We discuss consequences of a possible discrepancy between topological and martingale dimension for fractal spaces. We concentrate on the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick, which under some conditions may have positive two-dimensional Lebesgue measure, although their topological dimension is one. There are examples of such carpets for which the curl operator induced by the diffusion on the carpet is not closable, even more, its adjoint has a trivial domain. This can be generalized to the observation that on topologically one dimensional spaces with a strongly local regular Dirichlet form the exterior derivative operator taking $1$-forms into $2$-forms is not closable if the martingale dimension is larger than one.

(joint work with Alexander Teplyaev, University of Connecticut)