A Python Engine and Exact Lattice Construction for Rotational Tessellations from the Heronian 13–14–15 Triangle

We study the classical Heronian triangle with side lengths 13–14–15 and its role as a building block for exact planar tessellations with 12-fold rotational symmetry. Starting from explicit Cartesian coordinates, we construct a fundamental “kite” formed by the 13–14–15 triangle and its excentral reflection of equal area, and show how this tile generates a full lattice tessellation via two translation vectors of area 504. The local structure around the natural center of symmetry is organized by a fundamental hexagon whose edges are aligned with directions at integer multiples of 30 degrees, giving rise to a rotational “Heronian mandala” of order 12.

On the computational side, we implement a complete Python engine that reproduces the exact tessellation and its rotational copies, using a clean vector-based description. The code generates both geometric visualizations (showing the lattice structure and the fundamental hexagon) and dense, artistic renderings where each triangular tile is colored according to the polar angle of its centroid.

From an arithmetic viewpoint, we place this construction within the broader family of consecutive Heronian triangles (a−1, a, a+1), parametrized by the Pell-type equation k² − 3a² = −12. We formulate a natural uniqueness conjecture: among all consecutive Heronian triangles, the 13–14–15 triangle is the only one that admits an exact 12-fold rotational tessellation of the plane constructed from the triangle and its equal-area excentral reflection. We outline a strategy to attack this conjecture by combining the Pell parametrization with additional tiling constraints, which are expected to cut down the Pell surface to one or several elliptic curves of finite Mordell–Weil rank.

The preprint includes: (1) a self-contained geometric derivation of the lattice and the fundamental hexagon, (2) a parametric description of the privileged directions of the mandala, (3) a discussion of the arithmetic framework and the proposed uniqueness conjecture, and (4) the full Python code used to generate all figures.