From basic geometrical considerations, one can show that the (south-pole) stereographic projection of a complex number

onto the Riemann sphere (in

coordinates) is given by:

Now if we consider the projection of another point

, then plugging into the above formula gives

i.e. the points

and

project to diametrically opposite directions on the Riemann sphere.

We now recall from page 555 that the spin direction of a spinor with components

is the stereographic projection of the ratio

. For our two spinors

and

, these ratios are, respectively:

It is easy to see that

, and that therefore

and

represent diametrically opposite (antipodal) spin directions.