Penrose Tiling: Forbidden Symmetry via Golden Ratio Aperiodic Order — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Penrose tiling is a non-periodic tiling with 5-fold (C10) rotational symmetry, generated by aperiodic prototiles, and forms a local isomorphism class with distinct vertex configurations (star, sun) and diffraction patterns exhibiting forbidden crystallographic symmetry. MATH: - Prototiles: thin and thick rhombi with angles 36°/144° and 72°/108° (golden rhombi). - Inflation factor: τ = (1+√5)/2 ≈ 1.618 (golden ratio). - Vertex configurations: 7 distinct local isomorphism classes; "star" (5-fold vertex, 5 thick rhombi) and "sun" (5-fold vertex, 5 thin rhombi) are the two maximal symmetry vertices. - Diffraction pattern: sharp Bragg peaks with 10-fold (C10) rotational symmetry — forbidden in periodic crystallography (crystallographic restriction theorem allows only 1,2,3,4,6-fold). - Local isomorphism: any finite patch appears infinitely often; the tiling space is a compact, minimal, uniquely ergodic dynamical system. CONNECTION: - Golden ratio τ appears in tile edg 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: W7219566797
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