What question did this study set out to answer?

This research aims to explore the relationship between E8 lattice projection and the formation of icosahedral quasicrystals through the lens of the golden ratio.

July 23, 2026Open Access

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E8 Lattice Projection Yields Icosahedral Quasicrystals with Golden Ratio Phason Strain — E8 Intelligence Research

ACAndrew Stewart Caldin

Key Points

This research aims to explore the relationship between E8 lattice projection and the formation of icosahedral quasicrystals through the lens of the golden ratio.
E8 root system projected to 3D using a cut-and-project method.
Investigation of phason strain tensor associated with icosahedral symmetry and quaternion order parameter.
Analysis of Voronoi cell cross-sections for order-24 symmetry and diffraction patterns.
Phason strain tensor eigenvalues scale with the golden ratio φ, indicating a mathematical bridge to quasicrystalline order.
Emergence of ratios 0.618, 1.618, and 2.618 in quasicrystal diffraction patterns.
Solidification phase transition modeled via quaternion order parameter suggests its fundamental role in condensed matter physics.

Abstract

FINDING: E8 lattice projection yields 3D icosahedral quasicrystals with phason strain tensor scaling by golden ratio φ. | MATH: E8 root system (240 vertices, 6720 edges) projected to 3D via cut-and-project method; phason strain tensor eigenvalues proportional to φ = (1+√5)/2 ≈ 1.618; quaternion orientational order parameter Q = q₀ + q₁i + q₂j + q₃k with icosahedral symmetry group H₃ of order 120. | CONNECTION: Golden ratio φ appears as scaling factor in phason strain, linking E8 (8D exceptional Lie group) to icosahedral symmetry (H₃ Coxeter group). Voronoi cell cross-sections show order-24 symmetry, consistent with 24-cell (self-dual regular polytope in 4D). Ratios 0.618 (1/φ), 1.618 (φ), 2.618 (φ²) emerge in quasicrystal diffraction patterns. | DEPTH: 8 — Direct mathematical bridge between exceptional Lie algebra E8, quasicrystalline order, and golden ratio; solidification phase transition modeled via quaternion order parameter suggests fundamental role in condensed matter physics. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Andrew Stewart Caldin (Tue,) studied this question.

synapsesocial.com/papers/6a61b056faa9903c5116ae9d https://doi.org/https://doi.org/10.5281/zenodo.21467382

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