The minimal number of pieces realizing affine congruence by dissection of topological discs

Let be a group of affine transformations of the plane that contains a strict contraction and all translations. It is shown that any two topological discs are congruent dissection with respect to such that only three topological discs are used as pieces of dissection. Two pieces of dissection do not suffice in general even if consists of all affine transformations. This revised version was published online in August 2006 with corrections to the Cover Date.