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.

Examples of Fractals

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 ${\displaystyle \scriptstyle {\lambda _{\infty }=3.570}}$ , where the period doubling is infinite. This dimension is the same for any differentiable and unimodal function.[2]
${\displaystyle \log _{3}(2)}$  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 ${\displaystyle 2|\alpha |^{3s}+|\alpha |^{4s}=1}$  1.0933 1 Boundary of the Rauzy fractal   Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: ${\displaystyle 1\mapsto 12}$ , ${\displaystyle 2\mapsto 13}$  and ${\displaystyle 3\mapsto 1}$ .[4][page needed][5] ${\displaystyle \alpha }$  is one of the conjugated roots of ${\displaystyle z^{3}-z^{2}-z-1=0}$ .
${\displaystyle 2\log _{7}(3)}$  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.
${\displaystyle 3{\frac {\log(\varphi )}{\log \left(\displaystyle {\frac {3+{\sqrt {13}}}{2}}\right)}}}$  1.2083 1 Fibonacci word fractal 60°   Build from the Fibonacci word. See also the standard Fibonacci word fractal.

${\displaystyle \varphi =(1+{\sqrt {5}})/2}$  (golden ratio).

{\displaystyle {\begin{aligned}&2\log _{2}\left(\displaystyle {\frac {{\sqrt[{3}]{27-3{\sqrt {78}}}}+{\sqrt[{3}]{27+3{\sqrt {78}}}}}{3}}\right),\\&{\text{or root of }}2^{x}-1=2^{(2-x)/2}\end{aligned}}}  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.
${\displaystyle \log _{3}(4)}$  1.2619 1 Koch curve   3 Koch curves form the Koch snowflake or the anti-snowflake.
${\displaystyle \log _{3}(4)}$  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.
${\displaystyle \log _{3}(4)}$  1.2619 0 2D Cantor dust   Cantor set in 2 dimensions.
${\displaystyle \log _{3}(4)}$  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]
${\displaystyle \log _{3}(5)}$  1.4649 1 Vicsek fractal   Built by exchanging iteratively each square by a cross of 5 squares.
${\displaystyle 2-\log _{2}({\sqrt {2}})={\frac {3}{2}}}$  (conjectured exact) 1.5000 1 a Weierstrass function: ${\displaystyle \displaystyle f(x)=\sum _{k=1}^{\infty }{\frac {\sin(2^{k}x)}{{\sqrt {2}}^{k}}}}$    The Hausdorff dimension of the Weierstrass function ${\displaystyle f:[0,1]\to \mathbb {R} }$  defined by ${\displaystyle f(x)=\sum _{k=1}^{\infty }a^{-k}\sin(b^{k}x)}$  with ${\displaystyle 1  and ${\displaystyle b>1}$  has upper bound ${\displaystyle 2-\log _{b}(a)}$ . 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]
${\displaystyle \log _{4}(8)={\frac {3}{2}}}$  1.5000 1 Quadratic von Koch curve (type 2)   Also called "Minkowski sausage".
${\displaystyle \log _{2}\left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87}}}}+{\sqrt[{3}]{73+6{\sqrt {87}}}}}{3}}\right)}$  1.5236 1 Boundary of the Dragon curve   cf. Chang & Zhang.[10][8]
${\displaystyle \log _{2}\left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87}}}}+{\sqrt[{3}]{73+6{\sqrt {87}}}}}{3}}\right)}$  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]
${\displaystyle \log _{2}(3)}$  1.5849 1 Sierpinski gasket   Also the triangle of Pascal modulo 2.
${\displaystyle \log _{2}(3)}$  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.
${\displaystyle \log _{\sqrt[{\varphi }]{\varphi }}(\varphi )=\varphi }$  1.61803 1 a golden dragon   Built from two similarities of ratios ${\displaystyle r}$  and ${\displaystyle r^{2}}$ , with ${\displaystyle r=1/\varphi ^{1/\varphi }}$ . Its dimension equals ${\displaystyle \varphi }$  because ${\displaystyle ({r^{2}})^{\varphi }+r^{\varphi }=1}$ . With ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$  (Golden number).
${\displaystyle 1+\log _{3}(2)}$  1.6309 1 Pascal triangle modulo 3   For a triangle modulo k, if k is prime, the fractal dimension is ${\displaystyle \scriptstyle {1+\log _{k}\left({\frac {k+1}{2}}\right)}}$  (cf. Stephen Wolfram[11]).
