The Use of Shears to Construct Paradoxes in $R^2

The Use of Shears to Construct Paradoxes in $R^2

It is shown that the addition of a certain shear transformation to the planar isometry group is sufficient to allow a Banach-Tarski type paradox to be constructed in $\mathbf{R}^2$. This paradox is then combined with a result of Rosenblatt to obtain a characterization of two-dimensional Lebesgue mea...

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Veröffentlicht in:Proceedings of the American Mathematical Society 1982-07, Vol.85 (3), p.353-359, Article 353
1. Verfasser: Wagon, Stanley
Format: Artikel
Sprache:eng
Schlagworte:
Circles
Cubes
Lebesgue measures
Linear transformations
Mathematical congruence
Mathematical theorems
Paradoxes
Uniqueness
Wagons
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Zusammenfassung:It is shown that the addition of a certain shear transformation to the planar isometry group is sufficient to allow a Banach-Tarski type paradox to be constructed in $\mathbf{R}^2$. This paradox is then combined with a result of Rosenblatt to obtain a characterization of two-dimensional Lebesgue measure as a finitely additive measure.
ISSN:0002-9939
1088-6826
DOI:10.2307/2043845