According to Benoit Mandelbrot, "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, to illustrate what it means for a fractal to have a low or a high dimension.
Hausdorff dimension (exact value) | Hausdorff dimension (approx.) | Name | Illustration | Remarks | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Calculated | 0.538 | The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the logistic map for the critical parameter value λinfty=3.570 | ||||||||||||||||||||
log32 | 0.6309 | Built by removing the central third at each iteration. Nowhere dense and not a countable set. | ||||||||||||||||||||
| 0<D<1 | 1D generalized symmetric Cantor set | Built by removing the central interval of length \gammaln-1 ln-1=(1-\gamma)n-1/2n-1 \gamma=1/3 \gamma 0<D<1 | |||||||||||||||||||
log2\varphi=log2(1+\sqrt{5})-1 | 0.6942 | Built by removing the second quarter at each iteration.[3] \varphi=(1+\sqrt{5})/2 | ||||||||||||||||||||
log105=1-log102 | 0.69897 | Similar to the Cantor set.[4] | ||||||||||||||||||||
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] | |||||||||||||||||||
1 | 1 | Built by removing the central interval of length 2-2n | ||||||||||||||||||||
2+log2
=1 | 1 | Defined on the unit interval by f(x)=
2-ns(2nx) s(x) 1 f(x)=
wns(2nx) w=1/2 2+log2w w \left[1/2,1\right] | ||||||||||||||||||||
Calculated | 1.0812 | Julia set z2 + 1/4 | Julia set of f(z) = z2 + 1/4. | |||||||||||||||||||
Solution s 2 | \alpha | ^ + | \alpha | ^ = 1 | 1.0933 | Boundary of the Rauzy fractal | Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: 1\mapsto12 2\mapsto13 3\mapsto1 \alpha z3-z2-z-1=0 | |||||||||||||||
2log73 | 1.12915 | Term used by Mandelbrot (1977). The Gosper island is the limit of the Gosper curve. | ||||||||||||||||||||
Measured (box counting) | 1.2 | Julia set of f(z) = z2 + i. | ||||||||||||||||||||
{2}\right)} | 1.2083 | Build from the Fibonacci word. See also the standard Fibonacci word fractal. \varphi=(1+\sqrt{5})/2 | ||||||||||||||||||||
\begin{align}&
| 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).[9] [10] | |||||||||||||||||||
1.26 | 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. | |||||||||||||||||||||
log34 | 1.2619 | Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes. | ||||||||||||||||||||
log34 | 1.2619 | 3 Koch curves form the Koch snowflake or the anti-snowflake. | ||||||||||||||||||||
log34 | 1.2619 | L-system: same as dragon curve with angle = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle. | ||||||||||||||||||||
log34 | 1.2619 | Cantor set in 2 dimensions. | ||||||||||||||||||||
log34 | 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 of f(z) = z2 − 1. | |||||||||||||||||||
1.3057 | 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 [11] | |||||||||||||||||||||
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 [12] | ||||||||||||||||||||
log59 | 1.36521[13] | Quadratic von Koch island using the type 1 curve as generator | Also known as the Minkowski Sausage | |||||||||||||||||||
Calculated | 1.3934 | Julia set of f(z) = -0.123 + 0.745i | ||||||||||||||||||||
log35 | 1.4649 | Built by exchanging iteratively each square by a cross of 5 squares. | ||||||||||||||||||||
log35 | 1.4649 | Quadratic von Koch curve (type 1) | One can recognize the pattern of the Vicsek fractal (above). | |||||||||||||||||||
log\sqrt{5 | 1.4961 | Quadric cross | 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. | |||||||||||||||||||
2-log2\sqrt{2}=
| 1.5000 | a Weierstrass function: f(x)=
k | The Hausdorff dimension of the graph of the Weierstrass function f:[0,1]\toR f(x)=
a-k\sin(bkx) 1/b<a<1 b>1 2+logba | |||||||||||||||||||
log48=
| 1.5000 | Also called "Minkowski sausage". | ||||||||||||||||||||
| 1.5236 | cf. Chang & Zhang.[16] | ||||||||||||||||||||
| 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). | |||||||||||||||||||
log23 | 1.5850 | 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. | |||||||||||||||||||
log23 | 1.5850 | Also the limiting shape of Pascal's triangle modulo 2. | ||||||||||||||||||||
log23 | 1.5850 | Same limit as the triangle (above) but built with a one-dimensional curve. | ||||||||||||||||||||
log23 | 1.5850 | Boundary of the T-square fractal | The dimension of the fractal itself (not the boundary) is log24=2 | |||||||||||||||||||
log\sqrt[\varphi]{\varphi | 1.61803 | Built from two similarities of ratios r r2 r=1/\varphi1/\varphi \varphi ({r2})\varphi+r\varphi=1 \varphi=(1+\sqrt{5})/2 | ||||||||||||||||||||
1+log32 | 1.6309 | Pascal triangle modulo 3 | For a triangle modulo k, if k is prime, the fractal dimension is 1+
\right) | |||||||||||||||||||
1+log32 | 1.6309 | 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 | Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F23 = 28657 segments).[18] \varphi=(1+\sqrt{5})/2 | ||||||||||||||||||||
Solution of (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 n cn s
