clock and Clifford algebra clock. The Bott periodicity results then refine to a sequence of homotopy equivalences: For complex K-theory: For real and quaternionic KO- and KSp-theories: The resulting spaces are homotopy equivalent to the classical reductive symmetric spaces, and are the successive quotients of the terms of the Bott periodicity clock. These equivalences immediately yield the Bott periodicity theorems. The specific spaces are, (for groups, the principal homogeneous space is also listed): Proofs Bott's original proof used Morse theory, which had used earlier to study the homology of Lie groups. Many different proofs have been given. Notes References . An expository account of the theorem and the mathematics surrounding it. Topology of Lie groups Theorems in homotopy theory