnsley, Fractals Everywhere, Morgan Kaufmann; ISBN 0120790610
Bernard Sapoval, « Universalités et fractales », collection Champs, Flammarion.

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List of fractals by Hausdorff dimension

From Wikipedia, the free encyclopedia

Contents

[edit] Deterministic fractals

[edit] Random and natural fractals

[edit] References

[edit] See also

[edit] Bibliography

[edit] Internal links

[edit] External links

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