On continued fractions

Abstract

This paper on the geometry, algebra and arithmetics of continued fractions is based on a lecture for students, teachers and a non-specialist audience, beginning with the history of the golden number and Fibonacci sequence, continued fractions of rational and irrational numbers, Lagrange theorem on periodicity of continued fractions for quadratic irrationals, Klein's geometric interpretation of the convergents as integer points, Jung-Hirzebruch continued fractions with negative signs and two dimensional singularities, higher dimensional generalizations, and ending with a result on a periodic generalized 3-dimensional continued fraction for a cubic irrational.