Markov Numbers' Golden Ratio Growth and Fibonacci Connections — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Markov numbers form a tree of solutions to x²+y²+z²=3xyz, with growth rates linked to the golden ratio; Fibonacci numbers are a special case. | MATH: Markov equation: x²+y²+z² = 3xyz. Fibonacci solutions: (1, F₂ₙ₋₁, F₂ₙ₊₁) satisfy the equation. Tree structure via Vieta involutions: (x,y,z) → (x,y,3xy−z). Growth rate of Markov numbers along the tree: asymptotic to C·φ^(3n) where φ = (1+√5)/2 ≈ 1.618 (known result: Markov numbers grow exponentially with exponent related to log φ). | CONNECTION: The golden ratio φ appears as the dominant eigenvalue of the Markov tree's growth — the tree's branching structure is isomorphic to the Farey tree / Stern-Brocot tree, which encodes rational approximations and is deeply tied to modular group PSL(2,Z) and its cusp geometry. The ratio 1/φ ≈ 0.618 and φ² = φ+1 ≈ 2.618 appear in the recurrence structure. The Markov tree's vertices correspond to simple closed geodesics on the modular torus — a crystallographic/lattice symmetry (the modular gro
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026markov,
title = {{Markov Numbers' Golden Ratio Growth and Fibonacci Connections — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635357},
url = {https://zenodo.org/records/23635357},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Markov Numbers' Golden Ratio Growth and Fibonacci Connections — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635357
UR - https://zenodo.org/records/23635357
AB - FINDING: Markov numbers form a tree of solutions to x²+y²+z²=3xyz, with growth rates linked to the golden ratio; Fibonacci numbers are a special case. | MATH: Markov equation: x²+y²+z² = 3xyz. Fibonacci solutions: (1, F₂ₙ₋₁, F₂ₙ₊₁) satisfy the equation. Tree structure via Vieta involutions: (x,y,z) → (x,y,3xy−z). Growth rate of Markov numbers along the tree: asymptotic to C·φ^(3n) where φ = (1+√5)/2 ≈ 1.618 (known result: Markov numbers grow exponentially with exponent related to log φ). | CONNECTION: The golden ratio φ appears as the dominant eigenvalue of the Markov tree's growth — the tree's branching structure is isomorphic to the Farey tree / Stern-Brocot tree, which encodes rational approximations and is deeply tied to modular group PSL(2,Z) and its cusp geometry. The ratio 1/φ ≈ 0.618 and φ² = φ+1 ≈ 2.618 appear in the recurrence structure. The Markov tree's vertices correspond to simple closed geodesics on the modular torus — a crystallographic/lattice symmetry (the modular gro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -