Palindromes and periodic continued fractions

TL;DR

This paper explores the geometry of continued fractions, focusing on quadratic irrationalities, and establishes a criterion for when their continued fractions have symmetric periods, enhancing understanding of their structure.

Contribution

It provides a detailed geometric exposition and introduces a new criterion for symmetric periods in continued fractions of quadratic surds.

Findings

Established a criterion for symmetric periods in quadratic surds' continued fractions

Enhanced geometric understanding of continued fractions for quadratic irrationalities

Connected geometric techniques with properties of quadratic surds

Abstract

This paper is devoted to a detailed exposition of geometry of continued fractions. We pay particular interest to the case of quadratic irrationalities and use the technique described to prove a criterion for the continued fraction of a quadratic surd to have a symmetric period.