The conic-gearing image of a complex number and a spinor-born surface geometry

TL;DR

This paper introduces a novel geometric representation of complex numbers using quaternion mathematics, revealing the spinor structure as Lamé coefficients that connect base and tangent surfaces, with implications for physical laws.

Contribution

It presents a new conic-gearing geometric image of complex numbers based on quaternion spinors, linking algebraic structures to surface geometry and physical equations.

Findings

Quaternion units have an internal spinor structure.

Spinors act as Lamé coefficients coupling surfaces.

The approach provides a geometric interpretation of complex numbers and physical laws.

Abstract

Quaternion (Q-) mathematics formally contains many fragments of physical laws; in particular, the Hamiltonian for the Pauli equation automatically emerges in a space with Q-metric. The eigenfunction method shows that any Q-unit has an interior structure consisting of spinor functions; this helps us to represent any complex number in an orthogonal form associated with a novel geometric image (the conic-gearing picture). Fundamental Q-unit-spinor relations are found, revealing the geometric meaning of spinors as Lam\'e coefficients (dyads) locally coupling the base and tangent surfaces.