Hyperbolic Pentagons, Fibonacci Anyons, and Quantum Group Invariants — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results converge on a deep structural link between 5-fold symmetry, hyperbolic geometry, and quantum invariants — specifically, that pentagonal tiling (impossible in Euclidean plane) becomes natural in hyperbolic space, and this connects to Fibonacci anyons and quantum group invariants. | MATH: (1) Fibonacci anyons obey fusion rule τ × τ = 1 + τ, with quantum dimension φ = (1+√5)/2 ≈ 1.618. (2) Hyperbolic tiling {5,4} (pentagons, 4 meeting at vertex) has curvature -1, area per pentagon = π/5. (3) Binet formula: F_n = (φ^n − (−φ)^−n)/√5. (4) Quantum invariant Z(M) for hyperbolic 3-manifolds relates to volume via Witten–Reshetikhin–Turaev invariants at root of unity q = e^{2πi/(k+2)}. | CONNECTION: **CRITICAL** — 5-fold symmetry (forbidden in Euclidean crystals, Penrose tilings) is *allowed* in hyperbolic plane. The golden ratio φ appears as: (a) quantum dimension of Fibonacci anyon, (b) scaling factor in hyperbolic pentagon tilings, (c) the ratio of diagonal to side
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026hyperbolic,
title = {{Hyperbolic Pentagons, Fibonacci Anyons, and Quantum Group Invariants — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587117},
url = {https://zenodo.org/records/23587117},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hyperbolic Pentagons, Fibonacci Anyons, and Quantum Group Invariants — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587117
UR - https://zenodo.org/records/23587117
AB - FINDING: The search results converge on a deep structural link between 5-fold symmetry, hyperbolic geometry, and quantum invariants — specifically, that pentagonal tiling (impossible in Euclidean plane) becomes natural in hyperbolic space, and this connects to Fibonacci anyons and quantum group invariants. | MATH: (1) Fibonacci anyons obey fusion rule τ × τ = 1 + τ, with quantum dimension φ = (1+√5)/2 ≈ 1.618. (2) Hyperbolic tiling {5,4} (pentagons, 4 meeting at vertex) has curvature -1, area per pentagon = π/5. (3) Binet formula: F_n = (φ^n − (−φ)^−n)/√5. (4) Quantum invariant Z(M) for hyperbolic 3-manifolds relates to volume via Witten–Reshetikhin–Turaev invariants at root of unity q = e^{2πi/(k+2)}. | CONNECTION: **CRITICAL** — 5-fold symmetry (forbidden in Euclidean crystals, Penrose tilings) is *allowed* in hyperbolic plane. The golden ratio φ appears as: (a) quantum dimension of Fibonacci anyon, (b) scaling factor in hyperbolic pentagon tilings, (c) the ratio of diagonal to side Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -