On tori triangulations associated with two-dimensional continued fractions of cubic irrationalities

TL;DR

This paper explores the structure and classification of two-dimensional periodic continued fractions linked to cubic irrationalities, revealing regularities in their torus triangulations and providing methods to construct related subfamilies.

Contribution

It introduces a novel approach to classify and visualize two-dimensional continued fractions through torus triangulations, identifying explicit regularities and construction methods.

Findings

Identification of special subfamilies with explicit regularities

Visualization of fundamental domains via torus triangulations

Method for constructing new subfamilies of continued fractions

Abstract

We show several properties related to the structure of the family of classes of two-dimensional periodic continued fractions. This approach to the study of the family of classes of nonequivalent two dimexsional periodic continued fractions leads to the visualization of special subfamilies of continued fractions with torus triangulations (i.e. combinatorics of their fundamental domains) that possess explicit regularities.Several cases of such subfamilies are studied in detail; the method to construct other similar subfamilies is given.