Golden Ratio Eigenvalues Link Penrose Tiling to Fibonacci Tile Counts — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Penrose tiling's kites-and-darts inflation rule is a substitution system whose growth matrix eigenvalues are powers of the golden ratio, directly encoding Fibonacci numbers in tile counts. | MATH: Inflation substitution matrix for kites (K) and darts (D): K→K+D, D→K (or equivalent). Matrix M = [[1,1],[1,0]]; eigenvalues φ = (1+√5)/2 ≈ 1.618 and −1/φ ≈ −0.618. After n inflations, tile counts follow Fibonacci: F_{n+1} kites, F_n darts. The ratio of tile counts → φ. The minimal inflation multiplier for the CAST class (arXiv:1606.06858) is also φ for the Penrose case, with vertices in the 10th cyclotomic field ℚ(ζ₁₀). | CONNECTION: φ, 1/φ = 0.618, φ² = 2.618 all appear as eigenvalues and tile-count ratios. The 5-fold rotational symmetry (crystallographically forbidden in periodic lattices) is realized via the cyclotomic field ℚ(ζ₅) = ℚ(ζ₁₀), linking to the golden ratio's role in pentagonal/quasicrystalline order. The substitution rule is a 1D Fibonacci word inflation (A→AB, B→A) l
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 Eigenvalues Link Penrose Tiling to Fibonacci Tile Counts — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23430953},
url = {https://zenodo.org/records/23430953},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Ratio Eigenvalues Link Penrose Tiling to Fibonacci Tile Counts — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23430953
UR - https://zenodo.org/records/23430953
AB - FINDING: Penrose tiling's kites-and-darts inflation rule is a substitution system whose growth matrix eigenvalues are powers of the golden ratio, directly encoding Fibonacci numbers in tile counts. | MATH: Inflation substitution matrix for kites (K) and darts (D): K→K+D, D→K (or equivalent). Matrix M = [[1,1],[1,0]]; eigenvalues φ = (1+√5)/2 ≈ 1.618 and −1/φ ≈ −0.618. After n inflations, tile counts follow Fibonacci: F_{n+1} kites, F_n darts. The ratio of tile counts → φ. The minimal inflation multiplier for the CAST class (arXiv:1606.06858) is also φ for the Penrose case, with vertices in the 10th cyclotomic field ℚ(ζ₁₀). | CONNECTION: φ, 1/φ = 0.618, φ² = 2.618 all appear as eigenvalues and tile-count ratios. The 5-fold rotational symmetry (crystallographically forbidden in periodic lattices) is realized via the cyclotomic field ℚ(ζ₅) = ℚ(ζ₁₀), linking to the golden ratio's role in pentagonal/quasicrystalline order. The substitution rule is a 1D Fibonacci word inflation (A→AB, B→A) l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -