MERLIN SCIENCE — A_n Lattice Projections Unify Rhombus and Triangle Tilings via Voronoi — E8 Intelligence Research

Today's finding is this: the Voronoi and Delone cells of the A_n root lattice project cleanly onto the plane to produce both rhombus and equilateral triangle prototiles, and this single mechanism unifies the geometry of hexagonal tilings with the algebraic structure of higher-dimensional lattices. Let me give you the context. In crystallography and materials science, you see hexagonal symmetry everywhere—graphene, boron nitride, the densest packing of disks. But the mathematical reason *why* the hexagon is so special often gets buried in computation. We know the hexagonal lattice is the A_2 root lattice, but the bridge from that algebraic object to the physical tiling has been underappreciated. This work makes that bridge explicit and extends it to all dimensions. Here is the mechanism. The A_n root lattice is defined as all integer vectors in n+1 dimensions whose coordinates sum to zero. Take the Voronoi cell—the set of points closer to a given lattice point than to any other. Its f Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com