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List of fractals by Hausdorff dimension

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.

Contents

Examples of FractalsEdit

Hausdorff dimension
(exact value)
Hausdorff dimension
(approx.)
Topological Dimension Name Illustration Remarks
Calculated 0.538 0 Feigenbaum attractor   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 0 Cantor set   Built by removing the central third at each iteration. Nowhere dense and not a countable set.
Calculated 1.0812 1 Julia set z² + 1/4   Julia set for c = 1/4.[3]
Solution s of   1.0933 1 Boundary of the Rauzy fractal   Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism:  ,   and  .[4][page needed][5]   is one of the conjugated roots of  .
  1.12915 1 contour of the Gosper island   Term used by Mandelbrot (1977).[6] The Gosper island is the limit of the Gosper curve.
Measured (box counting) 1.2 1 Dendrite Julia set   Julia set for parameters: Real = 0 and Imaginary = 1.
  1.2083 1 Fibonacci word fractal 60°   Build from the Fibonacci word. See also the standard Fibonacci word fractal.

  (golden ratio).

  1.2108 1 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).[7][8]
1.26 1 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.
  1.2619 1 Koch curve   3 Koch curves form the Koch snowflake or the anti-snowflake.
  1.2619 1 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 0 2D Cantor dust   Cantor set in 2 dimensions.
  1.2619 1 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 1 Julia set z2 − 1   Julia set for c = −1.[3]
1.3057 1 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[3]
Calculated 1.3934 1 Douady rabbit   Julia set for c = −0,123 + 0.745i.[3]
  1.4649 1 Vicsek fractal   Built by exchanging iteratively each square by a cross of 5 squares.
  (conjectured exact) 1.5000 1 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.[9]
  1.5000 1 Quadratic von Koch curve (type 2)   Also called "Minkowski sausage".
  1.5236 1 Boundary of the Dragon curve   cf. Chang & Zhang.[10][8]
  1.5236 1 Boundary of the twindragon 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).[7]
  1.5849 1 Sierpinski gasket   Also the triangle of Pascal modulo 2.
  1.5849 1 Boundary of the T-square fractal   The dimension of boundary of the T-square fractal is the same as that of the Sierpinski gasket.
  1.61803 1 a golden dragon   Built from two similarities of ratios   and  , with  . Its dimension equals   because  . With   (Golden number).
  1.6309 1 Pascal triangle modulo 3   For a triangle modulo k, if k is prime, the fractal dimension is   (cf. Stephen Wolfram[11]).
  1.6309 1 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 1 Fibonacci word fractal   Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F23 = 28657 segments).[12]   (golden ratio).
Solution of   1.6402 1 Attractor of IFS with 3 similarities 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:  .[9]
Measured (box-counting) 1.7 1 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.[13]
  1.7227 1 Pinwheel fractal   Built with Conway's Pinwheel tile.
  1.7712 1 Sphinx fractal Built with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes [14].
  1.7712 1 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).
  1.8617 1 Pentaflake   Built by exchanging iteratively each pentagon by a flake of 6 pentagons.   (golden ratio).
solution of   1.8687 1 Monkey 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  .[15]
  1.8928 1 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 0 3D Cantor dust   Cantor set in 3 dimensions.
Estimated 1.9340 1 Boundary of the Lévy C curve   Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
  2 1 Boundary of the Mandelbrot set   The boundary interestingly has a fractal dimension 1 unit greater than its topological dimension.[16]
  2 1 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.[17]
  2 1 Sierpiński tetrahedron   The 3D analogue of the Sierpenski gasket. Each tetrahedron is replaced by 4 tetrahedra. Note that each of the four faces of the Sierpinski tetrahedron is a Sierpinski triangle.
Measured 2.01 ±0.01 1 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.[18]
Measured 2.06 ±0.01 1 Lorenz attractor   For parameters  , =16 and   . See McGuinness (1983)[19]
  2.3296 2 Dodecahedron fractal   Each dodecahedron is replaced by 20 dodecahedra.   (golden ratio).
  2.3347 2 3D quadratic Koch surface (type 1)   Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
2.4739 2 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.[20]
  2.50 2 3D quadratic Koch surface (type 2)   Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
  2.529 2 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 2 Icosahedron fractal   Each icosahedron is replaced by 12 icosahedra.   (golden ratio).
  2.5849 2 Octahedron fractal   Each octahedron is replaced by 6 octahedra.
  2.5849 2 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".

  2.7095 2 Von Koch in 3D   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.[21]
  2.7268 2 Menger sponge   The 3D analogue of the Sierpinski carpet. Each cube is replaced with 20 cubes. Note that each of the six faces of the Menger sponge is a Sierpinski carpet.

See alsoEdit

Notes and referencesEdit

  1. ^ Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman and Company, New York (1982); p. 15
  2. ^ Aurell, Erik (1987). "On the metric properties of the Feigenbaum attractor", SpringerLink.com. Accessed: 27 October 2018.
  3. ^ a b c d McMullen, Curtis T. (3 October 1997). "Hausdorff dimension and conformal dynamics III: Computation of dimension", Abel.Math.Harvard.edu. Accessed: 27 October 2018.
  4. ^ Messaoudi, Ali. Frontième de numération complexe", matwbn.icm.edu.pl. (in French) Accessed: 27 October 2018.
  5. ^ 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
  6. ^ Weisstein, Eric W. "Gosper Island". MathWorld. Retrieved 27 October 2018.
  7. ^ a b Ngai, Sirvent, Veerman, and Wang (October 2000). "On 2-Reptiles in the Plane 1999", Geometriae Dedicata, Volume 82. Accessed: 29 October 2018.
  8. ^ a b Duda, Jarek (March 2011). "The Boundary of Periodic Iterated Function Systems", Wolfram.com.
  9. ^ a b Falconer, Kenneth (1990–2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd. xxv. ISBN 0-470-84862-6.
  10. ^ Fractal dimension of the boundary of the dragon fractal
  11. ^ Fractal dimension of the Pascal triangle modulo k
  12. ^ The Fibonacci word fractal
  13. ^ Estimating Fractal dimension
  14. ^ W. Trump, G. Huber, C. Knecht, R. Ziff, to be published
  15. ^ Monkeys tree fractal curve Archived 21 September 2002 at Archive.today
  16. ^ Fractal dimension of the boundary of the Mandelbrot set
  17. ^ Fractal dimension of certain Julia sets
  18. ^ Fractals and the Rössler attractor
  19. ^ The fractal dimension of the Lorenz attractor, Mc Guinness (1983)
  20. ^ The Fractal dimension of the apollonian sphere packing Archived 6 May 2016 at the Wayback Machine
  21. ^ [1]

Further readingEdit

  • 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 linksEdit