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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク List of fractals by Hausdorff dimension ： ウィキペディア英語版 List of fractals by Hausdorff dimension 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. ==Deterministic fractals== )-1 || align="right" | 0.6942 || Asymmetric Cantor set || align="center" |200px || The dimension is not $\backslash frac$, as would be expected from the generalized Cantor set with γ=1/4, which has the same length at each stage. Built by removing the second quarter at each iteration. Nowhere dense and not a countable set. $\backslash scriptstyle\backslash varphi\; =\; (1+\backslash sqrt)/2$ (golden ratio). |- | $\backslash log\_(5)=1-\backslash log\_(2)$ || align="right" | 0.69897 || Real numbers whose base 10 digits are even || align="center" |200px || Similar to the Cantor set.〔 |- | $\backslash log(1+\backslash sqrt)$ || align="right" | 0.88137 || Spectrum of Fibonacci Hamiltonian|| align="center" | || 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.〔(Fractal dimension of the spectrum of the Fibonacci Hamiltonian )〕 |- | $\backslash frac\backslash right)\}$ || align="right" | 0200px || Built by removing at the $m$th iteration the central interval of length $\backslash gamma\backslash ,l\_$ from each remaining segment (of length $l\_=(1-\backslash gamma)^/2^$). At $\backslash scriptstyle\backslash gamma=1/3$ one obtains the usual Cantor set. Varying $\backslash scriptstyle\backslash gamma$ between 0 and 1 yields any fractal dimension .〔(The scattering from generalized Cantor fractals )〕 |- | $1$ || align="right" | 1 || Smith–Volterra–Cantor set || align="center" |200px || Built by removing a central interval of length $2^$ of each remaining interval at the ''n''th iteration. Nowhere dense but has a Lebesgue measure of ½. |- | $2+\backslash log\_2\backslash left(\backslash frac\backslash right)=1$ || align="right" | 1 || Takagi or Blancmange curve || align="center" |150px || Defined on the unit interval by $f(x)\; =\; \backslash sum\_^\backslash infty\; 2^s(2^x)$, where $s(x)$ is the sawtooth function. Special case of the Takahi-Landsberg curve: $f(x)\; =\; \backslash sum\_^\backslash infty\; w^n\; s(2^n\; x)$ with $w\; =\; 1/2$. The Hausdorff dimension equals $2+log\_2(w)$ for $w$ in $\backslash left()$. (Hunt cited by Mandelbrot). |- | Calculated|| align="right" | 1.0812 || Julia set z² + 1/4 || align="center" |100px || Julia set for ''c'' = 1/4.〔(fractal dimension of the Julia set for c = 1/4 )〕 |- | Solution ''s'' of $2|\backslash alpha|^+|\backslash alpha|^=1$|| align="right" | 1.0933 || Boundary of the Rauzy fractal|| align="center" |150px || Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: $1\backslash mapsto12$, $2\backslash mapsto13$ and $3\backslash mapsto1$.〔(Boundary of the Rauzy fractal )〕 $\backslash alpha$ is one of the conjugated roots of $z^3-z^2-z-1=0$. |- | $2\backslash log\_7(3)$ || align="right" | 1.12915 || contour of the Gosper island || align="center" |100px || Term used by Mandelbrot (1977).〔(Gosper island on Mathworld )〕 The Gosper island is the limit of the Gosper curve. |- | Measured (box counting) || align="right" | 1.2 || Dendrite Julia set || align="center" |150px || Julia set for parameters: Real = 0 and Imaginary = 1. |- | $3\backslash frac\backslash right)\}$ || align="right" | 1.2083 || Fibonacci word fractal 60° || align="center" | 200px || Build from the Fibonacci word. See also the standard Fibonacci word fractal. $\backslash varphi\; =\; (1+\backslash sqrt)/2$ (golden ratio). |- | ||| align="right" | 1.2108 || Boundary of the tame twindragon || align="center" |150px || One of the six 2-rep-tiles in the plane (can be tiled by two copies of itself, of equal size).