Let events in the history of some particle described by their phase angle . And recall that the unit vectors , and 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
And similarly the ordinate axis is composed from multiples of
These new vectors together with are called a Cartesian basis after the work of René Descartes . Cartesian axes can be visualized by rotating the electric and magnetic axes by degrees around the polar axis. The factor of two means that and make two complete turns as goes through each cycle, one turn for quarks of each phase. These definitions can be rearranged to give and .