Squaring the circle via affine congruence by dissection with smooth pieces

Christian Richter

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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