| [[Cantor set]] || [[File:Cantor set in seven iterations.svg|200px]] || || || || {{tick}} || || || || {{tick}}|| || || <math>\log_3(2)</math> || 0.6309 || || 2 || Built by removing the central third at each iteration. [[Nowhere dense]] and not a [[countable set]].
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| Asymmetric [[Cantor set]] || [[File:AsymmCantor.png|200px]] || {{tick}} || || || || || ||{{tick}} || || || 2 || <math>\log_2(\varphi)=\log_2(1+\sqrt{5})-1</math> || 0.6942 || || 2 || The dimension is not <math>\frac{\ln2}{\ln\frac83}</math>, which is the generalized Cantor set with γ=1/4, which has the same length at each stage.<ref>{{Cite journal|author=Tsang, K. Y. |title=Dimensionality of Strange Attractors Determined Analytically |journal=Phys. Rev. Lett. |volume=57|issue=12|pages=1390–1393 |year=1986|pmid=10033437 |doi=10.1103/PhysRevLett.57.1390|bibcode=1986PhRvL..57.1390T}}</ref> Built by removing the second quarter at each iteration. [[Nowhere dense]] and not a [[countable set]]. <math>\scriptstyle\varphi = \frac{1+\sqrt5}2</math> ([[golden cut]]).
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| [[Real number]]s whose base 10 digits are even || [[File:Even digits.png|200px]] || || || || {{tick}} || || {{tick}} || || || || 2 || <math>\log_{10}(5)=1-\log_{10}(2)</math> || 0.69897 || || 2 || Similar to the [[Cantor set]].<ref name="Falconer">{{Cite book | last = Falconer | first = Kenneth | author-link=Kenneth Falconer (mathematician) | title = Fractal Geometry: Mathematical Foundations and Applications | isbn = 978-0-470-84862-3 | no-pp = true | page = xxv}}</ref>
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| Spectrum of Fibonacci Hamiltonian || || || || {{tick}} || || || || || || || || <math> \log(1+\sqrt{2})</math> || 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.<ref>{{cite journal |last1=Damanik |first1=D. |last2=Embree |first2=M. |last3=Gorodetski |first3=A. |first4=S. |last4=Tcheremchantse |title=The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian |journal=Commun. Math. Phys. |volume=280 |issue=2 |pages=499–516 |year=2008 |doi=10.1007/s00220-008-0451-3 |arxiv=0705.0338|bibcode=2008CMaPh.280..499D |s2cid=12245755 }}</ref>{{page needed|date=October 2018}}
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| Generalized Cantor set || [[File:generalized cantor set.png|200px]] || || || {{tick}} || || {{tick}} || || || || || 2 || <math>\frac{-\log(2)}{\log\left(\displaystyle\frac{1-\gamma}{2}\right)}</math> || 0<D<1 || || 2 || Built by removing at the <math>m</math><sup>th</sup> iteration the central interval of length <math>\gamma\,l_{m-1}</math> from each remaining segment (of length <math>l_{m-1}=(1-\gamma)^{m-1}/2^{m-1}</math>). At <math>\scriptstyle\gamma=1/3</math> one obtains the usual [[Cantor set]]. Varying <math>\scriptstyle\gamma</math> between 0 and 1 yields any fractal dimension <math>\scriptstyle 0\,<\,D\,<\,1</math>.<ref>{{cite journal |first1=A. Yu |last1=Cherny |first2=E.M. |last2=Anitas |first3=A.I. |last3=Kuklin |first4=M. |last4=Balasoiu |first5=V.A. |last5=Osipov |title=The scattering from generalized Cantor fractals |journal=J. Appl. Crystallogr. |volume=43 |issue= 4|pages=790–7 |year=2010 |doi=10.1107/S0021889810014184 |arxiv=0911.2497 |s2cid=94779870 }}</ref>
