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Geometry of multidimensional Farey summation algorithm and frieze patterns

Oleg Karpenkov, Matty van Son

TL;DR

This work develops a geometric framework for multidimensional subtractive continued fractions by constructing Farey polyhedra and their sails, extending Klein’s sails to higher dimensions. It introduces prismatic diagrams as complete invariants of Farey polyhedra, and defines three-dimensional sails and LLS sequences to encode continued-fraction data; these lead to higher-dimensional frieze patterns governed by generalized Ptolemy relations with lambda-length determinants equal to 1. The Meester and Farey summation algorithms are shown to be dual, with convergence properties and a dense divergence set, and the nose-stretching procedure provides a constructive method to recover Farey polyhedra from continued fractions. Together, these results extend Conway-Coxeter frieze patterns to higher dimensions and reveal deep connections between integer geometry, combinatorics, and geometric continued fractions, opening avenues for further generalisation to other subtractive algorithms and higher-dimensional frieze theories.

Abstract

In this paper we develop a new geometric approach to subtractive continued fraction algorithms in high dimensions. We adapt a version of Farey summation to the geometric techniques proposed by F. Klein in 1895. More specifically we introduce Farey polyhedra and their sails that generalise respectively Klein polyhedra and their sails, and show similar duality properties of the Farey sail integer invariants. The construction of Farey sails is based on the multidimensional generalisation of the Farey tessellation provided by a modification of the continued fraction algorithm introduced by R. W. J. Meester. We classify Farey polyhedra in the combinatorial terms of prismatic diagrams. Prismatic diagrams extend boat polygons introduced by S. Morier-Genoud and V. Ovsienko in the two-dimensional case. As one of the applications of the new theory we get a multidimensional version of Conway-Coxeter frieze patterns. We show that multidimensional frieze patterns satisfy generalised Ptolemy relations.

Geometry of multidimensional Farey summation algorithm and frieze patterns

TL;DR

This work develops a geometric framework for multidimensional subtractive continued fractions by constructing Farey polyhedra and their sails, extending Klein’s sails to higher dimensions. It introduces prismatic diagrams as complete invariants of Farey polyhedra, and defines three-dimensional sails and LLS sequences to encode continued-fraction data; these lead to higher-dimensional frieze patterns governed by generalized Ptolemy relations with lambda-length determinants equal to 1. The Meester and Farey summation algorithms are shown to be dual, with convergence properties and a dense divergence set, and the nose-stretching procedure provides a constructive method to recover Farey polyhedra from continued fractions. Together, these results extend Conway-Coxeter frieze patterns to higher dimensions and reveal deep connections between integer geometry, combinatorics, and geometric continued fractions, opening avenues for further generalisation to other subtractive algorithms and higher-dimensional frieze theories.

Abstract

In this paper we develop a new geometric approach to subtractive continued fraction algorithms in high dimensions. We adapt a version of Farey summation to the geometric techniques proposed by F. Klein in 1895. More specifically we introduce Farey polyhedra and their sails that generalise respectively Klein polyhedra and their sails, and show similar duality properties of the Farey sail integer invariants. The construction of Farey sails is based on the multidimensional generalisation of the Farey tessellation provided by a modification of the continued fraction algorithm introduced by R. W. J. Meester. We classify Farey polyhedra in the combinatorial terms of prismatic diagrams. Prismatic diagrams extend boat polygons introduced by S. Morier-Genoud and V. Ovsienko in the two-dimensional case. As one of the applications of the new theory we get a multidimensional version of Conway-Coxeter frieze patterns. We show that multidimensional frieze patterns satisfy generalised Ptolemy relations.

Paper Structure

This paper contains 45 sections, 22 theorems, 97 equations, 9 figures.

Table of Contents

  1. Introduction
  2. History and background
  3. Main results
  4. Farey tessellation and related algorithms
  5. A few words on integer geometry
  6. Farey tessellation in higher-dimensions
  7. Construction of the Farey tessellation
  8. Farey and quasi-Farey nets
  9. Description of the Farey summation algorithm
  10. Definition of Farey summation continued fractions
  11. Enumeration of vertices
  12. Tabulation of the algorithm
  13. Meester algorithm
  14. A general definition
  15. Extended Meester continued fraction
  16. ...and 30 more sections

Key Result

Proposition 2.11

The following holds:

Figures (9)

  • Figure 1: Unit integer circle (left) and Farey tessellation of the entire lattice (right).
  • Figure 2: The $i$-th step in the Farey summation algorithm.
  • Figure 3: Farey summation algorithm for $(5,7,8)$ (left) and its central projection to the plane $x+y+z=1$.
  • Figure 4: Farey summation algorithm for $(5,7)$.
  • Figure 5: Meester algorithm divergence set (several iterations)
  • ...and 4 more figures

Theorems & Definitions (142)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 132 more