Affine congruence by dissection of intervals

Tarski’s circle squaring problem (see [8]) has motivated the following question: Can a circular disc be dissected into finitely many topological discs such that images of these pieces under suitable Euclidean motions form a dissection of a square? Dubins, Hirsch, and Karush give a negative answer in [1]. However, one can get positive results if the group of Euclidean motions is replaced by suitable other groups of affine maps of the plane (see [3, 5, 6, 7]). The general concept behind these phenomena is the congruence by dissection of discs with respect to some fixed group of affine transformations of R2. Let d denote the Euclidean distance in the plane R2. We recall that a topological disc D is the image of the closed unit disc {x ∈ R2 : d(x, 0) ≤ 1} under a homeomorphism of the plane onto itself. We say that D is dissected into the discs D1, . . . , Dn if D = D1∪. . .∪Dn and int(Di ) ∩ int(D j ) = ∅ for 1 ≤ i < j ≤ n, int(Di ) denoting the interior of Di . Given a group G of affine transformations of R2, two topological discs D, E are called congruent by dissection with respect to G if and only if there exist dissections of D and E into the same finite number n ≥ 1 of subdiscs D1, . . . , Dn and E1, . . . , En , respectively,