  We present a new approach to the theory of k-forms on self-similar fractals.
We work out the details for two examples, the standard Sierpinski gasket and
the 3-dimensional Sierpinski gasket, but the method is expected to be effective
for many PCF fractals and also infinitely ramified fractals such as the
Sierpinski carpet. Our approach is to construct k-forms and de Rham
differential operators d and delta for a sequence of graphs approximating the
fractal and then to pass to the limit with suitable renormalization, in
imitation of Kigami's approach to constructing Laplacians on functions. One of
our results is that our Laplacian on 0-forms is equal to Kigami's Laplacian on
functions. We give explicit construction of harmonic 1-forms for our examples.
We also prove that the measures on line segments provided by 1-forms are not
absolutely continuous with respect to Lebesgue measures.
