# 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.

## Deterministic fractalsEdit

Hausdorff dimension
(exact value)
Hausdorff dimension
(approx.)
Name Illustration Remarks
Calculated 0.538 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 Cantor set   Built by removing the central third at each iteration. Nowhere dense and not a countable set.
${\displaystyle \log _{2}(\varphi )=\log _{2}(1+{\sqrt {5}})-1}$  0.6942 Asymmetric Cantor set   The dimension is not ${\displaystyle {\frac {\ln 2}{\ln {\frac {8}{3}}}}}$ , which is the generalized Cantor set with γ=1/4, which has the same length at each stage.[3]

Built by removing the second quarter at each iteration. Nowhere dense and not a countable set. ${\displaystyle \scriptstyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$  (golden cut).

${\displaystyle \log _{10}(5)=1-\log _{10}(2)}$  0.69897 Real numbers whose base 10 digits are even   Similar to the Cantor set.[4]
${\displaystyle \log(1+{\sqrt {2}})}$  0.88137 Spectrum of Fibonacci Hamiltonian The study of the spectrum of the Fibonacci Hamiltonian proves upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that the spectrum converges to an explicit constant.[5]
${\displaystyle {\frac {-\log(2)}{\log \left(\displaystyle {\frac {1-\gamma }{2}}\right)}}}$  0<D<1 Generalized Cantor set   Built by removing at the ${\displaystyle m}$ th iteration the central interval of length ${\displaystyle \gamma \,l_{m-1}}$  from each remaining segment (of length ${\displaystyle l_{m-1}=(1-\gamma )^{m-1}/2^{m-1}}$ ). At ${\displaystyle \scriptstyle \gamma =1/3}$  one obtains the usual Cantor set. Varying ${\displaystyle \scriptstyle \gamma }$  between 0 and 1 yields any fractal dimension ${\displaystyle \scriptstyle 0\,<\,D\,<\,1}$ .[6]
${\displaystyle 1}$  1 Smith–Volterra–Cantor set   Built by removing a central interval of length ${\displaystyle 2^{-2n}}$  of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of ½.
${\displaystyle 2+\log _{2}\left({\frac {1}{2}}\right)=1}$  1 Takagi or Blancmange curve   Defined on the unit interval by ${\displaystyle f(x)=\sum _{n=0}^{\infty }2^{-n}s(2^{n}x)}$ , where ${\displaystyle s(x)}$ is the triangle wave function. Special case of the Takahi-Landsberg curve: ${\displaystyle f(x)=\sum _{n=0}^{\infty }w^{n}s(2^{n}x)}$  with ${\displaystyle w=1/2}$ . The Hausdorff dimension equals ${\displaystyle 2+\log _{2}(w)}$  for ${\displaystyle w}$  in ${\displaystyle \left[1/2,1\right]}$ . (Hunt cited by Mandelbrot[7]).
Calculated 1.0812 Julia set z² + 1/4   Julia set for c = 1/4.[8]
Solution s of ${\displaystyle 2|\alpha |^{3s}+|\alpha |^{4s}=1}$  1.0933 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 ${\displaystyle 3\mapsto 1}$ .[9][10] ${\displaystyle \alpha }$  is one of the conjugated roots of ${\displaystyle z^{3}-z^{2}-z-1=0}$ .
${\displaystyle 2\log _{7}(3)}$  1.12915 contour of the Gosper island   Term used by Mandelbrot (1977).[11] The Gosper island is the limit of the Gosper curve.
Measured (box counting) 1.2 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 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 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).[12][13]
1.26 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 Triflake   Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
${\displaystyle \log _{3}(4)}$  1.2619 Koch curve   3 Koch curves form the Koch snowflake or the anti-snowflake.
${\displaystyle \log _{3}(4)}$  1.2619 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 2D Cantor dust   Cantor set in 2 dimensions.
${\displaystyle \log _{3}(4)}$  1.2619 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 Julia set z2 − 1   Julia set for c = −1.[14]
1.3057 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[15]
1.328 5 circles inversion fractal   The limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an Apollonian packing. See[16]
Calculated 1.3934 Douady rabbit   Julia set for c = −0,123 + 0.745i.[17]
${\displaystyle \log _{3}(5)}$  1.4649 Vicsek fractal   Built by exchanging iteratively each square by a cross of 5 squares.
${\displaystyle \log _{3}(5)}$  1.4649 Quadratic von Koch curve (type 1)   One can recognize the pattern of the Vicsek fractal (above).
${\displaystyle \log _{\sqrt {5}}\left({\frac {10}{3}}\right)}$  1.4961 Quadric cross