〔(On 2-reptiles in the plane, Ngai, 1999 )〕〔(Recurrent construction of the boundary of the dragon curve (for n=2, D=1) )〕 |- | || align="right" | 1.26 || Hénon map || align="center" |100px || 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. |- | $\backslash log\_3(4)$ || align="right" | 1.261859507 || Triflake || align="center" | 150px || Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes. |- | $\backslash log\_3(4)$ || align="right" | 1.2619 || Koch curve || align="center" | 200px || 3 Koch curves form the Koch snowflake or the anti-snowflake. |- | $\backslash log\_3(4)$ || align="right" | 1.2619 || boundary of Terdragon curve || align="center" |150px || L-system: same as dragon curve with angle = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle. |- | $\backslash log\_3(4)$ || align="right" | 1.2619 || 2D Cantor dust || align="center" |100px || Cantor set in 2 dimensions. |- | $\backslash log\_3(4)$ || align="right" | 1.2619 || 2D L-system branch || align="center" |200px || 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|| align="right" | 1.2683 || Julia set z2 − 1 || align="center" |200px || Julia set for ''c'' = −1.〔(fractal dimension of the z²-1 Julia set )〕 |- | || align="right" | 1.3057 || Apollonian gasket || align="center" |100px || 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〔(fractal dimension of the apollonian gasket )〕 |- | || align="right" | 1.328 || 5 circles inversion fractal || align="center" |100px || The limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an Apollonian packing. See〔(fractal dimension of the 5 circles inversion fractal )〕 |- | Calculated|| align="right" | 1.3934 || Douady rabbit || align="center" |150px || Julia set for ''c'' = −0,123 + 0.745i.〔(fractal dimension of the Douady rabbit )〕 |- | $\backslash log\_3(5)$|| align="right" | 1.4649 || Vicsek fractal || align="center" |100px || Built by exchanging iteratively each square by a cross of 5 squares. |- | $\backslash log\_3(5)$|| align="right" | 1.4649 || Quadratic von Koch curve (type 1)|| align="center" |150px || One can recognize the pattern of the Vicsek fractal (above). |- | $\backslash log\_\backslash right)$|| align="right" | 1.49 ||Quadric cross || align="center" |150px || |- |$2-\backslash log\_2(\backslash sqrt)=\backslash frac$ (conjectured exact)|| align="right" | 1.5000 || a Weierstrass function: $\backslash displaystyle\; f(x)=\backslash sum\_^\backslash infty\; \backslash frac$ || align="center" |150px || The Hausdorff dimension of the Weierstrass function $f:()\backslash to\backslash mathbb$ defined by $f(x)=\backslash sum\_^\backslash infty\; a^\backslash sin(b^k\; x)$ with |- |$\backslash log\_4(8)=\backslash frac$|| align="right" | 1.5000 || Quadratic von Koch curve (type 2) || align="center" |150px || Also called "Minkowski sausage". |- |$\backslash log\_2\backslash left(\backslash frac\}\}\backslash right)$ || align="right" | 1.5236 || Boundary of the Dragon curve || align="center" | 150px|| cf. Chang & Zhang.〔(Fractal dimension of the boundary of the dragon fractal )〕〔(Recurrent construction of the boundary of the dragon curve (for n=2, D=2) )〕 |- |$\backslash log\_2\backslash left(\backslash frac\}\}\backslash right)$ || align="right" | 1.5236 || Boundary of the twindragon curve|| align="center" |150px || 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).