Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Banach-Tarski theorem on polygons, and the cancellation law

Authors: R. J. Gardner and M. Laczkovich
Journal: Proc. Amer. Math. Soc. 109 (1990), 1097-1102
MSC: Primary 52A10; Secondary 28C10, 52A20
MathSciNet review: 1017001
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Banach-Tarski theorem on polygons in $ {{\mathbf{R}}^2}$ implies that two polygons are equidecomposable if and only if they are equidissectable. The possibility of strengthening this result in various ways is investigated. We show that if two polytopes in $ {{\mathbf{R}}^d}$ are equidecomposable under a finite set of isometries which generates a discrete group, then they are equidissectable using the same isometries. We then give a simple example in $ {\mathbf{R}}$ showing that this is not true for arbitrary finite sets of isometries. A modification of this example is used to answer a question of S. Wagon concerning the cancellation law.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A10, 28C10, 52A20

Retrieve articles in all journals with MSC: 52A10, 28C10, 52A20

Additional Information

Keywords: Polygon, convex body, equidecomposable, equidissectable, circlesquaring, matching, cancellation law
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society