LAUSR

Abstract The coset diagram for each orbit under the action of the modular group on Q ( n ) ⁎ = Q ( n ) ∪ { ∞ } contains a circuit C i . For any α ∈ Q ( n ) , the path leading to the circuit C i and the circuit itself are obtained through continued fractions in this paper. We show that the structure of the continued fractions of a reduced quadratic irrational element is weaved with the structure or type of the circuit. The three types of circuits of the action of V 4 on Q ( n ) ⁎ are also interconnected with the structure of continued fractions. The action of the modular group on Q ( 5 ) ⁎ is chosen specifically because a circuit of it is related to the ratio of the Fibonacci numbers being the solution to the continued fractions of the golden ratio.