Golden Ratio Projection: 5D Lattice to Penrose Tilings — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Cut-and-project method generates Penrose tilings from 5D lattice via irrational plane, linking golden ratio inflation/deflation to quasicrystalline order. | MATH: 5D hypercubic lattice \(\mathbb{Z}^5\); irrational 2D plane \(E^\parallel\) with slope \(\tau = (1+\sqrt{5})/2\) (golden ratio); projection window = unit 5-cube shadow onto \(E^\perp\); de Bruijn's grid method yields Penrose tiling with vertex stars of 5-fold symmetry; inflation factor \(\tau^2 = \phi^2 = \phi+1 = 2.618\); deflation ratio \(1/\phi = 0.618\). | CONNECTION: Direct — golden ratio \(\phi = 1.618\) is the irrational slope; \(\phi^2 = 2.618\), \(\phi^{-1} = 0.618\), \(\phi^{-2} = 0.382\) all appear in tile edge ratios and inflation/deflation sequences. The 5D lattice projects to 2D with 5-fold (crystallographically forbidden) symmetry — a direct violation of classical crystallographic restriction, resolved by quasiperiodicity. | DEPTH: 9 — This is the foundational bridge between higher-dimensional lattices 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: W7208761609
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