User:Datumizer/Sandbox/Table of fractals

Name	Illustration	Iterated function system	Strange attractor	L-system	Escape-time fractal	Random fractal	Natural fractal	Finite subdivision rule	Exact self-similarity	Quasi-self-similarity	Statistical self-similarity	Hausdorff dimension (exact)	Hausdorff dimension (approx)	Self-similarity dimension	Spatial dimension	Remarks
Feigenbaum attractor			Y			Y			Y			Calculated	0.538		2	The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the logistic function for the critical parameter value $\scriptstyle {\lambda _{\infty }=3.570}$, where the period doubling is infinite. This dimension is the same for any differentiable and unimodal function.
Cantor set					Y				Y			$\log _{3}(2)$	0.6309		2	Built by removing the central third at each iteration. Nowhere dense and not a countable set.
Asymmetric Cantor set		Y						Y				$\log _{2}(\varphi )=\log _{2}(1+{\sqrt {5}})-1$	0.6942		2	The dimension is not ${\frac {\ln 2}{\ln {\frac {8}{3}}}}$, which is 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. $\scriptstyle \varphi ={\frac {1+{\sqrt {5}}}{2}}$ (golden cut).
Real numbers whose base 10 digits are even					Y		Y					$\log _{10}(5)=1-\log _{10}(2)$	0.69897		2	Similar to the Cantor set.
Spectrum of Fibonacci Hamiltonian				Y								$\log(1+{\sqrt {2}})$	0.88137		2	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.
Generalized Cantor set				Y		Y						${\frac {-\log(2)}{\log \left(\displaystyle {\frac {1-\gamma }{2}}\right)}}$	0<D<1		2	Built by removing at the $m$th iteration the central interval of length $\gamma \,l_{m-1}$ from each remaining segment (of length $l_{m-1}=(1-\gamma )^{m-1}/2^{m-1}$). At $\scriptstyle \gamma =1/3$ one obtains the usual Cantor set. Varying $\scriptstyle \gamma$ between 0 and 1 yields any fractal dimension $\scriptstyle 0\,<\,D\,<\,1$.