The quadric cross is made by scaling the 3-segment generator unit by 51/2 then adding 3 full scaled units, one to each original segment, plus a third of a scaled unit (blue) to increase the length of the pedestal of the starting 3-segment unit (purple).
Built by replacing each end segment with a cross segment scaled by a factor of 51/2, consisting of 3 1/3 new segments, as illustrated in the inset.

Images generated with Fractal Generator for ImageJ.

${\displaystyle 2-\log _{2}({\sqrt {2}})={\frac {3}{2}}}$  (conjectured exact) 1.5000 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.[4]
${\displaystyle \log _{4}(8)={\frac {3}{2}}}$  1.5000 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 Boundary of the Dragon curve   cf. Chang & Zhang.[18][19]
${\displaystyle \log _{2}\left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87}}}}+{\sqrt[{3}]{73+6{\sqrt {87}}}}}{3}}\right)}$  1.5236 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).[12]
${\displaystyle \log _{2}(3)}$  1.5849 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.
${\displaystyle \log _{2}(3)}$  1.5849 Sierpinski triangle   Also the triangle of Pascal modulo 2.
${\displaystyle \log _{2}(3)}$  1.5849 Sierpiński arrowhead curve   Same limit as the triangle (above) but built with a one-dimensional curve.
${\displaystyle \log _{2}(3)}$  1.5849 Boundary of the T-Square fractal   The dimension of the fractal itself (not the boundary) is ${\displaystyle \log _{2}(4)=2}$ [20]
${\displaystyle \log _{\sqrt[{\varphi }]{\varphi }}(\varphi )=\varphi }$  1.61803 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 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[21]).
${\displaystyle 1+\log _{3}(2)}$  1.6309 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 Fibonacci word fractal   Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F23 = 28657 segments).[22] ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$  (golden ratio).
Solution of ${\displaystyle (1/3)^{s}+(1/2)^{s}+(2/3)^{s}=1}$  1.6402 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: ${\displaystyle \sum _{k=1}^{n}c_{k}^{s}=1}$ .[4]
${\displaystyle \log _{8}(32)={\frac {5}{3}}}$  1.6667 32-segment quadric fractal (1/8 scaling rule)  [2]

Generator for 32 segment 1/8 scale quadric fractal.
Built by scaling the 32 segment generator (see inset) by 1/8 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 32/log 8 = 1.6667. Images generated with Fractal Generator for ImageJ.
${\displaystyle 1+\log _{5}(3)}$  1.6826 Pascal triangle modulo 5   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[21]).
Measured (box-counting) 1.7 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.[23]
${\displaystyle 1+\log _{10}(5)}$  1.6990 50 segment quadric fractal (1/10 scaling rule)   Built by scaling the 50 segment generator (see inset) by 1/10 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 50/log 10 = 1.6990. Images generated with Fractal Generator for ImageJ[24].

