TL;DR
This paper investigates the properties of Conway and Coxeter's number friezes and their connections to continued fractions, Farey sequences, and the modular group's action on the hyperbolic plane, providing educational insights.
Contribution
It elucidates the relationships between number friezes, continued fractions, Farey sequences, and hyperbolic geometry, offering a comprehensive overview suitable for undergraduate education.
Findings
Characterization of number friezes and their properties
Connections between friezes and continued fraction decompositions
Insights into the modular group's action on hyperbolic space
Abstract
We explore basic properties of number friezes, due to Conway and Coxeter, and their relations to decompositions of rational numbers into continued fractions, Farey sequences, and the modular group acting on the hyperbolic plane. These are notes from a mini-course for undergraduate students given at the 19th Summer School ``Modern Mathematics'', Dubna, Russia, July 18-29, 2019.
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