Fibonacci Numbers Emerge from Penrose Tiling and Hat Tile Substitutions — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The Fibonacci numbers emerge naturally from Penrose tiling inflation rules, whose substitution matrix eigenvalues are powers of the golden ratio; the "Einstein" hat tile also encodes Fibonacci structure via its substitution dynamics. | MATH: Penrose kites/darts substitution matrix \(M = \begin{pmatrix}2&1\\1&1\end{pmatrix}\), eigenvalues \(\lambda_\pm = \frac{3\pm\sqrt{5}}{2} = \varphi^2, \varphi^{-2}\) where \(\varphi = \frac{1+\sqrt{5}}{2} \approx 1.618\). Fibonacci word substitution \(0\mapsto 01, 1\mapsto 0\) has substitution matrix \(\begin{pmatrix}1&1\\1&0\end{pmatrix}\) with eigenvalues \(\varphi, -\varphi^{-1}\). Inflation multiplier for Penrose tilings is \(\varphi^2 = 2.618\) (or \(\varphi\) for some variants). The CAST class (arXiv:1606.06858) generalizes this: substitution matrices have eigenvalues in cyclotomic fields \(\mathbb{Q}(\zeta_{2n})\), with minimal inflation multipliers being algebraic integers of degree \(\phi(2n)/2\). | CONNECTION: Direct geometric har 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: W7215873254
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