Generator for 50 Segment Fractal.
${\displaystyle 4\log _{5}(2)}$  1.7227 Pinwheel fractal   Built with Conway's Pinwheel tile.
${\displaystyle \log _{3}(7)}$  1.7712 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).
${\displaystyle {\frac {\log(4)}{\log(2+2\cos(85^{\circ }))}}}$  1.7848 Von Koch curve 85°   Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then ${\displaystyle {\frac {\log(4)}{\log(2+2\cos(a))}}\in [1,2]}$ .
${\displaystyle \log _{2}\left(3^{0.63}+2^{0.63}\right)}$  1.8272 A self-affine fractal set   Build iteratively from a ${\displaystyle p\times q}$  array on a square, with ${\displaystyle p\leq q}$ . Its Hausdorff dimension equals ${\displaystyle \log _{p}\left(\sum _{k=1}^{p}n_{k}^{a}\right)}$ [4] with ${\displaystyle a=\log _{q}(p)}$  and ${\displaystyle n_{k}}$  is the number of elements in the ${\displaystyle k}$ th column. The box-counting dimension yields a different formula, therefore, a different value. Unlike self-similar sets, the Hausdorff dimension of self-affine sets depends on the position of the iterated elements and there is no formula, so far, for the general case.
${\displaystyle {\frac {\log(6)}{\log(1+\varphi )}}}$  1.8617 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 Monkeys 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 ${\displaystyle 1/3{\sqrt {3}}}$ .[25]
${\displaystyle \log _{3}(8)}$  1.8928 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 3D Cantor dust   Cantor set in 3 dimensions.
${\displaystyle \log _{3}(4)+\log _{3}(2)={\frac {\log(4)}{\log(3)}}+{\frac {\log(2)}{\log(3)}}={\frac {\log(8)}{\log(3)}}}$  1.8928 Cartesian product of the von Koch curve and the Cantor set   Generalization : Let F×G be the cartesian product of two fractals sets F and G. Then ${\displaystyle Dim_{H}(F\times G)=Dim_{H}(F)+Dim_{H}(G)}$ .[4] See also the 2D Cantor dust and the Cantor cube.
Estimated 1.9340 Boundary of the Lévy C curve   Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
2 Penrose tiling   See Ramachandrarao, Sinha & Sanyal.[26]
${\displaystyle 2}$  2 Boundary of the Mandelbrot set   The boundary and the set itself have the same dimension.[27]
${\displaystyle 2}$  2 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.[28]
${\displaystyle 2}$  2 Sierpiński curve   Every Peano curve filling the plane has a Hausdorff dimension of 2.
${\displaystyle 2}$  2 Hilbert curve
${\displaystyle 2}$  2 Peano curve   And a family of curves built in a similar way, such as the Wunderlich curves.
${\displaystyle 2}$  2 Moore curve   Can be extended in 3 dimensions.
2 Lebesgue curve or z-order curve   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.[29]
${\displaystyle \log _{\sqrt {2}}(2)=2}$  2 Dragon curve   And its boundary has a fractal dimension of 1.5236270862.[30]
2 Terdragon curve   L-system: F → F + F – F, angle = 120°.
${\displaystyle \log _{2}(4)=2}$  2 Gosper curve   Its boundary is the Gosper island.
Solution of ${\displaystyle 7({1/3})^{s}+6({1/3{\sqrt {3}}})^{s}=1}$  2 Curve filling the Koch snowflake   Proposed by Mandelbrot in 1982,[31] it fills the Koch snowflake. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio ${\displaystyle 1/3{\sqrt {3}}}$ .
${\displaystyle \log _{2}(4)=2}$  2 Sierpiński tetrahedron   Each tetrahedron is replaced by 4 tetrahedra.
${\displaystyle \log _{2}(4)=2}$  2 H-fractal   Also the Mandelbrot tree which has a similar pattern.
${\displaystyle {\frac {\log(2)}{\log(2/{\sqrt {2}})}}=2}$  2 Pythagoras tree (fractal)   Every square generates two squares with a reduction ratio of ${\displaystyle 1/{\sqrt {2}}}$ .
${\displaystyle \log _{2}(4)=2}$  2 2D Greek cross fractal   Each segment is replaced by a cross formed by 4 segments.
Measured 2.01 ±0.01 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.[32]
Measured 2.06 ±0.01 Lorenz attractor   For parameters ${\displaystyle \rho =40}$ ,${\displaystyle \sigma }$ =16 and ${\displaystyle \beta =4}$  . See McGuinness (1983)[33]
${\displaystyle \log _{2}(5)}$  2.3219 Fractal pyramid   Each square pyramid is replaced by 5 half-size square pyramids. (Different from the Sierpinski tetrahedron, which replaces each triangular pyramid with 4 half-size triangular pyramids).
${\displaystyle {\frac {\log(20)}{\log(2+\varphi )}}}$  2.3296 Dodecahedron fractal   Each dodecahedron is replaced by 20 dodecahedra. ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$  (golden ratio).
${\displaystyle 4+c^{D}+d^{D}=(c+d)^{D}}$  2<D<2.3 Pyramid surface   Each triangle is replaced by 6 triangles, of which 4 identical triangles form a diamond based pyramid and the remaining two remain flat with lengths ${\displaystyle c}$  and ${\displaystyle d}$  relative to the pyramid triangles. The dimension is a parameter, self-intersection occurs for values greater than 2.3.[34]
${\displaystyle \log _{3}(13)}$  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 Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.[35]
${\displaystyle \log _{4}(32)={\frac {5}{2}}}$  2.50 3D quadratic Koch surface (type 2)   Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
${\displaystyle \log _{3}(16)}$  2.5237 Cantor tesseract no image available Cantor set in 4 dimensions. Generalization: in a space of dimension n, the Cantor set has a Hausdorff dimension of ${\displaystyle n\log _{3}(2)}$ .
${\displaystyle {\frac {\log \left({\frac {\sqrt {7}}{6}}-{\frac {1}{3}}\right)}{\log({\sqrt {2}}-1)}}}$  2.529 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 Icosahedron fractal   Each icosahedron is replaced by 12 icosahedra. ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$  (golden ratio).
${\displaystyle 1+\log _{2}(3)}$  2.5849 3D Greek cross fractal   Each segment is replaced by a cross formed by 6 segments.
${\displaystyle 1+\log _{2}(3)}$  2.5849 Octahedron fractal   Each octahedron is replaced by 6 octahedra.
${\displaystyle 1+\log _{2}(3)}$  2.5849 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 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.[36]
${\displaystyle \log _{3}(20)}$  2.7268 Menger sponge   And its surface has a fractal dimension of ${\displaystyle \log _{3}(20)}$ , which is the same as that by volume.
${\displaystyle \log _{2}(8)=3}$  3 3D Hilbert curve   A Hilbert curve extended to 3 dimensions.
${\displaystyle \log _{2}(8)=3}$  3 3D Lebesgue curve   A Lebesgue curve extended to 3 dimensions.
${\displaystyle \log _{2}(8)=3}$  3 3D Moore curve   A Moore curve extended to 3 dimensions.
${\displaystyle \log _{2}(8)=3}$  3 3D H-fractal   A H-fractal extended to 3 dimensions.[37]
${\displaystyle 3}$  (conjectured) 3 (to be confirmed) Mandelbulb   Extension of the Mandelbrot set (power 8) in 3 dimensions[38][unreliable source?]

## Random and natural fractalsEdit

Hausdorff dimension
(exact value)
Hausdorff dimension
(approx.)
Name Illustration Remarks
1/2 0.5 Zeros of a Wiener process   The zeros of a Wiener process (Brownian motion) are a nowhere dense set of Lebesgue measure 0 with a fractal structure.[4][39]
Solution of ${\displaystyle \scriptstyle {E(C_{1}^{s}+C_{2}^{s})=1}}$  where ${\displaystyle \scriptstyle {E(C_{1})=0.5}}$  and ${\displaystyle \scriptstyle {E(C_{2})=0.3}}$  0.7499 a random Cantor set with 50% - 30%   Generalization : At each iteration, the length of the left interval is defined with a random variable ${\displaystyle C_{1}}$ , a variable percentage of the length of the original interval. Same for the right interval, with a random variable ${\displaystyle C_{2}}$ . Its Hausdorff Dimension ${\displaystyle s}$  satisfies : ${\displaystyle \scriptstyle {E(C_{1}^{s}+C_{2}^{s})=1}}$ . (${\displaystyle E(X)}$  is the expected value of ${\displaystyle X}$ ).[4]
Solution of ${\displaystyle s+1=12\cdot 2^{-(s+1)}-6\cdot 3^{-(s+1)}}$  1.144... von Koch curve with random interval   The length of the middle interval is a random variable with uniform distribution on the interval (0,1/3).[4]
Measured 1.22±0.02 Coastline of Ireland   Values for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault[40] at the University of Ulster and Theoretical Physics students at Trinity College, Dublin, under the supervision of S. Hutzler.[41]

Note that there are marked differences between Ireland's ragged west coast (fractal dimension of about 1.26) and the much smoother east coast (fractal dimension 1.10)[41]

Measured 1.25 Coastline of Great Britain   Fractal dimension of the west coast of Great Britain, as measured by Lewis Fry Richardson and cited by Benoît Mandelbrot.[42]
${\displaystyle \textstyle {\frac {\log(4)}{\log(3)}}}$  1.2619 von Koch curve with random orientation   One introduces here an element of randomness which does not affect the dimension, by choosing, at each iteration, to place the equilateral triangle above or below the curve.[4]
${\displaystyle \textstyle {\frac {4}{3}}}$  1.333 Boundary of Brownian motion   (cf. Mandelbrot, Lawler, Schramm, Werner).[43]
${\displaystyle \textstyle {\frac {4}{3}}}$  1.333 2D polymer Similar to the brownian motion in 2D with non self-intersection.[44]
${\displaystyle \textstyle {\frac {4}{3}}}$  1.333 Percolation front in 2D, Corrosion front in 2D   Fractal dimension of the percolation-by-invasion front (accessible perimeter), at the percolation threshold (59.3%). It's also the fractal dimension of a stopped corrosion front.[44]
1.40 Clusters of clusters 2D When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.[44]
${\displaystyle \textstyle {2-{\frac {1}{2}}}}$  1.5 Graph of a regular Brownian function (Wiener process)   Graph of a function f such that, for any two positive reals x and x+h, the difference of their images ${\displaystyle f(x+h)-f(x)}$  has the centered gaussian distribution with variance = h. Generalization : The fractional Brownian motion of index ${\displaystyle \alpha }$  follows the same definition but with a variance ${\displaystyle =h^{2\alpha }}$ , in that case its Hausdorff dimension =${\displaystyle 2-\alpha }$ .[4]
Measured 1.52 Coastline of Norway   See J. Feder.[45]
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.
${\displaystyle \textstyle {\frac {5}{3}}}$  1.66 3D polymer Similar to the brownian motion in a cubic lattice, but without self-intersection.[44]
1.70 2D DLA Cluster   In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.[44]
${\displaystyle \textstyle {\frac {\log(9\cdot 0.75)}{\log(3)}}}$  1.7381 Fractal percolation with 75% probability   The fractal percolation model is constructed by the progressive replacement of each square by a 3x3 grid in which is placed a random collection of sub-squares, each sub-square being retained with probability p. The "almost sure" Hausdorff dimension equals ${\displaystyle \textstyle {\frac {\log(9p)}{\log(3)}}}$ .[4]
7/4 1.75 2D percolation cluster hull   The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk,[46] or by Schramm-Loewner Evolution.
${\displaystyle \textstyle {\frac {91}{48}}}$  1.8958 2D percolation cluster   In a square lattice, under the site percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48.[44][47] Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings".
${\displaystyle \textstyle {{\frac {\log(2)}{\log({\sqrt {2}})}}=2}}$  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").
Measured Around 2 Distribution of galaxy clusters   From the 2005 results of the Sloan Digital Sky Survey.[48]
2.5 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.[49] 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.[44]
2.50 Lichtenberg figure   Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.[44]
${\displaystyle \textstyle {3-{\frac {1}{2}}}}$  2.5 regular Brownian surface   A function ${\displaystyle \scriptstyle {f:\mathbb {R} ^{2}->\mathbb {R} }}$ , gives the height of a point ${\displaystyle (x,y)}$  such that, for two given positive increments ${\displaystyle h}$  and ${\displaystyle k}$ , then ${\displaystyle \scriptstyle {f(x+h,y+k)-f(x,y)}}$  has a centered Gaussian distribution with variance = ${\displaystyle \scriptstyle {\sqrt {h^{2}+k^{2}}}}$ . Generalization : The fractional Brownian surface of index ${\displaystyle \alpha }$  follows the same definition but with a variance = ${\displaystyle (h^{2}+k^{2})^{\alpha }}$ , in that case its Hausdorff dimension = ${\displaystyle 3-\alpha }$ .[4]
Measured 2.52 3D percolation cluster   In a cubic lattice, at the site percolation threshold (31.1%), the 3D percolation-by-invasion cluster has a fractal dimension of around 2.52.[47] Beyond that threshold, the cluster is infinite.
Measured and calculated ~2.7 Broccoli   San-Hoon Kim used a direct scanning method and a cross section analysis of a broccoli to conclude that the fractal dimension of it is ~2.7.[50]
2.79 Surface of human brain   [51]
Measured and calculated ~2.8 Cauliflower   San-Hoon Kim used a direct scanning method and a mathematical analysis of the cross section of a cauliflower to conclude that the fractal dimension of it is ~2.8.[50]
2.97 Lung surface   The alveoli of a lung form a fractal surface close to 3.[44]
Calculated ${\displaystyle \textstyle {\in (0,2)}}$  Multiplicative cascade   This is an example of a multifractal distribution. However, by choosing its parameters in a particular way we can force the distribution to become a monofractal.[52]

