Congruence by dissection of topological discs - An elementary approach to Tarski's circle squaring problem

Preprint series: 01-11, Reports on Algebra and Geometry

Preprint series: , Reports on Algebra and Geometry

MSC:
52B45 Dissections and valuations (Hilbert's third problem, etc.)
52A10 Convex sets in $2$ dimensions (including convex curves), See also {53A04}
51M04 Elementary problems in Euclidean geometries

Abstract: Let ${\bf G}$ be a group of affine transformations of the plane ${\mathbb R}^2$ and let the family ${\mathcal F} \subseteq 2^{{\mathbb R}^2}$ consist of all topological discs whose boundary is subject to some smoothness condition (general, rectifiable, piecewise differentiable). Can any two members $D,E \in {\mathcal F}$ be cut into finitely many topological discs $D_1,D_2,\ldots,D_n \in {\mathcal F}$ and $E_1,E_2,\ldots,E_n \in {\mathcal F}$, respectively, such that $E_i$ is the image of $D_i$ under some map from ${\bf G}$, $1 \le i \le n$? Besides the general setting we consider the particular case where $D$ and $E$ are a circular disc and a square, respectively.

Keywords: topological disc, congruence by dissection, scissor congruence, Euclidean motion, homothety, similarity, equiaffine map, affine map, squaring the circle