The use of shears to construct paradoxes in 𝑅²

Wagon, Stanley

doi:10.1090/S0002-9939-1982-0656101-5

The use of shears to construct paradoxes in $\textbf {R}^{2}$
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by Stanley Wagon

Proc. Amer. Math. Soc. 85 (1982), 353-359

DOI: https://doi.org/10.1090/S0002-9939-1982-0656101-5

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Abstract:

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.

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