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The method of counting loops for calculating the partition function of theIsing model on the two dimensional square lattice is extended to lacunaryplanar lattices, especially scale invariant fractal lattices, the Sierpi\'nskycarpets with different values of the scale invariance ratio and of the numberof deleted sites. The critical temperature of the Ising model on these latticesis exactly calculated for finite iteration steps, and for a range of the scaleinvariance ratio $n$ from 3 to 1000 and of the number of deleted sites from$(n-2)^2$ to $(n-10)^2$. The critical temperature at the limit of an infinitenumber of iteration of the segmentation process is asymptotically extrapolated.Comparison is made with results obtained previously by numerical methods.Thermodynamical functions are also calculated and the fractal spectra of theIsing partition functions on several examples of Sierpi\'nski carpets areillustrated.

The Ising model in planar lacunary and fractal lattices, a path counting approach