## List of fractals by hausdorff dimension – wikipedia gas vs diesel towing

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The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the logistic function for the critical parameter value λ ∞ = 3.570 {\displaystyle \scriptstyle {\lambda _{\infty }=3.570}} , where the period doubling is infinite. This dimension is the same for any differentiable and unimodal function. [2]

Built by removing at the m {\displaystyle m} th iteration the central interval of length γ l m − 1 {\displaystyle \gamma \,l_{m-1}} from each remaining segment (of length l m − 1 = ( 1 − γ ) m − 1 / 2 m − 1 {\displaystyle l_{m-1}=(1-\gamma )^{m-1}/2^{m-1}} ). At γ = 1 / 3 {\displaystyle \scriptstyle \gamma =1/3} one obtains the usual Cantor set. Varying γ {\displaystyle \scriptstyle \gamma } between 0 and 1 yields any fractal dimension 0 < D < 1 {\displaystyle \scriptstyle 0\,<\,D\,<\,1} . [6]

Defined on the unit interval by f ( x ) = ∑ n = 0 ∞ 2 − n s ( 2 n x ) {\displaystyle f(x)=\sum _{n=0}^{\infty }2^{-n}s(2^{n}x)} , where s ( x ) {\displaystyle s(x)} is the sawtooth function. Special case of the Takahi-Landsberg curve: f ( x ) = ∑ n = 0 ∞ w n s ( 2 n x ) {\displaystyle f(x)=\sum _{n=0}^{\infty }w^{n}s(2^{n}x)} with w = 1 / 2 {\displaystyle w=1/2} . The Hausdorff dimension equals 2 + log 2 ⁡ ( w ) {\displaystyle 2+\log _{2}(w)} for w {\displaystyle w} in [ 1 / 2 , 1 ] {\displaystyle \left[1/2,1\right]} . (Hunt cited by Mandelbrot [7]).

Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: 1 ↦ 12 {\displaystyle 1\mapsto 12} , 2 ↦ 13 {\displaystyle 2\mapsto 13} and 3 ↦ 1 {\displaystyle 3\mapsto 1} . [9] [10] α {\displaystyle \alpha } is one of the conjugated roots of z 3 − z 2 − z − 1 = 0 {\displaystyle z^{3}-z^{2}-z-1=0} .

2 log 2 ⁡ ( 27 − 3 78 3 + 27 + 3 78 3 3 ) , or root of 2 x − 1 = 2 ( 2 − x ) / 2 {\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}}}

The quadric cross is made by scaling the 3-segment generator unit by 5 1/2 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).

The Hausdorff dimension of the Weierstrass function f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\to \mathbb {R} } defined by f ( x ) = ∑ k = 1 ∞ a − k sin ⁡ ( b k x ) {\displaystyle f(x)=\sum _{k=1}^{\infty }a^{-k}\sin(b^{k}x)} with 1 1 {\displaystyle b>1} has upper bound 2 − log b ⁡ ( a ) {\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]

Built from two similarities of ratios r {\displaystyle r} and r 2 {\displaystyle r^{2}} , with r = 1 / φ 1 / φ {\displaystyle r=1/\varphi ^{1/\varphi }} . Its dimension equals φ {\displaystyle \varphi } because ( r 2 ) φ + r φ = 1 {\displaystyle ({r^{2}})^{\varphi }+r^{\varphi }=1} . With φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} ( Golden number).

Generalization : Providing the open set condition holds, the attractor of an iterated function system consisting of n {\displaystyle n} similarities of ratios c n {\displaystyle c_{n}} , has Hausdorff dimension s {\displaystyle s} , solution of the equation coinciding with the iteration function of the Euclidean contraction factor: ∑ k = 1 n c k s = 1 {\displaystyle \sum _{k=1}^{n}c_{k}^{s}=1} . [4]

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.

For parameters a=1, b=0.9, k=0.4 and p=6 in the Ikeda map z n + 1 = a + b z n exp ⁡ [ i [ k − p / ( 1 + ⌊ z n ⌋ 2 ) ] ] {\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]

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] .

Build iteratively from a p × q {\displaystyle p\times q} array on a square, with p ≤ q {\displaystyle p\leq q} . Its Hausdorff dimension equals log p ⁡ ( ∑ k = 1 p n k a ) {\displaystyle \log _{p}\left(\sum _{k=1}^{p}n_{k}^{a}\right)} [4] with a = log q ⁡ ( p ) {\displaystyle a=\log _{q}(p)} and n k {\displaystyle n_{k}} is the number of elements in the k {\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.

Generalization : Let F×G be the cartesian product of two fractals sets F and G. Then D i m H ( F × G ) = D i m H ( F ) + D i m H ( G ) {\displaystyle Dim_{H}(F\times G)=Dim_{H}(F)+Dim_{H}(G)} . [4] See also the 2D Cantor dust and the Cantor cube.

2 R {\displaystyle \scriptstyle {f:\mathbb {R} ^{2}->\mathbb {R} }} , gives the height of a point ( x , y ) {\displaystyle (x,y)} such that, for two given positive increments h {\displaystyle h} and k {\displaystyle k} , then f ( x + h , y + k ) − f ( x , y ) {\displaystyle \scriptstyle {f(x+h,y+k)-f(x,y)}} has a centered Gaussian distribution with variance = h 2 + k 2 {\displaystyle \scriptstyle {\sqrt {h^{2}+k^{2}}}} . Generalization : The fractional Brownian surface of index α {\displaystyle \alpha } follows the same definition but with a variance = ( h 2 + k 2 ) α {\displaystyle (h^{2}+k^{2})^{\alpha }} , in that case its Hausdorff dimension = 3 − α {\displaystyle 3-\alpha } . [4]