?
(exact value) |
?
(value) |
Name |
Illustration |
Remarks |
| \textstyle{\frac {\log(2)}{\log(\delta)}?} |
0.4498? |
Logistic map bifurcations |
|
In the bifurcation diagram, when approaching the chaotic zone, a succession of period doubling appears, in a geometric progression tending to 1/?. (?=Feigenbaum constant=4.6692.) |
| \textstyle{\frac {\log(2)}{\log(3)}} |
0.6309 |
Cantor set |
|
Built by removing the central third at each iteration. Nowhere dense and not a countable set. |
| \log{(1+\sqrt{2})} |
0.88137 |
Spectrum of Fibonacci Hamiltonian |
|
The study 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.[1] |
| \textstyle{1} |
1 |
Smith-Volterra-Cantor set |
|
Built by removing a central interval of length 1/2^{2n} of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of ½. |
| \textstyle{\frac {\log(8)} {\log(7)}} |
1.0686 |
contour of the Gosper island |
|
|
| Measured (box counting) |
1.2 |
Dendrite Julia set |
|
Julia set for parameters: Real=0 and Imaginary=1. |
| \textstyle{3\frac{\log(\phi)}{\log (\frac{3+\sqrt{13}}{2})}} |
1.2083 |
Fibonacci fractal 60° |
|
Build from the Fibonacci word. See also the standard Fibonacci fractal. |
|
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 ? values. |
| \textstyle{\frac {\log(4)} {\log(3)}} |
1.2619 |
Koch curve |
|
3 von Koch curves form the Koch snowflake or the anti-snowflake. |
| \textstyle{\frac {\log(4)} {\log(3)}} |
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. |
| \textstyle{\frac {\log(4)} {\log(3)}} |
1.2619 |
2D Cantor dust |
|
Cantor set in 2 dimensions. |
|
1.3057 |
Apollonian gasket |
|
|
| \textstyle{\frac {\log(5)} {\log(3)}} |
1.4649 |
Box fractal |
|
Built by exchanging iteratively each square by a cross of 5 squares. |
| \textstyle{\frac {\log(5)} {\log(3)}} |
1.4649 |
Quadratic von Koch curve (type 1) |
|
One can recognize the pattern of the box fractal (above). |
| \textstyle{\frac {\log(8)} {\log(4)} = \frac{3}{2}} |
1.5000 |
Quadratic von Koch curve (type 2) |
|
Also called "Minkowski sausage". |
| \textstyle{\frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)}
{\log(2)}} |
1.5236 |
Dragon curve boundary |
|
Cf Chang & Zhang.[2][3] |
| \textstyle{\frac {\log(3)} {\log(2)}} |
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. |
| \textstyle{\frac {\log(3)} {\log(2)}} |
1.5850 |
Sierpinski triangle |
|
Also the triangle of Pascal modulo 2. |
| \textstyle{\frac {\log(3)} {\log(2)}} |
1.5850 |
Sierpi?ski arrowhead curve |
|
Same limit as the triangle (above) but built with a one-dimensional curve. |
| \textstyle{1+\frac{\log 2}{\log 3}} |
1.6309 |
Pascal triangle modulo 3 |
|
For a triangle modulo k, if k is prime, the fractal dimension is \scriptstyle{1 + \log_k(\frac{k+1}{2})} (Cf Stephen Wolfram[4]). |
| \textstyle{3\frac{\log(\phi)}{\log (1+\sqrt{2})}} |
1.6379 |
Fibonacci fractal |
|
Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal after F23 (28657) steps. [5]. |
| \textstyle{1+\frac{\log 3}{\log 5}} |
1.6826 |
Pascal triangle modulo 5 |
|
For a triangle modulo k, if k is prime, the fractal dimension is \scriptstyle{1 + \log_k(\frac{k+1}{2})} (Cf Stephen Wolfram[6]). |
| \textstyle{\frac {\log(7)} {\log(3)}} |
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). |
| \textstyle{\frac {\log(4)} {\log(2(1+\cos(85^\circ)))}} |
1.7848 |
Von Koch curve 85°, Cesaro fractal |
|
Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then \scriptstyle{\frac{\log(4)}{\log(2(1+\cos(a)))}}. The Cesaro fractal is based on this pattern. |
| \textstyle{\frac {\log(6)} {\log(1+\phi)}} |
1.8617 |
Pentaflake |
|
