A_n Lattice Projections Unify Rhombus and Triangle Tilings via Voronoi-Delone Cells — E8 Intelligence Research

Abstract

FINDING: The Voronoi/Delone cells of the A_n root lattice project to rhombus and equilateral triangle prototiles, unifying hexagonal tiling geometry with higher-dimensional lattice structure. | MATH: A_n root lattice: {x ∈ ℤ^(n+1) : Σx_i = 0}; Voronoi cell facets are rhombuses (angles 60°/120°); Delone cell facets are equilateral triangles; orthogonal projection yields 2D tilings with rhombus:triangle area ratio = √3 : 1 (or 1 : √3); hexagonal tiling emerges as A_2 Voronoi cell (regular hexagon, side s, area = (3√3/2)s²). | CONNECTION: Direct crystallographic link — A_2 is the hexagonal lattice; its Voronoi cell is the regular hexagon (angles 120°, dihedral symmetry D_6). The rhombus prototile has diagonals in ratio √3:1, related to 1.732, not directly to 0.618/1.618, but the hexagonal packing density π/(3√3) ≈ 0.6046 (not golden). However, the A_n family connects to E_8 (via A_8 ⊂ E_8) and to the Leech lattice via A_24 — deep lattice hierarchy. | DEPTH: 8 — This is a rigorous geometri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com