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

*A.P. Yefremov*^{1}

(1) Institute of Gravitation and Cosmology of Peoples' Friendship University of Russia 6 Miklukho-Maklaya St., Moscow 117198, Russia

### 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 Lame coefficients (dyads) locally coupling the base and tangent surfaces.

### References

- P. Fjelstad, Extending special relativity via the perplex numbers, Am. J. Phys. 54 (5), 416-422 (1986).
- A. P. Yefremov, Quaternions: Algebra, geometry, and physical theories, Hypercomplex Numbers in Geometry and Physics, 1, 104-119 (2004). Available online in < http://hypercomplex.xpsweb.com/articles/147/en/pdf/01-10-e.pdf>.
- A. P. Yefremov, Quaternion model of relativity: solutions for non-inertial motions and new effects, Adv. Sci. Lett. 1, 179-186 (2008).
- W. R. Hamilton, Lectures on Quaternions (Dublin, Hodges & Smith, 1853). Available online in < http://www.archive.org/details/lecturesonquater00hami>.
- A. P. Yefremov, Structure of hypercomplex units and exotic numbers as sections of bi-quaternions, Adv. Sci. Lett. 3, 537-542 (2010).
- R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comm. Math. Helv. 4, 9-20 (1932).
- A. P. Yefremov, Quaternionic multiplication rule and a local Q-metric, Lett. Nuovo Cim. 37 (8), 315-316 (1983).

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