## Abstract

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

Original language | English (US) |
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Pages (from-to) | 2151-2186 |

Number of pages | 36 |

Journal | International Journal of Number Theory |

Volume | 10 |

Issue number | 8 |

DOIs | |

State | Published - Dec 25 2014 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

## Keywords

- Hermite problem
- Jacobi-Perron algorithm
- Multidimensional continued fractions
- Triangle map