Continued Fractions

Every number can be written as a continued fraction: a whole number plus one over a whole number plus one over… These expansions give the best possible fractional approximations, which explains the ancient approximations of π, the rules for leap years, why music uses 12 notes, why sunflowers use the golden angle, how to solve Pell's equation, how Huygens chose his gears, and when eclipses repeat.