Hexagonal A2 Lattice: 30° Reciprocal Rotation and 120° Coxeter Symmetry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The A2 root lattice (hexagonal) and its reciprocal lattice are related by a 30° rotation, with the Weyl group's Coxeter element generating 120° rotations — a fundamental crystallographic symmetry. | MATH: A2 root lattice: hexagonal, basis vectors a₁, a₂ with |a₁|=|a₂|, angle 120°. Reciprocal lattice: bᵢ·aⱼ = 2πδᵢⱼ → reciprocal basis rotated 30° relative to direct, with |b| = (4π)/(√3·|a|). Coxeter element (product of two reflections) has eigenvalue e^(2πi/3) → rotation by 120°. | CONNECTION: 30° rotation = π/6; 120° = 2π/3. Ratio of reciprocal to direct lattice spacing: |b|/|a| = 4π/(√3·a²) — contains √3 (hexagonal symmetry). The 30° offset is the classic hexagonal-close-packed stacking geometry. No golden ratio appears directly, but the 120° angle is a 3-fold symmetry axis — related to the 2π/3 phase in complex roots of unity. | DEPTH: 7 — This is textbook crystallography, but the A2 lattice is the root system of SU(3), the gauge group of the strong force. The 120° Coxeter ro
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026hexagonal,
title = {{Hexagonal A2 Lattice: 30° Reciprocal Rotation and 120° Coxeter Symmetry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658211},
url = {https://zenodo.org/records/23658211},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hexagonal A2 Lattice: 30° Reciprocal Rotation and 120° Coxeter Symmetry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658211
UR - https://zenodo.org/records/23658211
AB - FINDING: The A2 root lattice (hexagonal) and its reciprocal lattice are related by a 30° rotation, with the Weyl group's Coxeter element generating 120° rotations — a fundamental crystallographic symmetry. | MATH: A2 root lattice: hexagonal, basis vectors a₁, a₂ with |a₁|=|a₂|, angle 120°. Reciprocal lattice: bᵢ·aⱼ = 2πδᵢⱼ → reciprocal basis rotated 30° relative to direct, with |b| = (4π)/(√3·|a|). Coxeter element (product of two reflections) has eigenvalue e^(2πi/3) → rotation by 120°. | CONNECTION: 30° rotation = π/6; 120° = 2π/3. Ratio of reciprocal to direct lattice spacing: |b|/|a| = 4π/(√3·a²) — contains √3 (hexagonal symmetry). The 30° offset is the classic hexagonal-close-packed stacking geometry. No golden ratio appears directly, but the 120° angle is a 3-fold symmetry axis — related to the 2π/3 phase in complex roots of unity. | DEPTH: 7 — This is textbook crystallography, but the A2 lattice is the root system of SU(3), the gauge group of the strong force. The 120° Coxeter ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -