On symmetries of 3-dimensional algebraic continued fractions

TL;DR

This paper establishes criteria for palindromic symmetries in 3-dimensional algebraic continued fractions, providing new proofs and extending the understanding of symmetries in multidimensional continued fractions like Klein polyhedra.

Contribution

It offers detailed criteria for symmetries in 3D algebraic continued fractions and introduces new proofs and generalizations for these symmetries.

Findings

Criteria for proper palindromic symmetry in dimension 4

New proof for cyclic palindromic symmetry criterion

Extension to Klein polyhedra as multidimensional continued fractions

Abstract

In this paper we prove in detail a criterion for an algebraic continued fraction to have a proper palindromic symmetry in dimension $4$. We also present a new proof of the criterion for an algebraic continued fraction to have a proper cyclic palindromic symmetry in dimension $4$. As a multidimensional generalization of continued fractions, we consider Klein polyhedra.