LIST OF FRACTALS BY HAUSDORFF DIMENSION

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.

Contents

Deterministic fractals

Random and natural fractals

References

See also

Bibliography

Internal links

External links

Deterministic fractals

{| border="0" cellpadding="4" rules="all" style="border: 1px solid #999; background-color:#FFFFFF"
|- align="center" bgcolor="#cccccc"
! δ
(exact value) || δ
(value) || Name || Illustration || width="40%" | Remarks
|-
| || align="right" | 0.4498? || Logistic map bifurcations || align="center" | || 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).
|-
| || align="right" | 0.6309 || Cantor set || align="center" | || Built by removing the central third at each iteration. Nowhere dense and not a countable set
|-
| $extstyle\{1\}$ || align="right" | 1 || Smith-Volterra-Cantor set || align="center" | || Built by removing a central interval of length 1/2^{2n} of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of ½.
|-
| || align="right" | 1.0686 || contour of the Gosper island || align="center" | ||
|-
| Measured (box counting) || align="right" | 1.2 || Dendrite Julia set || align="center" | || Julia set for parameters: Real=0 and Imaginary=1.
|-
| || align="right" | 1.26 || Hénon map || align="center" | || 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.
|-
| || align="right" | 1.2619 || Koch curve || align="center" | || 3 von Koch curves form the Koch snowflake or the anti-snowflake.
|-
| || align="right" | 1.2619 || boundary of Terdragon curve || align="center" | || L-system: same as dragon curve with angle=30°. The Fudgeflake is based on 3 initial segments placed in a triangle.
|-
| || align="right" | 1.2619 || 2D Cantor dust || align="center" | || Cantor set in 2 dimensions.
|-
| || align="right" | 1.3057 || Apollonian gasket || align="center" | ||
|-
| || align="right" | 1.4649 || Box fractal || align="center" | || Built by exchanging iteratively each square by a cross of 5 squares.
|-
| || align="right" | 1.4649 || Quadratic von Koch curve (type 1)|| align="center" | || One can recognize the pattern of the box fractal (above).
|-
||| align="right" | 1.5000 || Quadratic von Koch curve (type 2) || align="center" | || Also called "Minkowski sausage".
|-
| || align="right" | 1.5236 || Dragon curve boundary || align="center" | || Cf Chang & Zhang.^[1]
|-
| || align="right" | 1.5850 || 3-branches tree || align="center" | || 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.
|-
| || align="right" | 1.5850 || Sierpinski triangle || align="center" | || It’s also the triangle of Pascal modulo 2.
|-
| || align="right" | 1.5850 || Arrowhead Sierpinski curve || align="center" | || Same limit as the triangle (above) but built with a one-dimensional curve.
|-
| || align="right" | 1.6309 || Pascal triangle modulo 3 || align="center" | || For a triangle modulo k, if k is prime, the fractal dimension is (Cf Stephen Wolfram^[2])
|-
| || align="right" | 1.6826 || Pascal triangle modulo 5 || align="center" | || For a triangle modulo k, if k is prime, the fractal dimension is (Cf Stephen Wolfram^[2])
|-
| || align="right" | 1.7712 || Hexaflake || align="center" | || 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).
|-
| || align="right" | 1.7848 || Von Koch curve 85°, Cesaro fractal || align="center" | || Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then . The Cesaro fractal is based on this pattern.
|-
| || align="right" | 1.8617 || Pentaflake || align="center" | || Built by exchanging iteratively each pentagon by a flake of 6 pentagons. $phi$ = golden number =
|-
| || align="right" | 1.8928 || Sierpinski carpet || align="center" | ||
|-
| || align="right" | 1.8928 || 3D Cantor dust || align="center" | || Cantor set in 3 dimensions.
|-
|Estimated || align="right" | 1.9340 || Boundary of the Lévy C curve || align="center" | || Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
|-
| || align="right" | 1.974 || Penrose tiling || align="center" | || See Ramachandrarao, Sinha & Sanyal^[4]
|-
| $extstyle\{2\}$ || align="right" | 2 || Mandelbrot set || align="center" | || 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.
|-
| $extstyle\{2\}$ || align="right" | 2 || Sierpiński curve || align="center" | || Every Peano curve filling the plane has a Hausdorff dimension of 2.
|-
| $extstyle\{2\}$ || align="right" | 2 || Hilbert curve || align="center" | || Built in a similar way: the Moore curve
|-
| $extstyle\{2\}$ || align="right" | 2 || Peano curve || align="center" | || And a family of curves built in a similar way, such as the Wunderlich curves.
|-
| || align="right" | 2 || Lebesgue curve or z-order curve || align="center" | || 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.^[5]
|-
| || align="right" | 2 || Dragon curve || align="center" | || And its boundary has a fractal dimension of 1.5236.
|-
| || align="right" | 2 || Terdragon curve || align="center" | || L-System: F-> F+F-F. angle=120°.
|-
| || align="right" | 2 || T-Square || align="center" | ||
|-
| || align="right" | 2 || Gosper curve || align="center" | || Its boundary is the Gosper island.
|-
| || align="right" | 2 || Sierpinski tetrahedron || align="center" | || Each tetrahedron is replaced by 4 tetrahedra.
|-
| || align="right" | 2 || H-fractal || align="center" ||| Also the « Mandelbrot tree » which has a similar pattern.
|-
| || align="right" | || Pythagoras tree || align="center" ||| Every square generates 2 squares with a reduction ratio of sqrt(2)/2.
|-
| || align="right" | 2 || 2D Greek cross fractal || align="center" | || Each segment is replaced by a cross formed by 4 segments.
|-
| || align="right" | 2.06 || Lorenz attractor || align="center" | || For precise values of parameters.
|-
| || align="right" | 2.3296 || Dodecahedron fractal || align="center" ||| Each dodecahedron is replaced by 20 dodecahedra.
|-
| || align="right" | 2.3347 || 3D quadratic Koch surface (type 1) || align="center" ||| Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
|-
| || align="right" | 2.4739 || Apollonian sphere packing || align="center" | || The interstice left by the apollolian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.^[6]
|-
| || align="right" | 2.50 || 3D quadratic Koch surface (type 2) || align="center" ||| Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the first iteration.
|-
| || align="right" | 2.5237 || Cantor tesseract || align="center" | || Cantor set in 4 dimensions. Generalization: in a space of dimension n, the Cantor set has a Hausdorff dimension of
|-
| || align="right" | 2.5819 || Icosahedron fractal || align="center" ||| Each icosahedron is replaced by 12 icosahedra.
|-
| || align="right" | 2.5849 || 3D Greek cross fractal || align="center" ||| Each segment is replaced by a cross formed by 6 segments.
|-
| || align="right" | 2.5849 || Octahedron fractal || align="center" ||| Each octahedron is replaced by 6 octahedra.
|-
| || align="right" | 2.7268 || Menger sponge || align="center" | || And its surface has a fractal dimension of .
|-
| || align="right" | 3 || 3D Hilbert curve || align="center" | || A Hilbert curve extended to 3 dimensions.
|-
| || align="right" | 3 || 3D Lebesgue curve || align="center" | || A Lebesgue curve extended to 3 dimensions.
|-
| $extstyle\{log\_\{2\}(\{f\_\{1\}^\{2\}+f\_\{2\}^\{2\}+f\_\{3\}^\{2\}+f\_\{4\}^\{2\}\})\}$ || align="right" | $extstyle\{in(-infty,2)\}$ || Multiplicative Cascade || align="center" | || This is an example of a multifractal distribution, as it is generally not exactly self similar. However by choosing $extstyle\{f\_\{1\},f\_\{2\},f\_\{3\},f\_\{4\}\}$ in a particular way we can force the distribution to become a monofractal^[7].
|}

