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{{Short description|Condition for self-similar fractals}}
[[File:Open set condition.png|thumb|an open set covering of the [[sierpinski triangle]] along with one of its mappings ψ<sub>''i''</sub>.]]
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ<sub>''i''</sub>, the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
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{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>
==Computing Hausdorff
When the open set condition holds and each ψ<sub>''i''</sub> is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose [[Hausdorff dimension]] is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>▼
▲When the open set condition holds and each ψ<sub>''i''</sub> is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>
:<math> \sum_{i=1}^m r_i^s = 1. </math>
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==See also==
*[[Cantor set]]
*[[List of fractals by Hausdorff dimension]]
*[[Packing dimension]]
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