Continued fractions and lines across the Stern–Brocot diagram

This paper concerns the relationships between continued fractions and the geometry of the Stern–Brocot diagram. Each rational number can be expressed as a continued fraction [a_0; a₁,…,a_n] whose terms a_i are integers and are positive if i > 1. Select an index i ∈{ 1;,…, n} and replace a_i with an integer m to obtain a continued fraction expansion for an extended rational (Formula Presented). This paper shows that the vertices of the Stern–Brocot diagram corresponding to the numbers {α_m}_m∈ℤ lie on a pair of (extended) Euclidean lines across the diagram. The slopes of these two lines differ only by a sign change and they meet at the point L = ([a₀;a₁,…,a_i-1]; 0) ∈ ℝ². Moreover, as |m| → ∞, the associated vertices move down these lines and converge to L. This paper concludes with a discussion which interprets this result in the context of 2-bridge link complements and Thurston’s work on hyperbolic Dehn surgery.