Affine congruence by dissection of intervals
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.
scientific article; zbMATH DE number 5356287
scientific article