  We classify the Fibonacci chains (F-chains) by their index sequences and
construct an approximately finite dimensional (AF) $C^*$-algebra on the space
of F-chains as Connes did on the space of Penrose tiling. The K-theory on this
AF-algebra suggests a connection between the noncommutative torus and the space
of F-chains. A noncommutative torus, which can be regarded as the $C^*$-algebra
of a foliation on the torus, is explicitly embedded into the AF-algebra on the
space of F-chains. As a counterpart of that, we obtain a relation between the
space of F-chains and the leaf space of Kronecker foliation on the torus using
the cut-procedure of constructing F-chains.
