According to Falconer, one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its 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.
Contents
- 1 Deterministic fractals
- 2 Random and natural fractals
- 3 See also
- 4 Notes and references
- 5 Further reading
- 6 External links
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Hausdorff dimension (approx.) Name
0.538 Feigenbaum attractor
0.6309 Cantor set
0.6942 Asymmetric Cantor set
0.69897 Real numbers with even digits
0.88137 Spectrum of Fibonacci Hamiltonian
0<D<1 Generalized Cantor set
1 Smith-Volterra-Cantor set
1 Takagi or Blancmange curve
1.0812 Julia set z² + 1/4
1.0933 Boundary of the Rauzy fractal
1.12915 contour of the Gosper island
1.2 Dendrite Julia set
1.2083 Fibonacci word fractal 60°
1.2108 Boundary of the tame twindragon
1.26 Hénon map
1.2619 Koch curve
1.2619 boundary of Terdragon curve
1.2619 2D Cantor dust
1.2683 Julia set z2 ? 1
1.3057 Apollonian gasket
1.328 5 circles inversion fractal
1.3934 Douady rabbit
1.4649 Vicsek fractal
1.4649 Quadratic von Koch curve (type 1)
1.5000 a Weierstrass function:
1.5000 Quadratic von Koch curve (type 2)
1.5236 Boundary of the Dragon curve
1.5236 Boundary of the twindragon curve
1.5849 3-branches tree
1.5849 Sierpinski triangle
1.5849 Sierpinski arrowhead curve
1.5849 Boundary of the T-Square fractal
1.61803 a golden dragon
1.6309 Pascal triangle modulo 3
1.6309 Sierpinski Hexagon
1.6379 Fibonacci word fractal
1.6402 Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3
1.6826 Pascal triangle modulo 5
1.7 Ikeda map attractor
1.7227 Pinwheel fractal
1.7712 Hexaflake
1.7848 Von Koch curve 85°
1.8272 A self-affine fractal set
1.8617 Pentaflake
1.8687 Monkeys tree
1.8928 Sierpinski carpet
1.8928 3D Cantor dust
1.8928 Cartesian product of the von Koch curve and the Cantor set
1.9340 Boundary of the Lévy C curve
1.974 Penrose tiling
2 Boundary of the Mandelbrot set
2 Julia set
2 Sierpinski curve
2 Hilbert curve
2 Peano curve
2 Moore curve
2 Lebesgue curve or z-order curve
2 Dragon curve
2 Terdragon curve
2 Gosper curve
2 Curve filling the Koch snowflake
2 Sierpinski tetrahedron
2 H-fractal
2 Pythagoras tree (fractal)
2 2D Greek cross fractal
2.01 ±0.01 Rössler attractor
2.06 ±0.01 Lorenz attractor
2.3219 Fractal pyramid
2.3296 Dodecahedron fractal
2.3347 3D quadratic Koch surface (type 1)
2.4739 Apollonian sphere packing
2.50 3D quadratic Koch surface (type 2)
2.5237 Cantor tesseract
2.5819 Icosahedron fractal
2.5849 3D Greek cross fractal
2.5849 Octahedron fractal
2.5849 von Koch surface
2.7268 Menger sponge
3 3D Hilbert curve
3 3D Lebesgue curve
3 3D Moore curve
3 3D H-fractal
3 Mandelbulb
0.5 Zeros of a Wiener process
0.7499 a random Cantor set with 50% - 30%
1.144... von Koch curve with random interval
1.25 Coastline of Great Britain
1.2619 von Koch curve with random orientation
1.333 Boundary of Brownian motion
1.333 2D polymer
1.333 Percolation front in 2D, Corrosion front in 2D
1.40 Clusters of clusters 2D
1.5 Graph of a regular Brownian function (Wiener process)
1.52 Coastline of Norway
1.55 Random walk with no self-intersection
1.66 3D polymer
1.70 2D DLA Cluster
1.7381 Fractal percolation with 75% probability
1.75 2D percolation cluster hull
1.8958 2D percolation cluster
2 Brownian motion
Around 2 Distribution of galaxy clusters
2.5 Balls of crumpled paper
2.50 3D DLA Cluster
2.50 Lichtenberg figure
2.5 regular Brownian surface
2.52 3D percolation cluster
2.66 Broccoli
2.79 Surface of human brain
2.97 Lung surface
ε(0,2) Multiplicative cascade