  We construct quantum metric structures on unital AF algebras with a faithful
tracial state, and prove that for such metrics, AF algebras are limits of their
defining inductive sequences of finite dimensional C*-algebras for the quantum
propinquity. We then study the geometry, for the quantum propinquity, of three
natural classes of AF algebras equipped with our quantum metrics: the UHF
algebras, the Effros-Shen AF algebras associated with continued fraction
expansions of irrationals, and the Cantor space, on which our construction
recovers traditional ultrametrics. We also exhibit several compact classes of
AF algebras for the quantum propinquity and show continuity of our family of
Lip-norms on a fixed AF algebra. Our work thus brings AF algebras into the
realm of noncommutative metric geometry.
