Eigenvalues of Substitution Matrices Encode Quasicrystal Inflation Ratios — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Eigenvalues of substitution matrices in cut-and-project quasicrystals directly encode the inflation/deflation ratios that generate Bragg peak positions, linking linear algebra to non-crystalline diffraction patterns. | MATH: For a substitution matrix \( S \) (e.g., 2×2 for Penrose tiling), eigenvalues \( \lambda_1, \lambda_2 \) satisfy \( \lambda_1 = \tau^2 \) and \( \lambda_2 = \tau^{-2} \), where \( \tau = (1+\sqrt{5})/2 \approx 1.618 \). The cut-and-project method uses a higher-dimensional lattice (e.g., \( \mathbb{Z}^5 \) for icosahedral quasicrystals) projected onto a physical subspace, with Bragg peak positions indexed by integer combinations of basis vectors scaled by \( \tau \). | CONNECTION: Eigenvalues \( \tau^2 \approx 2.618 \) and \( \tau^{-2} \approx 0.382 \) are direct geometric harmony ratios. The substitution matrix's Perron-Frobenius eigenvalue \( \tau^2 \) governs self-similarity scaling, while the smaller eigenvalue \( \tau^{-2} \) relates to the golden rati 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: W7170704432
Cites:
Cited by:
Related to: