The minimal number of pieces realizing affine congruence by dissection of topological discs

Preprint series: 02-02, Reports on Algebra and Geometry

Preprint series: , Reports on Algebra and Geometry

MSC:
52B45 Dissections and valuations (Hilbert's third problem, etc.)

Abstract: Let $\mathcal G$ be a group of affine
transformations of the plane that contains a strict
contraction and all translations. It is shown that any
two topological discs $D,E \subseteq {\mathbb R}^2$
are congruent by dissection with respect to
$\mathcal G$ such that only three topological discs
are used as pieces of dissection. Two pieces of
dissection do not suffice in general even if
$\mathcal G$ consists of all affine transformations.

Keywords: Congruence by dissection, topological disc, minimal number of pieces, homothety, similarity, affine map, Tarski's circle squaring problem.