Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 48, No. 2, pp. 423-434 (2007)

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Squaring the circle via affine congruence by dissection with smooth pieces

Christian Richter

Mathematisches Institut, Friedrich-Schiller-Universität, D-07737 Jena, Germany,

Abstract: For every $k \ge 2$, we construct a dissection of a square into finitely many topological discs, each having a piecewise $k$ times differentiable boundary, such that images of these pieces under appropriate maps of the group $\bf{A}ff_+$ of orientation preserving affine transformations of the plane form a dissection of a circular disc. The dissections consist of six pieces in the case $k=2$ and of $14$ pieces for every $k \ge 3$.

Keywords: squaring the circle, congruence by dissection, topological disc with piecewise smooth boundary, affine transformation

Classification (MSC2000): 52B45

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Electronic version published on: 7 Sep 2007. This page was last modified: 28 Jun 2010.

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