  We construct periodic approximations to the free energies of Ising models on
fractal lattices of dimension smaller than two, in the case of zero external
magnetic field, using a generalization of the combinatorial method of Feynman
and Vodvickenko. Our procedure is applicable to any fractal obtained by the
removal of sites of a periodic two dimensional lattice. As a first application,
we compute estimates for the critical temperatures of many different Sierpinski
carpets and we compare them to known Monte Carlo estimates. The results show
that our method is capable of determining the critical temperature with,
possibly, arbitrary accuracy and paves the way to determine $T_c$ for any
fractal of dimension below two. Critical exponents are more difficult to
determine since the free energy of any periodic approximation still has a
logarithmic singularity at the critical point implying $\alpha = 0$. We also
compute the correlation length as a function of the temperature and extract the
relative critical exponent. We find $\nu=1$ for all periodic approximation, as
expected from universality.
