Gravitation and Cosmology

, Volume 17, Issue 1, pp 1–6 | Cite as

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

  • A. P. YefremovEmail author


Quaternion (Q-) mathematics formally containsmany 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 conicgearing picture). Fundamental Q-unit-spinor relations are found, revealing the geometric meaning of the spinors as Lamé coefficients (dyads) locally coupling the base and tangent surfaces.


Tangent Plane Interior Structure Spinor Function Geometric Image Tangent Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Fjelstad, Extending special relativity via the perplex numbers, Am. J. Phys. 54(5), 416 (1986).MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    A. P. Yefremov, Quaternions: Algebra, geometry, and physical theories, Hypercomplex Numbers in Geometry and Physics, 1, 104 (2004). Available online in 〈〉.Google Scholar
  3. 3.
    A. P. Yefremov, Quaternion model of relativity: solutions for non-inertial motions and new effects, Adv. Sci. Lett. 1, 179 (2008).CrossRefGoogle Scholar
  4. 4.
    W. R. Hamilton, Lectures on Quaternions (Dublin, Hodges & Smith, 1853). Available online in 〈〉.Google Scholar
  5. 5.
    A. P. Yefremov, Structure of hypercomplex units and exotic numbers as sections of bi-quaternions, Adv. Sci. Lett. 3, 537 (2010).CrossRefGoogle Scholar
  6. 6.
    R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comm. Math. Helv. 4, 9 (1932).MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. P. Yefremov, Quaternionic multiplication rule and a local Q-metric, Lett. Nuovo Cim. 37(8), 315 (1983).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Institute of Gravitation and Cosmology of Peoples’ Friendship University of RussiaMoscowRussia

Personalised recommendations