〔 |- | $\backslash log\_2(3)$ || align="right" | 1.5849 || 3-branches tree || align="center" | 110px110px || 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. |- | $\backslash log\_2(3)$ || align="right" | 1.5849 || Sierpinski triangle || align="center" | 100px || Also the triangle of Pascal modulo 2. |- | $\backslash log\_2(3)$ || align="right" | 1.5849 || Sierpiński arrowhead curve || align="center" | 100px || Same limit as the triangle (above) but built with a one-dimensional curve. |- | $\backslash log\_2(3)$ || align="right" | 1.5849 || Boundary of the T-Square fractal || align="center" | 200px || The dimension of the fractal itself (not the boundary) is $\backslash log\_2(4)=2$〔T-Square (fractal)〕 |- | $\backslash log\_$. Its dimension equals $\backslash varphi$ because $()^\backslash varphi+r^\backslash varphi\; =\; 1$. With $\backslash varphi\; =\; (1+\backslash sqrt)/2$ (Golden number). |- | $1+\backslash log\_3(2)$ || align="right" | 1.6309 || Pascal triangle modulo 3 || align="center" | 160px || For a triangle modulo ''k'', if ''k'' is prime, the fractal dimension is $\backslash scriptstyle\backslash right)\}$ (cf. Stephen Wolfram〔(Fractal dimension of the Pascal triangle modulo k )〕). |- | $1+\backslash log\_3(2)$ || align="right" | 1.6309 || Sierpinski Hexagon || align="center" | 150px || 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. |- | $3\backslash frac$ || align="right" | 1.6379 || Fibonacci word fractal || align="center" | 150px || Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (''F''23 = 28657 segments).〔(The Fibonacci word fractal )〕 $\backslash varphi\; =\; (1+\backslash sqrt)/2$ (golden ratio). |- | Solution of $(1/3)^s\; +\; (1/2)^s\; +\; (2/3)^s\; =\; 1$ || align="right" | 1.6402 || Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3 || align="center" | 200px || Generalization : Providing the open set condition holds, the attractor of an iterated function system consisting of $n$ similarities of ratios $c\_n$, has Hausdorff dimension $s$, solution of the equation coinciding with the iteration function of the Euclidean contraction factor: $\backslash sum\_^n\; c\_k^s\; =\; 1$.〔 |- | $1+\backslash log\_5(3)$ || align="right" | 1.6826 || Pascal triangle modulo 5 || align="center" | 160px || For a triangle modulo ''k'', if ''k'' is prime, the fractal dimension is $\backslash scriptstyle\backslash right)\}$ (cf. Stephen Wolfram〔). |- | Measured (box-counting) || align="right" | 1.7 || Ikeda map attractor || align="center" | 100px || For parameters a=1, b=0.9, k=0.4 and p=6 in the Ikeda map $z\_\; =\; a\; +\; bz\_n\; \backslash exp\backslash left$. It derives from a model of the plane-wave interactivity field in an optical ring laser. Different parameters yield different values.〔(Estimating Fractal dimension )〕 |- | $1+\backslash log\_(5)$ || align="right" | 1.7 || 50 segment quadric fractal || align="center" | 150px || Built with ImageJ〔(Fractal Generator for ImageJ ).〕 |- | $4\backslash log\_5(2)$ || align="right" | 1.7227 || Pinwheel fractal || align="center" | 150px || Built with Conway's Pinwheel tile. |- | $\backslash log\_3(7)$ || align="right" | 1.7712 || Hexaflake || align="center" | 100px || 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). |- | $\backslash frac$ || align="right" | 1.7848 || Von Koch curve 85° || align="center" | 150px || Generalizing the von Koch curve with an angle ''a'' chosen between 0 and 90°. The fractal dimension is then $\backslash frac\; \backslash in\; ()$. |- | $\backslash log\_2\backslash left(3^+2^\backslash right)$ || align="right" | 1.8272 || A self-affine fractal set || align="center" | 200px || Build iteratively from a $p\; \backslash times\; q$ array on a square, with $p\; \backslash le\; q$. Its Hausdorff dimension equals $\backslash log\_p\backslash left(\backslash sum\_^p\; n\_k^a\backslash right)$〔 with $a=\backslash log\_q(p)$ and $n\_k$ is the number of elements in the $k$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. |- | $\backslash frac$ || align="right" | 1.8617 || Pentaflake || align="center" | 100px || Built by exchanging iteratively each pentagon by a flake of 6 pentagons. $\backslash varphi=(1+\backslash sqrt)/2$ (golden ratio). |- | solution of $6(1/3)^s+5^s=1$ || align="right" | 1.8687 || Monkeys tree || align="center" | 100px || This curve appeared in Benoit Mandelbrot's "Fractal geometry of Nature" (1983). It is based on 6 similarities of ratio $1/3$ and 5 similarities of ratio $1/3\backslash sqrt$.