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

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 .A n 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).