Random and natural fractals

{| border="0" cellpadding="4" rules="all" style="border: 1px solid #999; background-color:#FFFFFF"
|- align="center" bgcolor="#cccccc"
! δ
(exact value) || δ
(value) || Name || Illustration || width="40%" | Remarks
|-
|Measured||align="right"|1.24||Coastline of Great Britain||align="center"| ||
|-
| || align="right" | 1.33 || Boundary of Brownian motion || align="center" | || (Cf Wendelin Werner).^[8]
|-
| || align="right" | 1.33 || 2D Polymer || align="center" | || Similar to the brownian motion in 2D with non self-intersection. (Cf Sapoval).
|-
| || align="right" | 1.33 || Percolation front in 2D, Corrosion front in 2D || align="center" | || 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).
|-
| || align="right" | 1.40 || Clusters of clusters 2D || align="center" | || When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4. (Cf Sapoval)
|-
| Measured|| align="right" | 1.52|| Coastline of Norway || align="center" | ||
|-
| Measured|| align="right" | 1.55 || Random walk with no self-intersection || align="center" | || Self-avoiding random walk in a square lattice, with a « go-back » routine for avoiding dead ends.
|-
| || align="right" | 1.66|| 3D Polymer || align="center" | || Similar to the brownian motion in a cubic lattice, but without self-intersection (Cf Sapoval).
|-
| || align="right" | 1.70 || 2D DLA Cluster || align="center" | || In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70 (Cf Sapoval).
|-
| || align="right" | 1.8958 || 2D Percolation cluster || align="center" | || 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 ».
|-
| || align="right" | 2 || Brownian motion || align="center" | || Or random walk. The Hausdorff dimensions equals 2 in 2D, in 3D and in all greater dimensions (K.Falconer "The geometry of fractal sets").
|-
| || align="right" | 2.33 || Cauliflower || align="center" | || Every branch carries around 13 branches 3 times smaller.
|-
| || align="right" | 2.5 || Balls of crumpled paper || align="center" | || 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)).
|-
| || align="right" | 2.50 || 3D DLA Cluster || align="center" | || In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50 (Cf Sapoval).
|-
| Measured|| align="right" | 2.66 || Broccoli || align="center" | ||^[9]
|-
| || align="right" | 2.79 || Surface of human brain || align="center" | ||^[10]
|-
| || align="right" | 2.97 || Lung surface || align="center" | || The alveoli of a lung form a fractal surface close to 3 (Cf Sapoval).
|-
|Calculated || align="right" | 3 || Quantum string drifting randomnly|| align="center" | || Hausdorff dimension of a quantum string whose representative point randomly drifts through loop space.^[11]
|}

References

1. Fractal dimension of the boundary of the dragon fractal
2. Fractal dimension of the Pascal triangle modulo k
3. Fractal dimension of the Pascal triangle modulo k
4. Fractal dimension of a penrose tiling
5. Lebesgue curve variants
6. Fractal dimension of the apollonian sphere packing
7. [Meakin (1987)]
8. Fractal dimension of the brownian motion boundary
9. Fractal dimension of the broccoli
10. Fractal dimension of the surface of the human brain
11. The Hausdorf dimension of a quantum string

See also

Bibliography

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

★ 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.

Internal links

★ Fractal dimension

★ Hausdorff dimension

★ Scale invariance

External links

★ 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 softwares

★ Fractals unleashed