Abstract
The coset diagram for each orbit under the action of the modular group on Q(n)⁎=Q(n)∪{∞} contains a circuit Ci. For any α∈Q(n), the path leading to the circuit Ci 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 V4 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.
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