6D Lattice Projection Linking H3 Roots to Icosahedral Quasicrystals — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The search results confirm the well-established mathematical link between 6D hypercubic lattices, the H3 root system, and 3D icosahedral quasicrystals via cut-and-project methods, with the rhombic hexecontahedron as the canonical projection cell. | MATH: H3 root system has 15 roots, 30 edges; icosahedral group order 120; rhombic hexecontahedron has 30 rhombi, 60 vertices; dihedral angles arccos(1/√5) ≈ 63.43° and 116.57°; golden ratio φ = 1.618, inverse φ⁻¹ = 0.618; projection from A6/hypercubic lattice to 3D yields Penrose-like tilings with forbidden 5-fold symmetry. | CONNECTION: Direct — the golden ratio appears as the scaling factor (φ^n) in the projection matrix; the dihedral angles arccos(1/√5) are the exact angles of the rhombic hexecontahedron, linking to 0.618 and 1.618; the 6D lattice projection is the non-crystallographic H3 root system realization. | DEPTH: 8 — This is a profound structural insight: the "forbidden" 5-fold symmetry emerges from higher-dimensional pe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Materials Chemistry
Field: Materials Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212295435
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