Clifford periodicity from finite groups

We deduce the periodicity eight for the type of Pin and Spin representation s of the orthogonal groups O(n) from simple combinatorial properties of the finite Clifford groups generated by the gamma matrices. We also include th e case of arbitrary signature O(p, q). The changes in the type of represent ation 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-c lockwise sense, but for each (-) sign in the metric, the rotation is clockw ise.