The minimal number of pieces realizing affine congruence by dissection of topological discs
As a variation of the set-theoretical reformulation of the classical circle squaring problem, the following is considered: Can one dissect a circular disc into finitely many topological discs such that congruent images of these pieces form a dissection of a square? In a previous paper \textit{E. Hertel} and \textit{C. Richter} [Beitr. Algebra Geom. 44, 47--55 (2003; Zbl 1040.52006)] proved that a circular disc and a square are congruent by dissection with respect to the group of dyadic homotheties such that only four topological discs with rectifiable boundary are used as pieces of dissection. In the present interesting paper the author shows that, for any two topological discs \(D,E\in\mathbb{R}^2\), and for any group \(\mathcal{G}\) of affine transformations that contains all translations and a strict contraction, there is a congruence by dissection between \(D\) and \(E\) with respect to \(\mathcal {G}\) realized by only three pieces of dissection. This result is optimal, since it is also proved that two pieces do not suffice in the general case, for instance when an affine congruence by dissection is realized between a strictly convex disc \(D\) and a disc \(E\) whose boundary consist only of line segments. This includes the circle squaring problem.
scientific article; zbMATH DE number 2099073