Fermat's spiral and the line between Yin and Yang

TL;DR

This paper characterizes Fermat's spiral as the unique smooth algebraic curve in polar coordinates that divides a disk into two perfect, symmetric sets with specific geometric properties related to Yin and Yang symbolism.

Contribution

It proves Fermat's spiral is the only smooth algebraic curve in polar form satisfying the Yin-Yang dividing properties in a unit disk.

Findings

Fermat's spiral uniquely satisfies the Yin-Yang dividing conditions.

The curve splits the disk into two congruent perfect sets with symmetry properties.

It crosses each concentric circle twice and each radius once.

Abstract

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.