The Geometry of Continued Fractions

Continued fractions provide a natural representation of real numbers with deep connections to number theory, hyperbolic geometry, and dynamics. We explore the Stern-Brocot tree — an infinite binary tree containing every positive rational number exactly once — and show how walking this tree corresponds to computing continued fraction expansions. Geometric interpretations via Ford circles and the Farey tessellation of the hyperbolic plane illuminate why continued fractions are optimal approximations \in the sense of Hurwitz's theorem: $|p/q - \alpha| < 1/(\sqrt{5},q^2)$ for infinitely many $p/q$.