Open set condition

When the open set condition holds and each ${\displaystyle \psi _{i}}$ is a similitude (that is, a composition of an isometry and a dilation around some point), then the unique fixed point of ${\displaystyle \psi }$ is a set whose Hausdorff dimension is the unique solution for s of the following:^[5]

${\displaystyle \sum _{i=1}^{m}r_{i}^{s}=1.}$

where r_i is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three non-collinear points a₁, a₂, a₃ in the plane R² and let ${\displaystyle \psi _{i}}$ be the dilation of ratio 1/2 around a_i. The unique non-empty fixed point of the corresponding mapping ${\displaystyle \psi }$ is a Sierpinski gasket, and the dimension s is the unique solution of

${\displaystyle \left({\frac {1}{2}}\right)^{s}+\left({\frac {1}{2}}\right)^{s}+\left({\frac {1}{2}}\right)^{s}=3\left({\frac {1}{2}}\right)^{s}=1.}$

Taking natural logarithms of both sides of the above equation, we can solve for s, that is: s = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.