an open set covering of the sierpinski triangle along with one of its mappings ψi.
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.[1] Specifically, given an iterated function system of contractive mappings , the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
The sets are pairwise disjoint.
Introduced in 1946 by P.A.P Moran,[2] the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.[3]
An equivalent statement of the open set condition is to require that the s-dimensional Hausdorff measure of the set is greater than zero.[4]
^Bandt, Christoph; Viet Hung, Nguyen; Rao, Hui (2006). "On the Open Set Condition for Self-Similar Fractals". Proceedings of the American Mathematical Society. 134 (5): 1369–74.
^Moran, P.A.P. (1946). "Additive Functions of Intervals and Hausdorff Measure". Proceedings of the Cambridge Philosophical Society. 42: 15–23. doi:10.1017/S0305004100022684.
^Llorente, Marta; Mera, M. Eugenia; Moran, Manuel. "On the Packing Measure of the Sierpinski Gasket" (PDF). University of Madrid.
^
Wen, Zhi-ying. "Open set condition for self-similar structure" (PDF). Tsinghua University. Retrieved 1 February 2022.
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