The Hexagram as a Geometric Forcing 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 — force the complete yin-yang geometry as a necessary consequence. The boundary curve is not a design choice. It is the unique solution satisfying two simultaneous constraints imposed by the hexagram itself: three-fold symmetry and equal area division. These two constraints together identify a unique set of points; the boundary 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). Independent researcher. Not peer reviewed.