Fermat's spiral and the line between Yin and Yang

Details Fermat's spiral and the line between Yin and Yang Fermat's spiral and the line between Yin and Yang

2009-02-09

arxiv.org/abs/0902.1556v2

Amer. Math. Monthly. 117:9 (2010), 786-800

Mathematics

Combinatorics

20 pages, 7 figures. In the 2nd version, a minor correction is made

Scientific paper

Let $D$ denote a disk of unit area. We call a subset $A$ of $D$ perfect if it has measure 1/2 and, with respect to any axial symmetry of $D$, the maximal symmetric subset of $A$ has measure 1/4. We call a curve $\beta$ in $D$ an yin-yang line if $\beta$ splits $D$ into two congruent perfect sets, $\beta$ crosses each concentric circle of $D$ twice, $\beta$ crosses each radius of $D$ once. We prove that Fermat's spiral is a unique yin-yang line in the class of smooth curves algebraic in polar coordinates.

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