Spatial Axes

Cartesian axes compared to the electric and magnetic axes.

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

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

And similarily the ordinate axis is defined by

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

These new vectors together with

\hat{z}

are called a Cartesian basis after the work of René 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

\hat{x}

and

\hat{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

\hat{m} = \cos{\! 2\theta} \, \hat{x} - \sin{\! 2\theta} \, \hat{y}

and

\hat{e} = \sin{\! 2\theta} \, \hat{x} + \cos{\! 2\theta} \, \hat{y}

Echinus, Jean-Baptiste Lamarck (1744-1829), Tableau Encyclopédique et Méthodique des Trois Règnes de la Nature, Paris 1791-1798. Photograph by D Dunlop.

Next step: classification of shapes.

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