We have seen that if we take a vector and reflect it with respect to the vector and then to the vector (in this order) we obtain a rotation of the vector of an angle double the one between and (in that order).

Algebraically it is a question of applying reflection twice with respect to a vector, with the known sandwich product:

So if we want a rotation equal to we will have to use a rotor which was named *spinor* by Wolfgang Pauli for the reason that we will see shortly.

So we have this strange equality:

from which we deduce that the two rotations e represent the same rotation but in two opposite directions:

- represents the counter-clockwise rotation in the range
- represents the clockwise rotation in the interval given that:

But so spinors have a period !

If spinors seem strange to you … it is because they are!

It is carved into everyone’s beliefs that an object returns to its initial state after a rotation of , but this time we are dealing with an object that returns to itself after *two* rotations.

This weirdness can be demonstrated with Dirac’s trick of the saucer or the belt, which will leave the spectators astounded as if it were a magic show.

This makes the half-integer spin of electrons a little less strange, after all.