E₈ Lattice Encodes 3D Icosahedral Quasicrystal Order via Quaternion Parameters — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: E₈ lattice's 240 vertices (Gosset 4₂₁ polytope) encode 3D icosahedral quasicrystal order via quaternion orientational parameters, linking 8D exceptional symmetry to 3D physical solidification. | MATH: E₈ root system: 240 vectors, norm²=2; 4₂₁ polytope: 240 vertices, 6720 edges; Weyl group order = 696,729,600. Quaternion order parameter q ∈ H, |q|=1, maps to SO(3) via q·x·q⁻¹. Icosahedral group A₅ order 60, embedded in E₈ via icosian quaternions (24 units + 96 others = 120, doubled to 240). | CONNECTION: Golden ratio φ = (1+√5)/2 = 1.618 appears in icosian coordinates (a+b i+c j+d k with a,b,c,d ∈ {0,±½,±φ/2,±1/2}); icosahedral vertices at (±1,±φ,0) permutations. Quasicrystal diffraction peaks scale by φ² = 2.618 and φ⁻¹ = 0.618. E₈ projection to 3D along icosian axes yields Penrose-like tiling with 5-fold symmetry. | DEPTH: 9 — Directly unifies exceptional Lie theory (E₈), quaternion algebra, icosahedral group simplicity, and condensed matter quasicrystal formation; the 240→12
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026e₈,
title = {{E₈ Lattice Encodes 3D Icosahedral Quasicrystal Order via Quaternion Parameters — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635389},
url = {https://zenodo.org/records/23635389},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - E₈ Lattice Encodes 3D Icosahedral Quasicrystal Order via Quaternion Parameters — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635389
UR - https://zenodo.org/records/23635389
AB - FINDING: E₈ lattice's 240 vertices (Gosset 4₂₁ polytope) encode 3D icosahedral quasicrystal order via quaternion orientational parameters, linking 8D exceptional symmetry to 3D physical solidification. | MATH: E₈ root system: 240 vectors, norm²=2; 4₂₁ polytope: 240 vertices, 6720 edges; Weyl group order = 696,729,600. Quaternion order parameter q ∈ H, |q|=1, maps to SO(3) via q·x·q⁻¹. Icosahedral group A₅ order 60, embedded in E₈ via icosian quaternions (24 units + 96 others = 120, doubled to 240). | CONNECTION: Golden ratio φ = (1+√5)/2 = 1.618 appears in icosian coordinates (a+b i+c j+d k with a,b,c,d ∈ {0,±½,±φ/2,±1/2}); icosahedral vertices at (±1,±φ,0) permutations. Quasicrystal diffraction peaks scale by φ² = 2.618 and φ⁻¹ = 0.618. E₈ projection to 3D along icosian axes yields Penrose-like tiling with 5-fold symmetry. | DEPTH: 9 — Directly unifies exceptional Lie theory (E₈), quaternion algebra, icosahedral group simplicity, and condensed matter quasicrystal formation; the 240→12 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -