A_n Lattice Projections Unify Rhombus and Triangle Tilings via Voronoi-Delone Cells — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
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
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BibTeX
@misc{caldin2026lattice,
title = {{A_n Lattice Projections Unify Rhombus and Triangle Tilings via Voronoi-Delone Cells — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23410003},
url = {https://zenodo.org/records/23410003},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A_n Lattice Projections Unify Rhombus and Triangle Tilings via Voronoi-Delone Cells — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23410003
UR - https://zenodo.org/records/23410003
AB - 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
ER -