The minimal number of pieces realizing affine congruence by dissection of topological discs
type: Item
instance of: scholarly article
title: The minimal number of pieces realizing affine congruence by dissection of topological discs
zbMATH Open document ID: 1049.52014
DOI: 10.1023/A:1025944327967
author: Christian Richter
published in: Periodica Mathematica Hungarica
publication date: 2004-09-07T00:00:00Z
review text: 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.
reviewed by: Paolo Dulio
Mathematics Subject Classification ID: 52B45
zbMATH DE Number: 2099073
zbMATH Keywords: congruence by dissection
zbMATH Keywords: topological disc
zbMATH Keywords: minimal number of pieces
zbMATH Keywords: homothety
zbMATH Keywords: similarity
zbMATH Keywords: affine map
zbMATH Keywords: Tarski's circle squaring problem
MaRDI profile type: Publication
cites work: Scissor congruence
cites work: Squaring the circle by dissection
cites work: Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem
cites work: Congruence by dissection of topological discs --- an elementary approach to Tarski's circle squaring problem
cites work: Q3687182
full work available at URL: a:1025944327967
OpenAlex ID: W1742501961
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description: scientific article; zbMATH DE number 2099073