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August 14, 2026Open Access

Golden Ratio's Geometric Construction and Number-Theoretic Links — E8 Intelligence Research

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Authors

ACAndrew Stewart Caldin

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Implication

Theoretical analysis reveals geometric constructions and number-theoretic identities of the golden ratio, highlighting its role in pentagonal symmetry and quasicrystals.

Key Points

  • Investigate the mathematical properties connecting geometric constructions of the golden ratio in pentagons to number-theoretic functions and quasicrystal symmetries.
  • Evaluated geometric constructions of the golden ratio (φ) in golden rectangles and regular pentagons using circle divisions, central angles, and chord lengths.
  • Derived algebraic and trigonometric properties, irrationality proofs via diagonal incommensurability, and analytic links involving the Möbius function μ(n) and Euler totient φ(n).
  • Confirmed fundamental identities where φ = (1+√5)/2 ≈ 1.6180339, φ⁻¹ = φ − 1 ≈ 0.618, and φ² = φ + 1 ≈ 2.618, all matching diagonal and side ratios within regular pentagons.
  • Demonstrated that φ = 2cos(π/5) directly governs 5-fold rotational symmetry central to quasicrystals while linking to number-theoretic identities involving logarithms and totient functions.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a7ee587b70b84ec8b9146e5https://doi.org/10.5281/zenodo.21898357
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  1. 1Golden Ratio's Geometric Construction and Number-Theoretic Links — E8 Intelligence Research2026
  2. 2Golden Ratio φ Arises from Pentagonal Symmetry via 36°-72°-72° Triangle — E8 Intelligence Research2026
  3. 3Golden Ratio Emerges from Pentagonal Symmetry via 36° and 72° Angles — E8 Intelligence Research2026
  4. 4Golden Ratio φ Emerges from the Regular Pentagon's Diagonal and 36-72-72 Triangle — E8 Intelligence Research2026
  5. 5Golden Ratio φ: Algebraic Integer, Fibonacci Limit, and Icosahedral Quasicrystal Links — E8 Intelligence Research2026