In learning about the K-theory of $C^*$-algebras, I have encountered the following 3 proofs of Bott periodicity:
$\bullet$ An argument based on Moyal quantization found in "Elements of Noncommutative Geometry."
$\bullet$ An argument based on Toeplitz algebras in Murphy's book.
$\bullet$ A seemingly brute force (but elementary) argument in Rordam, Larsen and Laustsen's book where they prove a variety of density results about projections in matrix algebras.
Which of these proofs is the most "geometrical" in the sense that it has a nice geometric interpretation when we restrict our attention to commutative $C^*$-algebras? As a student interested in noncommutative topology and geometry, if I wanted to study one of these proofs in gory detail, which should it be?
If there are other proofs that are more geometrical or more essential for understanding the phenomenon of Bott periodicity, I would be happy to hear about those too.