Penrose Quasicrystals and Chern-Simons Theory: A Bridge via q-Deformation — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results converge on two distinct but potentially interlocking mathematical structures: (1) the aperiodic 5-fold symmetry of Penrose tilings/quasicrystals, and (2) Chern-Simons theory with its knot invariants and Wilson/Seifert loops — with a possible bridge via q-deformation and non-commutative geometry. | MATH: Penrose tiling: inflation/deflation ratio = φ = (1+√5)/2 ≈ 1.618; area ratios of fat/thin rhombi = φ; vertex star angles are multiples of 36° (2π/10). Quasicrystal Fourier transform yields sharp Bragg peaks at positions generated by Z-module of rank 4 (in 2D) with irrational indices — e.g., wavevectors k = m₁ + m₂φ (mᵢ ∈ ℤ). Chern-Simons action: S_CS = (k/4π) ∫ Tr(A ∧ dA + (2/3) A ∧ A ∧ A), with level k ∈ ℤ. Wilson loop invariant: W_R(C) = Tr_R [P exp(∮_C A)]. q-deformation: q = exp(2πi/(k+2)) for SU(2)_k; knot invariant (Jones polynomial) V_L(q) satisfies skein relation q²V₊ − q⁻²V₋ = (q − q⁻¹)V₀. Seifert loops relate to surgery on 3-manifolds, linking numb
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026penrose,
title = {{Penrose Quasicrystals and Chern-Simons Theory: A Bridge via q-Deformation — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409626},
url = {https://zenodo.org/records/23409626},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Penrose Quasicrystals and Chern-Simons Theory: A Bridge via q-Deformation — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409626
UR - https://zenodo.org/records/23409626
AB - FINDING: The search results converge on two distinct but potentially interlocking mathematical structures: (1) the aperiodic 5-fold symmetry of Penrose tilings/quasicrystals, and (2) Chern-Simons theory with its knot invariants and Wilson/Seifert loops — with a possible bridge via q-deformation and non-commutative geometry. | MATH: Penrose tiling: inflation/deflation ratio = φ = (1+√5)/2 ≈ 1.618; area ratios of fat/thin rhombi = φ; vertex star angles are multiples of 36° (2π/10). Quasicrystal Fourier transform yields sharp Bragg peaks at positions generated by Z-module of rank 4 (in 2D) with irrational indices — e.g., wavevectors k = m₁ + m₂φ (mᵢ ∈ ℤ). Chern-Simons action: S_CS = (k/4π) ∫ Tr(A ∧ dA + (2/3) A ∧ A ∧ A), with level k ∈ ℤ. Wilson loop invariant: W_R(C) = Tr_R [P exp(∮_C A)]. q-deformation: q = exp(2πi/(k+2)) for SU(2)_k; knot invariant (Jones polynomial) V_L(q) satisfies skein relation q²V₊ − q⁻²V₋ = (q − q⁻¹)V₀. Seifert loops relate to surgery on 3-manifolds, linking numb Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -