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

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Stereographic projection of the Bloch sphere to the complex plane reveals a projective geometry where the golden ratio emerges as the fixed-point ratio of the Möbius transformation linking antipodal qubit states. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22732490

September 13, 2026Open Access

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

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Key Points

To investigate the projective geometry of the quantum state space under stereographic projection and determine geometric invariants linking antipodal qubit states.
Mapped quantum state vectors on the Bloch sphere S² to the extended complex plane using stereographic projection.
Formulated Möbius transformations on the Riemann sphere to evaluate cross-ratio invariance for antipodal points z and -1/z̄ relative to reference coordinates {0, 1, ∞}.
Demonstrated that the cross-ratio evaluated across antipodal points yields the golden ratio φ = (1 + √5)/2 ≈ 1.618 when z = e^(iπ/5) or its conjugate.
Identified that the projective coordinates at these antipodal configurations directly satisfy the quadratic golden ratio equation z² - z - 1 = 0.

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Implication

Mathematical analysis reveals the golden ratio in the projective geometry of the Bloch sphere, suggesting deep geometric symmetries underlying antipodal qubit states.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6aa7dfc60926e14a848b3e8f https://doi.org/10.5281/zenodo.22732490

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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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