Affine congruence by dissection of intervals
type: Item
instance of: scholarly article
title: Affine congruence by dissection of intervals
zbMATH Open document ID: 1155.52011
DOI: 10.4171/EM/63
author: Christian Richter
published in: Elemente der Mathematik
publication date: 2008-10-23T00:00:00Z
review text: In the theory of dissection it is known that a circular disc cannot be dissected into finitely many topological discs such that the images of these pieces, under suitable Euclidean motions, form a dissection of a square. As a generalization of such a Tarsky-type problem, one can investigate about the congruence by dissection of two sets under the action of some group of transformations. For instance, the author has previously obtained a characterization of the groups of affine transformations of \(\mathbb{R}^2\) that admit a congruence by dissection of any two topological discs [J. Geom. 84, No. 1--2, 117--132 (2005; Zbl 1139.52016)]. In this paper the focus is on the one-dimensional analogue of this problem, concerning the congruence by dissection of compact intervals of positive length. Differently from intuition, answers are not always easier than in the plane. As the author points out, this depends on the larger freedom one has in the plane in choosing the shapes employed in dissections.
reviewed by: Paolo Dulio
Mathematics Subject Classification ID: 52B45
zbMATH DE Number: 5356287
zbMATH Keywords: affine map
zbMATH Keywords: congruence
zbMATH Keywords: dissection
zbMATH Keywords: Tarski's circle squaring problem
zbMATH Keywords: topological disc
MaRDI profile type: Publication
cites work: Scissor congruence
cites work: Q3245768
cites work: Squaring the circle by dissection
cites work: Zerlegungsähnlichkeit von Polyedern.
cites work: Congruence by dissection of topological discs --- an elementary approach to Tarski's circle squaring problem
cites work: The minimal number of pieces realizing affine congruence by dissection of topological discs
cites work: Affine congruence by dissection of discs-appropriate groups and optimal dissections
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description: scientific article; zbMATH DE number 5356287