Fermat's Spiral and the Line Between Yin and Yang

Let D denote a disk of unit area. We call a set A ⊂ 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 β in D an yin-yang line if • β splits D into two congruent perfect sets, • β crosses each concentric circle of D twice, • β 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.