Derivation of Spherical to Cartesian Coordinates from the Bloch Sphere Geometry

Venkateswaran Kasirajan

Cameron Brown

The Bias Pressure Logic (BPL) system of the AIMS framework currently represents relational states as points on a one-dimensional arc (0–360°, 8 arc positions at 45° intervals). We propose and develop a three-dimensional extension that maps BPL states to the surface of and interior of a Bloch sphere, grounding the extension in Plimpton 322 sexagesimal geometry and demonstrating structural identity with standard quantum gate representations... We further connect the 3D BPL system to TurboQuant (Aghajanyan et al. 2025, arXiv:2504.19874), which achieves near-optimal vector quantization distortion rates by replacing flat-plane quantization with angular/spherical representations. The transition from 2D BPL (flat arc) to 3D BPL (Bloch sphere) is structurally parallel to TurboQuant's core innovation, suggesting that 3D BPL may achieve analogous efficiency gains in relational state representation. We also address the relationship to meta-learning coordinate systems, arguing that the shift from Cartesian to spherical-angular coordinates is both motivated by and consistent with the BPL's existing base-60 angular structure.

Zhong-Shuai 中帅 Zhang 张, Shi-Fa Xiao, Damao Xun +1 more

For a non-relativistic particle that freely moves on a curved surface, the fundamental commutation relations between positions and momenta are insufficient to uniquely determine the operator form of the momenta. With introduction of more commutation relations between positions and Hamiltonian and those between momenta and Hamiltonian, our recent sequential studies imply that the Cartesian system of coordinates is physically preferable, consistent with Dirac' observation. In present paper, we study quantization problem of the motion constrained on the two-dimensional sphere and develop a discriminant that can be used to show how the quantization within the intrinsic geometry is improper. Two kinds of parameterization of the spherical surface are explicitly invoked to investigate the quantization problem within the intrinsic geometry.

Andrew Stewart Caldin

FINDING: Stereographic projection maps the Bloch sphere (quantum state space) to the complex plane, revealing a projective geometry where the golden ratio emerges as the fixed-point ratio of the Möbius transformation linking antipodal qubit states. | MATH: Stereographic projection: \( \pi(x_1,x_2,x_3) = \frac{x_1 + i x_2}{1 - x_3} \) for sphere \( S^2 \to \mathbb{C} \cup \{\infty\} \). Bloch sphere: \( |\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle \). Möbius transformation on the Riemann sphere: \( z \mapsto \frac{az+b}{cz+d} \), \( ad-bc \neq 0 \). Cross-ratio invariance: \( \frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} \) is preserved under Möbius maps. For antipodal points \( z \) and \( -1/\bar{z} \), the cross-ratio with \( \{0,1,\infty\} \) yields the golden ratio \( \varphi = (1+\sqrt{5})/2 \approx 1.618 \) when \( z = e^{i\pi/5} \) (or its conjugate), satisfying \( z^2 - z - 1 = 0 \) in the projective coordinate. | CONNECTION: The golden ratio \( \va Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com