Proper cyclic symmetries of multidimensional continued fractions

TL;DR

This paper proves the existence of palindromic continued fractions in any dimension and establishes a criterion for proper cyclic palindromic symmetry in four dimensions, using Klein polyhedra as a key tool.

Contribution

It introduces new results on the symmetry properties of multidimensional continued fractions and provides a specific criterion for four-dimensional cases.

Findings

Existence of palindromic continued fractions in arbitrary dimensions

A criterion for proper cyclic palindromic symmetry when n=4

Use of Klein polyhedra as a multidimensional generalization

Abstract

This work is devoted to the proof of the statement about the existence of palindromic continued fractions in an arbitrary dimension. In addition, it is proved the criterion that an algebraic continued fraction has proper cyclic palindromic symmetry in the case $n = 4$. Klein polyhedra are considered as a multidimensional generalization of continued fractions.