What question did this study set out to answer?

This research aims to understand how E8 lattice structures project into a 3D icosahedral quasicrystal using quaternion mathematics.

July 23, 2026Open Access

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E8 Lattice to Icosahedral Quasicrystal via Quaternion Order Parameter — E8 Intelligence Research

ACAndrew Stewart Caldin

Key Points

This research aims to understand how E8 lattice structures project into a 3D icosahedral quasicrystal using quaternion mathematics.
Analyzed E8 root system composed of 240 minimal vectors and their projections into quaternion and icosahedral forms.
Utilized Hopf fibration for dimensional projection from 8D to 3D.
Developed Landau-type free energy models to study solidification transitions in quaternion invariants.
Successfully projected the E8 lattice to reveal a 3D icosahedral quasicrystal characterized by six fivefold symmetry axes.
Identified critical exponents related to the solidification phase transition in conjunction with quaternion order parameters.
Displayed the occurrence of the golden ratio φ within the geometry of the icosahedral vertices, linking to E8 structures.

Abstract

FINDING: E8 lattice (240 root vectors) projects to 3D icosahedral quasicrystal via quaternion orientational order parameter; solidification phase transition classified by quaternion order parameter. MATH: - E8 root system: 240 minimal vectors in 8D; 112 of form (±1,±1,0,0,0,0,0,0) and permutations, 128 of form (±½,±½,±½,±½,±½,±½,±½,±½) with even number of minus signs. - Quaternion order parameter: q ∈ ℍ (4D), normalized |q|=1, maps to SO(3) rotation via double cover SU(2). - Icosahedral group H₃: order 120, generated by quaternions representing 5-fold rotations (golden ratio φ = (1+√5)/2 ≈ 1.618). - Projection: 8D → 4D (quaternion space) → 3D via Hopf fibration or cut-and-project; yields 3D icosahedral quasicrystal with 6 fivefold axes. - Solidification order parameter: Landau-type free energy expanded in quaternion invariants; critical exponents linked to E8 root system. CONNECTION: - Golden ratio φ = 1.618 appears in icosahedral vertices: (0, ±1, ±φ) and permutations. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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E8 Lattice to Icosahedral Quasicrystal via Quaternion Order Parameter — E8 Intelligence Research

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