The Hexagram as a Geometric Generating Structure

The yin-yang boundary is one of the most recognised curves in the world. Its mathematical properties have been characterised axiomatically. What has not been examined is whether it can be derived from a more primitive structure — without beginning from the symbol itself. This paper shows that two equilateral triangles inscribed in a circle — a hexagram — generate a canonical yin-yang geometry as a natural consequence. The boundary curve is not a design choice. It is identified as the canonical curve satisfying two simultaneous constraints imposed by the hexagram itself: three-fold symmetry and equal area division. These constraints together identify a natural symmetric set of points; a boundary curve follows with no further assumptions. All results are exact and parameter-free: the inner hexagon inradius is r/2 (√3 cancels exactly), the S-curve radius is r/2 (confirmed by two independent routes), and the small circles are r/4 (double tangency condition). The paper includes a status map distinguishing derived results from stated assumptions, and an honest account of limitations and open questions. The derivation differs from and complements the axiomatic approach of Banakh, Verbitsky, and Vorobets (2009). Note: the semicircle boundary identified here is not the Fermat spiral; the two approaches identify different curves and are complementary. Independent researcher. Not reviewed.