=1 | |||||||||||||||||||
log832=
| 1.6667 | 32-segment quadric fractal (1/8 scaling rule) | see also: | 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. | ||||||||||||||||||
1+log53 | 1.6826 | Pascal triangle modulo 5 | For a triangle modulo k, if k is prime, the fractal dimension is 1+
\right) | |||||||||||||||||||
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 zn+1=a+bzn\exp\left[i\left[k-p/\left(1+\lfloorzn\rfloor2\right)\right]\right] | |||||||||||||||||||
1+log105 | 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[20] . | |||||||||||||||||||
4log52 | 1.7227 | Built with Conway's Pinwheel tile. | ||||||||||||||||||||
log37 | 1.7712 | Built with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes.[21] | ||||||||||||||||||||
log37 | 1.7712 | 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). | ||||||||||||||||||||
log37 | 1.7712 | Fractal H-I de Rivera | Starting from a unit square dividing its dimensions into three equal parts to form nine self-similar squares with the first square, two middle squares (the one that is above and the one below the central square) are removed in each of the seven squares not eliminated the process is repeated, so it continues indefinitely. | |||||||||||||||||||
| 1.7848 | Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then
\in[1,2] | ||||||||||||||||||||
+20.63\right) | 1.8272 | A self-affine fractal set | Build iteratively from a p-by-q array on a square, with p\leq logp\left(\sum\nolimits
a=logqp nk k | |||||||||||||||||||
| 1.8617 | Built by exchanging iteratively each pentagon by a flake of 6 pentagons. \varphi=(1+\sqrt{5})/2 | ||||||||||||||||||||
solution of 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 1/3 1/3\sqrt{3} | |||||||||||||||||||
log38 | 1.8928 | Each face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1). | ||||||||||||||||||||
log38 | 1.8928 | Cantor set in 3 dimensions. | ||||||||||||||||||||
log34+log32=
=
| Generalization : Let F×G be the cartesian product of two fractals sets F and G. Then \dimH(F x G)=\dimHF+\dimHG | |||||||||||||||||||||
2log2x x9-3x8+3x7-3x6+2x5+4x4-8x3+ 8x2-16x+8=0 | 1.9340 | Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2. | ||||||||||||||||||||
2 | See Ramachandrarao, Sinha & Sanyal.[23] | |||||||||||||||||||||
2 | 2 | The boundary and the set itself have the same Hausdorff dimension.[24] | ||||||||||||||||||||
2 | 2 | For determined values of c (including c belonging to the boundary of the Mandelbrot set), the Julia set has a dimension of 2. | ||||||||||||||||||||
2 | 2 | Every space-filling curve filling the plane has a Hausdorff dimension of 2. | ||||||||||||||||||||
2 | 2 | |||||||||||||||||||||
2 | 2 | And a family of curves built in a similar way, such as the Wunderlich curves. | ||||||||||||||||||||
2 | 2 | Can be extended in 3 dimensions. | ||||||||||||||||||||
2 | 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.[25] | |||||||||||||||||||||
log\sqrt{2 | 2 | And its boundary has a fractal dimension of 1.5236270862.[26] | ||||||||||||||||||||
2 | L-system: F → F + F - F, angle = 120°. | |||||||||||||||||||||
log24=2 | 2 | Its boundary is the Gosper island. | ||||||||||||||||||||
Solution of 7({1/3})s+6({1/3\sqrt{3}})s=1 | 2 | Proposed by Mandelbrot in 1982,[27] it fills the Koch snowflake. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio 1/3\sqrt{3} | ||||||||||||||||||||
log24=2 | 2 | Each tetrahedron is replaced by 4 tetrahedra. | ||||||||||||||||||||
log24=2 | 2 | Also the Mandelbrot tree which has a similar pattern. | ||||||||||||||||||||
)}=2 | Every square generates two squares with a reduction ratio of 1/\sqrt{2} | |||||||||||||||||||||
log24=2 | 2 | Each segment is replaced by a cross formed by 4 segments. | ||||||||||||||||||||
Measured | 2.01 ± 0.01 | 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.[28] | ||||||||||||||||||||
Measured | 2.06 ± 0.01 | For parameters \rho=40 \sigma=16 \beta=4 | ||||||||||||||||||||
4+cD+dD=(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 c d | |||||||||||||||||||
log25 | 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). | |||||||||||||||||||
| 2.3296 | Dodecahedron fractal | Each dodecahedron is replaced by 20 dodecahedra. \varphi=(1+\sqrt{5})/2 | |||||||||||||||||||
log313 | 2.3347 | Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the first (blue block), second (plus green blocks), third (plus yellow blocks) and fourth (plus clear blocks) iterations. | ||||||||||||||||||||
2.4739 | The interstice left by the Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.[31] | |||||||||||||||||||||
log432=
| 2.50 | Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration. | ||||||||||||||||||||
\right)}{log(\sqrt2-1)} | 2.529 | 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 \sqrt{2}-1 | ||||||||||||||||||||
| 2.5819 | Each icosahedron is replaced by 12 icosahedra. \varphi=(1+\sqrt{5})/2 | ||||||||||||||||||||
1+log23 | 2.5849 | Each segment is replaced by a cross formed by 6 segments. | ||||||||||||||||||||
1+log23 | 2.5849 | Each octahedron is replaced by 6 octahedra. | ||||||||||||||||||||
1+log23 | 2.5849 | 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 | 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.[32] | ||||||||||||||||||||
log320 | 2.7268 | And its surface has a fractal dimension of log320 | ||||||||||||||||||||
log28=3 | 3 | A Hilbert curve extended to 3 dimensions. | ||||||||||||||||||||
log28=3 | 3 | A Lebesgue curve extended to 3 dimensions. | ||||||||||||||||||||
log28=3 | 3 | A Moore curve extended to 3 dimensions. | ||||||||||||||||||||
log28=3 | 3 | A H-fractal extended to 3 dimensions.[33] | ||||||||||||||||||||
3 | (to be confirmed) | Extension of the Mandelbrot set (power 9) in 3 dimensions[34] |
Hausdorff dimension (exact value) | Hausdorff dimension (approx.) | Name | Illustration | Remarks | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.5 | The zeros of a Wiener process (Brownian motion) are a nowhere dense set of Lebesgue measure 0 with a fractal structure.[35] | ||||||||||||||||||||||||||
Solution of
+
E(C1)=0.5 E(C2)=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 C1 C2 s
+
E(X) X | |||||||||||||||||||||||||
Solution of s+1=12 ⋅ 2-(s+1)-6 ⋅ 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). | |||||||||||||||||||||||||
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[36] at the University of Ulster and Theoretical Physics students at Trinity College, Dublin, under the supervision of S. Hutzler.[37] 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) | |||||||||||||||||||||||||
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.[38] | |||||||||||||||||||||||||
| 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. | |||||||||||||||||||||||||
| 1.333 | Boundary of Brownian motion | (cf. Mandelbrot, Lawler, Schramm, Werner).[39] | |||||||||||||||||||||||||
| 1.333 | Polymer in 2D | Similar to the Brownian motion in 2D with non-self-intersection.[40] | |||||||||||||||||||||||||
| 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. | |||||||||||||||||||||||||
1.40 | When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4. | |||||||||||||||||||||||||||
| 1.5 | Graph of a regular Brownian function (Wiener process) | Graph of a function f x x+h f(x+h)-f(x) h \alpha h2\alpha 2-\alpha | |||||||||||||||||||||||||
Measured | 1.52 | See J. Feder.[41] | ||||||||||||||||||||||||||
Measured | 1.55 | Random walk in a square lattice that avoids visiting the same place twice, with a "go-back" routine for avoiding dead ends. | ||||||||||||||||||||||||||
| 1.66 | Polymer in 3D | Similar to the Brownian motion in a cubic lattice, but without self-intersection. | |||||||||||||||||||||||||
1.70 | In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70. | |||||||||||||||||||||||||||
| 1.7381 | Fractal percolation with 75% probability | The fractal percolation model is constructed by the progressive replacement of each square by a 3-by-3 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
| |||||||||||||||||||||||||
| 1.75 | 2D percolation cluster hull | The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk,[42] or by Schramm-Loewner Evolution. | |||||||||||||||||||||||||
| 1.8958 | In a square lattice, under the site percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48.[43] Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings". | ||||||||||||||||||||||||||
| 2 | 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.[44] | |||||||||||||||||||||||||
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.[45] Creases will form at all size scales (see Universality (dynamical systems)). | ||||||||||||||||||||||||||
2.50 | In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50. | |||||||||||||||||||||||||||
2.50 | Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA. | |||||||||||||||||||||||||||
| 2.5 | regular Brownian surface | A function f:R2\toR (x,y) h k f(x+h,y+k)-f(x,y) \sqrt{h2+k2} \alpha (h2+k2)\alpha 3-\alpha | |||||||||||||||||||||||||
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. Beyond that threshold, the cluster is infinite. | |||||||||||||||||||||||||
Measured and calculated | ~2.7 | 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.[46] | ||||||||||||||||||||||||||
Measured | ~2.8 | Measured with segmented three-dimensional high-resolution magnetic resonance images[47] | ||||||||||||||||||||||||||
Measured and calculated | ~2.8 | 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. | ||||||||||||||||||||||||||
2.97 | Lung surface | The alveoli of a lung form a fractal surface close to 3. | ||||||||||||||||||||||||||
Calculated | \in(0,2) | 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.[48] |