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     Abstract.  Let G be a group of affine transformations of the plane R  2  and let the family F consist of all topological discs in R  2  whose boundary is subject to some smoothness condition (general, rectifiable, piecewise C  1  , piecewise C  2  ). Are any two members D,E ∈  F congruent by dissection with respect to G such that all the pieces in the corresponding dissections of D and E belong to F as well? We give an affirmative answer if G contains all affine transformations and F consists of the discs whose boundary is piecewise C  1  . An example shows that C  1  cannot be replaced by C  2  . Moreover, if G is either the group of equiaffine transformations or the group of similarities, then congruence by dissection of two convex discs D and E turns out to be essentially equivalent to congruence by dissection of the boundaries bd(D ) and bd(E ).

Congruence by Dissection of Topological Discs—An Elementary Approach to Tarski's Circle Squaring Problem