Heronian Rotational Mandala: Geometry, Symmetry, and Exact Tessellations from the 13–14–15 Triangle

This work investigates the exceptional geometric and arithmetic properties of the Heronian triangle 13–14–15. By combining the triangle with its excentral reflected counterpart—both sharing the same area—we construct a fundamental kite-shaped tile that generates an exact rotational tessellation of order 121212. We present explicit coordinate geometry, transformation symmetries, and rotational structures underlying the mandala-like tiling.

The study suggests that the triangle 13–14–15 is unique within the family of consecutive Heronian triangles (a−1,a,a+1)(a-1,a,a+1)(a−1,a,a+1) in admitting such a perfect rotational tessellation. Motivated by connections with Pell equations and elliptic curves, we outline a research program toward a full mathematical proof, and invite collaboration from specialists in number theory and discrete geometry.