  We give a proof of the Bott periodicity theorem for topological K-theory of
C*-algebras based on Loring's treatment of Voiculescu's almost commuting
matrices and Atiyah's rotation trick. We also explain how this relates to the
Dirac operator on the circle; this uses Yu's localization algebra and an
associated explicit formula for the pairing between the first K-homology and
first K-theory groups of a (separable) C*-algebra.
