Cut-and-Project Construction of Icosahedral Quasicrystals from E8 Lattices — E8 Intelligence Research

FINDING: The cut-and-project method generates aperiodic quasicrystals by projecting a higher-dimensional periodic lattice (E8, or Z^n) onto a lower-dimensional irrational subspace, with the acceptance window in the perpendicular space controlling which points survive — and for icosahedral quasicrystals, the window is a projection of the 600-cell/icosahedral symmetry, yielding Fibonacci/φ-scaling in the diffraction pattern. MATH: - Cut-and-project: Λ = { π∥(x) : x ∈ L, π⊥(x) ∈ W }, where L ⊂ R^d is a lattice, π∥ projects onto physical space E∥ (dim n), π⊥ onto internal space E⊥ (dim d−n), W ⊂ E⊥ is the acceptance window (a bounded polytope). - Icosahedral case: L = Z^6 or the root lattice A4⊕A4 / D6; physical space E∥ is a 3D irrational slice; the icosahedral point group (order 60, ≅ A5) acts on the window. - E8 root lattice: 240 minimal vectors (roots), |roots| = 240 = 2·120; E8 is the unique even unimodular lattice in 8D; kissing number 240; determinant 1; Coxeter number h = 30; Weyl Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com