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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
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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.

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