  The method of counting loops for calculating the partition function of the
Ising model on the two dimensional square lattice is extended to lacunary
planar lattices, especially scale invariant fractal lattices, the Sierpi\'nsky
carpets with different values of the scale invariance ratio and of the number
of deleted sites. The critical temperature of the Ising model on these lattices
is exactly calculated for finite iteration steps, and for a range of the scale
invariance 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 infinite
number 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 the
Ising partition functions on several examples of Sierpi\'nski carpets are
illustrated.
