The Sierpinski gasket and other self-similar fractal subsets of $\mathbb R^d$,
$d\ge 2$, can be mapped by quasiconformal self-maps of $\mathbb R^d$ onto sets
of Hausdorff dimension arbitrarily close to one. In $\mathbb R^2$ we construct
explicit mappings. In $\mathbb R^d$, $d\ge 3$, the results follow from general
theorems on the equivalence of invariant sets for iterated function systems
under quasisymmetric maps and global quasiconformal maps. More specifically, we
present geometric conditions ensuring that (i) isomorphic systems have
quasisymmetrically equivalent invariant sets, and (ii) one-parameter isotopies
of systems have invariant sets which are equivalent under global quasiconformal
maps.