Golden Ratio Inflation and K-Theory in Aperiodic Penrose Quasicrystals — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Penrose tiling and quasicrystals are governed by golden-ratio inflation/deflation rules, with topological invariants classified via K-theory; a 5D projection model generates aperiodic order. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.618; reciprocal φ⁻¹ = φ−1 ≈ 0.618; φ² = φ+1 ≈ 2.618. Penrose tiling uses two rhombi with acute angles 36° and 72° (π/5 and 2π/5), enforcing fivefold rotational symmetry (C₅, crystallographically forbidden in periodic lattices). Inflation rule: substitution matrix [[2,1],[1,1]] has eigenvalues φ² and φ⁻². K-theory: topological invariants for quasicrystals live in K₀(C*-algebra of the hull), with gap labels in ℤ[φ] — a rank-2 abelian group. 5D projection: quasicrystal points = intersection of 5D hypercubic lattice ℤ⁵ with a 2D (or 3D) irrational plane, slope components in ℚ(√5). | CONNECTION: Direct — φ, φ⁻¹, φ² appear as scaling factors in inflation; 0.618 = 1/φ is the ratio of tile areas (fat:thin = φ:1). Fivefold symmetry relates to icosahedral root s
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026golden,
title = {{Golden Ratio Inflation and K-Theory in Aperiodic Penrose Quasicrystals — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23446959},
url = {https://zenodo.org/records/23446959},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Ratio Inflation and K-Theory in Aperiodic Penrose Quasicrystals — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23446959
UR - https://zenodo.org/records/23446959
AB - FINDING: Penrose tiling and quasicrystals are governed by golden-ratio inflation/deflation rules, with topological invariants classified via K-theory; a 5D projection model generates aperiodic order. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.618; reciprocal φ⁻¹ = φ−1 ≈ 0.618; φ² = φ+1 ≈ 2.618. Penrose tiling uses two rhombi with acute angles 36° and 72° (π/5 and 2π/5), enforcing fivefold rotational symmetry (C₅, crystallographically forbidden in periodic lattices). Inflation rule: substitution matrix [[2,1],[1,1]] has eigenvalues φ² and φ⁻². K-theory: topological invariants for quasicrystals live in K₀(C*-algebra of the hull), with gap labels in ℤ[φ] — a rank-2 abelian group. 5D projection: quasicrystal points = intersection of 5D hypercubic lattice ℤ⁵ with a 2D (or 3D) irrational plane, slope components in ℚ(√5). | CONNECTION: Direct — φ, φ⁻¹, φ² appear as scaling factors in inflation; 0.618 = 1/φ is the ratio of tile areas (fat:thin = φ:1). Fivefold symmetry relates to icosahedral root s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -