Cyclotomic Aperiodic Substitution Tilings: Algebraic Integer Inflation Multipliers Encoding Rotational Symmetry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) — substitution matrices over ℤ[ζ₂ₙ] yield algebraic integer inflation multipliers whose eigenvalues encode rotational symmetries and self-similarity. | MATH: Let ζ₂ₙ = e^{2πi/2n}. CAST vertices lie in ℤ[ζ₂ₙ]. Substitution matrix M has Perron–Frobenius eigenvalue λ (inflation multiplier). λ is an algebraic integer in ℤ[ζ₂ₙ] (or its real subfield ℤ[ζ₂ₙ + ζ₂ₙ⁻¹]). Minimal polynomial of λ divides the characteristic polynomial of M. For n=5 (decagonal), λ often satisfies λ² − φλ − 1 = 0 or λ² − (φ+1)λ + φ = 0, where φ = (1+√5)/2 = 1.618…, and φ⁻¹ = 0.618…, φ⁻² = 0.382… . For n=7 (tetradecagonal), λ may be root of λ³ − (2cos(π/7))λ² − … = 0, with 2cos(π/7) ≈ 1.8019, 2cos(2π/7) ≈ 1.24698, 2cos(3π/7) ≈ 0.44504. | CONNECTION: Direct. The real subfield of ℤ[ζ₂ₙ] contains 2cos(2πk/2n). For n=5, 2cos(2π/5) = φ⁻¹ = 0.618, 2cos(π/5) = φ = 1.618, 2cos(4π/5) = −φ⁻¹, 2cos(3π/5) = −φ. For n=6 (dodecagonal), 2cos(π/6) = √3 ≈ 1.732, 2cos(π/3) = 1,
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026cyclotomic,
title = {{Cyclotomic Aperiodic Substitution Tilings: Algebraic Integer Inflation Multipliers Encoding Rotational Symmetry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23465018},
url = {https://zenodo.org/records/23465018},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cyclotomic Aperiodic Substitution Tilings: Algebraic Integer Inflation Multipliers Encoding Rotational Symmetry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23465018
UR - https://zenodo.org/records/23465018
AB - FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) — substitution matrices over ℤ[ζ₂ₙ] yield algebraic integer inflation multipliers whose eigenvalues encode rotational symmetries and self-similarity. | MATH: Let ζ₂ₙ = e^{2πi/2n}. CAST vertices lie in ℤ[ζ₂ₙ]. Substitution matrix M has Perron–Frobenius eigenvalue λ (inflation multiplier). λ is an algebraic integer in ℤ[ζ₂ₙ] (or its real subfield ℤ[ζ₂ₙ + ζ₂ₙ⁻¹]). Minimal polynomial of λ divides the characteristic polynomial of M. For n=5 (decagonal), λ often satisfies λ² − φλ − 1 = 0 or λ² − (φ+1)λ + φ = 0, where φ = (1+√5)/2 = 1.618…, and φ⁻¹ = 0.618…, φ⁻² = 0.382… . For n=7 (tetradecagonal), λ may be root of λ³ − (2cos(π/7))λ² − … = 0, with 2cos(π/7) ≈ 1.8019, 2cos(2π/7) ≈ 1.24698, 2cos(3π/7) ≈ 0.44504. | CONNECTION: Direct. The real subfield of ℤ[ζ₂ₙ] contains 2cos(2πk/2n). For n=5, 2cos(2π/5) = φ⁻¹ = 0.618, 2cos(π/5) = φ = 1.618, 2cos(4π/5) = −φ⁻¹, 2cos(3π/5) = −φ. For n=6 (dodecagonal), 2cos(π/6) = √3 ≈ 1.732, 2cos(π/3) = 1, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -