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." 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.
Examples of FractalsEdit
|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.|
|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.|
|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 .[page needed] is one of the conjugated roots of .|
|1.12915||1||contour of the Gosper island||Term used by Mandelbrot (1977). 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.
|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).|
|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.|
|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|
|Calculated||1.3934||1||Douady rabbit||Julia set for c = −0,123 + 0.745i.|
|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.|
|1.5000||1||Quadratic von Koch curve (type 2)||Also called "Minkowski sausage".|
|1.5236||1||Boundary of the Dragon curve||cf. Chang & Zhang.|
|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).|
|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).|
|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). (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: .|
|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.|
|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 .|
|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 .|
|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.|
|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.|
|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.|
|Measured||2.06 ±0.01||1||Lorenz attractor||For parameters , =16 and . See McGuinness (1983)|
|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.|
|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.|
|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.|
|Wikimedia Commons has media related to fractals.|
Notes and referencesEdit
- Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman and Company, New York (1982); p. 15
- Aurell, Erik (1987). "On the metric properties of the Feigenbaum attractor", SpringerLink.com. Accessed: 27 October 2018.
- McMullen, Curtis T. (3 October 1997). "Hausdorff dimension and conformal dynamics III: Computation of dimension", Abel.Math.Harvard.edu. Accessed: 27 October 2018.
- Messaoudi, Ali. Frontième de numération complexe", matwbn.icm.edu.pl. (in French) Accessed: 27 October 2018.
- 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
- Weisstein, Eric W. "Gosper Island". MathWorld. Retrieved 27 October 2018.
- Ngai, Sirvent, Veerman, and Wang (October 2000). "On 2-Reptiles in the Plane 1999", Geometriae Dedicata, Volume 82. Accessed: 29 October 2018.
- Duda, Jarek (March 2011). "The Boundary of Periodic Iterated Function Systems", Wolfram.com.
- Falconer, Kenneth (1990–2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd. xxv. ISBN 0-470-84862-6.
- Fractal dimension of the boundary of the dragon fractal
- Fractal dimension of the Pascal triangle modulo k
- The Fibonacci word fractal
- Estimating Fractal dimension
- W. Trump, G. Huber, C. Knecht, R. Ziff, to be published
- Monkeys tree fractal curve Archived 21 September 2002 at Archive.today
- Fractal dimension of the boundary of the Mandelbrot set
- Fractal dimension of certain Julia sets
- Fractals and the Rössler attractor
- The fractal dimension of the Lorenz attractor, Mc Guinness (1983)
- The Fractal dimension of the apollonian sphere packing Archived 6 May 2016 at the Wayback Machine
- 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).