  We deduce the periodicity 8 for the type of $Pin$ and $Spin$ representations
of the orthogonal groups $O(n)$ from simple combinatorial properties of the
finite Clifford groups generated by the gamma matrices. We also include the
case of arbitrary signature $O(p,q)$. The changes in the type of representation
can be seen as a rotation in the complex plane. The essential result is that
adding a $(+)$ dimension performs a rotation by $\pi/4$ in the counter
clock-wise sense, but for each $(-)$ sign in the metric, the rotation is
clockwise.