${\displaystyle 1+\log _{3}(2)}$  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.
${\displaystyle 3{\frac {\log(\varphi )}{\log(1+{\sqrt {2}})}}}$  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] ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$  (golden ratio).
Solution of ${\displaystyle (1/3)^{s}+(1/2)^{s}+(2/3)^{s}=1}$  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 ${\displaystyle n}$  similarities of ratios ${\displaystyle c_{n}}$ , has Hausdorff dimension ${\displaystyle s}$ , solution of the equation coinciding with the iteration function of the Euclidean contraction factor: ${\displaystyle \sum _{k=1}^{n}c_{k}^{s}=1}$ .[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 ${\displaystyle z_{n+1}=a+bz_{n}\exp \left[i\left[k-p/\left(1+\lfloor z_{n}\rfloor ^{2}\right)\right]\right]}$ . It derives from a model of the plane-wave interactivity field in an optical ring laser. Different parameters yield different values.[13]
${\displaystyle 4\log _{5}(2)}$  1.7227 1 Pinwheel fractal   Built with Conway's Pinwheel tile.
${\displaystyle \log _{3}(7)}$  1.7712 1 Sphinx fractal Built with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes [14].
${\displaystyle \log _{3}(7)}$  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).
${\displaystyle {\frac {\log(6)}{\log(1+\varphi )}}}$  1.8617 1 Pentaflake   Built by exchanging iteratively each pentagon by a flake of 6 pentagons. ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$  (golden ratio).
solution of ${\displaystyle 6(1/3)^{s}+5{(1/3{\sqrt {3}})}^{s}=1}$  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 ${\displaystyle 1/3}$  and 5 similarities of ratio ${\displaystyle 1/3{\sqrt {3}}}$ .[15]
${\displaystyle \log _{3}(8)}$  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).
${\displaystyle \log _{3}(8)}$  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.
${\displaystyle 2}$  2 1 Boundary of the Mandelbrot set   The boundary interestingly has a fractal dimension 1 unit greater than its topological dimension.[16]
${\displaystyle 2}$  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]
${\displaystyle \log _{2}(4)=2}$  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.[18]
Measured 2.06 ±0.01 1 Lorenz attractor   For parameters ${\displaystyle \rho =40}$ ,${\displaystyle \sigma }$ =16 and ${\displaystyle \beta =4}$  . See McGuinness (1983)[19]
${\displaystyle {\frac {\log(20)}{\log(2+\varphi )}}}$  2.3296 2 Dodecahedron fractal   Each dodecahedron is replaced by 20 dodecahedra. ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$  (golden ratio).
${\displaystyle \log _{3}(13)}$  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]
${\displaystyle \log _{4}(32)={\frac {5}{2}}}$  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.
${\displaystyle {\frac {\log \left({\frac {\sqrt {7}}{6}}-{\frac {1}{3}}\right)}{\log({\sqrt {2}}-1)}}}$  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 ${\displaystyle {\sqrt {2}}-1}$ .
${\displaystyle {\frac {\log(12)}{\log(1+\varphi )}}}$  2.5819 2 Icosahedron fractal   Each icosahedron is replaced by 12 icosahedra. ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$  (golden ratio).
${\displaystyle 1+\log _{2}(3)}$  2.5849 2 Octahedron fractal   Each octahedron is replaced by 6 octahedra.
${\displaystyle 1+\log _{2}(3)}$  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".

${\displaystyle {\frac {\log(3)}{\log(3/2)}}}$  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]
${\displaystyle \log _{3}(20)}$  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.

Notes and references

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. ^
21. ^ [1]