Symmetries of quadratic forms classes and of quadratic surds continued fractions. Part II: Classification of the periods' palindromes

TL;DR

This paper classifies the symmetries of quadratic surds' continued fraction periods by relating them to quadratic form classes, providing a comprehensive symmetry classification and geometric interpretation.

Contribution

It introduces a classification of period symmetries of quadratic surds based on quadratic form class symmetries, linking continued fractions with geometric and reduction theories.

Findings

Classified periods of quadratic surds by symmetry type.

Connected form class symmetries with geometric domains in the de Sitter model.

Linked continued fraction reduction with classical quadratic form reduction.

Abstract

The continue fractions of quadratic surds are periodic, according to a theorem by Lagrange. Their periods may have differing types of symmetries. This work relates these types of symmetries to the symmetries of the classes of the corresponding indefinite quadratic forms. This allows to classify the periods of quadratic surds and at the same time to find, for an arbitrary indefinite quadratic form, the symmetry type of its class and the number of integer points, for that class, contained in each domain of the Poincare' model of the de Sitter world, introduced in Part I. Moreover, we obtain the same information for every class of forms representing zero, by the finite continue fraction related to a special representative of that class. We will see finally the relation between the reduction procedure for indefinite quadratic forms, defined by the continued fractions, and the classical reduction theory, which acquires a geometrical description by the results of Part I.