 ... ball.^{1}
 If you want to be really concrete, imagine a spinning gyroscope
fitting snugly in a box. Rotate the box.
 ... (frames).^{2}
 In other words, a element of
determines a mapping of to . As it happens,
the action is faithful, i.e., the mapping determines the element of
.
 ... moment.^{3}
 A quick review: write for the equivalence
class of . We will associate either a complex number
or else with each class . If , then
, where . All pairs of the form belong to the
class . So we associate with if , and
with . Mapping the complex plane plus to the
Riemann sphere via the usual stereographic projection completes the trick.
Some sample points to bear in mind: the north pole is ; the south
pole is ; points on the equator have the form
.
(For the purist, the special treatment of rankles. It is not
singled out on either the complex projective line or on the Riemann sphere.
Later I will show how to set up the correspondence without this blemish.)
 ... simple^{4}
 In
the infinite dimensional case, you have to use the spectral decomposition
of instead of an orthonormal basis of eigenvectors; another reason why
spin is simpler than position.

... proportional
to .
^{5}
 Why wouldn't the electron simply snap into
alignment with the magnetic field? Answer: the spinning electron would act
like a gyroscope, and precess in response to the torque exerted by the
field. Thus it would maintain its angle of inclination to the field.