〔(Monkeys tree fractal curve )〕 |- | $\backslash log\_3(8)$ || align="right" | 1.8928 || Sierpinski carpet || align="center" | 100px || Each face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1). |- | $\backslash log\_3(8)$ || align="right" | 1.8928 || 3D Cantor dust || align="center" | 200px|| Cantor set in 3 dimensions. |- | $\backslash log\_3(4)+\backslash log\_3(2)=\backslash frac\; +\backslash frac\; =\backslash frac$ || align="right" | || Cartesian product of the von Koch curve and the Cantor set || align="center" | 150px|| Generalization : Let F×G be the cartesian product of two fractals sets F and G. Then $Dim\_H(F\; \backslash times\; G)\; =\; Dim\_H(F)\; +\; Dim\_H(G)$.〔 See also the 2D Cantor dust and the Cantor cube. |- |Estimated || align="right" | 1.9340 || Boundary of the Lévy C curve || align="center" | 100px || Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2. |- | || align="right" | 1.974 || Penrose tiling || align="center" |100px || See Ramachandrarao, Sinha & Sanyal.〔(Fractal dimension of a Penrose tiling )〕 |- | $2$ || align="right" | 2 || Boundary of the Mandelbrot set || align="center" | 100px || The boundary and the set itself have the same dimension.〔(Fractal dimension of the boundary of the Mandelbrot set )〕 |- | $2$ || align="right" | 2 || Julia set || align="center" | 150px || For determined values of ''c'' (including ''c'' belonging to the boundary of the Mandelbrot set), the Julia set has a dimension of 2.〔(Fractal dimension of certain Julia sets )〕 |- | $2$ || align="right" | 2 || Sierpiński curve || align="center" | 100px || Every Peano curve filling the plane has a Hausdorff dimension of 2. |- | $2$ || align="right" | 2 || Hilbert curve || align="center" | 100px|| |- | $2$ || align="right" | 2 || Peano curve || align="center" | 100px|| And a family of curves built in a similar way, such as the Wunderlich curves. |- | $2$ || align="right" | 2 || Moore curve || align="center" | 100px|| Can be extended in 3 dimensions. |- | || align="right" | 2 || Lebesgue curve or z-order curve || align="center" | 100px|| 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.〔(Lebesgue curve variants )〕 |- | $\backslash log\_\backslash sqrt(2)=2$ || align="right" | 2 || Dragon curve || align="center" | 150px|| And its boundary has a fractal dimension of 1.5236270862.〔(Complex base numeral systems )〕 |- | || align="right" | 2 || Terdragon curve || align="center" | 150px|| L-system: ''F'' → ''F'' + F – F, angle = 120°. |- | $\backslash log\_2(4)=2$ || align="right" | 2 || Gosper curve || align="center" | 100px|| Its boundary is the Gosper island. |- | Solution of $7()^s+6($. |- | $\backslash log\_2(4)=2$ || align="right" | 2 || Sierpiński tetrahedron || align="center" | 80px|| Each tetrahedron is replaced by 4 tetrahedra. |- | $\backslash log\_2(4)=2$ || align="right" | 2 || H-fractal || align="center" |150px|| Also the Mandelbrot tree which has a similar pattern. |- | $\backslash frac=2$ || align="right" | || Pythagoras tree (fractal) || align="center" |150px|| Every square generates two squares with a reduction ratio of $1/\backslash sqrt$. |- | $\backslash log\_2(4)=2$ || align="right" | 2 || 2D Greek cross fractal || align="center" |100px || Each segment is replaced by a cross formed by 4 segments. |- | Measured || align="right" | 2.01 ±0.01|| Rössler attractor || align="center" | 100px || 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.〔(Fractals and the Rössler attractor )〕 |- | Measured || align="right" | 2.06 ±0.01|| Lorenz attractor || align="center" |100px || For parameters v=40,$\backslash sigma$=16 and b=4 . See McGuinness (1983)〔(The fractal dimension of the Lorenz attractor, Mc Guinness (1983) )〕 |- | $\backslash log\_2(5)$ || align="right" | 2.3219 || Fractal pyramid || align="center" |100px|| 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). |- | $\backslash frac$ || align="right" | 2.3296 || Dodecahedron fractal || align="center" |100px|| Each dodecahedron is replaced by 20 dodecahedra. $\backslash varphi\; =\; (1+\backslash sqrt)/2$ (golden ratio). |- | $\backslash log\_3(13)$ || align="right" | 2.3347 || 3D quadratic Koch surface (type 1) || align="center" |150px|| Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration. |- | || align="right" | 2.4739 || Apollonian sphere packing || align="center" |100px || The interstice left by the Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.〔(Fractal dimension of the apollonian sphere packing )〕 |- | $\backslash log\_4(32)=\backslash frac$ || align="right" | 2.50 || 3D quadratic Koch surface (type 2) || align="center" |150px|| Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration. |- | $\backslash log\_3(16)$ || align="right" | 2.5237 || Cantor tesseract || align="center" | no image available || Cantor set in 4 dimensions. Generalization: in a space of dimension ''n'', the Cantor set has a Hausdorff dimension of $n\backslash log\_3(2)$. |- | $\backslash frac-\backslash frac\backslash right)\}$ || align="right" | 2.529 || Jerusalem cube || align="center" | 150px || 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 $\backslash sqrt-1$. |- | $\backslash frac$ || align="right" | 2.5819 || Icosahedron fractal || align="center" |100px|| Each icosahedron is replaced by 12 icosahedra. $\backslash varphi=(1+\backslash sqrt)/2$ (golden ratio). |- | $1+\backslash log\_2(3)$ || align="right" | 2.5849 || 3D Greek cross fractal || align="center" |200px|| Each segment is replaced by a cross formed by 6 segments. |- | $1+\backslash log\_2(3)$ || align="right" | 2.5849 || Octahedron fractal || align="center" |100px|| Each octahedron is replaced by 6 octahedra. |- | $1+\backslash log\_2(3)$ || align="right" | 2.5849 || von Koch surface || align="center" |150px|| 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". |- | $\backslash log\_3(20)$ || align="right" | 2.7268 || Menger sponge || align="center" | 100px || And its surface has a fractal dimension of $\backslash log\_3(20)$, which is the same as that by volume. |- | $\backslash log\_2(8)=3$ || align="right" | 3 || 3D Hilbert curve || align="center" | 100px|| A Hilbert curve extended to 3 dimensions. |- | $\backslash log\_2(8)=3$ || align="right" | 3 || 3D Lebesgue curve || align="center" | 100px|| A Lebesgue curve extended to 3 dimensions. |- | $\backslash log\_2(8)=3$ || align="right" | 3 || 3D Moore curve || align="center" | 100px|| A Moore curve extended to 3 dimensions. |- | $\backslash log\_2(8)=3$ || align="right" | 3 || 3D H-fractal || align="center" | 120px|| A H-fractal extended to 3 dimensions. |- | $3$ (conjectured) || align="right" | (to be confirmed) || Mandelbulb || align="center" |100px|| Extension of the Mandelbrot set (power 8) in 3 dimensions〔(Hausdorff dimension of the Mandelbulb )〕 |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of fractals by Hausdorff dimension」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. 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