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Spatial Axes

Spatial axes in a Cartesian coordinate system are shown in this diagram.
Cartesian axes compared to the electric and magnetic axes.

Let events in the history of some particle described by their phase angle  \theta. And recall that the unit vectors \widehat{m} \equiv (1, 0, 0), \widehat{e} \equiv (0, 1, 0) and \widehat{z} \equiv (0, 0, 1) mark the magnetic, electric and polar axes. These algebraic entities can be used to construct another set of vectors. To start, the axis of the abscissa is defined from all scalar multiples of

\widehat{x} \equiv \cos{\! 2\theta} \, \widehat{m} + \sin{\! 2\theta} \, \widehat{e}

And similarly the ordinate axis is composed from multiples of

\widehat{y} \equiv - \sin{\! 2\theta} \, \widehat{m} + \cos{\! 2\theta} \, \widehat{e}

These new vectors together with \widehat{z} are called a Cartesian basis after the work of RenΓ© Descartes . Cartesian axes can be visualized by rotating the electric and magnetic axes by 2 \theta degrees around the polar axis. The factor of two means that \widehat{x} and \widehat{y} make two complete turns as  \theta goes through each cycle, one turn for quarks of each phase. These definitions can be rearranged to give \widehat{m} = \cos{\! 2\theta} \, \widehat{x} - \sin{\! 2\theta} \, \widehat{y} and \widehat{e} = \sin{\! 2\theta} \, \widehat{x} + \cos{\! 2\theta} \, \widehat{y}.

Spatial axes are suggested by this French engraving of sea urchins.
Jean-Baptiste Lamarck, Echinus, Tableau Encyclopédique et Méthodique des Trois Règnes de la Nature, Paris 1791. Photograph by D Dunlop.