## Notes and referencesEdit

1. ^ Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman and Company, New York (1982); p. 15
2. ^ Fractal dimension of the Feigenbaum attractor
3. ^ Tsang, K. Y. (1986). "Dimensionality of Strange Attractors Determined Analytically". Phys. Rev. Lett. 57 (12): 1390–1393. Bibcode:1986PhRvL..57.1390T. doi:10.1103/PhysRevLett.57.1390. PMID 10033437.
4. Falconer, Kenneth (1990–2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd. xxv. ISBN 0-470-84862-6.
5. ^ Fractal dimension of the spectrum of the Fibonacci Hamiltonian
6. ^ The scattering from generalized Cantor fractals
7. ^ Mandelbrot, Benoit. Gaussian self-affinity and Fractals. ISBN 0-387-98993-5.
8. ^ fractal dimension of the Julia set for c = 1/4
9. ^ Boundary of the Rauzy fractal
10. ^ 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
11. ^ Gosper island on Mathworld
12. ^ a b On 2-reptiles in the plane, Ngai, 1999
13. ^ Recurrent construction of the boundary of the dragon curve (for n=2, D=1)
14. ^ fractal dimension of the z²-1 Julia set
15. ^ fractal dimension of the apollonian gasket
16. ^ fractal dimension of the 5 circles inversion fractal
17. ^ fractal dimension of the Douady rabbit
18. ^ Fractal dimension of the boundary of the dragon fractal
19. ^ Recurrent construction of the boundary of the dragon curve (for n=2, D=2)
20. ^ T-Square (fractal)
21. ^ a b Fractal dimension of the Pascal triangle modulo k
22. ^ The Fibonacci word fractal
23. ^ Estimating Fractal dimension
24. ^ Fractal Generator for ImageJ Archived 20 March 2012 at the Wayback Machine..
25. ^ Monkeys tree fractal curve Archived 21 September 2002 at Archive.is
26. ^ Fractal dimension of a Penrose tiling
27. ^ Fractal dimension of the boundary of the Mandelbrot set
28. ^ Fractal dimension of certain Julia sets
29. ^ Lebesgue curve variants
30. ^ Complex base numeral systems
31. ^ "Penser les mathématiques", Seuil ISBN 2-02-006061-2 (1982)
32. ^ Fractals and the Rössler attractor
33. ^ The fractal dimension of the Lorenz attractor, Mc Guinness (1983)
34. ^ Lowe, Thomas (2016-10-24). "Three Variable Dimension Surfaces". ResearchGate.
35. ^
36. ^ [1]
37. ^ B. Hou; H. Xie; W. Wen & P. Sheng (2008). "Three-dimensional metallic fractals and their photonic crystal characteristics". Phys. Rev. B 77, 125113.
38. ^ Hausdorff dimension of the Mandelbulb
39. ^ Peter Mörters, Yuval Peres, Oded Schramm, "Brownian Motion", Cambridge University Press, 2010
40. ^ McCartney, Mark; Abernethya, Gavin; Gaulta, Lisa (24 June 2010). "The Divider Dimension of the Irish Coast". Irish Geography. 43 (3): 277–284. doi:10.1080/00750778.2011.582632. Retrieved 4 December 2014.
41. ^ a b Hutzler, S. (2013). "Fractal Ireland". Science Spin. 58: 19–20. Retrieved 15 November 2016. (See contents page, archived July 26, 2013)
42. ^
43. ^ Fractal dimension of the brownian motion boundary
44. Bernard Sapoval "Universalités et fractales", Flammarion-Champs (2001), ISBN 2-08-081466-4
45. ^ Feder, J., "Fractals,", Plenum Press, New York, (1988).
46. ^ Hull-generating walks
47. ^ a b Applications of percolation theory by Muhammad Sahimi (1994)
48. ^ Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey
49. ^ "Power Law Relations". Yale. Archived from the original on 28 June 2010. Retrieved 29 July 2010
50. ^ a b Kim, Sang-Hoon (February 2, 2008). "Fractal dimensions of a green broccoli and a white cauliflower". arXiv:.
51. ^ Fractal dimension of the surface of the human brain
52. ^ [Meakin (1987)]