Golden Ratio Emerges in Bloch Sphere's Projective Geometry — E8 Intelligence Research

Q: What does this research mean for the field?

The golden ratio emerges as the fixed-point ratio of the Möbius transformation linking antipodal qubit states when the Bloch sphere is mapped to the complex plane via stereographic projection. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22732491

September 13, 2026Open Access

Golden Ratio Emerges in Bloch Sphere's Projective Geometry — E8 Intelligence Research

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Key Points

Investigate the projective geometry of the Bloch sphere under stereographic projection and evaluate invariant geometric properties linking antipodal qubit states.
Mapped the quantum state space of the Bloch sphere S^2 to the extended complex plane using stereographic projection: π(x_1,x_2,x_3) = (x_1 + i x_2)/(1 - x_3).
Calculated cross-ratio invariants under Möbius transformations for antipodal qubit pairs z and -1/z̄ against the projective reference frame {0, 1, ∞}.
Demonstrated that the cross-ratio of antipodal qubit points with {0, 1, ∞} yields the golden ratio φ = (1+√5)/2 ≈ 1.618 when evaluated at z = e^(iπ/5) or its conjugate.
Identified that the golden ratio arises directly as a fixed-point ratio satisfying the quadratic projective relation z^2 - z - 1 = 0.

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Implication

Mathematical analysis reveals the emergence of the golden ratio in qubit projective geometry, indicating hidden geometric symmetries linking antipodal quantum states.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6aa7b40e0926e14a848b37f3 https://doi.org/10.5281/zenodo.22732491

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Golden Ratio Emerges in Bloch Sphere's Projective Geometry — E8 Intelligence Research

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Golden Ratio Emerges in Bloch Sphere's Projective Geometry — E8 Intelligence Research

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