Built by exchanging iteratively each pentagon by a flake of 6 pentagons. \phi = golden number = \scriptstyle{\frac{1+\sqrt{5}}{2}}. |
| \textstyle{\frac {\log(8)} {\log(3)}} |
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). |
| \textstyle{\frac {\log(8)} {\log(3)}} |
1.8928 |
3D Cantor dust |
|
Cantor set in 3 dimensions. |
| Estimated |
1.9340 |
Boundary of the Lévy C curve |
|
Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2. |
|
1.974 |
Penrose tiling |
|
See Ramachandrarao, Sinha & Sanyal[7]. |
| \textstyle{2} |
2 |
Boundary of the Mandelbrot set |
|
The boundary and the set itself have the same dimension [8]. |
| \textstyle{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. [9]. |
| \textstyle{2} |
2 |
Sierpi?ski curve |
|
Every Peano curve filling the plane has a Hausdorff dimension of 2. |
| \textstyle{2} |
2 |
Hilbert curve |
|
| \textstyle{2} |
2 |
Peano curve |
|
And a family of curves built in a similar way, such as the Wunderlich curves. |
| \textstyle{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.[10] |
| \textstyle{\frac {\log(2)} {\log(\sqrt{2})} = 2} |
2 |
Dragon curve |
|
And its boundary has a fractal dimension of 1.5236270862[11]. |
|
2 |
Terdragon curve |
|
L-System: F?F+F–F, angle=120°. |
| \textstyle{\frac {\log(4)} {\log(2)} = 2} |
2 |
T-Square |
|
|
| \textstyle{\frac {\log(4)} {\log(2)} = 2} |
2 |
Gosper curve |
|
Its boundary is the Gosper island. |
| \textstyle{\frac {\log(4)} {\log(2)} = 2} |
2 |
Sierpi?ski tetrahedron |
|
Each tetrahedron is replaced by 4 tetrahedra. |
| \textstyle{\frac {\log(4)} {\log(2)} = 2} |
2 |
H-fractal |
|
Also the « Mandelbrot tree » which has a similar pattern. |
| \textstyle{\frac {\log(2)} {\log(2/\sqrt{2})} = 2} |
|
Pythagoras tree |
|
Every square generates 2 squares with a reduction ratio of sqrt(2)/2. |
| \textstyle{\frac {\log(4)} {\log(2)} = 2} |
2 |
2D Greek cross fractal |
|
Each segment is replaced by a cross formed by 4 segments. |
|
2.06 |
Lorenz attractor |
|
For precise values of parameters. |
| \textstyle{\frac {\log(20)} {\log(2+\phi)}} |
2.3296 |
Dodecahedron fractal |
|
Each dodecahedron is replaced by 20 dodecahedra. |
| \textstyle{\frac {\log(13)} {\log(3)}} |
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 apollolian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.[12] |
| \textstyle{\frac {\log(32)} {\log(4)} = \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. |
| \textstyle{\frac {\log(16)} {\log(3)}} |
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 \scriptstyle{n\frac{\log(2)}{\log(3)}}. |
| \textstyle{\frac {\log(12)} {\log(1+\phi)}} |
2.5819 |
Icosahedron fractal |
|
Each icosahedron is replaced by 12 icosahedra. |
| \textstyle{\frac {\log(6)} {\log(2)}} |
2.5849 |
3D Greek cross fractal |
|
Each segment is replaced by a cross formed by 6 segments. |
| \textstyle{\frac {\log(6)} {\log(2)}} |
2.5849 |
Octahedron fractal |
|
Each octahedron is replaced by 6 octahedra. |
| \textstyle{\frac {\log(6)} {\log(2)}} |
2.5849 |
von Koch surface |
|
Each equilateral triangle is replaced by 6 triangles, twice smaller. |
| \textstyle{\frac {\log(20)} {\log(3)}} |
2.7268 |
Menger sponge |
|
And its surface has a fractal dimension of \scriptstyle{\frac{\log(12)}{\log(3)} = 2.2618}. |
| \textstyle{\frac {\log(8)} {\log(2)} = 3} |
3 |
3D Hilbert curve |
|
A Hilbert curve extended to 3 dimensions. |
| \textstyle{\frac {\log(8)} {\log(2)} = 3} |
3 |
3D Lebesgue curve |
|
A Lebesgue curve extended to 3 dimensions. |
| \textstyle{\frac {\log(8)} {\log(2)} = 3} |
3 |
3D Moore curve |
|
A Moore curve extended to 3 dimensions. |
Encyclopedia
top
